Class IX-Mathematics-C.B.S.E.-Practice-Paper

IX Mathematics Practice Paper

Mathematics for Class 9 1. Number Systems Q 1 Is every real number is a rational number ? Mark (1) Q 2 Is 1.01001000100001 …… irrational? If so, why? Mark (1) Q 3 Is every whole number is a natural number ? Mark (1) Q 4 Look at the following examples of rational number in the form p/q (q 0),where p and q integers with no common factors other than 1 and having terminating decimal representations. Can you guess the property which satisfy q ?

Mark (1) Q 5 Is zero a rational number? Explain it. Mark (1)

Q 6 If

, then find x is rational or irrational number. Mark (1)

Q 7 Insert three rational numbers between

. Mark (1)

Q 8 Find two rational numbers between 1 and 2. Mark (1) Q 9 Is

a rational number? Mark (1)

Q 10 Is it true that every integer is a rational Number ? Mark (1) Q 11 Is every rational number is an Integer. Mark (1) Q 12 Is

a rational number?

IX Mathematics C.B.S.E. Practice Papers

Page 1

Mark (1) Q 13 Mark (1) Q 14 Is 2 a rational number? Can you write it in the form

, where p and q are integers? Mark (1)

Q 15 Mark (1)

Q 16 Find, whether

is a terminating or non terminating decimal number. Mark (1)

Q 17 Find the value of x , if 5x-2 = 125. Mark (1) Q 18 Simplify:

Marks (2) Q 19 Rationalize the denominators of the following:

Marks (2)

Q 20 Simplify

. Marks (2)

Q 21 Find two irrational numbers between 2 ad 2.5. Marks (2) Q 22 Insert a rational & an irrational number between 2 and 3. Marks (2) Q 23 Identify

as rational number or irrational number.

Marks (2) Q 24 Give examples of two irrational numbers the product of which is: i) a raional number


Page 2

ii) an irrational number. Marks (2) Q 25 identify

80 as rational number or irrational number. Marks (2)

Q 26 How to insert irrational numbers between two given rational numbers. Marks (2)

Q 27 Find the decimal representation of

. Marks (2)

Q 28 Express

in the decimal form by long division method. Marks (2)

Q 29 Find three rational numbers between -2 and 5. Marks (2)

Q 30 Insert 100 rational numbers between

. Marks (2)

Q 31 Insert 10 rational numbers between

. Marks (2)

Q 32 State whether the following statements are true or false. Give reasons for your answers. (i) (ii)

Every integer is a whole number Every rational number is a whole number. Marks (2)

Q 33 Find five rational numbers between

and

. Marks (2)

Q 34 Find six rational numbers between 3 and 4. Marks (2) Q 35 Find five rational numbers between 1 and 2. Marks (2) Q 36 Express 0.8888 ………..in the form of p/q where p and are integers and q

0.

Marks (2)

Q 37 Rationalise the denominator of

. Marks (2)


Page 3

Q 38 Rationalise the denominator in each of the following:

Marks (3)

Q 39 Simplify

. Marks (3)

Q 40 Rationalize the denominators of the following:

Marks (3) Q 41 Represent

on the number line. Marks (3)

Q 42 Visualize 3.765 on the number line using successive magnification. Marks (3) Q 43 State whether the following statements are true or false. Justify. i) Every irrational number is a real number. ii)

Every point on the number line is of the form number.

iii)

Every real number is an irrational number.

, where m is a natural

Marks (3) Q 44 Classify the following numbers as rational or irrational.

Marks (3)

Q 45 Find three different irrational numbers between the rational numbers


.

Page 4

Marks (3) Q 46 Prove that (3+

2 )2 is an irrational number.

Marks (3) Q 47 Are square roots of all the +ve integers irrational? If not, give an example of the square root of a number that is a rational number. Marks (3) Q 48 Simplify

Marks (4) Q 49 Simplify


Marks (4) Q 51 Rationalize the denominator of



Page 5

Marks (4) Q 53 Simplify:


Marks (4) Q 55 Represent

on a number line. Marks (4)

Q 56 Represent

on a number line. Marks (4)

Q 57 You know that . Can you predict the decimal expansions of without actually doing the long division? If so how? Marks (5) Q 58 Represent

on the number line. Marks (5)

Q 59 Construct the square root spiral. Marks (5) Q 60 Find:

Marks (6)


Page 6

Q 61 Simplify



Marks (6) Q 64 Examine, whether the following numbers are rational or irrational.

Marks (6)


Page 7

Q 65 Write the following in decimal form and find the type of decimal expansion.

Marks (6)

Q 66 Express the following in the form

, where p and q are integers and q

0.

Marks (6) Q 67 Identify the following as rational or irrational number.

Marks (6) Most Important Questions Q 1 Are all-rational numbers real numbers? Q 2 Is it possible to find a natural number between 1 and 2? Q 3 Is each point on the number line of the form Q 4 Find one rational number between 5 and 6.

m,where m is a natural numbers?

Q 5 Without actual division, find whether the following rational numbers are terminating or non-terminating repeating: 11/50 and 27/56. Q 6 Name the following: (a) The outer layer of the cell (b) The fluid like substance present outside the nucleus Q 7 Can photosynthesis take place outside the leaves? If yes, then where this process takes place?


Page 8

Q 8 Express the decimal expression

as a rational numbers.

Q 9 Insert four rational numbers between (1/3) and (1/4). Q 10 Give one example of each: (a) (b)

a parasitic plant a parasitic animal

(c)

a saprophyte

Q 11 Find two irrational numbers between 1.5 and 1.7. Q 12 (a) What is a parasite? (b) State a difference between total parasite and partial parasite? Q 13 Find an irrational number between (1/7) and (1/5). Q 14 (a) What is an insectivorous plant? (b) Give one example of insectivorous plant. (c) Give the structure and mode of nutrition of one insectivorous plant. Q 15 Rationalize the expression [1/{(2

3) +

7}].

Q 16 Find the value of the expression Q 17 What is the role of fungi in daily life?

Q 18 If

find the value of a and b.

Q 19 Give an example of two irrational numbers whose a) Sum is a rational number b)

Difference is a rational number.

c) Product is a rational number. Q 20 Classify the following expressions as rational or irrational.

Q 21 Prove that the following expression

Q 22


Page 9

Q 23 Simplify

Q 24 Simplify

.

Q 25 Is

.

Q 26 Simplify

.

Q 27 Identify as rational or irrational number.

a)

12 x

12 b)

4x

18

Q 28 Simplify (0.008)1/3.

Q 29 Find the value of x if Q 30 Find the value of

Q 31 Find the value of

.

.

Q 32 Find the value of x when

.

Q 33 Simplify the following:

Q 34 Show that Q 35 Express the following expression in the form of a rational number


Page 10

Q 36 Simplify


.

Page 11

2. Polynomials 3

Q 1 Check, whether 1 is the zero of the polynomial 9x - 5x + 20. Mark (1)

Q 2 Factorise

. Mark (1) 3

2

Q 3 Show that x = 1 is a zero of the polynomial 3x - 4x + 8x - 7. Mark (1)

Q 4 Is

a polynomial ? Mark (1)

Q 5 Check whether

is a polynomial. Mark (1) 5

2

Q 6 Find the degree of polynomial 30x - 15x + 40. Mark (1) Q 7 Find the degree of the polynomial 4x+5. Mark (1)

Q8 Mark (1) Q 9 Find the remainder when 6x - 5x + 2x - 9 is divided by (x - 1). 3

2

Mark (1) Q 10 Find the zeroes of the polynomial 6x2 – 3. Mark (1) 2

Q 11 Find the zeroes of the quadratic polynomial x – 4x + 3. Mark (1) Q 12 Find the zeroes of the quadratic polynomial x2 + 7x + 10. Mark (1) Q 13 Given a polynomial p(x). The graph of y = p(x) intersects the x-axis at three points. Find the number of zeroes of p(x). Mark (1) Q 14 Find the zeroes of polynomial 2x2 - 8. Marks (2) Q 15 Using a suitable identity, factorise the following expressions:

Marks (2)


Page 12

Q 16 Factorise: 4y2 - 4y + 1. Marks (2) Q 17 Show that x = 1 is a zero of the polynomial 2x3 - 3x2 + 7x - 6. Marks (2) Q 18 Give one example each of a binomial of degree 35 and a monomial of degree 100. Marks (2)

Q 19 Classify the following polynomials as linear, quadratic, cubic & bi-quadratic polynomials:

Marks (2) Q 20 Write the degrees of each of the following polynomials:

Marks (2) Q 21 Write the coefficient of x2 in each of the following:

Marks (2) Q 22 Find the zeroes of the quadratic polynomial x2 – 4x +3. Marks (2) 3

3

3

Q 23 Resolve into factors: 27x + y + z - 9xyz. Marks (3) 3

3

2

2

Q 24 Factorise: 64a - 27b - 144a b + 108ab . Marks (3) 3

3

Q 25 If p =2 - a, then show that a + 6ap + p - 8 = 0. Marks (3) Q 26 If x + y + z = 6, xy + yz + zx = 11. Find the value of x + y2 + z2. 2


Page 13

Marks (3) Q 27 Factorise: (i) 27y3 + 125z3 (ii) 64m3 - 343n3 Marks (3)

Q 28 Factorise: Marks (3) Q 29 Find the value of k, if (x - 1) is a factor of the following expression:

Marks (3) 4

3

Q 30 Divide the polynomial 3x - 4x - 3x - 1 by x - 1. Marks (3) 3

Q 31 Check whether 7 + 3x is a factor of 3x + 7x. Marks (3) Q 32 Find the zero of the polynomial in each of the following cases: (i) h(x) = 2x (ii) p(x) = cx + d, c (iii) p(x) = ax, a

0 0

Marks (3) Q 33 If x = 4/3 is a zero of the polynomial f(x) = 2x - 11x + kx - 20, find the value of k. 3

2

Marks (3) Q 34 Identify constant, linear, quadratic & cubic polynomials from the following polynomials:

Marks (3) Q 35 Give possible expressions for the length & breadth of the rectangle whose area is given by A = 25a2 - 35a + 12.


Page 14

Marks (4) 3

2

Q 36 Factorise the polynomial x - 23x + 142x - 120. Marks (4) Q 37 Factorise: (i) 12x2 - 7x + 1 (ii) f(x) = 2x2 + 7x + 3 Marks (4) Q 38 Write each of the following expressions, as a product of linear factors, with integer coefficient: (i) 5x2 + 16x + 3 (ii) 24p2 - 41p + 12 Marks (4) Q 39 Use the factor theorem to determine whether g(x) is a factor of f(x) in each of the following cases: (i) f(x) = x3 - 3x2 + 4x - 4, g(x) = x - 2

(ii)

Marks (4)

Q 40 Check whether the following polynomials have (x + 1) as a factor. (i) x3 +x2 + x + 1

(ii) Marks (4) 3

2

Q 41 Find the remainder when x + 3x + 3x + 1 is divided by (i) x + 1

(ii) Marks (4) Q 42 Which of the following expressions are polynomial and which are not? State reasons for your answer.

Marks (4) 3

2

Q 43 If f(x) = 2x - 13x + 17x + 12 , find (i) f(2) (ii) f(-3) Marks (4) Most Important Questions


Page 15

Q 1 Find the degree of the following polynomials. a) x3 + 4x5 + 3x - 7 b) x2 - 5x + 7 Q 2 Identify the following polynomials as linear, Quadratic or cubic. a) 5x+7 b) x3-8 Q 3 Identify which if the following are polynomials? a) x3 (2) x + 5x2 +7 b) x + (1/x) 2 Q 4 Find the coefficient of x in the following expressions. a) x3 - 2x2 + 5x - 9 b) -

x2 + x - 5

Q 5 Find the value of 2 x2 - 5x + 3 at x = 0. Q 6 Verify whether x = - 4 is a Zero of the polynomial x2 - 5x + 36.

Q 7 Verify if is a root (Zero) of the polynomial f(x) = 3x2 - 2. Q 8 If x = 2 is a root of the polynomial f(x) = 2x 2 - 3x + 7a, find the value of 'a'. Q 9 If x = - (1/2) is a zero of the polynomial f(x) = 2x3 + ax2 - 3x + 2 , find the value of „a‟. Q 10 Find one integral roots of the polynomial f(x) = x3 + 6x2 + 11x + 6. Q 11 If x = 0 and x = - 2 be the zeros of the polynomial p(x) = x3 - 2x2 + 3ax + b. Find the values of a and b. Q 12 Check whether 2, 3, - 1/2 are the rational roots of the polynomial 2x3 + 3x2 - 11x - 6. Q 13 If x = 4/3 is a root of the polynomial f(x) = 6x3 - 11x2 + kx - 20, find the value of k. Q 14 Define a polynomial. Are all algebraic expressions polynomials. Explain with the help of examples. Q 15 Find all the rational roots of the polynomial f(x) = 2x3 + x2 - 7x - 6. Q 16 Using remainder theorem find the remainder when x = 0 of the polynomial 2x 2 + 7x + 5. Q 17 Check if x + 3 is a factor of the polynomial 3x2 + 7x - 6. Q 18 Find the remainder when (x - 1/2) is divided by the polynomial 4x2 - 2x + 1. Q 19 Show that (x-1) is a factor of x10-1. Q 20 Find the remainder when x3 + 3x2 + 3x + 1 is divided by x + 3

2

Q 21 Find the remainder when x + 3x + 3x + 1 is divided by x + 3

. .

2

Q 22 Find a and b, if x + 1 and x + 2 are the factors of x + 3x - 2ax + b. Q 23 What should be subtracted from the polynomial x3 - 6x2 - 15x + 80 so that the result is exactly divisible by x 2 +x - 12. Q 24 What is the value of k, if x - 3 is a factor of the polynomial k2x3 - kx2 + 3kx - k? Q 25 The polynomials ax3 + 3x2 - 13 and 2x3 - 5x + a are divided by x + 2. If the remainder in each case is the same, find the value of „a‟. Q 26 Factorize x3 - 2x2y + x. Q 27 Evaluate: 233 - 173. Q 28 Write the expanded form of (3x + 2y - z)2. Q 29 Factorize the following 8a3 + b3 + 12a2b + 6ab2. Q 30 Factorize 3 - 12(a - b). Q 31 Factorize the following x3 + x - 3x2 - 3. Q 32 Factorize a2 + b2 + 2(ab + bc + ca).


Page 16

Q 33 Factorize 27a3 + 125b3. Q 34 Factorize 10x4y - 10xy4. Q 35 Factorize x3 - 12x(x - 4) - 64. Q 36 Factorize 8x3 + 27y3 + z3 - 18xyz. Q 37 Factorize (a2 - b2)3 + (b2 - c2)3 + (c2 - a2)3. Q 38 Factorize 32a3 + 108b3.

Q 39 Simplify the following Q 40 Factorize x8 - y8. Q 41 Using factor theorem factorize the following expression x3 - 6x2 + 3x +10.


Page 17

3. Coordinate Geometry Q 1 In which quadrant, the point P(x, y) will lie?, where x is a positive and y is a negative number. Mark (1) Q 2 Write the name of the point of intersection of coordinate axes. Mark (1) Q 3 Define quadrant. Mark (1) Q 4 Write the x-coordinate of a point which lies on y-axis. Mark (1) Q 5 What is the sign of the x-coordinate of a point in third quadrant. Mark (1) Q 6 If a point P(2,3) lies in first quadrant then what will be the coordinate of point Q opposite to it in fourth quadrant having equal distant from both the axes ? Mark (1) Q 7 In the fig. given below, name the point whose abscissa and ordinate both are positive and write the abscissa and ordinate of the point E :

Mark (1) Q 8 Write the answer of each of the following questions: (i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane? (ii) Write the coordinates of the point where these two lines (as described above) intersect. Marks (2) Q 9 A point is at a distance of 4 units from x axis and 5 units from the y axis. Represent the position of the point in the Cartesian plane and also write its Cartesian coordinates. Marks (2) Q 10 Find the value of x and y if: 1) (x-3, 7) = (5,7) 2) (2,2y-3)=(2,7) Marks (2)


Page 18

Q 11 From the graph below determine the coordinates of the points A, B, C and D.

Marks (2) Q 12 Plot the coordinates A (-4, 0) and B (3, 0) on the coordinate plane hence finds: 1) Distance of A from origin. 2) Distance of B from origin. 3) Distance between points. Marks (2) Q 13 In which quadrant does the following points lies? 1) A(-1, 2) 2) B(2, 2) 3) C(3,-2) 4) D(-2,-2) Marks (2) Q 14 Is Coordinate points (0, 5) and (5, 0) be same? Discuss. Marks (2) Q 15 If we plot the coordinate points A(-1, 0) , B(0, 1) , C(0,-1) and D(-1, 0) on Cartesian plane . Which figure comes out after joining the Points? Also find the side of figure. Marks (2) Q 16 What is the distance of a point (7, -6) from x-axis and y-axis? Marks (2) Q 17 Which of the following points: B(1, 0), C(0,1 ), E (-1, 0), F ( 0, -1), G (4, 0), H (0, -7) (i) lie on x –axis? (ii) lie on y – axis?


Page 19

Marks (3) Q 18 Locate the points (3, 0), (-2, 3), (2, -3), (-5, 4) and (-2, -4) in Cartesian plane. Also find the quadrant. Marks (3) Q 19 Find the value of x and y if: (2x-3/2, 4y-7/2 )=(5/2,3/2) Marks (3) Q 20 See figure, and write the following:

1) The points identified by the coordinates (1, 2) and (- 1, -2) 2) The coordinate of A, B, C and D. 3) Abscissa of point C. Marks (3) Q 21 Plot the points (x, y) given in the following table on the plane, choosing suitable units of distance on the axes: X

-1

2

0

3

2

y

3

-5

-3

-3

1

Marks (3) Q 22 What will be the position of point A(2, 1) if: 1) Abscissa is multiplied by -1? 2) Ordinate is multiplied by -2? 3) Point each coordinate is multiplied by -3? Marks (3) Q 23 Plot the following points in a Cartesian plane: (-2,4), (3,-1), (-1, 0), (1, 2) & (-3, -5) Marks (4) Q 24 Observe the fig. given below and answer the following:


Page 20

(i) (ii)

The coordinates of B. The Coordinates of C.

(iii) (iv)

The point identified by the coordinate (-3, -5). The abscissa of the point D.

(v)

The coordinates of H.

Marks (4) Q 25 Determine the quadrants in which the following points lie; (i)

A (1,1)

(ii) B (2,4) (iii) C (-3, -10) (iv) D (-1,2) (v) E (1,-1) (vi) F (-2,-4) (vii) G (-3, 10) (viii) H (1,-2) Marks (4) Q 26 A car starts from the center of city and in each consecutive hour it covers a distance of 15km (along north), 5 km (along east), 15 km (along south) and 10 km (along west) respectively. Assuming the centre of city to be the origin, north-south direction is along y axis and west-east direction is along x axis; show the various position of the car on the Cartesian plane. Also, find how far is the car from x and y axis respectively at its final position. Marks (4) Q 27 See the figure, and write the following: 1) The coordinates of B. 2) The point identified by the point (-3, -5). 3) The abscissa of point D. 4) The ordinate of the point E. 5) The point identified by the coordinates (2, -4). (4 marks) Marks (4)


Page 21

Most Important Questions Q 1 What is the name to the horizontal and vertical line in a coordinate system? Q 2 The origin is indicated by what coordinates? Q 3 How many Quadrants are there in the Cartesian Plane? Q 4 In which Quadrant will the coordinates (-2,3 ) lie? Q 5 in which Quadrant will the coordinates (-3, -4) lie? Q 6 What is the abscissa and the ordinate in the coordinates (3, -5) Q 7 Write the abscissa and the ordinate of the coordinates of the points ( 0,3) (3,0) (0, 0). Q 8 Plot the following point on the number line using a graph and join the points. a) (3, -4) b) ( -3, 2) Q 9 Which coordinates do the following points indicate?

Q 10 Plot the following coordinates on the Cartesian system : (2, 0), (-3, -4), (2, -2), (-4, 0), (-2, 3) and (1, 1.5). Q 11 Define the following: a) The Cartesian plane b) The coordinate axes c) The Origin.


Page 22

4. Linear Equations in Two Variables Q 1 State true or false : ax + by + c = 0, represents a closed curve. Mark (1) Q 2 Find the coefficient of x in the equation {

(a2 + b2) } x + {

(a2 - b2)} y =

(a2 b2). Mark (1)

Q 3 Write the coordinate axis represented by the line y = 0 . Mark (1) Q 4 Write the equation 12x +3y = 20 in the form of ax+by+c= 0 and find out the values of a, b and c. Mark (1) Q 5 Write 2x = 3y+5 in standard form of equation in two variables. Mark (1) Q 6 Write the equation of the line parallel to the y-axis. Mark (1) Q 7 Write the equation of the line parallel to x-axis. Mark (1) Q 8 Write the coordinate axis represented by the line x = 0 . Mark (1) Q 9 State true or false: A linear equation in two variables can have only two solutions. Mark (1) Q 10 State true or false: x = ay is the equation of line passing through origin. Mark (1) Q 11 State true or false: A linear equation of two variable can have infinitely many solutions. Mark (1) Q 12 State true or false: x = y + 2 represent a line passes through origin. Mark (1) Q 13 Find the value of k, if x=2 , y=1 is a solution of the equation 2x+3y=k. Marks (2)


Page 23

Q 14 The cost of a notebook is twice as the cost of a pen. Write a linear equation in two variables to represent this statement. Marks (2) Q 15 Find the value of k in the equation 2x+3y= k, where x = 2 and y = 3 is the solution of the equation. Marks (2) Q 16 Find the two different solutions of the equation 2x+y=4. Marks (2)

Q 17 If a = b = 3, then find the value of x from the equation

. Marks (2)

Q 18 Let y varies direct as x. If y=14, when x=7, then write a linear equation. What is the value of y when x= -2? Marks (2) Q 19 How many solution(s) of the equation 5x-3=3x+5 are there on the: (i) Number line (ii) Cartesian plane. Marks (2) Q 20 For what value of k the point (k,5) lies on the line 4x-5y=10 ? Marks (2) Q 21 If x = 1 and y = 1 is the solution of the equation 5x + 2ay = 3a, find the value of a. Marks (2)

Q 22 Marks (2) Q 23 The taxi fare in a city is as follows: For the first kilometre, the fare is Rs 8 and for the subsequent distance it is Rs 5 per km. Taking the distance covered as kilometres and total fare as rupees, write a linear equation for this information. Marks (3) Q 24 Find four different solutions of the equation x+2y=6. Marks (3) Q 25 Write each of the following equations in the form ax+by+c=0 and


Page 24

Indicate the values of a, b and c in each case: (i) (ii) (iii)

2x + 3y = 4.37 x-4 = 3y 4= 5x-3y Marks (3)

Q 26 Solve the equation 2x+1=x-3, and represent the solution(s) on (i) the number line. (ii) the Cartesian plane.

Marks (3)

Q 27 Evaluate: (5x+1)(x+3)-8=5(x+1)(x+2). Marks (3) Q 28 If the point (-1,-5) lies on the graphs of 3x=ay+7 and y=bx+7, find the value of a and b. Marks (3) Q 29 At what point does the graph of the linear equation 2x+3y=9 meet a line which is parallel to the y-axis, at a distance of 4 units from the origin and on the right of the y-axis. Marks (3) Q 30 If the point (4,3) lies on the graph of the equation 3x-ay=6, find whether (-2,-6) also lies on the same graph. Marks (3) Q 31 Give the equations of two lines passing through (-2,-4). How many more such lines are there, and why? Marks (3) Q 32 In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius: F=(9/5)C+32 Draw the graph of linear equation above using Celsius for x-axis and Fahrenheit for y-axis. Marks (4) Q 33 Write each of the following as an equation in two variables: (i) x=-5 (ii) y=2 (iii) 2x=3 (iv) 5y=2 Marks (4) Q 34 The taxi fare in a city is as follows: For the first kilometer, the fare is Rs.20 and for the subsequent distance it is Rs. 6 per km. Taking x km as the distance covered and Rs. y as the total fare, write a linear equation for this information and draw its graph. Marks (4)


Page 25

Q 35 The cost of a box is Rs.25. Taking x as the number of boxes and y, the total cost in rupees, construct a linear equation. Also, draw the graph. Marks (4) Q 36 The ratio of hydrogen and oxygen in water is 2:1. Set up an equation between hydrogen and oxygen and draw its graph. From the graph read the hydrogen if oxygen is 6 gram. Marks (4) Q 37 The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the acceleration produced is (i) 5 m/sec2 , (ii) 6 m/sec2.

Marks (4)

Q 38 The parking charges of a car at certain place in Delhi is Rs.50 for first one hour and Rs. 10 for subsequent hours. (a) Write down the equation and draw the graph for the data.(b) Read the charges from the graph (i) for 2 hours (ii) for 8 hours. Marks (4) Q 39 Two players A and B together scored 40 runs in a cricket match. If there is no extra run scored in their partnership, then represent this information in the form of linear equation in two variables. Draw graph of the linear equation. From the graph, find the runs recorded by player A if run scored by player B is 10. Marks (4)

Q 40 Marks (4) Q 41 If one-fourth of the sum of a number and seven is four less than three times the number, find the number. Marks (4) Q 42 Solve the equation 2x + 1 = x – 3, and represent the solution(s) on (i) the number line, (ii) the Cartesian plane. Marks (5) Q 43 Find two solutions for each of the following equations:

Marks (6) Most Important Questions


Page 26

Q 1 Represent the linear equation in two variables in its standard form. 2y – 3 + 2x =5 Q 2 What is the value of a, b, c in the given equation 3x – 5= 2y Q 3 Find the value of b and c in the equation 3x= 15 Q 4 Check whether (4,0) is a solution of the equation2x+3y =8

Q 5 Find if

2, -1 is a solution of the equation x + 3y =1

Q 6 Form an equation in two variables with the given information: the number of ducks is three more than three times number of hens, and the total of all hens and ducks is 156. Q 7 Find four solutions of the given equations 3x –y=4 Q 8 Find the value of k, when x = -1 and y =2 in the equation 3x –7y = 3k .

Q 9 Find two solutions for each of the following equations: (i)

4x + 3y =12

(ii)

2x + 5y = 0

Q 10 Express the following linear equations in the form ax + by + c = 0 and find the values of a , b, c.In the equation

Q 11 Write the following as linear equation in two variables.2x = 15 and –3y –4 =0 Q 12 Write five solutions of the given equation:

x + y = 7.

Q 13 Write the four solutions of the equation (2x - 1)/(3y - 5) = 1/3. Q 14 Write the solution of the equation x +y =4 Q 15 Lata and Gautami together contributed Rs. 100 for a donation camp. Represent this situation graphically. Q 16 The taxi fare in a city is as follows: For the first km, the fare is Rs.8 and for every subsequent Km it is Rs. 5. Taking the distance traveled as x and the total fare as y, represent the equation graphically. Q 17 Represent the equation 2 + 3y = 7x graphically. Q 18 Form an equation for the statement: The sum of cost of pens and twice the cost of pencils is Rs. 6 and represent the situation graphically. Q 19 Form the graph of the equation y = 2x Q 20 Plot the graph of the equation y – 2x = 4


Page 27

Q 21 Express y =4 as linear equation in two variables. Q 22 Give the representation of 2x +9 =0 as an equation in a) One variable b) Two variable Q 23 The temperature in degree Celsius is given by the following formula F = (9/5) C + 32 Answer the following questions

a.

What will be the temperature in degree Celsius if the temperature is 45

b. c.

If the temperature is 0 C, what is the temperature in Fahrenheit? Is there a temperature, which is numerically the same in both Fahrenheit and Celsius? If yes, find it.


F?

Page 28

5. Introduction to Euclids Geometry Q 1 Write Ecluid's definition of straight line. Mark (1) Q 2 State true of false : Two distinct lines intersect at more than one point. Mark (1) Q 3 Fill in the blank : A_____ is that which has no part. Mark (1) Q 4 How many points a line segment can have? Mark (1) Q 5 State true or false: Given two distinct points, there are two lines which passes through them. Mark (1) Q 6 Fill in the blank : Three or more lines are said to be _____if their common point lies on them. Mark (1) Q 7 According to Euclid, Name of geometrical figure which has only length and breadth. Mark (1) Q 8 State two equivalent versions of Ecluid's fifth postulate. Marks (2) Q 9 If A, B and C are three points on a line, and B lies between A and C then prove that AB + BC = AC. Marks (2) Q 10 If a point C lies between two points A and B such that AC = BC, then prove that AC = (1/2)AB. Marks (2) Q 11 Which axiom is related to comparison of things According to Euclid‟s axiom? Marks (2) Q 12 If a point C lies between two points A and B such that AC = BC, then find the relation between BC and AB. Marks (2) Q 13 As per Euclid‟s axiom, „If equals are added to equals, explain with examples. Marks (2)


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Q 14 What are the types of the boundaries of the surfaces? Marks (2) Q 15 Which proof was given by the great mathematician Thales about circle? Marks (2) Q 16 Which type of the the shape of altars used for household rituals in the Vedic period? Marks (2) Q 17 In Fig., if AC = BD then prove that AB = CD.

Marks (3) Q 18 If PS = RT as shown in the figure, then what will be the value of ST?

Marks (3) Q 19 Define the following according to book ‟Element” by Euclid: (i) Surface (ii) Point (iii)Straight line (iv) Line. Marks (3) Q 20 What is Playfair‟s Axiom? Marks (3) Q 21 What are two equivalent versions of Euclid‟s fifth postulate? Marks (3) Q 22 Define Postulate according to Euclid. Marks (3) Q 23 State Euclid‟s five postulates. Marks (4) Q 24 Define the following terms: (i) Intersecting lines (ii) Parallel lines


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(iii) Line segment (iV) Collinear points. Marks (4) Q 25 Define the following terms: (i) Axiom (ii) Theorem

Marks (4)

Q 26 Give seven Euclid‟s axioms. Marks (4) Q 27 Mention Five postulates of Eulid. Marks (4) Q 28 Prove that an equilateral triangle can be constructed on any given line segment. Marks (4) Q 29 Give the definition for each of the following terms:(i) Parallel lines, (ii) Perpendicular lines, (iii) Line segment, (iv) Radius of a circle,(v)Square. Marks (4) Most Important Questions

Q 1 Define the following terms a) Point b) Line c) Line Segment Q 2 Explain the following terms : a) Concurrent lines b) Collinear points c) Parallel lines d) Intersecting lines Q 3 How many lines can pass through a given point? Q 4 How many lines can pass through two given points? Q 5 How many line segments can pass through three collinear point A, B, C? Q 6 State True or false a) Two lines intersect at a point. b) A line segment has a fixed length. c) A ray has a fixed length. d) Only one line can pass through a given point. e) Two lines are coincident if they have only one point in common.


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Q 7 Fill in the blanks: a)Two distinct points in a plane determine a. …line. b) Given a line and a point, which is not on the line, there is …… line, which passes through the given point and is …… to the given line. c) Whole of anything is …… to the sum of its parts and …… than any one of them. d) If equals are subtracted from wholes the remainders are …… . Q 8 State the two equivalent version of Euclid‟s fifth postulate. Q 9 If C is a point which lies between two points A and B such that AC = BC, then prove that AC= ½ AB. Explain by drawing the line. Q 10 Prove that the mid-point of any line segment is unique.


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6. Lines and Angles Q 1 State corresponding angles axiom. Mark (1) Q 2 Define collinear points. Mark (1) Q 3 It is given that

XYZ = 64° and XY is produced to a point P. If ray YQ bisect

ZYP , find

XYQ and reflex

QYP.

Marks (2) Q 4 In figure if x + y = w + z, then prove that AOB is a line.

Marks (2) Q 5 In the figure, find the value of y°.

Marks (2) Q 6 In Fig, lines PQ and RS intersect each other at point O. If

POR :

ROQ = 5 : 7, find all the angles.


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Marks (2) Q 7 Find out the two pairs of adjacent angles in the fig. given below:

Marks (2) Q 8 The measure of an angle is twice the measure of its supplementary angle. Find its measure. Marks (2) Q 9 In Fig, lines PQ and RS intersect each other at point O.

If

POR :

ROQ = 2 : 3, find angle POR and angle ROQ.

Marks (3) Q 10 In Fig. two straight lines PQ and RS intersect each other at O. If POT = 75°,find the values of a,b and c.

Marks (3)


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Q 11 In figure, lines l1 and l2 intersect at O forming angles as shown in the figure. If a = 35° Find the value of b, c and d.

Marks (3) Q 12 If the angles of a triangle are in the ratio 2 : 3 : 4, find all the three angles. Marks (3) Q 13 In the figure, side QR of

PQR has been produced S, if

P : Q : R = 3 : 2 : 1 and RT

PR, then

TRS will be

Marks (3) Q 14 Find the correct figure having: a) adjacent angles but not linear pair, b) vertically opposite angles, c) linear pair and adjacent angles only

Fig-1


Fig-2

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Fig-3 Marks (3) Q 15 Two supplementary angles are in the ratio 4:5. Find the angles. Marks (3) Q 16 In figure, determine the value of y.

Marks (3) Q 17 In figure the sides AB and AC of are produced to points E and D BCD

respectively meet at point O, then prove that

respectively. If bisectors BO and CO of

BOC = 90° - (1/2)

CBE and

BAC

Marks (4)

Q 18 In figure OP||RS. Determine

PQR .

Marks (4) Q 19 ABCDE is a regular pentagon and bisector of


BAE meets CD

in M. IF bisector of

BCD meets AM at P find

CPM.

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Marks (4) Q 20 In figure, ray OS stands on a line POQ. Ray OR and Ray OT are angle bisectors of ROT.

POS and

SOQ, respectively. Find

Marks (4) Q 21 In Figure, OP,OQ,OR and OS are four rays.Prove that

POQ+

QOR+

SOR+

POS=360°.

Marks (4)


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Q 22 S is a point on side QR of

PQR such that PS=PR. Show that PQ>PS.

Marks (4) Q 23 In the given figure

Q>

R and M is a point QR such that PM

is the bisector of angle P . If the perpendicular from P on QR meets QR at N, then prove that

MPN = (1/2)(

Q-

R)

Marks (4) Most Important Questions Q 1 Find an angle which is 1/3rd its supplement. Q 2 Two supplementary angles differ by 34°. Find the angles. Q 3 In the given figure, OA and OB are the opposite rays and


AOC +

BOD = 90°. Find

COD.

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Q 4 In the given figure, if x + y = w + z, then prove that AOB is a line. Q 5 In the given figure, find the value of x

Q 6 In the given figure if a – 2b = 30, find the value of a and b. Also given that

Q 7 Ray OE bisects

AOB and Of is a ray opposite to OE. Show that


FOB =

AOC and

BOC form a linear pair.

FOA

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Q 8 In the given figure, find the value of x Q 9 Three lines intersect at a point „o‟, forming angles as shown in the figure. Find the value of x, y, z and u.

Q 10 In the given figure find the value of x , hence find all the three angles.


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Q 11 In the given figure find the value of OP, OQ, OR and OS be any four rays, Prove that

POQ +

QOR +

SOR +

POS =

360° Q 12 In the given figure

1 = 60 and

2 = (2/3)rd of a right angle. Prove that the line l

Q 13 In the given figure, OP RS. Determine

m.

PQR.

Q 14 If two parallel are intersected by a transversal, the bisectors of any pair of alternate interior angles are parallel.


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Q 15 The side BC of a

ABC is produced, such that D is on ray BC. The bisector of

figure, Prove that ABC +

ACD = 2

ALC

Q 16 In the given figure, find all the angles of

Q 17 In the given figure AM

Q 19 In the given figure, PQ

ABC

BC and AN is the bisector of

Q 18 In the given figure AB DE, find

A meets BC in L as shown in the given

A. If

B = 65° and

C = 33°, find

MAN.

AED.

PS, PQ SR,


SQR= 28° and

QRT= 65°, then find the values of x and y.

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Q 20 Find the value of x in the given figure, Q 21 In the given figure AB divides

DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.

Q 22 In the given figure, find the value of x.


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7. Triangles Q 1 In figure, OA = OB and OD = OC.

Show that (i)

AOD

(ii) AD

BOC

BC.

Marks (2) Q 2 ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal. Show that (i) ABE ACF (ii) AB = AC, i.e. ABC is an isosceles triangle.

Marks (2) Q 3 AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB.

Marks (2)


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Q 4 Triangle ABC is an isosceles triangle; CD is bisector to the base AB. Prove that the altitude, the bisector and the median to the base of triangle ABC match.

Marks (2) Q5

Marks (2) Q 6 ABCD is a parallelogram and BEFC is a square. Show that triangles ABE and DCF are congruent.


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Marks (2) Q 7 PQR and QST are two triangles such that

Prove that

4=

6

1=

3

4=

5

R=

T

Marks (2) Q 8 BD is a line segment. From D two line segments AD and DC are drawn such that AD = CD, also

3=

4. Prove that segment BD bisects

ABC.


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Marks (2)

Q 9 D is a point on side BC of AB > AD.

ABC such that AD = AC. Show that

Marks (3) Q 10 P is a point equidistant from two lines l and m intersecting at point A. Show that the line AP bisects the angle between them.

Marks (3) Q 11 In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD. Show that AD = AE.

Marks (3)


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Q 12 In

ABC, the bisector AD of

A is perpendicular to side BC. Show that AB = AC.

Marks (3) Q 13 AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that BAD = (i) (ii)

ABE and

DAP AD = BE

EPA =

DPB. Show that

EBP

Marks (3) Q 14 In figure, AC = AE, AB = AD and

BAD =

EAC. Show that BC = DE.

Marks (3) Q 15 Angles opposite to equal sides of an isosceles triangle are equal. Marks (3)


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Q 16 ABC and DBC are two isosceles triangles on the same base BC. Show that

ABD =

ACD.

Marks (3) Q 17 ABC and DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at P, show that (i)

ABD

ACD

(ii)

ABP

ACP

Marks (3) Q 18 AB is a line-segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B. Show that the line PQ is the perpendicular bisector of AB. Marks (4) Q 19 If D is the mid-point of the hypotenuse AC of a right triangle ABC, prove that BD = (1/2)AC.

Marks (4)


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Q 20 angle.

ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB. Show that

BCD is a right

Marks (4) Q 21 Prove that the perimeter of a triangle is greater than the sum of its altitudes. Marks (4) Q 22 In Figure, PR>PQ and PS bisects

QPR. Prove that

PSR>

PSQ.

Marks (4) Q 23 In figure, the side QR of that

QTR=(1/2)

PQR is produced to a point S. If the bisectors of

PQR and

PRS meet at point T, then prove

QPR.

Marks (4)


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Q 24 In Figure,

B<

A and

C<

D. Show that D
Marks (4) Most Important Questions

Q 1 In the given figure ABCD is a quadrilateral in which AD = BC and

(i)

ABD

DBA =

CBA Prove that

BAC

(ii) BD= AC

(iii)

ABD =

BAC

Q 2 Line segment AB is parallel to another line segment CD. O is the mid-point of AD. Show that

(i)

AOB

DOC (ii) O is also the mid-point of BC.


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Q 3 In the given figure it is given that

Q 4 In

A=

C and AB = BC. Prove that

ABD

CBE.

ABC, AB = AC, and the bisectors of angles B and C intersect at point O. Prove that BO=CO and the ray AO is the bisector

of angle BAC. Q5

ABC and

ABD =

DBC are two triangles on the same base BC such that AB= AC and DB= DC. Prove that

ACD.


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Q 6 Line l is the bisector of an angle

A and B is any point on l. BP and BQ are perpendiculars from B to the arms of

A. Show that (i)

APB

AQB

(ii) BP = BQ or B is equidistant from the arms of

A.

Q 7 P is a point on the bisector of ABC. If the line through P parallel to AB meets BC at Q, prove that the triangle BPQ is isosceles.

Q 8 In two right triangles one side and an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent. Q 9 AD and BE are respectively altitudes of an isosceles triangle ABC with AC = BC. Prove that AE = BD.

Q 10 If the bisector of the exterior vertical angle of a triangle is parallel to the base. Show that the triangle is isosceles. Q 11 If E and F are respectively the midpoints of equal sides AB and AC of a triangle ABC, Show that BF = CE. Q 12 In an isosceles triangle ABC with AB= AC, D and E are points on BC such that BE= CD, show that AD =AE. Q 13 ABC and DBC are two isosceles triangles on the same base BC. Show that


ABD =

ACD.

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Q 14 ABC is a right-angled triangle in which

A = 90

and AB = AC. Find

B and

C.

Q 15 ABC is an isosceles triangle with AB = AC. Show that B= C. Q 16 If BE and CF are equal altitudes of a triangle ABC. Prove that triangle ABC is isosceles. Q 17 AD is the altitude of an isosceles triangle in which AB = AC. Show that (i) AD bisects BC (ii) AD bisects A. Q 18 In the given figure QPR = PQR and M and N are respectively on sides QR and PR of PQR such that QM= PN. Prove that OP= OQ, were O is the point of intersection of PM and QN.

Q 19 Fill in the blanks: (i) Sides opposite to equal angles of a triangle are …… (ii) In an equilateral triangle all angles are …… and of …… degree. (iii)In right triangles ABC and DEF, if hypotenuse AB = EF and AC = DE, then (iv)If altitudes CE and BF of a triangle ABC are equal, then AB =…

ABC

…

(v)In triangle ABC if A = C then AB =… Q 20 State true or False (i)

If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.


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(ii)

The bisectors of two equal angles of a triangle are equal.

(iii)

If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(iv) (v)

The two altitudes corresponding to two equal sides of a triangle need not be equal. Two right triangles are congruent if hypotenuse and a side of the triangle are respectively equal to the hypotenuse and the side of the other triangle.

Q 21 Show that in a right angled triangle, the hypotenuse is the longest side. Q 22 Prove that any two sides of a triangle are together greater than twice the median drawn to the third side.

Q 23 In the given figure PQR is a triangle and S is any point in its interior, show that SQ + SR < PQ + PR.

Q 24 Prove that the perimeter of a triangle is greater than the sum of the three medians.


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Q 25 In the given figure

E>

A and

C>

D. Prove that AD > EC.

Q 26 In the given figure T is a point on the side QR of

PQR and S is a point such that RT = ST. Prove that PQ + PR > QS.

Q 27 Of all the line segments drawn from a point P to a line m not containing P, let PD be the shortest. If B and C are points on m such that D is the mid-point of BC, prove that PB = PC.


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Q 28 In the given figure AC > AB and D is the point on AC such that AB =AD. Prove that BC > CD.

Q 29 In the given figure prove that CD +DA +AB +BC >2AC

Q 30 Fill in the blanks: (i) In a right triangle the hypotenuse is the… side. (ii) The sum of three altitudes of a triangle is... than its perimeter. (iii) The sum of any two sides is …… than the third side. (iv) If two sides of a triangle are unequal, then the larger side has ….. angle opposite to it. (v) If two angles of a triangle are unequal, then the smaller angle has the ….. side opposite to it.


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8. Quadrilaterals Q 1 Name a quadrilateral whose each pair of opposite sides is equal. Mark (1) Q 2 What is the sum of two consecutive angles in a parallelogram? Mark (1) Q 3 The angles of quadrilateral are respectively 100

, 30

, 92 and x. Find the value of x. Marks (2) Q 4 The angles of quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral. Marks (2) Q 5 Thee sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a parallelogram. Marks (2) Q 6 In a triangle ABC, P,Q and R are the mid – points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ. Marks (2) Q 7 Find the four angles P, Q, R and S in the parallelogram PQRS as shown below.

Marks (2) Q 8 Two opposite angles of a parallelogram are (5x + 1)° and (49 – 3x)°. Find the measure of these opposite angles of the parallelogram. Marks (2) Q 9 Prove that each of the four sides of a rhombus is of the same length. Marks (2) Q 10 ABCD is a rhombus. Show that diagonals AC bisects angle A as well as angle C. Marks (2)


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Q 11 In the figure given below ,ABCD and PQRC are rectangles and Q is the mid – point of AC.

Prove that PR = ½ AC.

Marks (2) Q 12 Find the values of a and also find angles related to a as shown in the figure.

Marks (3) Q 13 Prove that angle bisectors of a parallelogram form a rectangle. Marks (3) Q 14 ABC is an isosceles triangle with AB = AC and let D, F, E be the mid-points of BC, CA and AB respectively. Show that AD is perpendicular to EF and AD bisects EF. Marks (3) Q 15 In a triangle ABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram. Marks (3) Q 16 ABCD is parallelogram. P is a point on AD such that AP = 1/3 AD and Q is a point on BC such that CQ = 1/3 BC. Prove that AQCP is a parallelogram. Marks (3)


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Q 17 In the figure given below , triangle ABC is right – angled at B. Given that AB = 9 cm, AC = 15 cm and D, E are the mid – points of the sides AB and AC respectively, calculate the area of trapezium DECB.

Marks (3) Q 18 ABCD is a rhombus. AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC. Marks (4) Q 19 In a quadrilateral ABCD, CO and DO are the bisectors of

C and

D respectively. Prove that

. Marks (4) Q 20 AD is the median of

ABC. E is the mid point of AD. BE produced meet AC at F. Show that AF=(1/3)AC.

Marks (4) Q 21 Show that the quadrilateral formed by joining the mid point of the consecutive sides of a rectangle is a rhombus. Marks (4) Q 22 P is the mid-point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R. prove that AR = 2BC. Marks (4) Q 23 P,Q,R are, respectively, the mid points of sides AB, BC and CA and of a triangle ABC. PR and AQ meet at X. BR and PQ meet at Y. Prove that XY = ¼ AB. Marks (4) Most Important Questions

Q1

Q2 Q 3 The sides BA and DC of a quadrilateral ABCD are produced as shown in fig. Prove that a + b = x + y.


Page 60

Q 4 The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. Q5

Q 6 In a parallelogram ABCD, prove that sum of any two consecutive angles is 180 0. Q7

Q 8 In the given figure, ABCD is a parallelogram. Compute the values of x and y.


Page 61

Q 9 In the given figure, AN and CP are perpendicular to the diagonal BD of a parallelogram ABCD.

Prove that :

Q 10 Q 11 In the given figure, find the four angles A, B, C and D in the parallelogram ABCD.


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Q 12 In the figure given below,find all the angles of triangle BCD.

Q 13 Prove that angle bisectors of a parallelogram forms a rectangle. Q 14 AB and CD are the two parallel lines which are cut by a transversal l in point X and Y respectively. The bisectors of interior angles intersect in P and Q. form a parallelogram. Is it a rectangle? Q 15 ABCD is a Rhombus AD is produced to E so that DE = DC and EC produced meets AB produced in F. prove that BF = BC. Q 16 In a quadrilateral ABCD, CO and DO are the bisector of C and D respectively. Prove that COD = (1/2)( A + B Q 17 ABC be an isosceles triangle with AB = AC and let D, E, F are the mid-points of BC, CA and AB respectively. Show that AD perpendicular to EF ad AD bisector of EF. Q 18 In triangle ABC, AD is the median through A and E is the mid-point of AD. BE produced meets AC in F proved that AF = 1/3 AC Q 19 Show that the quadrilateral formed by joining the mid point of the consecutive sides of a rectangle is a rhombus. Q 20 ABCD is parallelogram. P is a point on AD such that AP = 1/3 AD and Q is a point on BC such that CQ = 1/3 BP. Prove that AQCP is a parallelogram. Q 21 In a triangle ABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram. Q 22 P is the mid-point of side AB of a parallelogram ABCD. A line through B parallel to PD meets DC at Q and AD produced at R. prove that AR = 2BC. Q 23 P, Q, R are, respectively, the mid points of sides BC,CA and AB of a triangle ABC. PR and BQ meet at X. CR and PQ meet at Y. prove that XY = ¼ BC


Page 63

9. Areas of Parallelograms and Triangles Q 1 State true or false : A diagonal of a parallelogram divides it into two parts of equal areas. Mark (1) Q 2 State true or false: Parallelograms on the same base and between the same parallels are equal in area. Mark (1) Q 3 State true or false: A parallelogram and triangle on same base and between same parallel lines are equal in area. Mark (1) Q 4 ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show ar(

ACB) = ar(

ACF).

Marks (2) Q 5 BD is one of the diagonal of a quadrilateral ABCD. AM and CN are the perpendiculars from A and C, respectively, on BD. Show

that

. Marks (2)

Q 6 In fig. D and E are points on sides AB and AC respectively of BC.

ABC such that ar(

BCE) = ar(

BCD). Show that DE ||

Marks (2) Q 7 Prove that of all the parallelograms of which the sides are given, the parallelogram which is rectangle has the greatest area. Marks (2) Q 8 In figure,

RPQ = 90

, S is the mid-point of QR and SP= 2.5 cm. Compute the area of the triangle PQR.

Marks (2)


Page 64

Q 9 In the following figure, PQRS is a trapezium in which PQ || SR. Prove that ar(

QOR) = ar(

POS).

Marks (2) Q 10 In the given figure, PQR and QST are two quadrilateral triangles such that S is the mid-point of QR.

Marks (2) Q 11 The angles of a quadrilateral are in the ratio 1:2:3:4. Find all the angles of the quadrilateral. Marks (2) Q 12 The angles of a quadrilateral are in the ratio 2:4:5:7. Find the angles. Marks (2) Q 13 Prove that , the bisector of any two consecutive angles of parallelogram intersect at right angle. Marks (2) Q 14 Two opposite angles of a parallelogram are (3x-2)0 and (50-x)0. Find the measure of each angle of the parallelogram. Marks (2) Q 15 Prove that the area of triangle is half the product of any of its sides and the corresponding altitude.

Marks (3)


Page 65

Q 16 prove that the area of a trapezium is equal to sides and a, b are the measurement of parallel sides.

, where h is the perpendicular distance between parallel

Marks (3) Q 17 If in fig ABCD is a parallelogram, DE

AB and BF

AD. If AB = 16cm, DE = 8 cm and BF = 10 cm, find AD.

Marks (3) Q 18 ABCD is a trapezium in which AB = 5 cm , AD = BC = 4 cm and distance between parallel sides AB and DC is 3 cm. Find DC and area of trapezium ABCD. Marks (3) Q 19 O is any point on diagonal BD of the parallelogram ABCD. Prove that ar( Marks (3)

OAB) = ar(

OBC).

Q 20 ABCD is a Quadrilateral. A line through D, parallel to AC, meets BC produced in P as shown in figure. Prove that ar(

ABP)

= ar(Quad ABCD).

Marks (3) Q 21 XY is a line parallel to side BC of Show that ar(

ABE) = ar(

ABC. BE || AC and CF || AB meet XY (produced on both sides) in E and F respectively.

ACF). Marks (3)

Q 22 P is the point in the interior of a parallelogram ABCD. Show that

. Marks (4) Q 23 A quadrilateral ABCD is such that diagonal BD divides its area in two equal parts. Prove that BD bisect AC. Marks (4) Q 24 In a triangle ABC , D is the mid-point of AB. P is any point of BC . CQ || PD meets AB in Q. Show

that. Marks (4)


Page 66

Q 25 Prove that parallelogram on the same base and between the same parallels are equal in area. Marks (4) Q 26 The diagonal of a parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area. Marks (4) Q 27 A point O inside a rectangle ABCD is joined to the vertices. Prove that ar( Marks (4) Q 28 In (

AOB)+ar(

COD)=(1/2)ar( ||gm ABCD).

ABC, D is the mid-point of BC, E is the mid-point of BD. If „O‟ is the mid-point of AE, prove that ar (

BOE)=(1/8) ar

ABC). Marks (4)

Q 29 The side AB of a parallelogram ABCD is produced to any point P. A line through A parallel to CP meets CB produced in Q and the parallelogram PBQR is completed. Show that ar(||gm ABCD)= ar (||gm BPRQ). Marks (4) Q 30 D, E, F are the mid-points of the sides AB,BC and CA respectively of is half the area of

ABC. Prove that DBEF is a parallelogram whose area

ABC.

Marks (4) Most Important Questions Q 1 Prove that parallelograms on equal bases and between the same parallels are equal in area. Q 2 Prove that parallelograms on the same base and having equal areas lie between the same parallels. Q 3 The area of parallelogram PQRS is 152 cm2. Find the area of rectangle PQXY. If the base PQ = 19 cm, find the height of the parallelogram.

Q 4 Prove that the area of triangle is half as the area of parallelogram if a parallelogram and a triangle lie on the same base and between the same parallels. Q 5 Prove that the area of a triangle is half the product of its base and corresponding height.


Page 67

Q 6 Find the area of

PQR given that the area of the parallelogram PQRS is 25 cm2.

Q 7 Show that the area of rhombus is half the product of its diagonals. Q 8 Show that the area of trapezium is half the product of sum of parallel sides and perpendicular distance between parallel sides.

Q 9 Prove that area(

AFG) =

area(BDEF).

Q 10 Prove that two triangles having same base and equal areas lie between the same parallels. Q 11 Prove that a median of a triangle divides it into two triangles of equal area. Q 12 In the given figure, ABCD is a parallelogram and O is any point inside ABCD. Prove that

area(

AOB) + area(

OCD) =

area(ABCD).

Q 13 ABC is a triangle in which D is the mid-point of BC and E is the mid-point of AD. prove that triangle BED= 1/4 (area of triangle ABC) . Q 14 If D, E and F are the mid points of sides AB, BC and AC respectively then show that

(i)

area (

(ii) area (

ADE) =

area (AFE)

BDE) = area (

(iii) area ( ADEF) =

area (

CEF)

ABC)


Page 68

Q 15 In a trapezium PQRS prove that area (

POR) = area (

SOR)

Q 16 In the given figure, if BE||CF and area (ABCE) = area (BDEF) then prove that AD|| BE.

Q 17 If one diagonal of a quadrilateral bisect the other then prove that the first diagonal divides the quadrilateral into two triangles of equal area. Q 18 In the given figure E is the mid point of BC and D is the mid point of AE. PEDB and QEDC are parallelograms then show that

area (

PBE) + area(

QCE) =

area(


ABC).

Page 69

10.

Circles

Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.The perpendicular bisector of two chords of a circle intersect at centre of the circle. Mark (1) Q 3 True or False : If two arcs of a circle are congruent, then corresponding chords are unequal. Mark (1) Q 4 State the following statement as true or false . Give reasons also. Line segment joining the centre to any point on the circle is a radius of the circle. Mark (1) Q 5 O is the centre of a circle of radius 5 cm. OP PQ.

AB and OQ

CD,

AB

CD, AB = 6 cm and CD = 8 cm. Determine

Marks (2) Q 6 In figure, O is the centre of a circle and

ADC=1200. Find

BAC.

Marks (2) Q 7 In Figure,

ABC is equilateral . Find (i)

ABC (ii)

AEC.

Marks (2)


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Q 8 The diameter of a circle is 5 cm. If a chord is 4 cm long then find the distance between the centre and the chord of the circle. Marks (2) Q 9 The chord of a unit (in cm) circle subtends an angle of 120° at the centre. Find length of the chord in cm. Marks (2) Q 10 Find the value of

CAB in the figure given below.

Marks (2) Q 11 If ON and OM are perpendiculars to CD and AB respectively,then find the length of AM in the figure given below.

Marks (2) Q 12 Find the value of x in the figure given below, where BC = 6 cm and AB = 10 cm.

Marks (2) Q 13 Prove that diameter is the greatest chord in a circle. Marks (2)


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Q 14 ABCD is a cyclic quadrilateral in which AB||CD. If

B=650, then find other angles. Marks (2)

Q 15 Find the value of a and b in the figure given below where

SOT=30.

Marks (2) Q 16 Find the value of angle CAB in the figure given below.

Marks (2) Q 17 Find the value of x in the figure given below, where BC = 6 cm and AB = 10 cm.

Marks (2) Q 18 Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB. Marks (3) Q 19 O is the centre of a circle and the measure of arc ABC is 100°. Determine

ADC and

ABC.

Marks (3)


Page 72

Q 20 If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.

Marks (3) Q 21 Prove that the line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. Marks (3) Q 22 Prove that equal chords of a circle subtend equal angles at the centre. Marks (3)

Q 23 Bisector AD of

BAC of

ABC passes through the center O of the circumcircle of Marks (3)

Q 24 A, B, C and D are the four points on a circle. AC and BD intersect at point E such that

ABC. Prove that AB = AC.

BEC = 130° and

ECD = 20°. Find

BAC.

Marks (3) Q 25 A,B,C and D are four consecutive points on a circle such that AB = CD. Prove that AC = BD. Marks (3)


Page 73

Q 26 Find the value of a0 and b0 in the figure given below.

Marks (3) Q 27 If a chord of length 24 cm is at a distance of 5 cm from the centre in the circle, then find the area of the circle. Marks (3) Q 28 Prove that all the chords of a circle through a given point within it, the least is one which is bisected at that point. Marks (3) Q 29 AB and CD are the two chords of the circle such that AB = 6 cm, CD = 12 cm and AB CD is 3 cm, find the radius of the circle. Marks (4)

CD. If the distance between AB and

Q 30 A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each telephone. Marks (4) Q 31 Two equal chords AB and CD of circle with center O, when produced meet at a point E. Prove that BE = DE and AE = CE. Marks (4) Q 32 In a circle with centre O, chords AB and CD intersect inside the circle at E. Prove that Marks (4)

AOC +

BOD = 2

AEC.

Q 33 OA and OB are respectively perpendicular to chords CD and EF of a circle whose centre is O. If OA = OB, prove that CD = EF. Marks (4)


Page 74

Most Important Questions Q 1 What is a Circle? Q 2 What is the fixed point and fixed distance in a circle are called. Q 3 In how many parts a circle divides the plane on which it lies. Name them. Q 4 What is diameter and what is its relation with the radius . Q 5 Define Arc, Major arc and Minor arc, Semicircle. (a) The centre of a circle lies in the interior of the circle. (b) The longest chord of a circle is the diameter of circle. (c) Segment of a circle is the region between an arc and two radii. (d) Sector of a circle is the region between a chord and either of its arc. (e) Line from the centre to any point on the circle is radius. Q 7 If major arc and minor arc are equal. What we call these arcs, and the regions formed. Q 8 There are two equal chords of length 5 cm each in a circle. One chord subtends an angle of 50 0 at the centre. What is the angle subtended by other chord at the centre.

Q 9 Two chords make angles of 550 at the centre, if one chord is 7 cm long what is the length of another chord.

Q 10 In the given figure, OL perpendicular to AB, if AL = 3 cm, find BL.


Page 75

Q 11 In the given figure below AL = 5cm and BL = 5cm. Find OLA. Q 12 In a circle of radius 5cm how far from the centre will be a chord of length 6cm. Q 13 How many circles can be drawn through (a) One given point (b) Two given points (c) Three given points Q 14 If a circle is given, how can you find its centre using suitable construction. Q 15 An arc of a circle is given, complete the circle. Q 16 In the figure given below AB and CD are chords and PQ is the diameter.If

AEQ =

DEQ, prove that AB = CD.

Q 17 If two circles intersect at two points, prove that the perpendicular bisector of the common chord will pass through their centres. Q 18 If two circles intersect at two points, prove that the line through their centres is the perpendicular bisector of the common chord.


Page 76

Q 19 Two circles having radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of

the common chord. Q 20 Three boys are sitting on the circumference of a circular park with equal distance between them. If radius of the park is 20m find the linear distance between them.


Page 77

Q 21 A chord meets two concentric circles at points A, B, C & D as shown in the figure below. Prove that AC = BD.

Q 22 Two equal chords intersect within the circle, prove that the line joining the point of intersection and the centre makes equal angles with the chords. Q 23 In the given figure below,

DBC = 45

and

BAC = 45

. Find

BCD.

Q 24 In the figure shown, AD and AC are the diameters of the circles. Prove that the intersection point of circles lie on the third side

of the triangle ACD.


Page 78

Q 25 Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.

Q 26 In the given figure , find

BDC.

Q 27 A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc. Q 28 In the given figure, find

BAC.


Page 79

Q 29 In the figure below find

BCD. Further if AB = BC find

ECD.

Q 30 If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle. Q 31 If non-parallel sides of a trapezium are equal, prove that it is cyclic. Q 32 ABC and ADC are two right triangles with common hypotenuse AC. Prove that


CAD =

CBD.

Page 80

11. Q 1 Draw an angle of 135

Constructions

using ruler and compasses only.

Marks (2) Q 2 Draw a line segment of length 8 cm. Bisect it and measure the length of each part. Marks (3) Q 3 Construct an equilateral triangle whose altitude is 4 cm. Marks (3) Q 4 Constructed a triangle ABC in which AB = 5.8 cm BC+CA = 8.4 cm and

B = 60 degree.

Marks (3) Q 5 Construct a triangle ABC in which BC = 3.4 cm , AB–AC = 1.5 cm and Marks (3)

B = 45

.

Q 6 Construct an equilateral triangle whose altitude is 5 cm. Marks (3) Q 7 Construct a triangle ABC in which AB = 5.8 cm , BC + CA = 8.4 cm and Marks (4)

B = 45°.

Q 8 Construct a right angled triangle whose base is 5 cm and sum of its hypotenuse and other side is 8 cm. Marks (4) Q 9 Construct a triangle ABC in which BC = 3.4 cm , AB – AC = 1.5 cm

and

B = 30°. Marks (4)

Q 10 Write the steps of constructions for a triangle ABC whose perimeter and two base angles

B and

C are given.Marks (4)

Q 11 Using ruler and compasses only, construct a triangle ABC from the following data AB+BC+CA = 12 cm

B = 45

and

C= 60. Marks (4) Most Important Questions Q 1 Q 2 Construct an angle 45° at the initial point of a line segment PQ of length 6 cm. Q 3 Construct an angle 30° at the initial point of a line segment PQ of length 4 cm. Q 4 Construct an angle 15° at the initial point of a line segment PQ of length 6 cm. Q 5 Construct an angle 105° at the initial point of a line segment PQ of length 4 cm. Q 6 Construct an angle 135° at the initial point of a line segment PQ of length 5 cm. Q 7 Construct an angle 22 ° at the initial point of a line segment PQ of length 7 cm. Q 8 Construct an angle 75° at the initial point of a line segment PQ of length 5 cm. Q 9 Construct a triangle PQR, in which PQ Q 10 Construct a triangle PQR, in which PQ = 6 cm

P = 45° and PR + RQ = 10 cm.

Q 11 Construct a triangle PQR, in which PQ = 8 cm,

P = 45° and PR – RQ = 3 cm.

Q 12 Construct a triangle PQR, in which PQ = 7cm,

P = 60° and RQ – PR = 2.5 cm.

Q 13 Q 14 Construct a triangle PQR, in which PQ = 7cm,

Q 15 Construct a similar triangle PQR, in which Q 16 Construct a triangle PQR, in which

P = 30° and

P = 45° and


P = 30° and PR – RQ = 2 cm.

Q = 60° and PR + RQ + QP = 12 cm.

Q = 60° and PR + RQ + QP = 9 cm.

Page 81

12.

Heron's Formula

Q 1 Write Heron‟s formula to find the area of a triangle. Mark (1) Q 2 Write the area of the rhombus , if d 1 and d2 are the lengths of its diagonals . Mark (1) Q 3 What is the area of equilateral triangle whose side is a units ? Mark (1) Q 4 What is the area of an isosceles right angled triangle whose equal side is a units ? Mark (1) Q 5 What is the side of a rhombus whose diagonal is d1 and d2? Mark (1) Q 6 Find the area of a triangle whose sides are 13cm, 14cm and 15cm. Marks (2) Q 7 The perimeter of a triangular field is 450 m and its sides are in the ratio 13:12:15. Find the area of the triangle. Marks (2) Q 8 Find the area of a triangle whose two sides are 8 cm and 11cm and the perimeter is 32cm. Marks (2) Q 9 The lengths of the sides of a triangle are 5 cm, 12 cm and 13 cm. Find the length of the perpendicular from the opposite vertex to the side whose length is 13 cm. Marks (2) Q 10 If the sides of the triangle are 26 cm, 28 cm and 30 cm, find the area of the triangle. Marks (2) Q 11 The perimeter of a right triangle is 450 m. If its sides are in the ratio 13 : 12 : 5. Find the area of the triangle. Marks (2) Q 12 There is a slide in a park. One of its side walls has been painted in some colours with a message “KEEP THE PARK GREEN AND CLEAN”. If the sides of the wall are 15m, 11m and 6m, find the area painted in colour. Marks (2) Q 13 If the side of an equilateral triangle is „a‟, then find the altitude of the equilateral triangle. Marks (2) Q 14 Find the area of a triangle whose two sides are 8 cm and 11cm and the perimeter is 32cm. Marks (2) Q 15 A rhombus shaped field has green grass for 18 horses to graze. If each side of the rhombus is 30m and its longer diagonal is 48m, how much area of grass field will each Horse be grazing? Marks (2) Q 16 Find the area of a quadrilateral ABCD whose sides are 9m, 40m, 28m and 15m respectively and the angle between the first two sides is a right angle. Marks (2) Q 17 Find the area of a rhombus whose perimeter is 80m and one of diagonal is 24m. Marks (2) Q 18 A floral design on a floor is made up of 16 tiles that are triangular, the sides of the triangle being 9cm, 28cm and 35cm. Find the area of the floral design. Marks (2)


Page 82

Q 19 Compute the area of trapezium.

Marks (2) Q 20 In fig. given below, BD is the diagonal of quadrilateral ABCD. Find the area of ABCD.

Marks (3) Q 21 Compute the area of the following trapezium:

Marks (3) Q 22 Find area and perimeter of triangle whose sides are 8cm ,19cm and 15 cm. Marks (3) Q 23 Find the area of triangle whose sides are 5cm,12cm, 13 cm . Also find the Shortest altitude. Marks (3) Q 24 In a rectangular field of dimension 50 ft x 30 ft, a triangular park is constructed. If the dimension of the triangular park is 14 ft, 15 ft and 13 ft, find the area of the remaining field. Marks (3)


Page 83

Q 25 Sanya owns a piece of land which is in the shape of a rhombus. She wants her daughter and son to work on the land and produce different crops to suffice the needs of their family. She divided the land in two equal parts. If the perimeter of the land is 400 m and one of the diagonals is 160 m, how much area each of them will get? Marks (3) Q 26 An umbrella is made by stitching 10 triangular pieces of cloth of two different colours, each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella? Marks (3) Q 27 Find the area of triangle whose sides are 5cm, 12cm, 13 cm. Also find the shortest altitude. Marks (3) Q 28 A kite is in the shape of a square with diagonal 32 cm and an isosceles triangle of base 8cm and equal sides are 6cm. How much paper is required to build the kite?

Marks (3) Q 29 Find the area of the trapezium whose parallel sides are 25 cm, 13 cm and other sides are 15 cm and 15 cm. Marks (3) Q 30 The side of a quadrilateral taken in order are 5,12,14 and 15 metres respectively and the angle formed by the first two sides is a right angles find its area.

Marks (3) Q 31 Find the area and perimeter of triangle whose sides are 8cm, 19cm and 15 cm. Marks (3) Q 32 Find the area of a quadrilateral ABCD which AD = 24 cm, side is 26cm.

BAD = 90° and

BCD is an equilateral triangle whose each

Marks (4) Q 33 Mayank made a picture of an aeroplane with paper as shown in figure calculate total area of paper used.

Marks (4)


Page 84

Q 34 A kite in the shape of a square with diagonal 32 cm and an isosceles triangle of base 8 cm and equal sides are 6cm how much paper is required to build the kite. Marks (4) Q 35 The perimeter of a right triangle is 60 cm and its hypotenuse is 26 cm. Find area of triangle and its other two sides. Marks (4) Q 36 A trapezium PBCQ with its parallel sides QC and PB in the ratio 7:5 is cut from a rectangle ABCD, if area of trapezium is 4/7 part of the area of rectangle, find the length of QC and PB.

Marks (4) Q 37 A trapezium whose parallel sides are 25 cm and 10 cm. The non-parallel sides are 14 cm and 13 cm. find the area of the trapezium . Marks (4) Q 38 The sides of a quadrilateral ABCD, taken in order are 5 cm,12 cm,14 cm and 15 cm respectively, and angle contained between first two sides is a right angle. Find its area. Marks (4) Q 39 A rhombus sheet, whose perimeter is 32 cm and whose one diagonal is 10 cm long, is painted on both sides at the rate of 5 per sq. cm. Find the cost of painting. Marks (4) Q 40 The perimeter of a right triangle is 60 cm and its hypotenuse is 26 cm. Find other two sides and area of triangle? Marks (4) Q 41 A field is in the shape of trapezium whose parallel sides are 25m and 10m. The non-parallel sides are 14m and 13m. Find the area of the field.

Marks (4)


Page 85

Most Important Questions Q 1 What is the area of an equilateral triangle of side 2a. Q 2 What will be the area of a right angled triangle whose base is 12 cm and hypotenuse is 13 cm. Q 3 Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm. Q 4 The perimeter of a triangular field is 540 m and its sides are in the ratio 25:17:12. Find the area of the triangle. Q 5 A triangle and a parallelogram are on the same base and have the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram. Q 6 The perimeter of a triangular park is 240 m. If two of its sides are 78 m and 50 m, find the length of the perpendicular on the side of length 50 m from the opposite vertex. Q 7 Find the percentage increase in the area of a triangle if its each side is doubled. Q 8 A skirt is made by stitching 10 triangular pieces of cloth of two different colors, each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each color is required for the skirt. Q 9 Find the area of the quadrilateral ABCD in which AD = 24 cm,

BAD = 90° and BCD forms an equilateral triangle whose each

side is equal to 26 cm. Q 10 Find the area of a rhombus whose perimeter is 80 cm and one of its diagonal is 24 cm. Q 11 Two parallel sides of a trapezium are 60 cm and 77 cm and the others sides are 25 cm and 26 cm. Find the area of the trapezium. Q 12 Radha has a piece of land which is in the shape of a rhombus. She wants her son and daughter to cultivate equal areas of land, thus she divides the area of the rhombus equally amongst the two. If the perimeter of land is 400 m and of the diagonals is 160 m, How much area will each one get? Q 13 A floral design on the wall is made up of 16 tiles which are triangular, the sides are of the triangle are 9 cm, 28 cm and 35 cm. Find the cost of polishing the tiles at the rate of 50 paise per cm2.


Page 86

13. Surface Areas and Volumes Q 1 Find the area enclosed between two concentric circles of radii 4 cm and 3 cm. Marks (2) Q 2 The diameter of a garden roller is 1.4 m and it is 2 m long. How much area will it cover in 5 revolutions? Marks (2) Q 3 A cuboid has total surface area of 40 sq m and its lateral surface area is 26 sq m. Find the area of base. Marks (2) Q 4 Three metal cubes whose edges measure 3 cm, 4 cm and 5 cm respectively are melted to form a single cube. Find the edge of the new cube. Also find the surface area of the new cube. Marks (2) Q 5 An iron pipe 20 cm long has exterior diameter equal to 50 cm. If the thickness of the pipe is 1cm, find the whole surface area of the pipe. Marks (2) Q 6 The lateral surface of a cylinder is equal to the curved surface of a cone. If the radius is the same, find the ratio of the height of the cylinder and slant height of the cone. Marks (2) Q 7 A right circular cylinder just enclosed a sphere of radius r as shown in figure find the surface area of the sphere , curved surface area of the cylinder and also their ratio.

Marks (2) Q 8 A godown is in the form of a cuboid measuring 60 m x 40 m x 20 m. How many cuboidal boxes can be stored in it if the volume of one box 0.8 m3? Marks (2)


Page 87

Q 9 The diameter of a sphere is decreased by 50%. What is the ratio between initial and final curved surface areas? Marks (3) Q 10 Find the volume of the largest right circular cone that can be fitted in a cube whose edge is 14 cm. Marks (3) Q 11 The semi-circular sheet of metal of diameter 28 cm is bent into an open conical cup. Find the depth and the capacity of cup. Marks (3) Q 12 A well with 10 m inside diameter is dug 14 m deep. Earth taken out of it is spread all around to a width of 5 m to form an embankment. Find the height of embankment. Marks (3) Q 13 The radius and slant height of a cone are in the ratio 4 : 7. If its curved surface area is 792 sq cm, find its radius. Marks (3) Q 14 The diameter of 0.84 m long roller is 1.5 m. If it takes 100 complete revolutions to level a playground, find the cost of levelling it at the rate of 50 paise per square metre. Marks (3) Q 15 Three equal cubes are placed adjacently in a row. Find the ratio of total surface area of the new cuboid to that of sum of the surface areas of the three cubes. Marks (3) Q 16 The cost of papering four walls of a room at 90 paise per square metre is 157.50. The height of the room is 5 metres. Find the length and the breadth of the room if they are in the ratio 4:1. Marks (3) Q 17 A hollow cylinder is made of iron of height 1 m. Its inner diameter is 54 cm and thickness of iron sheet of cylinder is 9 cm. Find the weight of the hollow cylinder if 1 c.c. of iron weighs 8 gm. Marks (3) Q 18 A well with 8 m inner diameter is dug 21 m deep. Earth taken out of it is spread all around to a width of 3 m to form an embankment. Find the height of embankment. Marks (3) Q 19 A plastic box 1.25 m long,1.05 m wide and 75 cm deep is to be made. It is to be open at the top. Ignoring the thickness of the plastic sheet, determine the area of the sheet required for making the box and also find the cost of sheet for it, if a sheet measuring 1 sq m cost 20. Marks (3) Q 20 A reservoir is in the form of rectangular parallelepiped. Its length is 20 m. If 18 kl of water is removed from the reservoir, the water level goes down by 15 cm. Find the width of the reservoir. Marks (3)


Page 88

Q 21 If v is the volume of a cuboid of dimension a, b, c and s is its surface area, then prove that

Marks (3) Q 22 The area of three adjacent faces of a cuboid are x, y and z. If the volume is V, prove that V2 = xyz. Marks (3) Q 23 A wall of the length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm x 12 cm x 8 cm, how many bricks would be required? Marks (3) Q 24 How many litres of water flow out of a pipe having an area of cross-section of 5 sq cm. in one minute if the speed of the water in the pipe is 30 cm/sec? Marks (3) Q 25 A circular tent is cylindrical to a height of 3 metres and conical above it. If its diameter is 105 m and the slant height of the conical portion is 53 m, calculate the length of canvas 5 m wide to make the required tent. Marks (3) Q 26 The radius and slant height of a cone are in the ratio 4:7. If its curved surface area is 792 sq cm, find its radius. Marks (3) Q 27 A powder tin is in cylindrical shape, whose base has a diameter of 14 cm and height 20 cm. A label is wrapped around the surface of the container. If the label is pasted leaving 2 cm from the top and the bottom. What is the area of the label? Marks (3) Q 28 A sphere of diameter 7 cm is dropped in a right circular cylinder vessel partly filled with water. The diameter of the cylindrical vessel is 14 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? Marks (3) Q 29 The diameter of a sphere is decreased by 50%. By what percent will its curved surface area decrease? Marks (3) Q 30 Three equal cubes are placed adjacently in a row. Find the ratio of total surface area of the new cuboid to that of sum of the surface areas of the three cubes. Marks (3) Q 31 How many bricks will be required for a wall 8 m long, 6 m high and 22.5 cm thick if each brick measures 25 cm x 11.25 cm x 6 cm? Marks (3)


Page 89

Q 32 The internal measurements of a cuboidal room are 10 m

4m

6 m. Find the cost of white washing of the walls at the rate of

5 per square metre. Marks (3) Q 33 A cylinder is within the cube touching all the vertical faces. A cone is inside the cylinder. If the height and base of the cone is same as cylinder, find the ratio of their volumes. Marks (4) Q 34 The external length, breadth and height of a closed rectangular wooden box are 18 cm, 10 cm and 6 cm respectively and thickness of wood is ½ cm. When the box is empty, it weighs 15 kg and when filled with sand it weighs 100 kg. Find the weight of one cubic cm of wood and one cubic cm of sand. Marks (4) Q 35 How many litres of water flows out of a pipe having an area of cross-section of 5 sq. cm in one minute, if the speed of water in the pipe is 30 cm/sec? Marks (4) Q 36 A hemispherical bowl of internal diameter 40 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 2 cm and height 8 cm. How many bottles are required to empty the bowl? Marks (4) Q 37 An open box is made of wood 3 cm thick. Its external length, breadth and height are 1.48 m,1.16 m and 8.3 dm. Find the cost of painting the inner surface of 1000 per 10 sq metres. Marks (4) Q 38 The ratio of radii of two right circular cylinders is 2:5 and that of their heights is 1:4, find the ratio of their Volumes. Marks (4) Q 39 A wooden toy is in the form of a cone. The diameter of the base of the cone is 6 cm and the height of the cone is 4 cm. Find the cost of painting the toy at the rate of Rs. 5 per 100 cm2. Marks (4) Q 40 An open cuboidal box is made of 3 cm thick wood. Its external length, breadth and height are 1.48 m, 1.16 m and 8.3 dm respectively. Find the cost of painting the inner surface at the rate of Rs. 50 per square metre. Marks (4) Q 41 A conical vessel having internal radius 3 cm and height 25 cm is full of water. The water is emptied into a cylindrical vessel with internal radius of 10 cm. Find the height to which the water rises. Marks (4)


Page 90

Most Important Questions

Q 1 A small indoor greenhouse is made of entirely glass planes (including) bases together with tape. It is 30 cm long, 25 cm wide and 25 cm high. What is the area of the glass? How much tape is needed for all the 12 edges? Q 2 Find the surface area of a cube whose edge is 11 cm. Q 3 The dimensions of a cuboid are in the ratio 1:2:3 and its total surface area is 88 m2. Find the dimensions of the cuboid. Q 4 Three cubes each of side 5 cm are joined end to end. Find the surface area of the resulting cuboid. Q 5 cuboidal oil tin is 30 cm by 40 cm by 50 cm. Find the cost of the tin required for making 20 such tins if the cost of tin sheet is Rs 20 per square meter. Q 6 The floor of a rectangular hall has a perimeter of 250 m. Its height is 6 m. Find the cost of painting its four walls at the rate of Rs. 6 per square meter. Q 7 Agrawal sweets was placing an order for making cardboard boxes for packing their sweets. Two sizes of the boxes were required. The bigger of dimension 25 cm x 20 cm x 5 cm and the smaller of dimension 15 cm x 12 cm x 5 cm. 5 % of the total surface is required extra, for all the overlaps. If the cost of cardboard is Rs. 4 for 1000 cm 3, find the cost of cardboard required for supplying 250 boxes of each kind. Q 8 The curved surface area of a right circular cylinder of height 14 cm is 88 cm 2 find the diameter of the base. Q 9 The diameter of a garden roller is 1.4 m and it is 2 m long. How much area will it cover in 5 revolutions? Q 10 A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm , the outer diameter is 4.4 cm .Find a) Inner curved surface area b) Outer curved surface area c) Total surface area Q 11 A cylindrical vessel without lid, has to be tin coated on both the sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin- coating at the rate of Rs. 3.50 per 1000 cm2. Q 12 A lampshade is cylindrical in shape, it has to be covered with a decorative cloth, the frame has the base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm is to given to it for folding it over the top and the bottom of the frame. Find out how much cloth is needed for covering the lampshade. Q 13 The diameter of a cone is 14 cm and its slant height is 9 cm. Find the area of its curved surface. Q 14 The radius and the slant height are in the ratio 4:7. If its curved surface area is 792 cm2, find its radius. Q 15 The radius of the cone is 7 cm and its curved surface area is 176 cm2. Q 16 A jokers cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet needed to make 10 such caps.


Page 91

Q 17 A circus tent is cylindrical to a height of 3 m and conical above it, if its diameter is 105 m and the slant height of the conical portion is 53 m, calculate the length of the canvas 5 m wide required to make the tent. Q 18 There are two cones, the surface area of one cone is twice the surface area of the other cone. The slant height of the latter is twice that of the former. Find the ratio of their radii. Q 19 Find the surface area of a sphere of radius 7 cm. Q 20 The radius of a hemi-spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of the surface areas of the balloon in the two cases. Q 21 The internal and the external diameters of a hollow hemi-spherical vessel are 24 cm and 25 cm respectively. The cost of painting one square meter of the surface is 7 paise. Find the total cost of painting the vessel all over. Q 22 A storage tank consists of a circular cylinder, with a hemisphere adjoined at either ends. If the external diameter of the cylinder be 1.4 cm and its length be 5 m, what will be the cost of painting it on the outside at the rate of Rs. 10 per square meter? Q 23 A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 6 cm and its slant height is 5 cm. Find the cost of painting the toy at the rate of Rs 5 per 1000 cm 2. Q 24 A matchbox measures 4 cm x 2.5 cm x 1.5 cm. What will be the volume of a packet containing 12 such boxes? Q 25 The capacity of a tank, which is cuboidal in shape, is 50,000 l. Find the breadth of the tank if its length and depth are respectively 2.5 m and 10 m. Q 26 A cube of 9 cm edge is immersed completely in a rectangular vessel containing water. If the dimensions of the base are 15 cm and 12 cm. Find the rise in water level in the vessel. Q 27 A solid cube of side 12 cm is cut into 8 cubes of equal volume. What will be the side of the new cube? Also find the ratio between their surface areas. Q 28 How many cubic centimeters of iron are there in an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm, the iron being 1.5 cm thick throughout? If one cubic cm of iron weighs 15 g, find the weight of the empty box in kg. Q 29 A well with 10 m inside diameter is dug 14 m deep. Earth taken out of it is spread all around to a width of 5 m to form an embankment. Find the height of the embankment. Q 30 How many liters of water can flow out of a pipe having an area of cross-section of 5 cm2 in one minute, if the speed of water in the pipe is 30 cm/sec? Q 31 The ratio between the radii of the base and the height of the cylinder is 2 : 3 what is the Total surface area if the volume of the cylinder is 1617 cm3. Q 32 The trunk of a tree is cylindrical in shape and its circumference is 176 c m. If the length of the trunk is 3 m. Find the volume of timber that can be obtained from the trunk. Q 33 Find the length of 13.2 kg of copper wire of diameter 4 mm, when 1 cubic cm of copper weighs 8.4 gm. Q 34 The diameter of a right circular cone is 8 cm and its volume is 48

cm3. What is the height of the cone?

Q 35 The volume of a cone is 18480 cm3. If the height of the cone is 40 cm. Find the radius of the base.


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Q 36 A right triangle ABC with its sides 5 cm, 12 cm and 13 cm is revolved about its side of 12 cm. Find the volume of the right circular cone so formed. Q 37 A cone of radius 5 cm is filled with water. If the water is poured in a cylinder of radius 10 cm, the height of the water rises by 2 cm , find the height of the cone. Q 38 A solid cube of side 7 cm is melted to make a cone of height 5 cm, find the radius of the base of the cone. Q 39 Find the volume of the largest right circular cone that can be fitted in a cube of edge 14 cm. Q 40 A solid lead ball of radius 7 cm was melted and then drawn into a wire of diameter 0.2 cm. Find the length of the wire. Q 41 How many spherical bullets can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being 4 cm in diameter. Q 42 Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S‟. Find the radius r‟ of the new sphere ratio of S and S‟ Q 43 A dome of a building is in the form of hemisphere. From inside, it was white washed at the cost of Rs.498.96. If the cost of whitewashing it, is Rs.2.00 per square meter, find the inside surface area of the dome and the volume of air inside the dome.


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14.

Statistics

Q 1 Define array or arrayed data. Mark (1) Q 2 Define frequency. Mark (1) Q 3 Write the relation between class mark, lower limit and upper limit of a class interval. Mark (1) Q 4 Define primary data. Marks (2) Q 5 Define secondary data. How it is differ from the primary data? Marks (2) Q 6 Find the mode of the following data : Marks

Number of students

48

4

49

10

50

12

51

10

52

10

How many students are there whose marks are less than the modal value? Marks (2) Q 7 Following data represents the favourite fruit liked by 20 children.

P G A M M P A M G M A M M M M M A A P G. Make a frequency table to find how many more children chose apple as their favourite fruit than pomegranate. Marks (2) Q 8 Make a bar graph of the given data. Instrument

Frequency

clarinet

11

flute

18

trumpet

7

violin

5

Marks (2)

Q 9 Following frequency table represents the number of students in each section of class 9th of ABC school. Find the mean number of students per sections. Number of Student Per Section Section A

20

Section B

18

Section C

25

Section D

22

Section E

20 Marks (2)


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Q 10 The table below shows the age of seven students

participating in a music recital. Find the median and mode of the data.

Age (years) 12 10 9 10 11

8 13 Marks (2)

Q 11 Find the median and mode of the speeds displayed in the graph.

Marks (2) Q 12 Make the frequency polygon of the given data. Class-Intervals

Frequency

0 – 10

9

10 – 20

14

20 – 30

8

30 – 40

10

40 – 50

9 Marks (2)

Q 13 The weights of new born babies (in kg) in a hospital on a particular day are as follows: 2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4 1. 2.

Determine the range. How many babies have weight below 2.5 kg.

3. How many babies have weight more than 2.8 kg. Marks (3) Q 14 The class- marks of a distribution are 26,31,36,41,46,51,56,61,66,71.Find the true class limits. Marks (3) Q 15 The bar graph shown in figure represents the circulation of newspaper in 5 languages. Study the bar graph and answer the following questions:

(i) What is the total number of newspapers published in Hindi, English,Urdu, Punjabi and Bengali?


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(ii) State the language in which the largest number of news papers is

published.

(iii) State the language in which the number of newspapers published is minimum. Marks (3) Q 16 Prepare a frequency distribution from the following data by taking the class intervals. Mid points

Frequency

5

3

15

9

25

15

35

10

45

6

55

4 Total = 47 Marks (3)

Q 17 Mean of 18 numbers is 57. If 9 is added to each number, find the new mean. Marks (3) Q 18 Prove that the sum of the deviations of individual observations from Marks (3)

their mean is zero.

Q 19 The median of the following observation arranged in ascending order is 22. Find x. 8, 11, 13, 15, x+1, x+3, 30,35,40,43 Marks (3) Q 20 1. If the mean of the following data is 20.2, find the value of p: X

10

15

20

25

30

f

6

8

p

10

6

Marks (4) Q 21 The water bills of 32 houses in a colony for a period is given below : 56, 43, 32, 38, 56, 22 ,68, 85, 52, 47, 35, 58, 63, 74, 27, 84, 69, 35, 44, 75, 55, 30, 54, 65, 45, 67, 95, 72, 43, 65, 35, 5 9 . Tabulate the data and present the data as a cumulative frequency table using 70-79 as one of the class intervals. Marks (4) Q 22 Construct a frequency polygon for the following data: Age ( in years)

Frequency

0-2

4

2-4

2

4-6

12

6-8

18

8-10

25 Marks (4)

Q 23 The population of a state in different census years is as given below: Census year Population in Lakhs

1981

1982

1983

1984

1985

30

50

70

110

150

Represent the above information with the help of bar graph.


Marks (4)

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Q 24 The average marks of boys in an examination are 65 and that of girls is 74. If the average of marks of all candidates in that examination is 70, find the ratio of the number of boys to the number of girls that appeared in the examination. Marks (4) Q 25 1. The marks obtained (out of 100) by a class of 80 students are given below: Marks

Number of students

10-20

6

20-30

17

30-50

15

50-70

16

70-100

26

Construct a histogram to represent the data above. Marks (4) Q 26 1. Draw a histogram for the following distribution: Marks obtained

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

No. of Students

7

10

6

8

12

3

2

2

Marks (4) Most Important Questions Q 1 List the different types of statistical data. Q 2 List the different types of frequency distribution. Q 3 List the different ways for the presentation of raw data. Q 4 Given below are the ages of 25 students of class IX in a school. Prepare a discrete frequency distribution. 15, 16, 16, 14, 17, 17, 16, 15, 15, 16, 16, 17, 15, 16, 16, 14, 16, 15, 14, 15, 16, 16, 15, 14, 15. Q 5 What is data? Explain the types of statistical data? Q 6 The class marks of a distribution are 26, 31, 36, 41, 46, 51, 56, 61, 66, 71. Determine the true class limits. Q 7 Form a grouped frequency distribution from the following data by inclusive method taking 4 as the magnitude of class intervals. 31, 23, 19, 29, 22, 20, 16, 10, 13, 34, 38, 33, 28, 21, 15, 18, 36, 24, 18, 15, 12, 30, 27, 23, 20, 17, 14, 32, 26, 25, 18, 29, 24, 19, 16, 11, 22, 15, 17, 10. Q 8 The marks obtained by 40 students of Class IX in an examination are given below : 18, 8, 12, 6, 8, 16, 12, 5, 23, 2, 16, 23, 2, 10, 20, 12, 9, 7, 6, 5, 3, 5, 13, 21, 13, 15, 20, 24, 1, 7, 21, 16, 13, 18, 23, 7, 3, 18, 17, 16. Present the data in the form of a frequency distribution using the same class size, one such class being 20-25 (where 25 is not included). Q 9 The class marks of a distribution are : 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102. Determine the class size, the class limits and the true class limits.


Page 97

Q 10 100 plants each were planted in 100 schools during Van Mahotsava. After one month, the number of plants that survived were recorded as : 95 67 28 32 65 65 69 33 98 96 76 42 32 38 42 40 40 69 95 92 75 83 76 83 85 62 37 65 63 42 89 65 73 81 49 52 64 76 83 92 93 68 52 79 81 83 59 82 75 82 86 90 44 62 31 36 38 42 39 83 87 56 58 23 35 76 83 85 30 68 69 83 86 43 45 39 83 75 66 83 92 75 89 66 91 27 88 89 93 42 53 69 90 55 66 49 52 83 34 36 Represent the above data in a frequency distribution table. Q 11 Consider the marks obtained (out of 100 marks) by 30 students of Class IX of a school: 10 20 36 92 95 40 50 56 60 70 92 88 80 70 72 70 36 40 36 40 92 40 50 50 56 60 70 60 60 88 Construct a frequency distribution table. Q 12 For the following data of daily wages(in rupees) received by 30 labourers in a certain factory, construct a grouped frequency distribution table by dividing the range into class intervals of equal width, each corresponding to 2 rupees, in such a way that the midvalue of the first class interval corresponds to 12 rupees. 14, 16, 16, 14, 22, 13, 15, 24, 12, 23, 14, 20, 17, 21, 22, 18, 18, 19, 20,17, 16, 15, 11, 12, 21, 20 ,17, 18, 19, 23. Q 13 Form a discrete frequency distribution from the following scores : 15, 18, 16, 20, 25, 24, 25, 20, 16, 15, 18, 18, 16, 24, 15, 20, 28, 30, 27, 16, 24, 25, 20, 18, 28, 27, 25, 24, 24, 18, 18, 25, 20 ,16, 15, 20, 27, 28, 29, 16. Q 14 The weights in grams of 50 oranges picked at random from a consignment are as follows : 131, 113, 82, 75, 204, 81, 84, 118, 104, 110, 80, 107, 111, 141, 136, 123, 90, 78, 90, 115, 110, 98, 106, 99, 107, 84, 76, 186, 82, 100, 109, 128, 115, 107, 115, 119, 93, 187, 139, 129, 130, 68, 195, 123, 125, 111, 92, 86, 70, 126. Form the grouped frequency table by dividing the variable range into intervals of equal width, each corresponding to 20 gms in such a way that the mid-value of the first class corresponds to 70 gms. Q 15 The marks obtained by 35 students in an examination are given below:


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370, 290, 318, 175, 170, 410, 378, 405, 380, 375, 315, 305, 325, 275, 241, 288, 261, 355, 402, 380, 178, 253, 428, 240, 210, 175, 154, 405, 380, 370, 306, 460, 328, 440, 425. Form a cumulative frequency table with class intervals of length 50. Q 16 List the different ways for representing data graphically. Q 17 What is a histogram? Q 18 From the graph given below, read the temperature at 11 a.m. and 4 p.m.

Q 19 The following table gives the number of vehicles passing through a busy crossing in Noida in different time intervals on a particular day. Time

8-9

9 - 10

10 - 11

11 – 12

12 – 13

13 – 14

14 - 15

300

400

350

250

200

150

100

Interval No. of vehicles Represent the above data by a bar graph. Q 20 The population of a state in different census year is as given below: Census year

1981

1982

1983

1984

1985

Population in Lakhs

30

50

70

110

150

Represent the above information with the help of a histogram. Q 21 Number of children in seven different classes are given below: Class

VI

VII

VIII

IX

X

XI

XII

No. of children

80

65

75

100

120

80

90

Represent the data with the help of a bar graph. Q 22 Read the bar graph shown in the figure and answer the following questions :


Page 99

(a) What is the information given by the bar graph ? (b)

What is the order of the change of number of students over several years?

(c)

In which year is the increase of students maximum?

(d) State whether true or false : The enrolment during 1996 – 97 is twice that of 1995 – 96.

Q 23 The results of pass percentage of Class X and XII for 5 years are given below in the table: Year

1994 – 95

1995 – 96

1996 – 97

1997 - 98

1998 - 99

X

90

95

90

80

98

XII

95

80

85

90

95

Q 24 What is a bar graph? Q 25 The bar graph shown in figure represents the circulation of newspaper in 10 languages. Study the bar graph and answer the following questions: 1. 2.

What is the total number of newspapers published in Hindi, English, Urdu, Punjabi and Bengali? Name two pairs of languages which publish the same number of newspapers.

3.

State the language in which the largest number of news papers is published.


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4.

State the language in which the number of news papers published is between 2500 and 3600.

Q 26 The marks scored by 750 students in an examination are given in the form of a frequency distribution table : Marks

600 -640

640 -680

680 -720

720 -760

760 -800

800 -840

840 -880

No. of

16

45

156

284

172

59

18

students

Represent this data in the form of a histogram and construct a frequency polygon. Q 27 Construct a frequency polygon for the following data : Age (in

0 –2

2–4

4–6

6 –8

8 – 10

10 –12

12 –14

2

4

6

8

9

6

5

14 –16

16-18

years)

Frequency

3

1

Q 28 Construct a frequency polygon for the following data : Age (in years)

Frequency

0 –2

2–4

4–6

6 –8

8 – 10

10 –12

12 –14

2

4

6

8

9

6

5


14 –16

3

16-18

1

Page 101

Q 29 The following are the scores of two groups of class IV students in a test of reading ability. Scores

Group A

Group B

50 - 52

4

2

47 – 49

10

3

44 – 46

15

4

41 – 43

18

8

38 – 40

20

12

35 – 37

12

17

32 - 34

13

22

Total

92

68

Construct a frequency polygon for each of these two groups on the same axes. Q 30 Represent the following data by means of a histogram : Weekly Wages

10 – 15

15 – 20

20 – 25

25 – 30

30 – 40

40 – 60

60 - 80

7

9

8

5

12

12

8

(in Rs.) No. of workers (frequency) Q 31 Find the arithmetic mean of first 7 numbers whole numbers . Q 32 If the mean of 2, 4, 6 and p is 11, find the value of p. Q 33 Find the median of the following data : 37, 31, 42, 43, 46, 25, 39, 45, 32 Q 34 Find the median of the following data : 25, 34, 31, 23, 22, 26, 35, 28, 20, 32. Q 35 Five people were asked about the time in a week they spend in doing social work in their community. They said 10, 7, 13, 20 and 15 hours, respectively. Find the mean (or average) time in a week devoted by them for social work. Q 36 If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 11, find the mean of first three observations. Q 37 Give one example of a situation in which (a) The mean is an appropriate measure of central tendency. (b) The mean is not an appropriate measure of central tendency but the median is an appropriate measure of central tendency. Q 38 The median of the observations 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 is arranged in ascending order is 24. Find the value of x. Q 39 Find the mode of the following data : 110, 120, 130, 120, 110, 140, 130, 120, 140, 120. Q 40 The mean monthly salary of 10 members of a group is Rs. 1445, one more member whose monthly salary is Rs. 1500 has joined the group. Find the mean monthly salary of 11 members of the group. Q 41 Find the mode for the following series : 7.5, 7.3, 7.2, 7.2, 7.4, 7.7, 7.7, 7.5, 7.3, 7.2, 7.6, 7.2


Page 102

Q 42 Following table shows the weights of 12 students: Weight (in kgs.) Number of students

67 4

70 3

72 2

73 2

75 1

Find the mean weight. Q 43 The mean of 40 observations was 160. It was detected on rechecking that the value of 165 was wrongly copied as 125 for computation of mean. Find the correct mean. Q 44 The mean of 5 numbers is 18. If one numbers is excluded, their mean is 16. Find the excluded number. Q 45 Find the median of the following data : 19, 25, 59, 48, 35, 31, 30, 32, 51. If a student replaces 25 by 52 by mistake, what will be the new median? Q 46 The mean of 10 numbers is 20. If 5 is subtracted from every number, what will be the new mean? Q 47 If the mean of the following distribution is 6, find the value of p. x: f:

2 3

4 2

6 3

10 1

p+5 2

Q 48 Consider a small unit of a factory where there are 5 employees : A supervisor and four labourers. The labourers draw a salary of Rs 5,000 per month each while the supervisor gets Rs 15,000 per month. Calculate the mean, median and mode of the salaries of this unit of the factory. Interpret the findings.


Page 103

15.

Probability

Q 1 Tom draws a marble from a bag containing 12 marbles.There are 3 red marbles, 4 blue marbles and 5 green marbles, find the probability that he will draw a blue marble. Marks (2) Q 2 A basket has 5 apples, 10 oranges and 5 bananas. Find the probability of getting out an apple. Marks (2) Q 3 One card is drawn from a well-shuffled deck of 52 cards then find the probability that the card will be a king. Marks (2) Q 4 Mr. And Mrs. X stays in a house along with their seven children. The female to male ratio in the family is 1:2. Find the probability that all the children are either boy or girl. Marks (2) Q 5 The record of a weather station shows that out of the past 200 consecutives days, its weather forecasts were correct 150 times then find the probability that on a given day it was correct . Marks (2) Q 6 The record of a weather station shows that out of the past 200 consecutives days, its weather forecasts were correct 150 times then find the probability that on a given day it was not correct. Marks (2) Q 7 A drawer contains 8 red socks, 3 white socks and 5 blue socks. Without looking, Mayank draws out a pair of socks. Find the probability that the pair of socks is white. Marks (2) Q 8 Two coins are tossed simultaneously 100 times and we get the following outcomes:(i) One head = 20 (ii) Two heads = 50. Find the probability that there is no head. Marks (2) Q 9 Two coins are tossed simultaneously 100 times and we get the following outcomes:(i) No head = 30 (ii) Two heads = 50. Find the probability that there is only one head. Marks (2) Q 10 Two coins are tossed simultaneously 100 times and we get the following outcomes:(i) No head = 30 (ii) One head = 20. Find the probability of getting two heads. Marks (2) Q 11 A coin is tossed 23 times and observed that 10 times head comes up. Find the probability that a tail comes up. Marks (2)


Page 104

Q 12 A coin is tossed 100 times. It is observed that 60 times head comes up and 40 times tail comes up then find the probability that neither a head nor a tail comes up. Marks (2) Q 13 A coin is tossed 100 times with the following frequencies: Head: 45, tail: 55. Compute the probability for each event. Marks (3) Q 14 The percentage of marks obtained by a student in monthly unit tests is given below: Unit test

I

II

Percentage 69

71

III 73

IV

V

68

76

Find the probability that the student gets more than 68% marks.

Marks (3)

Q 15 A die is thrown 200 times with the following frequency, for the outcomes 1,2,3,4,5,6 , as given below: Outcomes :

1

2

3

4

5

6

Frequency:

30

40

50

20

30

30

Find the probability that the outcome is less than 5.

Marks (3)

Q 16 1000 families with 2 children were selected randomly and the following data was recorded: Number of girls in a family: Number of families

0

: 200

1

2

500

300

If a family is chosen at random, compute the probability that it has 1. no girl,

2. at least one girl.

Marks (3)

Q 17 To know the opinion of students about mathematics, a survey of 100 students was conducted. The data is recorded in the following table: Opinion:

Like

Number of Students:

70

Dislike 30

Find the probability that a student chosen at random 1. likes mathematics, 2. dislikes mathematics.


Marks (3)

Page 105

Q 18 8 bags of wheat flour, each marked 10 kg, actually contained the following weights of flour(in kg) 10.01, 9.97, 10.03, 9.96, 10.04, 10.06, 10.02, 9.98 Find the probability that any of these bags chosen at random contain less than 10 kg of wheat. Marks (3) Q 19 A cycle manufacturing company kept a record of recycling of tyres and maintained the record of distance covered by it. Table given below shows the record of 100 tyres: Distance (in km): 0-2000 2000-5000 5000-7000 7000-10000 Frequency:

30

50

10

10

What will be the probability to replace a tyre less than 5000 km ? Marks (3) Q 20 In a match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that he did not hit a boundary. Marks (3) Q 21 On one page of a telephone directory, there were 210 telephone numbers. The frequency distribution of their unit place is given as follows: Digits: Frequency:

0

1

2

3

4

5

10

20

30

40

10

20

6 30

7 5

8 30

9 15

Without looking at the page, the pencil is placed on one of these number and the number is chosen at random.What is the probability that the digit in its unit place is multiple of 3? Marks (3) Q 22 The record of a weather station shows that out of the past 200 consecutive days,its weather forecasts were correct 180 times. What is the probability that on a given day it was correct and also find the probability on a given day it was not correct? Marks (3) Q 23 A die is thrown, find the probability of getting (i) a prime number (ii) an even number. Marks (3) Q 24 10 bags of rice, each marked 6 kg, actually contained the following weights of rice (in kg): 5.96 6.07

6.06 5.99

6.09 6.11

6.04 5.93

6.00 6.00

Then find the probability that any of these bags chosen at random contains more than 6 kg. Marks (3)


Page 106

Q 25 The distance (in km) of 40 engineers from their residence to their place of work were found as follows. 5

3

10

20

25

11

13

7

12

31

19

10

12

17

18

11

32

17

16

2

7

9

7

8

3

5

12

15

18

3

12

14

2

9

6

15

15

7

6

12

What is the empirical probability that an engineer lives: (i) less than 7 km from her place of work? (ii) more than or equal to 7 km from her place of work? (iii) within ½ km from her place of work? Marks (4) Q 26 In Cherrapunji, it rains for 200 days in an ordinary year, find the probability that (i) there will not be rain in that year, (ii) there will be rain in that year. Marks (4) Q 27 In a cricket match, a batsman hits the ball 24 times out of 72 balls he plays. Find the probability that he did not hit the ball. Marks (4) Q 28 100 plants each, were planted in 100 schools during Van Mahotsava. After one month, the no. of plants that survived were recorded as in data below: No. of plants survived

less than 25

26-50

51-60

61-70

more than 70

total no. of schools

No. of schools =

15

20

30

30

5

100

frequency

When a school is selected of random for inspection what is the probability of (i) more than 25 plants survived in school? (ii)less than 61 plants survived in the school? Marks (4)


Page 107

Q 29 A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. Looking at their performances, she found that a few students got under 20 marks and a few got 70 marks or above. So she decided to group them into intervals of varying sizes as follows: 0 − 20, 20 − 30… 60 − 70, 70 − 100. Then she formed the following table: Marks

Number of student

0 − 20

7

20 − 30

10

30 − 40

10

40 − 50

20

50 − 60

20

60 − 70

15

70 − above

8

Total

90

(i) Find the probability that a student obtained less than 20 % in the mathematics test. (ii) Find the probability that a student obtained marks 60 or above. Marks (4) Q 30 To know the opinion of the students about the subject Maths, a survey of 200 students was conducted. The data is recorded in the following table. Opinion

Number of students

like

135

dislike

65

Find the probability that a student chosen at random (i) likes Maths, (ii) does not like it. Marks (4)


Page 108

Q 31 An organization selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed in the table below: Monthly income

Vehicles per family

(in Rs)

0

1

2

Above 2

Less than 7000

10

160

25

0

7000 − 10000

0

305

27

2

10000 − 13000

1

535

29

1

13000 − 16000

2

469

59

25

16000 or more

1

579

82

88

Suppose a family is chosen, find the probability that the family chosen is (i) earning Rs 10000 − 13000 per month and owning exactly 2 vehicles. (ii) earning Rs 16000 or more per month and owning exactly 1 vehicle. (iii) earning less than Rs 7000 per month and does not own any vehicle. (iv) earning Rs 13000 − 16000 per month and owning more than 2 vehicles. Marks (4) Q 32 1500 families with 2 children were selected randomly, and the following data were recorded: Number of girls in a family Number of families

2 475

1 814

0 211

Compute the probability of a family, chosen at random, having (i) 2 girls (ii) 1 girls (iii) No girl. Marks (4) Q 33 50 seeds were selected at random from each of 5 bags of seeds, and were kept under stadardised conditions favourable to germination. After 20 days, the number of seeds which had germinated in each collection were counted and recorded as follows: Bag

1

2

3

4

5

Number of seeds

40

48

42

29

41

germinated What is the probability of germination of (i)more than 40 seeds in a bag? (ii)49 seeds in a bag? (iii)more that 35 seeds in a bag? Marks (4)


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Q 34 A tyre manufacturing company kept a record of the distance covered before a tyre needed to bereplaced. The table shows the results of 1000 cases. Distance (in km)

less than 4000

4000 to 9000

9001 to 14000

more than 14000

Frequency

20

210

325

445

If you buy a tyre of this company, what is the probability that: (i) it will need to be replaced before it has covered before it has covered 4000 km? (ii) it will last more than 9000 km? (iii) it will need to be replaced after it has covered somewhere between 4000 km and 14000 km? Marks (4) Q 35 Two coins are tossed simultaneously 500 times, and we get two heads:105 times; One head:275;No head:120 times, find the probability of occurrence of each of these events. And check the sum of Probabilities of all events. Marks (4) Most Important Questions

Q 1 Write the formula of finding the probability of an event. Q 2 Name the various approaches to probability. Q 3 In a cricket match, a batsman hits a boundary 5 times out of the 25 balls he plays. Find the probability that he didn‟t hit a boundary. Q 4 The percentage of marks obtained by a student in the monthly unit tests are given below Unit test : I II III IV V Percentage of marks obtained : 69 67

73

68

74

Based on this data, find the probability that the student gets more than 70% marks in a unit test. Q 5 Two coins are tossed simultaneously 1000 times with the following

frequencies of different outcomes :

Two heads : 210 times One head : 550 times No head

: 240 times

Find the probability of occurrence of each of these events.


Page 110

Q 6 To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following table : Opinion Like

Dislike 135

65

Number of Students

Find the probability that a student chosen at random (a) like Maths. (b) Does not like Maths. Q 7 The record of a weather station shows that out of the past 400 consecutive days, its weather forecasts were correct 175 times. (i) What is the probability that on a given day it was correct? (ii) What is the probability that it was not correct on a given day? Q 8 A die is thrown 2000 times with the following frequencies for the outcomes 1, 2, 3, 4, 5 and 6 as shown below: Outcome : Frequency :

1

2

3

358

300

314

4

5

298

350

6 380

Find the probability of happening of each outcome. Q 9 Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes : Outcome : Frequency:

3 heads 22

2 heads 80

1 head 70

No head 28

Find the probability of getting (a) Three heads (b) Two heads and one tail (c) At least two heads


Page 111

Q 10 A tyre manufacturing company kept a record of the distance covered before a tyre needed to be replaced. The table shows the results of 1000 cases. Distance

Less than

4000 t0

9000 t0

More than

(in km)

4000

9000

14000

14000

Frequency

20

210

325

445

If you buy a tyre of this company, what is the probability that : (i) it will need to be replaced before it has covered 4000 km? (ii) it will last more than 9000 km? Q 11 Fifty seeds were selected at random from each of 5 bags of seeds, and were kept under standardized conditions favorable to germination. After 20 days, the number of seeds which had germinated in each collection were counted and recorded as follows: Bag

: 1

Number of seeds germinated : 50

2 38

3 40

4 41

5 41

What is the probability of germination of (i) more than 40 seeds in a bag? (ii) 50 seeds in a bag? (iii) more that 36 seeds in a bag? Q 12 Following frequency distirbution gives the weight of 38 students of a class Weigh

31 - 35

36 - 40

9

5

41 - 45

46 - 50

51 - 55

56 - 60

3

1

2

61 - 65

66 - 70

(in kg) Number of Student 14

2

2

Find the probability that the weight of the student of a class is : (i) not more than 45 kg?


(ii) at least 45 kg.

Page 112

Q 13 1000 families with 2 children were selected randomly, and the following data was recorded : No. of girls in a family :

0

Frequency :

300

1

2

560

140

If a family is chosen at random, find the probability that it has (a) No girl (b) One girl (c) Two girls Q 14 On one page of a telephone directory, there are 400 telephone numbers. The frequency distribution of their unit place digit (for example : in the number 2441150, the unit place digit is 0) is given in the table below: Digit

: 0

1

2

3

4

5

6

7

Frequency : 44 52 44 44 40 20

28

56

8

9

32 40

A number is chosen at random, find the probability that the digit at its unit‟s place is : (i) 6. (ii) a non-zero multiple of 3. (iii) an odd number. (iv) a non-zero even number. Q 15 The percentage of marks obtained by a student in the monthly unit tests are given below : Unit Test Percentage of marks obtained

: I II III IV V : 58 64 76 62 85

Find the probability that the student gets : (i)

a first class i.e. at least 60% marks

(ii)

marks between 70% and 80%.

(iii)

a distinction i.e. 75% or above.

(iv)

a second class i.e. between 50% and 60%.


Page 113

Q 16 The distances (in km) of 20 female engineers from their residence to their place of work were found as follows: 5

3

10

15

7

28

10

12

22

2

9

21

1

11

14

17

31

7

8

22

Find the probability that the distance of the work place female engineers is : (i)

Less than 20 km. from her place of work?

(ii)

At least 10 km. from her place of work?

(iii)

Within 2.5 km. from her place of work?

(iv)

At most 25 km. from her place of work.

Q 17 An insurance company selected 2000 drivers at random (i.e., without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table: Age of Drivers

Accidents in one year

(in year) 0

1

2

3

4

Over 4

18 - 29

440

160

110

61

35

20

30-50

505

125

60

22

18

9

Above 50

360

45

35

15

9

4

Find the probabilities of the following events for a driver chosen at random from the city: (i) being 18-29 years of age and having exactly 4 accidents in one year. (ii) being 30-50 years of age and having one or more accidents in a year.


(iii) having no accidents in one year.

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Class IX-Mathematics-C.B.S.E.-Practice-Paper

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