SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics 1.
Topic : Number System
DPPS. NO. – 01
If 0 < a < b < c < d where (a, b, c, d) are positive integers, which among the following has the least value ? (A)
ab cd
(B)
ad bc
(C)
cd ab
(D)
bd ac
2.
Three boys P, Q, R agree to divide a bag of marbles as follows : P takes one more than half of the marbles; Q takes a third of the remaining marbles; R takes the marbles left out now in the bag. The original number of marbles found at the beginning in the bag must be (A) a multiple of 6 (B) one more than a multiple of 6 (C) Two more than a multiple of 6 (D) three more than a multiple of 6.
3.
If a – 1 = b + 2 = c – 3 = d + 4 then the largest among a, b, c, d is : (A) a (B) b (C) c
(D) d
4.
4ab5 is a four digit number divisible by 55 where a,b are unknown digits. Then b –a is : (A) 1 (B) 4 (C) 5 (D) 0
5.
(a – 1)2 + (b – 2)2 + (c – 3)2 + (d – 4)2 = 0. Then a b c d + 1 is : (A) 02 (B) 102 (C) 52
(D) 12 + 22 + 32 + 42 + 1
6.
Given that a,b,c and d are natural numbers and that a = bcd, b = cda, d = abc then (a + b + c + d ) 2 is : (A) 16 (B) 8 (C) 2 (D) 1
7.
The number of 3 digit numbers which end in 7 and are divisible by 11 is (A) 2 (B) 4 (C) 6
(D) 8
8.
The product of two numbers is 27. 33. 55. 73. Then the sum of these two numbers may be divisible by : (A) 16 (B) 9 (C) 25 (D) 49
9.
Let a,b,c,d be positive integers where a + b + c = 53, b + c + d = 51, c + d + a = 57, d + a + b = 58. Then the greatest and the smalllest numbers among a,b,c,d are respectively : (A) b and d (B) a and c (C) c and a (D) d and b
10.
It is given that 5 (A) 10
11.
3 1 b = 19 (where the two fractions are mixed fractions); then a + b = a 2 (B) 12 (C) 9 (D) 15
a and b are two primes of the form p and p +1 and M = a a + bb ; N = ab + ba then : (A) M and N are composite (B) M is a prime but N is composite (C) M and N are primes (D) M is composite, N is prime
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Page # 1
12.
What is the value of x if log3 x log9 x log27 x log81 x (A) 9
13.
(B) 27
(B) 54
(D) 51
(C) 1
(D) 0
Find the last digit in the finite decimal representation of the number 1/5 2003 ?
If a =
(B) 4
3 2
(A) 4 17.
(C) 50
(B) 3
(A) 2 16.
(D) None of these
If x and y are natural numbers, find the number pairs (x, y) for which x2 – y2 = 31. (A) 4
15.
(C) 81
The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice resulting in the incorrect sum 1998. What was the number of the page that was added twice ? (A) 45
14.
25 ? 4
–3
and B =
(C) 8
3 – 2
–3
(D) 6
, find the value of (a + 1)–1 + (b + 1)–1
(B) 3
(C) 1
Find the smallest positive number from the numbers below 10 – 3
(D) 0 11 , 3 11 – 10, 18 – 5
13 , 51 – 10 26 ,
10 26 – 51. (A) 51 – 10 26 18.
(D) 10 + 3 11
(B) 11
(C) 4
(D) 5
How many different four digit numbers are there in the octal (Base 8) system, expressed in octal system ? (A) 3584
20.
(C) 10 – 3 11
Find the number of two digit numbers divisible by the product of the digits. (A) 7
19.
(B) 10 26 – 51
(B) 2058
(C) 6000
(D) 7000
A hundred and twenty digit number is formed by writing the first x natural number in front of each other as 12345678910111213..... Find the remainder when this number is divided by 8. : (A) 6
(B) 7
(C) 2
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(D) 0
Page # 2
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
Topic : Algebra-1
DPPS. NO. – 02
1.
If (a-5)2 + (b-c)2 + (c-d)2 + (b+c+d-9)2 = 0, then the value of (a + b + c) (b + c + d) is : (A) 0 (B) 11 (C) 33 (D) 99
2.
If x +
1 1 = 3, then the value of x6 + 6 is : x x
(A) 927
(B) 114
(C) 364
(D) 322
3.
If in 3 + 3 5 , x = 3 and y = 3 5 , then its rationalising factor is (A) x + y (B) x – y (C) x5 + x4y + x3y2 + x2y3 + xy4 + y5 (D) x5 – x4y + x3y2 – x2y3 + xy4 – y5
4.
If a, b, c are positive, then (A) always smaller than (C) greater than
5.
If
37 =2+ 13
(A) 1,2,5
8.
(B) always greater than (D) greater than
a b
a only if a < b b
If a + b – c = 0, then the value o (a b c )2 is : (B) 2bc 1 x
If
(C) 4ab
(D) 4ac
where x,y,z are natural numbers then values of x,y,z are
1 y
7.
a b
a only if a > b b
(A) 2ab
6.
ac is : bc
1 z
(B) 1,5,2
(C) 5,2,1
(D) 2,5,1
1 1 1 1 + + = where (a+b+c)0 and abc 0. What is the value of (a+b) (b+c) (c+a) ? a b c a b c
(A) 0
(B) 1
If, x +
1 1 = 1 and y – = 1, then the value of xyz is y z
(A) 1
(B) –1
(C) –1
(D) 2
(C) 0
(D) –2
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9. 10.
If x + y + z = 1, x² + y² + z² = 2 and x³ + y³ + z³ = 3 then the value of xyz is: (A) 1/5 (B) 1/6 (C) 1/7
(D) 1/8
The polynomials ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by (x – 4) leaves remainders R1 & R2 respectively then value of ‘a’ if 2R1 – R2 = 0. (A) –
18 127
(B)
18 127
(C)
17 127
(D) –
17 127
11.
, , are zeros of cubic polynomial x3 – 12x2 + 44x + c. If , , are in A.P., find the value of c. (A) –48 (B) 24 (C) 48 (D) – 24
12.
If x2 – 4 is a factor of 2x3 + ax2 + bx + 12, where a and b are constant. Then the values of a and b are : (A) – 3, 8 (B) 3, 8 (C) –3, – 8 (D) 3, – 8
13.
If xy + yz + zx = 1, then the expression 1
(A) x y z
1
(B) xyz
xy yz zx 1 xy + 1 yz + 1 zx is equal to
(C) x + y + z
(D) xyz
14.
For which values of 'a' and 'b' does the following pair of linear equations have an infinite number of solutions: 2x + 3y = 7, (a – b)x + (a + b)y = 3a + b – 2 (A) a = 5 , b = 1 (B) a = 4, b = 2 (C) a = 1, b = 5 (D) a = 2, b = 4
15.
If the sum of the roots of the equation ax2 + bx + c = 0 is equal to product of their reciprocal then, (A) a2 + bc = 0 (B) b2 + ca = 0 (C) c2 + ab = 0 (D) b + c = 0
16.
If the sum of the two roots of the equation (A) 0
17.
(B)
a2 b 2 2
1 1 1 is zero, then the product of the two roots is: xa xb c
(C)
ab 2
If 4x +3y = 123, find how many positive integer solutions are possible ? (A) 9 (B) 10 (C) infinite
(D) –
a
2
b2 2
(D) can’t be detremined
18.
A fast train takes 3 hours less than a slow train for a journey of 900 km. If the speed of slow train is 15 km/ hr. less than that the fast train. Find the sum of speeds of the two trains. (A) 120 km/hr (B) 125 km/hr (C) 135 km/hr (D) 150 km/hr
19.
The quadratic equation a x2 + bx + c = 0 has real roots and . If a, b, c real and of the same sign, then (A) and are both positive (B) and are both negative (C) and are of opposite sign (D) nothing can be said about the signs of and as the information is insufficient.
20.
Number of integral values of ‘p’ for which the quadratic equation x2 – px + 1 = 0 has no real roots is : (A) 2 (B) 3 (C) 5 (D) infinite
21.
If 2a = b, the pair of equations ax + by = 2a2 – 3b2, x + 2y = 2a – 6b possess : (A) no solution (B) only one solution (C) only two solutions (D) an infinite number of solutions
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Page # 2
22.
23.
The equations 2x – 3y + 5 = 0 and 6y – 4x= 10, when solved simultaneously , have (A) only one solution (B) no solution (C) only two solutions (D) infinite number of solutions and If and are the roots of the equation x2 - 9x + 5 = 0. then find the equation whose whose roots are 2 . 2
(A) 10x2 - 122 x - 61 = 0
24.
(C) 20x2 - 122x - 61 = 0
If the roots of the equation px2 + rx + r = 0 are in the ratio a : b, then value of
(A)
25.
(B) x2 - 122x - 61 = 0
r p
(B) –
r p
(C)
1 p
(D) None of these b a is : a b
(D)
1 r
If x2 + ax + b = 0 and x2 + bx + a = 0, a b , have a common root 'a' then which of the following is true ? (A) a + b = 1 (B) a + 1 = 0 (C) a = 0 (D) a + b + 1 = 0
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Page # 3
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
1.
If y =
1 x x2 1 x x2
Topic : Algebra- 2
and x is real, then y can lies between :
(A) 1 and 3 2.
DPPS. NO. – 03
(B) –3 and –
1 3
(C)
1 and 3 3
(D) None of these
Find the value of x if : (5 2 6 )x
2
3
(5 2 6 )x
(A) 2, 2
2
3
10
(B) 2, 2
(C) 2,
2
3
If the roots of the equation px2 + qx + r = 0 are in the ratio : m then : (A) ( + m)2 pq = mr2 (B) ( + m)2 pr = mq 2 2 (C) ( + m) pr = mq (D) none
4.
If the roots of the equation (m2 + 1)x2 + 2amx + a2 – b2 = 0 be equal, then (A) a2 + b2(m2 + 1) = 0 (B) b2 + a2(m2 + 1) = 0 2 2 2 (C) a – b (m + 1) = 0 (D) b2 – a2(m2 + 1) = 0
5.
Let , be the roots of the equation x2 – px + r = 0 and
(D) 2 2 , 2
, 2 be the roots of the equation x2 – qx + r = 0. Then the value 2
of r is (A)
2 (p – q) (2q – p) 9
(B)
2 (q – 2p) (2p – q) 9
(C)
2 (q – 2p) (2q –p) 9
(D)
2 (2p – q) (2q – p) 9
6
If x2 + ax + 10 = 0 and x 2 + bx – 10 = 0 have a common root, then a 2 – b2 is equal to (A) 10 (B) 20 (C) 30 (D) 40
7.
Two friends tried solving a quadratic equation x 2 + bx + c = 0. One started with the wrong value of b and got the roots as 4 and 14; the other started with the wrong value of c and got the roots as 17 and – 2. Find the actual roots. (A) 7, 8 (B) 28, 2 (C) 19, 2 (D) 13, 2
8.
The quadratic equation a x2 + bx + c = 0 has real roots and . If a, b, c are real and of the same sign, then (A) and are both positive (B) and are both negative (C) and are of opposite sign (D) nothing can be said about the signs of and as the information is insufficient.
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9.
10.
For what value of m does the equation x2 – x + m2 = 0 possess no real roots : 1 1 (A) , 2 U 2 ,
1 (B) , 1 U 2 ,
(C) , 2 U , 1
(D) 1, U (– , 1)
If the roots of the equation px2 + rx + r = 0 are in the ratio a : b, then value of (A)
11.
r p
(B) –
r p
(C)
1 p
b a is : a b
(D)
1 r
If , are the roots of x2 + x + 1 = 0 and , are the root of x2 + 3x + 1 = 0, then = (A) 2
(B) 4
(C) 6
(D) 8
12.
A fast train takes 3 hours less than a slow train for a journey of 900 km. If the speed of slow train is 15 km/ hr. less than that of the fast train. Find the sum of speeds of the two trains. (A) 120 km/hr (B) 125 km/hr (C) 135 km/hr (D) 150 km/hr
13.
If the difference of the roots of equation x2 – bx + c = 0 be 1 then (A) b2 – 4c – 1 = 0 (B) b2 – 4c = 0 (C) b2 – 4c + 1 = 0
(D) b2 + 4c – 1 = 0
14.
The number of integral values of ‘m’ less than 50, so that the roots of the quadratic equation mx2 + (2m – 1) x + (m – 2) = 0 are rational, are (A) (B) 7 (C) 8 (D) None of these
15.
If the equation x2 + x + = 0 has equal roots and one root of the equation x2 + x – 12 = 0 is 2, then ( ,) = (A) (4, 4) (B) (–4, 4) (C) (4, –4) (D) (–4, –4)
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Page # 2
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics 1.
4.
1
(B) 2
1
(C) 2
(D) 4
The inverse of the sum of the following series up to n terms can be written as
(A)
3
DPPS. NO. – 04
The sum of the third and seventh terms of an A.P. is 6 and their product is 8, then common difference is : (A) 1
2.
Topic : Progression
(n 1)2 n2 2n
(B)
n2 2n (n 1)2
(C)
n2 2n (n 1)2
3 5 7 +... 4 36 144
(D)
(n 1)2 n2 2n
If m arithmetic means are inserted between 1 and 31, so that the ratio of the 7 th and (m – 1)th means is 5 : 9, then the value of m is (A) 12 (B) 14 (C) 16 (D) None of these 1 1 1 1 1 + ....... equals 1 3 6 10 15 (A) 2 (B) 3
(D)
(C) 5
5.
Let Sn denote the sum of the first 'n' terms of an A.P. and S2n = 3Sn. Then, the ratio S3n : Sn is equal to : (A) 4 : 1 (B) 6 : 1 (C) 8 : 1 (D) 10 : 1
6.
Suppose a, b, c are in A.P. and a 2, b2, c2 are in G.P. if a < b < c and a + b + c =
(A)
7.
If
1 2 2
1 2 3
(C)
1 1 – 3 2
(D)
1 1 – 2 2
bc –a ca–b ab–c , and are in A.P. and a + b + c 0, then : a b c
(A) b = 8.
(B)
3 , then the value of a is : 2
ac ac
(B) b =
2ac ac
(C) b =
ac 2
(D) b =
ac
If the ratio of the sum of n terms of two A.P’s is (3n + 4) : (5n + 6), then the ratio of their 5 th term is (A)
21 31
(B)
31 41
(C)
31 51
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(D)
11 31
Page # 1
9.
If a2 + b2 + c2 = 1 and p = ab + bc + ca, then : (A)
10.
11.
1 p2 2
(B) –
1 1 p 2 2
(C) –
1 p 1 2
(D) – 1 p
1 2
If b1, b2, b3.......belongs to A.P. such that b1 + b4 + b7 + ..... + b28 = 220, then the value of b1 + b2 + b3 ........+.........+b28 equals (A) 616 (B) 308 (C) 2,200
(D) 1,232
If a2(b + c), b2(c + a), c2 (a + b) are in A.P., then either a,b,c are in A.P. or (A) ab + bc + ca = 0 (B) a + b + c = 0 (C) a – b – c = 0
(D) a – b + c = 0
12.
If sum of n terms of a sequence is given by S n = 2n 2 + 3n, find its 50 th term. (A) 250 (B) 225 (C) 201 (D) 205
13.
Divide 600 biscuits among 5 boys so that their shares are in Arithmetic progression and the two smallest shares together make one-seventh of what the other three boys get. What is the sum of the shares of the two boys who are getting lesser number of biscuits, than the remaining three ? (A) 75 (B) 85 (C) 185 (D) 90
14.
An A.P. consists of 21 terms. The sum of three terms in the middle is 129 and of the last three terms is 237. Then the A.P. is : (A) 3, 7, 11, 15 ......... (B) 2, 7, 12, ........ (C) 5, 9, 13, 15 ........ (D) none of these
15.
Find the value of (A)
8 33
2 2 2 2 2 + + + + ....... + 15 35 63 99 9999
(B)
2 11
(C)
98 303
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(D)
222 909
Page # 2
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
Topic : Geometry - 1
DPPS. NO. – 05
1.
Through D, the mid-point of the side BC of a triangle ABC, a straight line is drawn to meet AC at E and AB produced at F so that AE = AF. Then the ratio BF : CE is : (A) 1 : 2 (B) 2 : 1 (C) 1 : 3 (D) None of these
2.
A triangle whose sides are integers has a perimeter 8. Find the area of the triangle . (A)
3.
8
(B)
10
(C)
7
Two sides of a triangle are 10 cm and 5 cm in length and the length of the median to the third side is 6 The area of the triangle is 6 x cm2. The value of x is (A) 12 (B) 13 (C) 14
4.
(D) 15
Area of DEF Area of ABC
(A) 3 : 1
(B) 9 : 1
(C) 1 : 1
ABCD is a parallelogram P is a point on AD such that Then
(D) None of these
AP 1 = Q is the point of inter section of AC and BP.. AD 2013
AQ = AC
(A) 1 : 2013 6.
1 cm. 2
In the adjoining figure. ABC is equilateral. AD, BE and CF are respectively perpendicular to AB, BC and AC. Then
5.
(D) None of these
(B) 1 : 2015
(C) 1 : 2014
(D) None of these
ABC is an isosceles triangle with B = C = 78°. D and E are points on AB, AC respectively such that BCD = 24° and angle CBE = 51°. Find BED. (A) 26º
(B) 12º
(C) 15º
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(D) None of these Page # 1
7.
The internal bisector AE of the angle A of triangle is (A) not greater than the median through A for all triangles. (B) not greater than the median through A for only acute angled triangles. (C) not greater than the median through A for only obtuse angled triangles. (D) not less than the median through A for all triangles.
8.
P is a point inside an equilateral triangle of side 2010 units. The sum of the length of the perpendiculars drawn from P to the sides is equal to (A) 2010
(B) 2010 3
(C)1005 3
(D)
2010 3
A
y R 3
9.
Q
x 7
B
P
C
In the figure given above, points P and Q are mid points on the sides AC and BP respectively. Area of each part is shown in the figure, then find the value of x + y. (A) 11 (B) 4 (C) 7 (D) 18 10.
11.
Find the number of integer sided triangle with perimeter 100. (A) 207 (B) 204 (C) 144
(D) 208
Two sides of a triangle are 3 cms and Find the third side. (A) 2 (B) 2
(D) Cannot be determined
2 cms. The medians to these sides are perpendicular to each other..
(C) 1
12.
In ABC, BC = 20 , medians BE = 18 and median CF = 24 (E, F are midpoints of AC, AB respectively). Find the area of ABC. (A) 244 sq. units (B) 288 sq. units (C) 144 sq. units (D) none of these
13.
The measures of length of the sides of a triangle are integers and that of its area is also an integer. One side is 21 and the perimeter is 48. Find the measure of the shortest side. (A) 13 units (B) 10 unit (C) 12 unit (D) None of these
14.
Let ABC be an acute angled triangle and CD be the altitude through C. If AB = 8 and CD = 6, find the distance between the mid-points of AD and BC. (A) 1 unit (B) 2 unit (C) 5 unit (D) 10 unit
15.
In a triangle ABC, medians AD and BE are drawn. If AD = 4, DAB =
and ABE = , then the area of the 6 3
ABC is : (A) 16.
8 3
(B)
16 3
(C)
32 3 3
(D)
64 . 3
In a triangle ABC, if a : b : c = 3 : 7 : 8, then Circumradius (R) : inradius (r) is equal to (A) 2 : 7 (B) 7 : 2 (C) 3 : 7 (D) 7 : 3
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Page # 2
17. 18.
Let ABC be an equilateral triangle and AD be the altitude through A. Then (A) AD2 = 3BD2 (B) AD2 = 5BD2 (C) AB2 + AC2 = BC2
In triangle ABC with angle A = 90 . The bisectors of the angle B and C meet at P. The distance from P to the hypotenuse is 4 2 . Find the distance AP.. (A) 4 (B) 6
19.
(C) 8
(D) None of these
Let ABC be a triangle with AB = AC = 6, if the circumradius of the triangle is 5 then find BC (A) 10
20.
(D) AD2 = 2BD2
48
(B) 9
(C) 5
(D) None of these
In a triangle ABC , AB = AC = 20 cm and D and E are two points on AB and AC and AD = AE = 12 cm , If the area of ADFE is 24 cm2 . find the area of the triangle BFC in sq. cm (A) 24
40
(B) 20
(C) 3
(D) None of these
21.
ABC is an isosceles triangle with mA = 20° and AB = AC. D and E are points on AB and AC such that AD = AE. I is the midpoint of the segment DE. If BD = ID, then the angle of IBC are : (A) 110°, 35°, 35° (B) 100°, 40°,40° (C) 80°,50°,50° (D) 90°,45°,45°
22.
A, B, C and D are points on a line. E is a point outside this line. Given that AE = BE = AB = BC and CE = CD, we find that the measure of DEA is :
(A) 90° 23.
(D) 150°
(B) 15°
(C) 20°
(D) 50°
2
In triangle ABC, BD bisects angle B. If A = 3 B and B = 3C, then BDC is :
(A) 75° 25.
(C) 120°
In the adjoining figure find the size of ACE, given AD = DB and DE = DC.
(A) 30° 24.
(B) 105°
(B) 105°
(C) 90°
(D) 120°
Triangle ABC is isosceles with base AC. Points P and Q are respectively in CB and AB such that AC = AP = PQ = QB. Find the measure of B . . (A) 26
5 º 7
(B) 25
5 º 7
(C) 45º
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(D) None of these Page # 3
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics 1.
Topic : Geometry - 2
DPPS. NO. – 06
In the adjoining figure O is the centre of the circle. AOD = 120°. If the radius of the circle be ‘r’ then find the sum of the areas of the quadrilateral AODP and OBQC :
(A)
3 2 r 2
(C) 3r 2
(B) 3 3 r2
(D) None of these
2.
Chords AB and PQ meet at K and are perpendicular to one another. If AK = 4, KB = 6 and PK = 2, find the area of the circle. (A) 25sq. units (B) 20sq. units (C) 100sq. units (D) 50sq. units
3.
ABC is a right angled triangle, right angled at B such that BC = 6 cm and AB = 8 cm. A circle with centre O is inscribed in ABC. The radius of the circle is : (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm
4.
If a regular polygon of sides ‘n’ is circumscribed about a given circle of radius R then the length of side of the polygon is (A) 2R tan
5.
n
(B) 2R sin
n
(C) 2R cot
n
(D) 2R sec
n
The common tangents to the circle and with centres P and Q meet the line joining P and Q at ‘O’ as shown. Given that the length OP = 28 cm and the diameters of and are in the ratio 4 : 3. The radius of the circle is
(A) 2 cm
(B) 3 cm
(C) 4 cm
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(D) 5 cm
Page # 1
6.
BC is the diameter of a semicircle. The sides AB and AC of a triangle ABC meet the semicircle in P and Q respectively. PQ subtends 140° at the centre of the semi-circle. Then A is
(A) 10° 7.
(B) 20°
(C) 30°
(D) 40°
In the given diagram shown below, if PB = 8 cm, AB = 4 cm, PD = 6 cm, then CD = ? A
4 cm
B
8 cm P
?
C
(A)
16 cm 3
D
6 cm
(B) 10 cm
(C) 6 cm
(D) 7 cm
8.
E is the midpoint of diagonal BD of a parallelogram ABCD. If the point E is joined to a point F on DA such that DF = 1 DA, then the ratio of the area of DEF to the area of quadrilateral ABEF is : 3 (A) 1 : 3 (B) 1: 4 (C) 1 : 5 (D) 2 : 5
9.
In the figure, ABCD is a square and P, Q, R, S are the midpoints of the sides. D
R
S
A
C
Q P
B
What is the area of the shaded part, as a fraction of the area of the whole square ? (A) 10.
1 3
(B)
1 4
(C)
1 5
(D)
1 6
In the figure, PQ is a chord of a circle with centre O and PT is the tangent at P such that QPT = 70º. Then the measure of PRQ is equal to
(A) 135º 11.
(B) 150º
(C) 120º
(D) 110º
One of the side of a triangle is divided into line segment of lengths 6cm and 8cm by the point of tangency of the incircle of the triangle. If the radius of the incircle is 4cm, then the length (in cm) of the longer of the two remaining sides of the triangle is (A) 12 (B) 13 (C) 15 (D) 16
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Page # 2
12.
A square board side 10 centimeters, standing vertically, is tilted to the left so that the bottom-right corner is raised 6 centimeters from the ground.
6 cm
By what distance is the top-left corner lowered from its original position ? (A) 1 cm (B) 2 cm (C) 3 cm 13.
(D) 0.5 cm
In the figure, AB || DC, AD > BC, and DC : AB = k, where k>1. The ratio (AD2 – BC2) : (DB2 – AC2) is : A
B
D
(A)
1 k 1
(B)
C
k 1 k –1
(C)
k –1 k 1
(D)
k2 1 k2 1
14.
ABCD is square AB produced to Q such that AC intersect DQ at P, BPQ intersect BC at R. Such that DP = 3, PR = 2. Find RQ (A) 2.5 (B) 3.2 (C) 3.5 (D) 4
15.
In the adjoining figure O is the centre of the circle. ACOB is a square with A on the circle. Through B a line parallel to OA is drawn to cut the circle at D nearer to A. Then BOD =
B
O
D A
(A) 20º 16.
(B) 18º
C
(C) 15º
(D) 25º
In a rectangle ABCD the lengths of sides AB, BC, CD and DA are (5x + 2y + 2) cm, (x + y + 4) cm, (2x + 5y – 7) cm and (3x + 2y – 11) cm respectively. Which of the following statements is /are true ? (A) One of the sides of the rectangle is 15 cm long. (B) Each diagonal of the rectangle is 39 cm long. (C) Perimeter of the rectangle is 102 cm. (D) All of these
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Page # 3
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
1.
1 1 + a b
(B)
5.
(B) 2
cos cos m cos
If tan = 125 78
If cos + sin =
(B)
(C) 3
(D) 4
sin cos (C) a sin m
sin cos (D) a cos m
(B) equal to 1 (D) greater than 1 but less than 2
130 3
(C)
125 102
(D) None of these
(C)
2 cos sin
(D) none of these.
(C)
n sin cos ec
(D) n
2 cos, then cos – sin = ?
(B) 2 sin
If sin + cosec = 2, then sinn + cosecn = (A) n(sin + cosec )
8.
(D) a – b
4 5 sin 7 cos , where 0 < < 90º, then the value of is : 15 6 cos 3 sin
(A) 2 tan 7.
sin sin (B) a sin m
The value of (cos2 + sec2 ) is always : (A) less than 1 (C) greater than or equal to 2
(A) 6.
(C) a + b
The shadow of a pole standing on a horizontal plane is a metre longer when the sun’s elevation is than when it is . The height of the pole of will be : (A) a
4.
1 1 – a b
1 3 The value of sin10 cos 10 is : (A) 1
3.
DPPS. NO. – 07
If tanA + tanB = a and cot A + cotB = b, then the value of cot (A+ B) is : (A)
2.
Topic : Trigonometry
If 0 < < 90 and (A) 30º
(B) 2
sin sin = 4, then the value of is : 1 cos 1 cos (B) 45º (C) 60º
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(D) none
Page # 1
9.
The angles of elevation of an aeroplane flying vertically above the ground as observed from two consecutive stones 1 km apart are 45º and 60º. The height of the aeroplane above the ground in km is
3 1 2
(A)
10.
(B)
3 3 2
(D)
(C) r = p q
(D) p = q = r
(C) 1
(D) 2
2
1 – sin x 1 – sin x cos x ,q= ,r= , then 1 sin x cos x 1 sin x
If p =
Which one of the following statement is correct ? (A) p = q r (B) q = r p 11.
1 2 2
(C) 3 3
The trigonometric expression sec – 1
sin – 1
cot2 1 sin + sec2 1 sec has the value (A) –1 12.
13.
(B) 0
The expression (1 – tan A + sec A) (1 – cot A + cosec A) has value : (A) – 1 (B) 0 (C) + 1 If each of , and is a positive acute angle such that sin ( + – ) = = 1, then the values of , and is : (A) 45º, 45º & 90º
14.
15.
16.
17.
tan 23° + tan 22° + tan 23° tan 22° = (A) – 1 (B) 0
(C) 37 1 , 45° & 52 1
(D) none
(C) 1
(D) 2
2
2
(D) None
The value of sin( +) sin ( – ) cosec2 is equal to : (A) – 1 (B) 0 (C) sin
(D) none of these
If in an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle, then the area of the triangle is :
If tanx =
(A)
19.
1 1 , cos( + – ) = and tan ( + – ) 2 2
If x = r sin cos , y = r sin sin , z = r cos then the value of x2 + y2 + z2 is : (A) 0 (B) 1 (C) r2
(A) 4 + 2 3
18.
(B) 60º, 45º & 75º
(D) + 2
(B) 6 + 4 3
(C) 12 +
7 3 4
(D) 3 +
7 3 4
p , then the value of (p cos 2x + q sin2x) is : q
q(3q2 p 2 ) (p 2 q 2 )
(B)
p(3q2 p 2 ) (p 2 q 2 )
(C) p
If a cos + b sin = 3 & a sin b cos = 4 then a 2 + b2 has the value = (A) 25 (B) 14 (C) 7
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(D) q
(D) none Page # 2
20.
If 4 sin x = 12 sin 2 x – 1 where 0 x 90º then the value of (A)
21.
2 3 3
(B) 2 3
(B) 2100 m.
(C) 4200 m.
If sin x + sin y = a and cos x – cos y = b.Then find the value of (A) cos(x + y)
23.
(D) 3 3
(C) 3
An aeroplane is flying horizontally at a height of 3150 m above a horizontal plane ground. At a particular instant it passes another aeroplane vertically below it. At this instant, the angles of elevation of the planes from a point on the ground are 30º and 60º. Hence, the distance between the two planes at that instant is : (A) 1050 m.
22.
tan x tan x is equal to : – sec x 1 sec x 1
(B) cos(x – y)
(D) 5250 m.
1 (2 – a2 – b2) 2
(C) sin(x + y)
(D) sin(x – y)
A person on the top of a tower observes a scooter moving with uniform velocity towards the base of the tower. He finds that the angle of depression changes from 30º to 60º in 18 minutes. The scooter will reach the base of the tower in next :
24.
(A) 9 minutes
(B) 18 / ( 3 – 1) minutes
(C) 6 3 minutes
(D) the time depends upon the height of the tower
From a lighthouse 100 m high, it is observed that two ships are approaching it from west and south. If angles of depression of the two ships are 300 and 450 respectively the distance between the ships, in meters, is :
(A) 100 3 1
(B) 100 3 1
2
(C) 200
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(D) 400
Page # 3
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
Topic : Co-ordinate Geometry
DPPS. NO. – 08
1.
The cordinates of point P are (a, b) where a is positive b is negative. In which quadrant does the point. (–2a, –2b) be : (A) I Quadrant (B) II Quadrant (C) III Quadrant (D) IV Quadrant
2.
The diagonals of a rhombus lie along the coordinate axis. Its area is 24 sq. units. If two of the vertices of rhombus are ( 3, 0), then the other two vertices are : (A) (0, 8)
(B) ( 6, 0)
(C) (0, 4)
(D) None of these
3.
Three vertices of a parallelogram are (a + b, a – b), (2a + b, 2a – b) and (a – b, a + b). Find the fourth vertex. (A) (–b, b) (B) (–a, a). (C) (–b, a). (D) (b, b).
4.
Area of the triangle whose vertices are (t, t–2), (t+2, t+2) and (t+3, t) is : (A) Dependent of t (B) Independent of t (C) Can’t say anything
5.
If the points (k, 2 – 2k), (1 – k, 2k) and (–k –4, 6 – 2k) be collinear, the possible values of k are (A) –
6.
(D) Zero
1 2
(B)
1 2
(C) 1
(D) –1
21 The line segment joining (–3, –1) and (–8, –9), divided at the point 5, in the ratio: 5
(A) 2 : 3 externally
(B) 3 : 2 externally
(C) 2 : 3 internally
(D) 3 : 2 internally
7.
If the area of the triangle formed by the points (x, 2x), (–2, 6) and (3, 1) is 5 square units, then x = : 2 (A) (B) 2 (C) Both (A) and (B) (D) None of these 3
8.
The ratio in which the line 3x+y–9 = 0 divides the segment joining the points (1,3) and (2, 7) is : (A) 3 : 4 (B) 1 : 2 (C) 2 : 3 (D) 3 : 7
9.
If a vertex of a triangle be (1, 1) and the middle points of two sides through it be (– 2, 3) and (5, 2) then the centroid of the triangle is : (A) 5,
3 2
(B)
5 ,3 3
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(C)
1 3 , 2 2
(D) None of these
Page # 1
10.
The line segment joining the points (3, – 4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, – 2) and (5/3, q) respectively, then the value of p and q is : (A) p = 7/3, q = – 3 (B) p = 7/3, q = 0 (C) p = 5/3, q = 0 (D) None of these
11.
The points (– a, – b), (0, 0), (a, b) and (a2, ab) are : (A) Collinear (C) Vertices of a rectangle
(B) Vertices of a parallelogram (D) None of these.
12.
If the centroid and circumcentre of a triangle are (3, 3) and (6, 2) respectively, then the orthocentre is (A) (–3, 5) (B) (–3, 1) (C) (3, –1) (D) (9, 5)
13.
Area of the quadrilateral with vertices P(–1, 6), Q(–3, –9), R(5, –8) and S(3, 9) is (in sq. units) (A) 48 (B) 96 (C) 24 (D) none of these
14.
f P (1, 2), Q (4, 6), R (5, 7) & S (a, b) are the vertices of a parallelogram PQRS, then : (A) a = 2, b = 4 (B) a = 3, b = 4 (C) a = 2, b = 3
15.
If two opposite vertices of a square are (5, 4) and (1, –6), then the coordinates of its remaining two vertices is : (A) (–2, 2) & (5, 3)
16.
(D) a = 3, b = 5
(B) (8, –3) & (–2, 1)
(C) (8, 6) & (3, 5)
(D) (1, –3) & (2, 5)
Let A, B, C, D be collinear points in that order. Suppose AB : CD = 3 : 2 and BC : AD = 1 : 5. Then AC : BD is : (A) 1 : 1
(B) 11 : 10
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(C) 16 : 11
(D) 17 : 13
Page # 2
SUMMER WORKSHOP (IJSO)
Daily Practice Problem Sheet Subject : Mathematics
Topic : Mensuration
DPPS. NO. – 09
1.
In a right angled triangle, if the square of the hypotenuse is twice the product of the other two sides, then which of the following is true . (A) triangle is isoscels (B) Triangle is equilateral (C) Triangle is scalene (D) None of these
2.
Find the ratio of the area of the equilateral triangle inscribed in a circle to that of a regular hexagon inscribed in the same circle. (A) 1 : 2 (B) 1 : 4 (C) 2 : 3 (D) None of these
3.
In this figure, AOB is a quarter circle of radius 10 and PQRO is a rectangle of perimeter 26. The perimeter of the shaded region is : B
Q
R
O
(A) 13 + 5 4.
(B) 17 + 5
P
A
(C) 7 + 10
(D) 7 + 5
In the figure shown, the bigger circle has radius 1 unit. Therefore, the radius of smaller circle must be (A) (B)
(C)
(D)
2 +1
1 2
1 2 1 2 1
5.
A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn meeting the circumference of the semicircle at D. Given that AC = 2 cm and CD = 6 cm, the area of the semicircle is : (A) 32 (B) 50 (C) 40 (D) None of these
6.
Three parallel lines 1, 2 and 3 are drawn through the vertices A, B and C of a square ABCD. If the distance between 1 and 2 is 7 and between 2 and 3 is 12, then the area of the square ABCD is : (A) 193 (B) 169 (C) 196 (D) 225
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Page # 1
7.
In a right angled triangle ABC, right angled at C, a + b = 17 unit and c = 3 unit, then find the area of the triangle. (A) 1 sq.unit (B) 2 sq.unit (C) 3 sq.unit (D) 4 sq.unit
8.
Two circles seen in the figure are concentric. Chord AB of the larger circle is tangent to the smaller circle and its length is equal to 16. The area of the shaded region, is : (A) 32 (B) 64 (C) 32 (D) 16 2 – 16
9.
If in the figure, each circle is of radius 2 cm, then the width AD of the rectangle ABCD is : D C
A
(A) 10 cm
B
(C) 4 ( 3 1) cm
(B) 12 cm
(D) 4 ( 3 1) cm
10.
The circumference of the circumcircle of the triangle formed by x -axis, y-axis and graph of 3x + 4y=12 is: (A) 3 units (B) 4 units (C) 5 units (D) 6.25 units
11.
In the figure AD = DB, BE =
1 1 EC and CF = AF. If the area of ABC = 120 cm2, the area (in cm2) of DEF is 3 2 A
D B (A) 21
F C
E
(B) 35
(C) 40
(D) 45
12.
A circle is inscribed in an equilateral triangle of side 'a' cm. The area (in cm 2) of a square inscribed in the circle is : (A) a2/6 (B) a2/3 (C) 3a2/4 (D) a2/12
13.
Each side of an equilateral triangle is a cm. Find out the ratio of areas of the circumcircle and the incircle of the triangle (A) 1 : 2 (B) 2 : 1 (C) 4 : 1 (D) 1 : 4
14.
A circle is drawn in a sector of a larger circle of radius, r, as shown in the adjacent figure. The smaller circle is tangent to the two bounding radii and the arc of the sector. The radius of the small circle is
(A)
r 2
(B)
r 3
(C)
2 3r 5
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(D)
r 2 Page # 2
15.
The sides of a quadrilateral are all positive integers and three of them are 5, 10, 20. How many possible value are there for the fourth side ? (A) 29 (B) 31 (C) 32 (D) 34
16.
A circle is inscribed in an isosceles trapezium ABCD in which AB is parallel DC. If AB = 10 and the area of the circle. (A) 45 (B) 50 (C) 60 (D) 75
17.
A sphere and cube have equal surface areas.The ratio of their volumes is (A)
6
(B)
6
6
(C)
(D)
DC = 30. Find
3
18.
A cylinder of radius 6 cm and height h cm is filled with ice cream. The ice cream is then distributed among 10 children in identical cones having hemispherical tops. The radius of the base of the cone is 3 cm and its height is 12cm. Then the height h of the cylinder must be : (A) 100/7 cm (B) 18 cm (C) 15 cm (D) 200 / 11 cm
19.
A round pencil has length 8 units when unstreched and diameter 1/4. It is sharpened perfectly so that it remains 8 units long with 7 units section still cylindrical and remaining 1 unit giving a conical tip. Volume of the pencil now is (A)
11 96
(B)
37 192
(C)
7 64
(D) None
20.
Water flows into a tank 150 metres long and 100 metres broad through a pipe whose cross-section is 2 dm by 1.5 dm at the speed of 15 km per hour. In what time, will the water be 3 metres deep ? (A) 100hr (B) 10hr (C) 50 hr (D)25 hr
21.
The radii of three cylindrical jars of equal height are in the ratio 1 : 2 : 3. Second jar is full of water which is first poured into the first jar. After filling the first jar, water is poured into the third jar. Which of the following statements is ture ? (A) Third jar is half filled (B) Third jar is one third filled (C) Third jar is two thirds filled (D) Third jar is four ninths filled.
22.
From a 25 cm × 35 cm rectangular cardboard, an open box is to be made by cutting out identical squares of area 25 cm 2 from each corner and turning up the sides. The volume of the box is : (A) 3000 cm3
(B) 1875 cm3
(C) 21875 cm3
(D) 1250 cm3
23.
The surface water in a swimming pool forms a rectangle of length 40 m and breadth 15m. The depth of water increases uniformly from 1.2m at one end to 2.4m at the other end. The volume (in m 3) of water in the pool is : (A)1080 (B) 720 (C) 600 (D) 540
24.
A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A metal sphere is now lowered into the water. The size of the sphere is such that when it touches the inner surface, it just gets immersed. The fraction of water that overflows from the conical vessel is: (A)
25.
3 8
(B)
5 8
(C)
7 8
(D)
5 16
The diameter of one of the bases of a truncated cone is 100 mm. If the diameter of this base is increased by 21% such that it still remains a truncated cone with the height and the other base unchanged, the volume also increases by 21%. The radius the other base (in mm) is (A) 65 (B) 55 (C) 45 (D) 35
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Page # 3
SUMMER WORKSHOP (IJSO) Daily Practice Problem Sheet ANSWER KEY DPPS -1. Number System Ques Ans. Ques Ans.
1 A 16 C
2 C 17 A
3 C 18 D
4 A 19 A
5 C 20 A
6 A
7 D
8 C
9 B
10 A
11 C
12 D
13 A
14 C
15 C
10 A 25 B
11 A
12 A
13 B
14 A
15 A
10 B
11 D
12 C
13 A
14 A
15 A
10 A
11 A
12 C
13 A
14 A
15 C
10 D 25 B
11 C
12 B
13 B
14 C
15 C
10 D
11 C
12 B
13 B
14 A
15 C
10 D
11 B
12 D
13 C
14 C
15 C
DPPS - 2. Algebra-1 Ques Ans. Ques Ans.
1 D 16 D
2 D 17 B
3 D 18 B
4 D 19 B
5 C 20 A
6 B 21 A
7 C 22 A
8 A 23 C
9 B 24 B
DPPS - 3. Algebra-2 Ques Ans.
1 C
2 A
3 C
4 C
5 D
6 D
7 A
8 B
9 A
DPPS - 4. Progression Ques Ans.
1 C
2 D
3 B
4 A
5 B
6 D
7 B
8 C
9 C
DPPS - 5. Geometry-1 Ques Ans. Ques Ans.
1 D 16 B
2 A 17 A
3 C 18 C
4 A 19 C
5 B 20 C
6
7
21 B
22 B
8 C 23 C
9 D 24 B
DPPS - 6. Geometry-2 Ques Ans. Ques Ans.
1 C 16 D
2 D
3 B
4 A
5 B
6 B
7 B
8 C
9 C
DPPS - 7. Trigonometry Ques Ans. Ques Ans.
1 B 16 A
2 D 17 B
3 B 18 B
4 C 19 A
5 A 20 B
6 B 21 B
7 B 22 A
8 A 23 A
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9 B 24 C
Page # 1
DPPS - 8. Co-ordinate Geometry Ques Ans. Ques Ans.
1 B 16 D
2 C
3 A
4 B
5 B
6 C
7 B
8 A
9 B
10 B
11 A
12 A
13 B
14 C
15 B
10 C 25 B
11 B
12 A
13 C
14 B
15 D
DPPS - 9. Mensuration Ques Ans. Ques Ans.
1 A 16 D
2 A 17 C
3 B 18 C
4 D 19 A
5 B 20 A
6 A 21 B
7 B 22 B
8 B 23 A
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9 D 24 A
Page # 2
