Intermediate Previous papers
Mathematics-IIB
MAY–2012 SECTION-A (10 × 2 = 20) 1) Find the other end of the diameter of the circle x 2 + y 2 − 8 x − 8 y + 27 = 0 if one end of it is (2, 3) 2) Find the equation of the sphere whose center is (2, –3, and 4) and radius is 5. 3) Find the equation of the parabola whose focus is S (1, –7) and vertex is A (1, –2). 4) Show that the angle between the two asymptotes of a hyperbola
x2 a
(
2
−
)
y2 b
2
5) Find the n derivative of f ( x ) = log 8 x 3 + 36 x 2 + 54 x + 27 for all x > − th
6) Evaluate
∫ π
8) Evaluate
1
1 x + x dx 1 − 2 e x 2
sin 2 x − cos 2 x
∫ sin 0
3
x + cos3 x
dx
7) Evaluate
1
∫ ( x + 3)
x+2
b −1 or 2 s ec (c) a
−1 = 1 is 2 tan
3 . 2
dx on I ⊂ ( −2, ∞ )
1∕ 3 1∕ 2 d2 y dy + 2 9) Find order and degree of dx dx
1
4
=0
10) Find the area bounded between the curves y 2 − 1 = 2 x and x = 0 . SECTION-B (5 × 4 = 20) 11)
Find the condition that the tangents drawn from the exterior point ( g , f ) to s = x 2 + y 2 + 2 gx + 2 fy + c = 0
12)
are perpendicular to each other. Find the equation of the parabola whose axis is parallel to Y-axis and which passes through the points (4, 5), (–2, 11) and (–4, 21).
13)
Find the eccentricity, foci and equation of the directrices of the hyperbola 5 x 2 − 4 y 2 + 20 x + 8 y = 4
14)
If PP′ and QQ′ are two perpendicular focal chords of a conic, prove that
1
+
1
( SP )( SP′ ) ( SQ )( SQ′ )
is
constant. 15)
Evaluate
∫x
1 + x − x 2 dx
16) Solve
dy − x tan ( y − x ) = 1 dx
(
17) Solve x + 2 y
3
) ddyx = y
SECTION-C (5 × 7 = 35) 18) Find the equation and center of the circle passing through the points (–2, 3), (2, –1) and (4, 0). 19)
Find the equation of the circle which cuts the circles x 2 + y 2 + 2 x + 4 y + 1 = 0 ,
2 x 2 + 2 y 2 + 6 x + 8 y − 3 = 0 and x 2 + y 2 − 2 x + 6 y − 3 = 0 orthogonally. 20)
The tangent and normal to the ellipse x 2 + 4 y 2 = 4 at a point P (θ ) on it meets the major axis in Q and R respectively. If 0 < θ < π
21)
If y =
(
sin h −1 x 1 + x2
2
( 3) .
and QR = 2 , then show that θ = cos −1 2
(
)
then show that 1 + x 2 y2 + 3 xy1 + y = 0 and hence by using Leibnitz theorem, deduce
)
that 1 + x 2 yn + 2 + (2n + 3) xyn +1 + (n + 1) 2 yn = 0 22) 24)
Evaluate
∫ sin x +
1
1 3 cos x
dx
23)
Evaluate
∫ 0
log (1 + x ) 1 + x2
dx
The velocity of a train which starts from rest is given by the following table, the time being recorded in minutes from the start and the speed in kilometers. Estimate approximately the total distance run in 20 minutes by Simpson’s rule and Trapezoidal rule. Minutes 2 4 6 8 10 12 14 16 18 20 Kmph 10 18 25 29 32 20 11 5 2 0
1
Intermediate Previous papers
Mathematics-IIB
MARCH–2012 SECTION-A (10 × 2 = 20) 1) Find the equation of the circle passing through (3, 4) and having the centre at (–3, 4). 2) Find the centre and radius of the sphere x 2 + y 2 + z 2 − 2 x + 4 y − 6 z − 2 = 0 3) Find the value of ‘K’ if points (1, 2), (K, –1) are conjugate with respect to the parabola y 2 = 8 x . 4) If the eccentricity of a hyperbola is 5/4, then find the eccentricity of its conjugate hyperbola. th 5) Find the n derivative of f ( x ) = sin 7 x cos x ∀x ∈ R .
3
1 6) Evaluate x + dx , x > 0 x
∫
7) Evaluate
∫
3
dx ( x + 1)( x + 2 )
8) Evaluate
2x
∫1+ x
2
dx
2
9) Find the area of the region enclosed between y = x + 3, y = 0, x = −1, x = 2 3
10) Form the differential equation corresponding to y = cx − 2c , where ' c ' is a parameter. 2
SECTION-B (5 × 4 = 20) 11)
Find the angle between the tangents drawn from (3, 2) to the circle x + y − 6 x + 4 y − 2 = 0
12)
Find the condition for the line y = mx + c to be a tangent to the parabola x = 4ay .
13)
Find the pole of the line 21x − 16 y − 12 = 0 with respect to the ellipse 3x + 4 y = 12 .
14)
If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that 1 1 2 + = SP SQ l
15)
Evaluate
16)
Solve the differential equation (2 x − y ) dy = (2 y − x) dx
17)
Solve the differential equation
2
2
2
2
2
dx
∫ 5 + 4 cos x dy + y tan x = sin x dx
SECTION-C (5 × 7 = 35) 18) Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on 4 x + 3 y − 24 = 0 . 19)
Find the coordinates of the limiting points of the coaxial system to which the circles
x 2 + y 2 + 10 x − 4 y − 1 = 0 and x 2 + y 2 + 5x + y + 4 = 0 are two members. 20)
Show that the poles of the tangents to the circle x + y = a + b with respect to the ellipse 2
x2 a 21)
2
+
y2 b
2
= 1 lies on
x2 a
4
+
y2 b
4
=
1 a + b2 2
2
2
2
.
If y = cos(m log x), x > 0 , then show that x y2 + xy1 + m y = 0 and hence deduce that 2
2
x 2 yn+ 2 + (2n + 1) xyn +1 + (m2 + n2 ) yn = 0 22)
∫
Obtain reduction formula for I n = (tan x)dx, n being a positive integer, n ≥ 2 and deduce the value
∫
n
6
of (tan x)dx . π
23)
Show that
2
x
∫ sin x + cos x dx = 2 0
π 2
log( 2 + 1) . 5
24)
Calculate the approximate value of
dx
∫ 1 + x , by taking n = 4 in the Simpson’s rule. 1
2
Intermediate Previous papers
Mathematics-IIB
MAY–2011 SECTION-A (10 × 2 = 20) 1) If x + y + 2 gx + 2 fy − 12 = 0 represents a circle with center (2, 3), then find g , f and its radius. 2
2
2) Find the equation of the sphere that passes through the point (4, 3, –1) and having center at (3, 8, 1). 3) Find pole of the line 2 x + 3 y + 4 = 0 with respect to y = 8x . 2
4) Find the equation of hyperbola whose foci are at ( ±5, 0) and with transverse axis length 8. th
5) Find n derivative of sin 5 x.sin 3 x 6) Evaluate
∫
1 2 + 1 − x2 1 + x2
dx
7) Evaluate
∫
1 dx ( x + 1)( x + 2)
2
∫ 1 − x dx
8) Evaluate
0
9
9) Calculate the approximate value of
∫ x dx using Trapezoidal rule with 4 parts. 2
1
6
d 2 y dy 5 10) Find order and degree of differential equation 2 + = 6 y dx dx SECTION-B (5 × 4 = 20) 11)
Find length of chord intercepted by the circle x + y − x + 3 y − 22 = 0 on the line y = x − 3
12)
Find the equation of the tangent and normal to the parabola y = 8x at (2, 4).
13)
Find eccentricity, foci, length of latusrectum and equation directrices of the hyperbola x − 4 y = 4 .
14)
Find the condition that straight line
15)
Evaluate
2
2
2
2
∫
1 + 3x − x 2 dx
2
k = A cos θ + B sin θ may touch the circle r = 2a cos θ . r dy 16) Solve ( x 2 − y 2 ) 17) Solve = xy dx
dy + y tan x = cos3 x dx SECTION-C (5 × 7 = 35) 18) If (1, 2) (3, –4), (5, -–6) and (c, 8) are concyclic, find “c”. 19)
If (3, 5) is a limiting point of coaxial system of circle of which x + y + 2 x + 2 y − 24 = 0 find other
20)
limiting point. Show that the points of intersection of perpendicular tangents to an ellipse lies on a circle.
21)
If y = esin
2
−1
x
2
, then show that (1 − x ) y2 − xy1 − y = 0 and hence deduce that 2
(1 − x 2 ) yn + 2 − (2n + 1) xyn +1 − (n2 + 1) yn = 0 . 22)
∫
π
23)
n
Find reduction formula of (cot x)dx , hence find
Find
∫ (cot
4
x)dx
x
∫ 1 + sin x dx 0
24)
Find area enclosed between curves y = 4 x, y = 4(4 − x) 2
2
3
Intermediate Previous papers
Mathematics-IIB
MARCH–2011 SECTION-A (10 × 2 = 20) 1) Obtain the parametric equation of the circle ( x − 3) 2 + ( y − 4) 2 = 82 . 2) If (2, 3, 5) is one end of a diameter of the sphere x 2 + y 2 + z 2 − 6 x − 12 y − 2 z + 20 = 0 , then find the coordinates of the other end of the diameter. 3) Find the points on the parabola y 2 = 8 x whose focal distance is 10. 4) If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola, then prove that
1 e
2
+
1 e12
=1.
5) Find the 3 derivative of e x cos x . rd
6) Evaluate
sin 4 x
∫ cos
6
x
dx
7) Evaluate
∫
1 dx x log x[log(log x)]
1
8) Evaluate
∫x
x3 2
0
+1
dx
9) Find the area of the region enclosed by the curves x = 4 − y , x = 0 . 2
10) Form the differential equation of the family of all circles with their centers at the origin and also find its order. SECTION-B (5 × 4 = 20) 11) Find the equation of the circle with center (–2, 3) cutting a chord length 2 units on 3 x + 4 y + 4 = 0 . 12)
If the polar of P with respect to the parabola y 2 = 4ax touches the circle x 2 + y 2 = 4a 2 , then show that P lies on the curve x 2 − y 2 = 4a 2 .
13)
Find the equations of the tangents to the hyperbola x 2 − 4 y 2 = 4 which are: i) parallel to and ii) perpendicular to the line x + 2 y = 0 .
14)
Find the area of triangle formed by points with the polar coordinates ( a, θ ) , 2a,θ +
15)
Evaluate
16)
Solve xdy = y + x cos
π
2π , 3a, θ + 3 3
.
dx
∫ 5 + 4 cos 2 x
2
y dx x
17) Solve
dy 2 3 ( x y + xy ) = 1 dx
SECTION-C (5 × 7 = 35) 18) 19) 20)
Show that the circles x 2 + y 2 − 6 x − 2 y + 1 = 0 , x 2 + y 2 + 2 x − 8 y + 13 = 0 touch each other. Find the point of contact and the equation of the common tangent at their point of contact. Find the equation of the circle which passes through the origin and belongs to the coaxial system of which the limiting points are (1, 2) and (4, 3). Find the length of major axis, minor axis, latusrectum, eccentricity, coordinates of center, foci and the equation of directrices of the ellipse 4 x 2 + y 2 − 8 x + 2 y + 1 = 0 .
21)
If y = e m sin
22)
Evaluate
−1
Evaluate
2sin x + 3cos x + 4 x sin 3 x
∫ 1 + cos 0
24)
, then prove that (1 − x 2 ) yn + 2 − (2n + 1) xyn +1 − (n 2 + m 2 ) yn = 0 .
∫ 3sin x + 4cos x + 5 dx π
23)
x
2
x
dx
A curve is drawn to pass through the points given by the following table. Using Simpson's rule, find the approximate area bounded by the curve, the X-axis and the lines x = 1 and x = 4 . x 1 1.5 2 2.5 3 3.5 4 y 2 2.4 2.7 2.8 3 2.6 2.1
4
Intermediate Previous papers
Mathematics-IIB
MAY–2010 SECTION-A (10 × 2 = 20) 1) Find the centre and radius of the circle 1 + m 2 ( x 2 + y 2 ) − 2cx − 2mcy = 0 . 2) Find the equation of the sphere that passes through the point (4, 3, –1) and having its centre (3, 8, 1).
1 , 2 is one extremity of a focal chord of the parabola y 2 = 8x , find the coordinates of the other extremity. 2
3) If
4) Find the eccentricity of the Ellipse (in Standard Form) whose length of the latusrectum is half of its minor axis. th 5) Find the n derivative of y = cos x
2
6) Evaluate
e x (1 + x )
∫ cos ( xe 2
x
)
dx
7) Evaluate
1
∫ ( log x ) dx
8) Evaluate
x2 dx 2 0
∫ 1 + x
9) Find the area bounded by y = x + 3, X − axis, x = −1, x = 2 3
10) Form the differential equation corresponding to y = A cos 3 x + B sin 3 x where A < B are parameters. SECTION-B (5 × 4 = 20) 11)
Show that the tangent at (–1, 2) of the circle x + y − 4 x − 8 y + 7 = 0 touches the circle 2
2
x 2 + y 2 + 4 x + 6 y = 0 and also find its point of tangency. 12)
Find the value of k if the lines x + y + 2 = 0 and x − 2 y + k = 0 are conjugate w.r.t y + 4 x − 2 y − 3 = 0 .
13)
Find eccentricity, coordinates of foci and equations of directrices of the ellipse 16 y − 9 x = 144 .
14)
If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that
2
2
2
1 1 2 + = SP SQ l
1
∫ 2 − 3cos x dx
15)
Evaluate
16)
Solve (e + 1) ydy + ( y + 1)dx = 0 x
17) Solve
dy + y sec x = tan x dx
SECTION-C (5 × 7 = 35) 18) Find the equation of a circle which passes through the points (5, 7), (8, 1) and (1, 3). 19) Find the coordinates of the limiting points of the coaxial system to which the circles
x 2 + y 2 + 10 x − 4 y − 1 = 0 and x 2 + y 2 + 5x + y + 4 = 0 are two members. 20)
Show that the equation of the parabola in standard form is y = 4ax .
21)
If y = e m sin
22)
Find the reduction formula for (sin x) dx (n ≥ 2) and hence find
2
−1
x
2
∫
π
23)
, then show that (1 − x ) yn+ 2 − (2n + 1) xyn +1 − (n + m ) yn = 0 .
Show that
2
x
∫ sin x + cos x dx = 2 0
24)
2
n
π 2
2
2
∫ (sin
4
x)dx
log( 2 + 1)
Show that the area of the region bounded by circle x + y = a
2
x2 a2
+
y2 b2
= 1 (ellipse) is π ab . Also deduce the area of the
2
5
Intermediate Previous papers
Mathematics-IIB
MARCH–2010 SECTION-A (10 × 2 = 20) 1) Obtain the parametric equation of each of the following circle x + y − 6 x + 4 y − 12 = 0 . 2
2
2) Find the center and radius of the sphere x + y + z − 2 x − 4 y − 6 z = 11 . 2
2
2
3) Find the value of k if the lines 2 x + 3 y + 4 = 0 and x + y + k = 0 are conjugate w.r.t. y = 8x . 2
4) Find the eccentricity of the hyperbola xy = 1 . th 5) Find the n derivative of log(4 − x )
2
π
1 + x log x dx on ( 0,∞ ) x
6) Evaluate
∫
8) Evaluate
∫ cos ( xe ) dx on I ⊂ R − { x ∈ R : cos( xe ) = 0}
7) Evaluate
2
∫ cos
5
x sin 4 xdx
0
e (1 + x ) x
2
x
x
9) Find the area of the region enclosed by the given curves x = 4 − y , x = 0 2
dy 2 + 1 + 10) Find the order and degree of dx 2 dx d2y
53
=0
SECTION-B (5 × 4 = 20) 11) Find the equation of the circle whose center lies on X-axis and passing through the points (–2, 3), (4, 5). 12)
Show that the equations of the common tangents to the circle x + y = 2a and the parabola y = 8ax 2
2
2
2
are y = ± ( x + 2a ) 13)
Find the eccentricity, foci and the equations of directrices of the following ellipse:
4 x2 + y 2 − 8x + 2 y + 1 = 0 14)
Show that the polar equation of a conic in the standard form is
l = 1 + e cos θ . r
(‘ l ’ is semi-latusrectum, ‘ e ’ is eccentricity) 15)
Evaluate
(
dx
∫ 5 + 4 cos x
(
)
17) Solve 1 + x
16) Solve x 2 + y 2 dy = 2 xydx
2
+y=e ) dy dx
tan −1 x
SECTION-C (5 × 7 = 35) 18)
Show that the circles x + y − 6 x − 2 y + 1 = 0 ; x + y + 2 x − 8 y + 13 = 0 touch each other. Find the
19)
point of contact and the equation of common tangent at the point of contact. Find the limiting points of the coaxial system determined by the circles
2
2
2
2
x 2 + y 2 + 10 x − 4 y − 1 = 0, x 2 + y 2 + 5x + y + 4 = 0 20)
If the polar of P with respect to the parabola y = 4ax touches the circle x + y = 4a , then show that 2
2
2
2
P lies on the curve x − y = 4a . 2
21)
2
2
If y = cos(m log x), x > 0 , then show that x y2 + xy1 + m y = 0 and hence deduce that 2
2
x 2 yn+ 2 + (2n + 1) xyn +1 + (m2 + n2 ) yn = 0 22)
Evaluate
∫ (x
x +1 2
+ 3x + 12
1
)
dx
23) Evaluate
0 1
24)
Find the approximate value of π from
∫
1
∫ 1+ x
2
log(1 + x) 1 + x2
dx
dx by using Simpson’s rule by dividing [0, 1] into 4 equal
0
parts.
6
Intermediate Previous papers
Mathematics-IIB
MAY–2009 SECTION-A (10 × 2 = 20) 1) If the center of the circle x 2 + y 2 + ax + by − 12 = 0 is (2, 3), find the values of a , b and the radius of the circle. 2) Find the equation of the sphere that passes through the point (4, 3, –1) and having its centre (3, 8, 1). 3) Find the coordinates of the points on the parabola y 2 = 2 x whose focal distance is 5/2. 4) Find the equations of the tangents to the hyperbola 3 x 2 − 4 y 2 = 12 which is parallel to the line y = x − 7 . 5) Find the n derivative of f ( x ) = log(8 x 3 + 36 x 2 + 54 x + 27) th
6) Evaluate
∫(
2
2
)
sec x.cos ec x dx
7) Evaluate
e x (1 + x)
∫ (2 + x)
2
π
dx
8) Evaluate −
2
∫ π
(sin 2 x.cos 4 x)dx 2
9) Find the area of the enclosed by the curve f ( x) = sin x in the interval [ 0, 2π ] . 10) Form the differential equation corresponding to y = cx − 2c 2 , where ‘ c ’ is a parameter. SECTION-B (5 × 4 = 20) 11)
Show that x + y + 1 = 0 touches the circle x 2 + y 2 − 3 x + 7 y + 14 = 0 and find the point of contact.
12)
Prove that the poles of tangents to the parabola y 2 = 4ax w.r.t the parabola y 2 = 4bx lie on parabola.
13)
One focus of hyperbola located at (1, –3) and corresponding directrix in the line y = 2 . Find the equation of hyperbola if its eccentricity is 3/2.
14)
If PSQ is chord passing through the focus S of a conic and ‘ l ’ is semi latusrectum, show that
1 1 2 + = SP SQ l
1
∫ (1 − x)(4 + x ) dx
15)
Evaluate
16)
Solve ( x 2 − y 2 ) dx − xydy = 0
2
17) Solve (1 + y 2 ) dx = (tan −1 y − x )dy
SECTION-C (5 × 7 = 35) 18)
Find the equation of the circle whose centre lies on X-axis and passing through the points (−2,3), (4,5) .
19)
In the limiting points of the coaxial system determined by the circles x 2 + y 2 + 2 x − 6 y = 0 and
2 x 2 + 2 y 2 − 10 y + 5 = 0 . 20)
Find eccentricity, coordinates of foci and equations of directories of the ellipse
9 x 2 + 16 y 2 − 36 x + 32 y − 92 = 0 . 21)
If y =
sin h −1 x 1+ x
(
2
(
)
then show that 1 + x 2 y2 + 3 xy1 + y = 0 and hence by using Leibnitz theorem, deduce
)
that 1 + x 2 yn + 2 + (2n + 3) xyn +1 + (n + 1) 2 yn = 0 .
22)
∫
Obtain the reduction formula for I = sin xdx , n being a positive integer, n ≥ 2 and deduce the value
∫
n
4
of sin xdx . π
23)
Show that
2
x
∫ sin x + cos x dx = 2 0
π 2
log( 2 + 1) 6
24)
Dividing [0, 6] into 6 equal parts, evaluate
∫ x dx approximately by using Trapezoidal rule and 3
0
Simpson's rule.
7
Intermediate Previous papers
Mathematics-IIB
MARCH–2009 SECTION-A (10 × 2 = 20) 1) If the equation x 2 + y 2 − 4 x + 6 y + c = 0 represents a circle with radius 6, find c . 2) Find the centre and radius of the sphere x 2 + y 2 + z 2 − 2 x − 4 y − 6 z = 11 . 3) Find the coordinates of the points on the parabola y 2 = 2 x whose focal distance is 5/2. 4) Find the equation of the Hyperbola whose foci are (4, 2), (8, 2) and eccentricity is 2. 5) If y = aenx + be − nx , then show that y2 = n 2 y . 6) Evaluate
∫
1 − cos 2xdx
7) Evaluate
x8
∫ 1+ x
18
2
dx on ℝ
8) Find the value of
∫ 1 − x dx 0
9) Find the area bounded by the parabola y = x , the x-axis and the lines x = −1, x = 2 . 2
6
d 2 y dy 3 5 10) Find the order and degree of 2 + = 6y . dx dx SECTION-B (5 × 4 = 20) 11)
If a point P is moving such that the lengths of tangents drawn from P to x 2 + y 2 + 6 x + 18 y + 26 = 0 are in the ratio 2:3, then find the equation of the locus of P.
12)
Show that the equations of the common tangents to the circle x 2 + y 2 = 2a 2 and the parabola y 2 = 8ax are y = ± ( x + 2a ) .
13)
Find eccentricity, coordinates of foci and equations of directrices of the ellipse
9 x 2 + 16 y 2 − 36 x + 32 y − 92 = 0 .
π π and 3, form an equilateral triangle 2 6
14)
Show that the points with polar coordinates ( 0, 0 ) , 3,
15)
Evaluate ( x cos
16)
Solve
∫
−1
x)dx
1 + x 2 dx + 1 + y 2 dy = 0
17) Solve
dy − y tan x = e x sec x dx
SECTION-C (5 × 7 = 35) 18)
Show that the circles x 2 + y 2 − 6 x − 2 y + 1 = 0 , x 2 + y 2 + 2 x − 8 y + 13 = 0 touch each other. Find the point of contact and the equation of common tangent at the point of contact.
19)
Find the limiting points of the coaxial system determined by the circles x 2 + y 2 + 10 x − 4 y − 1 = 0 ,
x2 + y 2 + 5x + y + 4 = 0 . 20)
If the polar of P with respect to the Parabola y 2 = 4ax touches the circle x 2 + y 2 = 4a 2 , then show that P lies on the curve x 2 − y 2 = 4a 2 .
21)
If y =
(
sin h −1 x 1+ x
2
(
)
then show that 1 + x 2 y2 + 3 xy1 + y = 0 and hence by using Leibnitz theorem, deduce
)
that 1 + x 2 yn + 2 + (2n + 3) xyn +1 + (n + 1) 2 yn = 0 . π
22)
Evaluate
∫
(6 x + 5) 6 − 2 x 2 + xdx
23) Show that
2
0
1
24)
Find the approximate value of π from
1
∫ 1+ x
2
x
∫ sin x + cos x dx = 2
π 2
log( 2 + 1)
dx by using Simpson’s rule by dividing [0, 1] into 4 equal
0
parts.
8
Intermediate Previous papers
Mathematics-IIB
MAY–2008 SECTION-A (10 × 2 = 20) 1) Find the equation of the circle passing through (2, –1) and having the center at (2, 3). 2) Find the centre and radius of the sphere x 2 + y 2 + z 2 − 2 x − 4 y − 6 z = 11 . 3) If (1, 2) and ( k , −1) are conjugate points w.r.t to the parabola y 2 = 8 x , then find k . 4) If the eccentricity of a hyperbola be 5/4, then find the eccentricity of its conjugate hyperbola. 5) Find the n derivative of log(4 x 2 − 9) . th
6) Evaluate
∫
1 + x log x e dx on (0, ∞ ) x x
7) Evaluate
2 x2
∫ 1+ x
8
4
dx
8) Evaluate
∫ 2 − x dx 0
9) Find the area of the region enclosed by the given curves y = x + 3, y = 0, x = −1, y = 2 x . 3
10)
Form the differential equations of the following family of curves where parameters are given in brackets
y = ae3 x + be 4 x ; (a, b) SECTION-B (5 × 4 = 20) 11)
Find the equation of tangent and normal at (3, 2) of the circle x 2 + y 2 − x − y − 4 = 0 .
12)
If the focal chord of the parabola y 2 = 4ax meets it at P , Q and if S the focus then prove that
1 1 1 + = . SP SQ a 13)
Find the equation of tangent to the ellipse 2 x 2 + y 2 = 8 which is parallel and perpendicular to x − 2 y = 4
14)
Show that the following points form an equilateral triangle ( 0, 0 ) , 5,
15)
Evaluate
17)
Solve
π 7π , 5, . 18 18
∫
1 2 x − 3x2 + 1
dx
16) Solve
dy y 2 + y + 1 + =0 dx x 2 + x + 1
dy + y sec x = tan x dx
SECTION-C (5 × 7 = 35) 18)
Find the equations of transverse common tangents of x 2 + y 2 − 4 x − 10 y + 28 = 0 and
x2 + y2 + 4 x − 6 y + 4 = 0 . 19)
Find the equation of the circle which is orthogonal to each of the following circles
x 2 + y 2 + 2 x + 17 y + 4 = 0 , x 2 + y 2 + 7 x + 6 y + 11 = 0 , x 2 + y 2 − x + 22 y + 3 = 0 . 20)
Derive the equation of the hyperbola in standard form.
21)
If y = sin −1 x , then show that (1 − x 2 ) y 2 − xy1 = 0 .Hence show that
(1 − x 2 ) yn + 2 − (2n + 1) xyn +1 − n 2 yn = 0 22)
Evaluate
23)
Evaluate
2cos x + 3sin x
∫ 4 cos x + 5sin x dx 1
∫ 0 5
24)
Evaluate
log(1 + x) 1 + x2 dx
∫ 1+ x
dx
approximately by using the Simpson's rule with n = 4.
0
9
Intermediate Previous papers
Mathematics-IIB
MODEL QUESTION PAPER – 2013 SECTION-A (10 × 2 = 20) 1) If the equation x 2 + y 2 − 4 x + 6 y + c = 0 represents a circle with radius 6, find the value of c . 2) Find the equation of the directrix of the parabola 2 x 2 + 7 y = 0 3) Find the length of the latus rectum of the ellipse
x2 y2 + =1 16 8
4) Find the eccentricity of the hyperbola x 2 − 4 y 2 = 4
5) Find the distance between the two points in a plane whose polar coordinates are 2, 6) If y =
1 , then find yn . 2x + 5
8) Evaluate 10)
∫
esin
−1
7) Evaluate
, 3, . 6 4
1 + sin 2xdx
4
x
1− x
∫
π π
dx
2
9) Evaluate
∫ (x
x 2 − 1)dx
1
State the Simpson’s rule for Numerical Integration of a function f ( x) over the interval [ a, b ] by dividing
[a, b] into n sub-intervals. SECTION-B (5 × 4 = 20)
x2
y2
11)
If the line y = mx + c touches the ellipse
12)
Find the equations of the tangents shown drawn from (–2, 1) to the hyperbola 2 x 2 − 3 y 2 = 6 .
13)
Transform the polar equation r cos
2
a2
+
b2
= 1 , (a > b) then show that c 2 = a 2 m 2 + b 2 .
θ = a , (a > 0) origin as pole and the axis as initial line, into 2
Cartesian form
log x (−1) n ∠n 1 1 1 , then show that yn = log x − 1 − 2 − 3 − ⋯⋯⋯ n . n +1 x x
14)
If y =
15)
Evaluate
16)
Solve ( x 2 + y 2 ) dx = 2 xydy
x6 − 1
∫ 1+ x
2
dx 17) Solve
dy 2 x + y + 3 = dx 2 y + x + 1
SECTION-C (5 × 7 = 35) 18)
Find the equation of the pair of tangents drawn from (3, 2) to the circle x 2 + y 2 − 6 x + 4 y − 2 = 0 .
19)
Find the equation of the circle passing through the points of intersection of the circles
x 2 + y 2 − 8 x − 6 y + 21 = 0 , x 2 + y 2 − 2 x − 15 = 0 and the point (1, 2). 20)
Find the equation of the circle passing through the origin and coaxial with the circles
x 2 + y 2 − 6 x + 4 y − 8 = 0 and x 2 + y 2 − 2 x + y + 4 = 0 . 21)
Find the pole of the line x + y + 2 = 0 with respect to the parabola y 2 + 4 x − 2 y − 3 = 0
22)
Evaluate
23)
Evaluate
24)
Find the area enclosed by the curves y = 3 x and y = 6 x − x 2 .
∫
3sin x + cos x + 7 dx sin x + cos x + 1
∫x
x1/ 4
1/2
+1
dx
10
