1) Find the other end of the diameter of the circle x2+y2−8x−8y+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
2 2 2 2 1 x y a −b = is 2 tan 1 b a − or 1 2 sec− ( )c
5) Find the nth derivative of f x
( )
=log 8(
x3+36x2+54x+27)
for all3
2
x
> −
. 6) Evaluate 1 2 1 1 e d x x x x + − ∫
7) Evaluate(
)
1 d 3 2 x x+ x+∫
on I⊂ − ∞(
2,)
8) Evaluate 2 3 2 2 3 0 sin cos d sin cos x x x x x π − +∫
9) Find order and degree of1 1 3 4 1 2 2 2
d
d
0
d
d
y
y
x
x
+
=
∕ ∕10) Find the area bounded between the curves y2− =1 2xand
x
=
0
. SECTION-B (5 × 4 = 20)11) Find the condition that the tangents drawn from the exterior point
(
g f,)
to s=x2+y2+2gx+2fy c+ =0 are perpendicular to each other.12) 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 5x2−4y2+20x+8y=4 14) If PP′ and
′
are two perpendicular focal chords of a conic, prove that( )( ) ( )( )
SP1SP′ + SQ1SQ′is constant.
15) Evaluate
∫
x 1+ −x x2dx16) Solve
d
tan
(
)
1
d
y
x
y
x
x
−
−
=
17) Solve(
)
3d
2
d
y
x
y
y
x
+
=
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 circlesx2+y2+2x+4y+ =1 0,
2 2
2x +2y +6x+8y− =3 0and x2+y2−2x+6y− =3 0orthogonally.
20) The tangent and normal to the ellipse x2+4y2 =4 at a point P
( )
θ
on it meets the major axis in Q and R respectively. If 0< <θ
π
2 andQR
=
2
, then show that cos 1( )
23
θ
= − . 21) If 1 2 sin 1 h x y x − =+ then show that
(
)
2
2 1
1+x y +3xy + =y 0and hence by using Leibnitz theorem, deduce that
(
1+x2)
yn+2+(2n+3)xyn+1+ +(n 1)2yn =0 22) Evaluate1
d
sin
x
+
3 cos
x
x
∫
23) Evaluate(
)
1 2 0 log 1 d 1 x x x + +∫
24) 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
2) Find the centre and radius of the sphere x2+y2+z2−2x+4y−6z− =2 0
3) Find the value of ‘K’ if points (1, 2), (K, –1) are conjugate with respect to the parabolay2=8x
.
4) If the eccentricity of a hyperbola is 5/4, then find the eccentricity of its conjugate hyperbola. 5) Find the nth derivative off x( )
=sin 7 cosx x ∀ ∈x R.6) Evaluate 3 1 d x x x +
∫
,
x>0 7) Evaluate(
1)(
d 2)
x x+ x+∫
8) Evaluate 3 2 2 2 d 1 x x x +∫
9) Find the area of the region enclosed between
y
=
x
3+
3,
y
=
0,
x
= −
1,
x
=
2
10) Form the differential equation corresponding to
y
= −
cx
2
c
2, where ' 'c is a parameter. SECTION-B (5 × 4 = 20)11) Find the angle between the tangents drawn from (3, 2) to the circle
x
2+
y
2− +
6
x
4
y
− =
2
0
12) Find the condition for the line y=mx+c to be a tangent to the parabolax
2=
4
ay
.13) Find the pole of the line
21
x
−
16
y
− =
12
0
with respect to the ellipse3
x
2+
4
y
2=
12
. 14) If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that1 1 2 SP+SQ= l 15) Evaluate d 5 4 cos x x +
∫
16) Solve the differential equation(2x−y dy) =(2y−x dx) 17) Solve the differential equation dy ytanx sinx
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
2 2
10
4
1
0
x
+
y
+
x
−
y
− =
andx
2+
y
2+ + + =
5
x
y
4
0
are two members.20) Show that the poles of the tangents to the circle
x
2+
y
2=
a
2+
b
2 with respect to the ellipse2 2 2 2
1
x
y
a
+
b
=
lies on 2 2 4 4 2 21
x
y
a
+
b
=
a
+
b
.21) If
y
=
cos( log ),
m
x x
>
0
, then show thatx y
2 2+
xy
1+
m y
2=
0
and hence deduce that2 2 2
2
(2
1)
1(
)
0
n n n
x y
++
n
+
xy
++
m
+
n
y
=
22) Obtain reduction formula for
I
n=
∫
(tan
nx dx
)
,
n being a positive integer, n≥2 and deduce the valueof
∫
(tan
6x dx
)
. 2x
ππ
=
+
∫
SECTION-A (10 × 2 = 20)
1) If
x
2+
y
2+
2
gx
+
2
fy
− =
12
0
represents a circle with center (2, 3), then find g f, and its radius. 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 line2
x
+
3
y
+ =
4
0
with respect toy
2=
8
x
.4) Find the equation of hyperbola whose foci are at ( 5, 0)± and with transverse axis length 8. 5) Find nth derivative of sin 5 .sin 3x x
6) Evaluate 2 2 1 2 1 1 dx x x + − +
∫
7) Evaluate1
(
x
1)(
x
2)
dx
+
+
∫
8) Evaluate 2 01 x dx
−
∫
9) Calculate the approximate value of
9 2 1
x dx
∫
using Trapezoidal rule with 4 parts. 10) Find order and degree of differential equation6 2 5 2
6
d y
dy
y
dx
dx
+
=
SECTION-B (5 × 4 = 20)11) Find length of chord intercepted by the circle
x
2+
y
2− +
x
3
y
−
22
=
0
on the liney
= −
x
3
12) Find the equation of the tangent and normal to the parabolay
2=
8
x
at (2, 4).13) Find eccentricity, foci, length of latusrectum and equation directrices of the hyperbola
x
2−
4
y
2=
4
. 14) Find the condition that straight line k Acos Bsinr =
θ
+θ
may touch the circler=2 cosaθ
. 15) Evaluate∫
1 3x+ −x dx2 16) Solve (x2 y2)dy xy dx − = 17) Solve 3 tan cos dy y x 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
2+
y
2+
2
x
+
2
y
−
24
=
0
find other limiting point.20) Show that the points of intersection of perpendicular tangents to an ellipse lies on a circle. 21) Ify=esin−1x, then show that
(1
−
x
2)
y
2−
xy
1− =
y
0
and hence deduce that2 2
2 1
(1
−
x
)
y
n+−
(2
n
+
1)
xy
n+−
(
n
+
1)
y
n=
0
.22) Find reduction formula of
∫
(cot
nx dx
)
, hence find∫
(cot
4x dx
)
23) Find 0
1 sin
x
dx
x
π+
∫
2) If (2, 3, 5) is one end of a diameter of the spherex2+y2+z2−6x−12y−2z+20=0, then find the coordinates of the other end of the diameter.
3) Find the points on the parabola y2 =8x whose focal distance is 10.
4) If e e, 1 are the eccentricities of a hyperbola and its conjugate hyperbola, then prove that
2 2 1
1
1
1
e
+
e
=
.5) Find the 3rd derivative ofexcosx.
6) Evaluate 4 6
sin
cos
x
dx
x
∫
7) Evaluate1
log [log(log )]
dx
x
x
x
∫
8) Evaluate 1 3 2 0 1 x dx x +∫
9) Find the area of the region enclosed by the curvesx= −4 y2,x=0.
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 on3x+4y+ =4 0. 12) If the polar of P with respect to the parabolay2 =4axtouches the circlex2+y2 =4a2, then show that P
lies on the curvex2−y2=4a2.
13) Find the equations of the tangents to the hyperbola x2−4y2 =4 which are: i) parallel to and ii) perpendicular to the linex+2y=0.
14) Find the area of triangle formed by points with the polar coordinates
( )
a,θ
,2 ,
3
a
θ
π
+
,2
3 ,
3
a
θ
π
+
. 15) Evaluate5 4 cos 2
dx
x
+
∫
16) Solve
xdy
y
x
cos
2y
dx
x
=
+
17) Solve 2 3(
)
1
dy
x y
xy
dx
+
=
SECTION-C (5 × 7 = 35)18) Show that the circlesx2+y2−6x−2y+ =1 0, x2+y2+2x−8y+13=0 touch each other. Find the point of contact and the equation of the common tangent at their point of contact.
19) 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).
20) Find the length of major axis, minor axis, latusrectum, eccentricity, coordinates of center, foci and the equation of directrices of the ellipse4x2+y2−8x+2y+ =1 0.
21) Ify=emsin−1x, then prove that(1−x2)yn+2−(2n+1)xyn+1−(n2+m2)yn =0. 22) Evaluate
2sin
3cos
4
3sin
4 cos
5
x
x
dx
x
x
+
+
+
+
∫
23) Evaluate 3 2 0sin
1 cos
x
x
dx
x
π+
∫
SECTION-A (10 × 2 = 20)
1) Find the centre and radius of the circle
1
+
m
2(
x
2+
y
2)
−
2
cx
−
2
mcy
=
0
.2) Find the equation of the sphere that passes through the point (4, 3, –1) and having its centre (3, 8, 1). 3) If
1
, 2
2
is one extremity of a focal chord of the parabola2
8
y
=
x
, find the coordinates of the other extremity. 4) Find the eccentricity of the Ellipse (in Standard Form) whose length of the latusrectum is half of its minor axis. 5) Find the nth derivative ofy
=
cos
2x
6) Evaluate 2 (1 ) cos ( ) x x e x dx xe +
∫
7) Evaluate∫
(
log x dx
)
8) Evaluate1 2 2 0 1 x dx x +
∫
9) Find the area bounded by
y
=
x
3+
3,
X
−
axis x
,
= −
1,
x
=
2
10) Form the differential equation corresponding to y=Acos 3x+Bsin 3x where A<B are parameters. SECTION-B (5 × 4 = 20)
11) Show that the tangent at (–1, 2) of the circle
x
2+
y
2− −
4
x
8
y
+ =
7
0
touches the circle2 2
4
6
0
x
+
y
+
x
+
y
=
and also find its point of tangency.12) Find the value of
k
if the lines x+ + =y 2 0 and x−2y+ =k 0are conjugate w.r.ty
2+
4
x
−
2
y
− =
3
0
. 13) Find eccentricity, coordinates of foci and equations of directrices of the ellipse16
y
2−
9
x
2=
144
. 14) If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that1
1
2
SP
+
SQ
=
l
15) Evaluate
1
2 3cos
−
x
dx
∫
16) Solve
(
e
x+
1)
ydy
+ +
(
y
1)
dx
=
0
17) Solvedy
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
2 2
10
4
1 0
x
+
y
+
x
−
y
− =
andx
2+
y
2+ + + =
5
x
y
4
0
are two members. 20) Show that the equation of the parabola in standard form isy
2=
4
ax
. 21) Ify=emsin−1x, then show that(1
−
x
2)
y
n+2−
(2
n
+
1)
xy
n+1−
(
n
2+
m
2)
y
n=
0
. 22) Find the reduction formula for∫
(sin
nx dx
)
(
n
≥
2)
and hence find∫
(sin
4x dx
)
23) Show that 2 0log( 2 1)
sin
cos
2 2
x
dx
x
x
ππ
=
+
+
∫
24) Show that the area of the region bounded by
2 2
2 2
1
x
y
a
+
b
=
(ellipse) is
π
ab
. Also deduce the area of the circlex
2+
y
2=
a
22) Find the center and radius of the sphere
x
2+
y
2+ −
z
22
x
−
4
y
−
6
z
=
11
.3) Find the value of
k
if the lines2x+3y+ =4 0 andx+ + =y k 0 are conjugate w.r.t.y
2=
8
x
. 4) Find the eccentricity of the hyperbolaxy=1.5) Find the nth derivative of
log(4
−
x
2)
6) Evaluate
1
x
log
x
dx
x
+
∫
on( )
0,
∞
7) Evaluate 2 5 4 0 cos xsin xdx π∫
8) Evaluate 2 (1 ) cos ( ) x x e x dx xe +∫
on I⊂ − ∈R{
x R: cos(xex)=0}
9) Find the area of the region enclosed by the given curves
x
= −
4
y
2,
x
=
0
10) Find the order and degree of
5 3 2 2 2
1
0
d y
dy
dx
dx
+ +
=
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
2+
y
2=
2
a
2and the parabolay
2=
8
ax
are
y
= ± +
(
x
2
a
)
13) Find the eccentricity, foci and the equations of directrices of the following ellipse:
2 2
4
x
+
y
− +
8
x
2
y
+ =
1
0
14) Show that the polar equation of a conic in the standard form isl 1 ecos r = +
θ
. (‘l
’ is semi-latusrectum, ‘e’ is eccentricity) 15) Evaluate5 4 cos
dx
x
+
∫
16) Solve(
x2+y2)
dy=2xydx 17) Solve( )
1
x
2dy
y
e
tan1xdx
−
+
+ =
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
2 2
10
4
1 0,
2 25
4
0
x
+
y
+
x
−
y
− =
x
+
y
+ + + =
x
y
20) If the polar of P with respect to the parabola
y
2=
4
ax
touches the circlex
2+
y
2=
4
a
2, then show that P lies on the curvex
2−
y
2=
4
a
2.21) If
y
=
cos( log ),
m
x x
>
0
, then show thatx y
2 2+
xy
1+
m y
2=
0
and hence deduce that2 2 2 2
(2
1)
1(
)
0
n n nx y
++
n
+
xy
++
m
+
n
y
=
22) Evaluate(
2)
1
3
12
x
dx
x
x
+
+
+
∫
23) Evaluate 1 2 log(1 ) 1 x dx x + +∫
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 y2 =2x whose focal distance is 5/2.
4) Find the equations of the tangents to the hyperbola 3x2−4y2 =12 which is parallel to the liney= −x 7. 5) Find the nth derivative of f x( )=log(8x3+36x2+54x+27)
6) Evaluate
∫
(
sec
2x
.cos
ec x dx
2)
7) Evaluate2
(1
)
(2
)
xe
x
dx
x
+
+
∫
8) Evaluate 2 2 4 2 (sin x.cos x dx) π π −∫
9) Find the area of the enclosed by the curve f x( )=sinx in the interval
[
0, 2π
]
.10) Form the differential equation corresponding toy=cx−2c2, where ‘c’ is a parameter. SECTION-B (5 × 4 = 20)
11) Show that x+ + =y 1 0 touches the circle x2+y2−3x+7y+14=0 and find the point of contact. 12) Prove that the poles of tangents to the parabola y2 =4axw.r.t the parabola y2 =4bxlie on parabola. 13) One focus of hyperbola located at (1, –3) and corresponding directrix in the liney=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 that1
1
2
SP
+
SQ
=
l
15) Evaluate 21
(1
−
x
)(4
+
x
)
dx
∫
16) Solve (x2−y2)dx−xydy=0 17) Solve (1+y2)dx=(tan−1y−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 x2+y2+2x−6y=0 and2 2
2x +2y −10y+ =5 0.
20) Find eccentricity, coordinates of foci and equations of directories of the ellipse
2 2 9x +16y −36x+32y−92=0. 21) If 1 2 sin 1 h x y x − =
+ then show that
(
)
2
2 1
1+x y +3xy + =y 0and hence by using Leibnitz theorem, deduce that
(
1+x2)
yn+2+(2n+3)xyn+1+ +(n 1)2yn =0.22) Obtain the reduction formula for
I
=
∫
sin
nxdx
, nbeing a positive integer, n≥2 and deduce the value of∫
sin xdx
4 . 23) Show that 2 0log( 2 1)
sin
cos
2 2
x
dx
x
x
ππ
=
+
+
∫
24) Dividing [0, 6] into 6 equal parts, evaluate
6 3 0
x dx
∫
approximately by using Trapezoidal rule and Simpson's rule.2) Find the centre and radius of the spherex2+y2+z2−2x−4y−6z=11.
3) Find the coordinates of the points on the parabolay2 =2x 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) Ify=aenx+be−nx, then show thaty2 =n y2 .
6) Evaluate
∫
1 cos 2xdx
−
7) Evaluate8 18
1
x
dx
x
+
∫
onℝ
8) Find the value of2
0
1 x dx−
∫
9) Find the area bounded by the parabolay=x2, the x-axis and the lines
x
= −
1,
x
=
2
. 10) Find the order and degree of6 3 2 5 2 6 d y dy y dx dx + = . SECTION-B (5 × 4 = 20)
11) If a point P is moving such that the lengths of tangents drawn from P to x2+y2+6x+18y+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 x2+y2=2a2 and the parabolay2 =8ax arey= ± +(x 2 )a .
13) Find eccentricity, coordinates of foci and equations of directrices of the ellipse
2 2
9x +16y −36x+32y−92=0.
14) Show that the points with polar coordinates
( )
0, 0 ,3,
2
π
and3,
6
π
form an equilateral triangle15) Evaluate
∫
( cosx −1x dx)16) Solve
1
+
x dx
2+
1
+
y dy
2=
0
17) Solvedy
y
tan
x
e
xsec
x
dx
−
=
SECTION-C (5 × 7 = 35)
18) Show that the circlesx2+y2−6x−2y+ =1 0, x2+y2+2x−8y+ =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 circlesx2+y2+10x−4y− =1 0,
2 2 5 4 0
x +y + x+ + =y .
20) If the polar of P with respect to the Parabola y2 =4ax touches the circlex2+y2 =4a2, then show that P lies on the curvex2−y2=4a2.
21) If 1 2 sin 1 h x y x − =
+ then show that
(
)
2
2 1
1+x y +3xy + =y 0and hence by using Leibnitz theorem, deduce that
(
1+x2)
yn+2+(2n+3)xyn+1+ +(n 1)2yn =0.∫
2x
ππ
=
+
∫
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 spherex2+y2+z2−2x−4y−6z=11.
3) If (1, 2)and( , 1)k − are conjugate points w.r.t to the parabolay2 =8x, then find
k
.4) If the eccentricity of a hyperbola be 5/4, then find the eccentricity of its conjugate hyperbola. 5) Find the nth derivative oflog(4x2−9).
6) Evaluate
e
x1
x
log
x
dx
x
+
∫
on(0, )∞ 7) Evaluate 2 82
1
x
dx
x
+
∫
8) Evaluate 4 0 2−x dx∫
9)Find the area of the region enclosed by the given curvesy=x3+3,y=0,x= −1,y=2x.
10) Form the differential equations of the following family of curves where parameters are given in brackets
3x 4x y=ae +be ;( , )a b SECTION-B (5 × 4 = 20)
11) Find the equation of tangent and normal at (3, 2) of the circlex2+y2− − − =x y 4 0. 12) If the focal chord of the parabolay2 =4axmeets it at
P
,Q and if Sthe focus then prove that1 1 1
SP+SQ =a .
13) Find the equation of tangent to the ellipse2x2+y2 =8which is parallel and perpendicular to x−2y=4 14) Show that the following points form an equilateral triangle
( )
0, 0 ,5,
18
π
,7
5,
18
π
. 15) Evaluate 2 1 2 3 1 dx x− x +∫
16) Solve 2 2 1 0 1 dy y y dx x x + + + = + +17) Solve
dy
y
sec
x
tan
x
dx
+
=
SECTION-C (5 × 7 = 35)
18) Find the equations of transverse common tangents ofx2+y2−4x−10y+28=0and
2 2
4 6 4 0 x +y + x− y+ = .
19) Find the equation of the circle which is orthogonal to each of the following circles
2 2 2 17 4 0
x +y + x+ y+ = ,x2+y2+7x+6y+ =11 0,x2+y2− +x 22y+ =3 0. 20) Derive the equation of the hyperbola in standard form.
21) Ify=sin−1x, then show that(1−x2)y2−xy1=0.Hence show that
2 2
2 1
(1−x )yn+ −(2n+1)xyn+ −n yn=0 22) Evaluate
2 cos
3sin
4 cos
5sin
x
x
dx
x
x
+
+
∫
23) Evaluate 1 2 0log(1
)
1
x
dx
x
+
+
∫
24) Evaluate 5 01
dx
x
+
∫
approximately by using the Simpson's rule with n = 4.1) If the equation x2+y2−4x+6y+ =c 0represents a circle with radius 6, find the value ofc. 2) Find the equation of the directrix of the parabola 2x2+7y=0
3) Find the length of the latus rectum of the ellipse
2 2
1
16
8
x
y
+
=
4) Find the eccentricity of the hyperbola x2−4y2 =45) Find the distance between the two points in a plane whose polar coordinates are
2,
, 3,
6
4
π
π
. 6) If1
2
5
y
x
=
+
, then findyn . 7) Evaluate∫
1 sin 2xdx
+
8) Evaluate 1 sin 2
1
xe
dx
x
−−
∫
9) Evaluate 4 2 1 (x x −1)dx∫
10) 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)
11) If the line
y
=
mx
+
c
touches the ellipse2 2
2 2 1
x y
a +b =
, (a>b) then show thatc2 =a m2 2+b2. 12) Find the equations of the tangents shown drawn from (–2, 1) to the hyperbola2x2−3y2 =6. 13) Transform the polar equation
cos
22
r
θ
=
a
, (a>0)origin as pole and the axis as initial line, intoCartesian form 14) Ify log x
x
= , then show that
1 ( 1) 1 1 1 log 1 2 3 n n n n y x n x + − ∠ = − − − − ⋯⋯⋯ . 15) Evaluate 6 2 1 1 x dx x − +
∫
16) Solve (x2+y2)dx=2xydy 17) Solve 2 3
2 1 dy x y dx y x + + = + + SECTION-C (5 × 7 = 35)
18) Find the equation of the pair of tangents drawn from (3, 2) to the circlex2+y2−6x+4y− =2 0. 19) Find the equation of the circle passing through the points of intersection of the circles
2 2 8 6 21 0
x +y − x− y+ = , x2+y2−2x− =15 0 and the point (1, 2).
20) Find the equation of the circle passing through the origin and coaxial with the circles
2 2
6 4 8 0
x +y − x+ y− = andx2+y2−2x+ + =y 4 0.
21) Find the pole of the line x+ + =y 2 0 with respect to the parabola y2+4x−2y− =3 0
22) Evaluate 3sin cos 7
sin cos 1 x x dx x x + + + +
