Maths IIB - FIITJEE Hyderabad
`K U K A T P A L L Y
CENTRE
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN
MATHS-IIB 2017-18
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INTERMEDIATE PUBLIC EXAMINATION, MARCH 2017 Total No. of Questions - 24
Reg.
Total No. of Printed Pages - 2
No.
Part - III MATHEMATICS, Paper-II (B) (English Version) Time : 3 Hours]
[Max. Marks : 75
SECTION - A
10
2 = 20 M
I.
Very Short Answer Type questions:
1.
Obtain the parametric equation of the circle 4( x 2
2.
Find the value of k, if the points (4, 2) and ( k , 3) are conjugate points with respect to the circle x2
3.
y2
5x
8y
y2 )
9.
0.
6
Find the angle between the circles given by the equations x 2 x
2
y
2
4x
6y
Find the coordinates of the points on the parabola y 2
5.
If 3x
6.
Evaluate
7.
Evaluate
8.
Evaluate
k
0 is a tangent to the hyperbola x 2
1 cosh x
e x (1
sinh x
6y
0,
41
4y 2
8x whose focal distance is 10. 5 . Find the value of k.
dx on R.
x)
dx on I cos ( xe ) 2
12 x
0.
59
4.
4y
y2
x
R \{x R : cos(xex )
0}
2
sin x dx 2
3
9.
Evaluate 0
10.
x x2
16
dx
Find the order of the differential equation of the family of all the circles with their centres at the origin.
SECTION – B II.
5
4 = 20 M
Short Answer Type questions: (i) Attempt any five questions (ii) Each question carries four marks
11.
If a point P is moving such that the lengths of tangents drawn from P to the circles x2
y2
4x
6y
12
0 and x 2
y2
6x
18y
26
0 are in the ratio 2 : 3 then find the
equation of the locus of P . 12.
Find the equation and the length of the common chord of the following circles: x2
13.
y2
2x
2y
1
0 ; x2
y2
4x
3y
2
0
Find the equation of ellipse in the standard form, if it passes through the points ( 2, 2) and (3, 1) .
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Find the equation of the tangents to the ellipse 2x 2
14.
(i) parallel to x
2y
4
y2
(ii) perpendicular to x
0
8 which are
y
2
0
15.
If e , e 1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that
16.
Find the area of the region bounded by the parabola y 2
17.
Solve the following differential equation ( x
y
1)
dy dx
4x and x 2
1 e12
1.
4y .
1
SECTION – C III.
1 e2
5
7 = 35 M
Long Answer Type questions: (i) Attempt any five questions (ii) Each question carries seven marks
18.
If (2, 0) , (0,1) , (4, 5) and (0, c ) are concyclic the find c .
19.
Find the transverse common tangents of the circles x 2 x2
y2
4x
6y
4
y2
Derive the equation of a parabola in the standard form y 2
21.
Evaluate
22.
If I n
cos n xdx , then show that I n
0 and
4 ax with diagram.
1 cosn n
1
x sin x
n
1 n
In
2 and
for n
2 deduce the value
cos 4 x dx .
Show that 0
24.
28
9 cos x sin x dx . 4isnx 5 cos x
2
23.
10 y
0.
20.
of
4x
x sin x
cos x
dx
2 2
Solve the differential equation (x
1) .
log( 2
y)dy
(x
y
1)dx
BLUE PRINT (MATHS-IIB) S.No.
Chapter Name
Weightage Marks
Coordinate Geometry 1.
Circles
15 (2 + 2 + 4 + 7)
2.
System of Circles
3.
Parabola
9 (2 + 7)
4.
Ellipse
8 (4 + 4)
5.
Hyperbola
6 (2 + 4)
13 (2 + 4 + 7)
Calculus 6.
Indefinite Integration
18 (2 + 2 + 7 + 7)
7.
Definite Integration & Areas
15 (2 + 2 + 4 + 7)
8.
Differential Equations
13 (2 + 4 + 7)
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VERY SHORT ANSWER QUESTIONS
1. A.
Find the equation of circle with centre (1, 4) and radius 5.
B.
Find the centre and radius of the circle x 2
C.
Find the centre and radius of the circle 3x 2
D.
Find the equation of the circle whose centre is (-1, 2) and which passes through (5, 6)
E.
Find the equation of the circle passing through (2, 3) and concentric with the circle x2
y2
8x
12 y
If the circle x 2
F.
15
y2
y2
2x
4y
3y 2
6x
4
0
4y
4
0
0
ax
by
12
0 has the centre at (2, 3) then find a, b and the radius of the
circle. G.
Find the equation of the circle whose extremities of a diameter are (1, 2) and (4, 5)
H.
Find the other end of the diameter of the circle x 2
I.
Obtain the parametric equation of the circle represented by x 2
J.
Find the values of a , b if ax 2
bxy
3y 2
5x
y2
2y
8x
8y
0 if one end of it is (2, 3)
27 y2
6x
8y
96
0
0 represents a circle. Also find the radius
3
and centre of the circle. m2 x 2
y2
K.
Find the centre and radius of each of the circles
L.
Obtain the parametric equation of each of the following circles. (i) x 2
M.
y2
2
82
lies on the circle x 2
y2
(ii) x
4
Show that A 3, 1
3
2
y
4
1
2x
2cx
2mcy
0
0 . Also find the other end of the
4y
diameter through A.
2. A.
Locate the position of the point (2, 4) with respect to the circle x 2
y2
4x
6y
B.
Find the length of the tangent from (1, 3) to the circle x 2
4y
11
0.
C.
Find the equation of the tangent to x 2
D.
If the parametric values of two points A and B lying on the circle x 2
y2
6x
y2
6x
4y
y2
12
2x
11
0.
4y
12
0 are
to
the
circle
0 at ( 1,1)
30 and 60 respectively then find the equation of the chord joining A and B . E.
Find x2
the y2
area
22x
4y
of
the
triangle
formed
by
the
normal
If 4, k and 2, 3 are conjugate points with respect to the circle x 2
G.
Find the power of the point P with respect to the circle S
5, 6 and S
3, 4
0 with the coordinate axes.
25
F.
(i) P
at
x2
y2
8x
12 y
y2
17 then find k .
0 when
15
H.
If the length of the tangent from (2, 5) to the circle x 2
I.
Find the length of the chord formed by x 2
J.
Find the value of k if the points 4,2 and k , 3 are conjugate points with respect to the circle x2
y2
5x
8y
6
y2
y2
5x
4y
a2 on the line x cos
k
0 is y sin
37 then find k . p.
0.
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Find the equation of the normal to the circle x 2
K.
y2
4x
6y
0 and 3, 2 . Also find the
11
other point where the normal meets the circles. L.
Find the equation of the pair of tangents from 10, 4 to the circle x 2
M.
Find the equation of the normal at P of the circles S (i) P
3, 4 ,
S
x
2
y
2
x
y
Fin the polar of 1,2 with respect to x 2
O.
Find the pole of ax
by
c
P.
Find the pole of 3x
4y
45
Q.
Find the value of k if the point x2
y2
25 .
0 where P and S are given by
24
N.
y2
7.
0 with respect to x 2
0 c
y2
0 with respect to x 2
and
1, 3
y2
y2 6x
r2 . 8y
5
0
are conjugate with respect to the circle
2, k
35 .
R.
Find the angle between the tangents drawn from 3, 2 to the circle x 2
S.
Find the number of possible common tangents that exist for the following pairs of circles y2
2x
4y
If the angle between the circles x 2
y2
12 x
x2
y2
6x
6y
0 , x2
14
4
y2
6x
4y
2
0.
0
3. A.
6y
0 and x 2
41
y2
kx
6y
59
0 is 45
find k . B.
Find k if the following pairs of circles are orthogonal (i) x 2
C.
E.
y2
0, x 2
k
y2
2 ax
8
12x
6y
0, x 2
41
y2
0
4x
6y
59
0
3 . 4 Find the equation of the common tangent of the following circles at their point of contact
Show that angle between the circles x 2
x2
F.
2by
Find the angle between the circles given by the equations x2
D.
y2
y2
10x
2y
0 , x2
22
y2
y2
2x
a2 , x 2
8y
8
y2
ax
ay is
0
Find the equation of the radical axis of the following circles (i)
x2
(ii) x 2 G.
y2
y2
2x
2x
4y
1
0 , x2
4y
1
0 , x2
y2
y2
4x
4x
6y
y
5
0
0
Find the equation of the common chord of the following pair of circles. (i) x 2 (ii)
y2
x
a
4x 2
4y
y
b
3 2
0 , x2
c2 ,
y2
x b
5x 2
y
6y
a
4 2
0
c2 a
b
4. A.
Find the equation of the parabola whose vertex is 3, 2 and focus is 3,1 .
B.
Find the coordinates of the points on the parabola y 2
C.
Find the vertex and focus of 4 y 2
12 x
20 y
67
2x whose focal distance is
5 . 2
0.
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D.
Find the equation of the parabola whose focus is S(1, 7) and vertex is A(1, 2) .
E.
Find the coordinates of the points on the parabola y 2
F.
If (1/2, 2) is one extremity of a focal chord of the parabola y 2
8x whose focal distance is 10. 8x . Find the coordinate of the
other extremity. k is a tangent to the parabola y 2
G.
Find the value of k if the line 2 y
H.
Find the equation of the normal to the parabola y 2
I.
Find the vertex and focus of x 2
J.
Find the equations of axis and directrix of the parabola y 2
K.
Find the equation of the parabola whose focus is S(3, 5) and vertex is A(1, 3) .
L.
Find the equation of the parabola whose latus rectum is the line segment joining the points ( 3, 2) and ( 3,1) .
M.
Find the equations of the tangent and normal to the parabola y 2
5x
6x
6y
6
6x .
4x which is parallel to y
2x
0.
5
0. 6y
2x
0.
5
6x at the positive end of the
latus rectum. N.
Find the equation of the tangent and normal to the parabola x 2
O.
Show that the line 2 x
y
2
4x
0 is a tangent to the parabola y 2
8y
12
0 at 4,
3 . 2
16 x . Find the point of contact
also. Find the equation of tangent to the parabola y 2
P.
16 x inclined at an angle 60 with its axis and
also find the point of contact.
5. 1 e2
1 e12
A.
If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that
B.
Find the equations of the hyperbola whose foci are ( 5,0) , the transverse axis is of length 8.
C.
If 3x
D.
If the eccentricity of a hyperbola is
4y
k
0 is a tangent to x 2
4y 2
5 find the value of k .
E.
5 , then find the eccentricity of its conjugate hyperbola. 4 Find the equation to the hyperbola whose foci are (4, 2) and (8, 2) and eccentricity is 2.
F.
If the lx
G.
Find the equations of the tangents to the hyperbola 3x 2
my
(i) parallel and
1
1 is a normal to the hyperbola
x2 a2
y2 b
1 then, show that
2
(ii) perpendicular to the line y
x
4y 2
a2 l2
b2 m2
( a2
b2 )2
12 which are
7
H.
Define rectangular hyperbola. What is its eccentricity?
I.
Find the product of lengths of lengths of the perpendiculars from any point on the hyperbola
J.
x2 y 2 1 to its asymptotes. 16 9 Find the equation of the hyperbola whose asymptotes are 3x
K.
Find the equation of the normal at
L.
If the angle between the asymptotes is 300 then find its eccentricity.
/ 3 to the hyperbola 3x 2
5 y and the vertices are ( 5,0) .
4y 2
12 .
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6. 3
1 x
x
dx , x
A.
Evaluate
B.
Find
C.
Evaluate
x5 dx on 1 x 12
D.
Find
x 1 . e x2
E.
Evaluate
sin 2x dx on
1
1
( ax
F.
1 x
(x
bx )2
5) x
dx , ( a
4
sec2 x cos ec 2 x dx on I
H.
1 cos 2x dx on I 1 cosh x
1 and b
)
)
1) on
0, b
\ {n : n
[2n , (2n
dx on
sinh x
(0,
dx on ( 4,
0, a
G.
I.
.
dx on where I
dx
ax b x
0
}
(2 n
1) ], n
.
cot 2 x dx on
J.
Evaluate
K.
Evaluate
x6 1 dx for x 1 x2
L.
Evaluate
cos3 x sin x dx on
M.
Evaluate
sin 4 x dx , x 1 cos6 x
N.
Evaluate
sin 2 x dx on
O.
Evaluate
P.
Evaluate
Q.
Evaluate
elog(1
R.
Evaluate
sin 2 x dx on I 1 cos 2 x
\ (2n
1) : n
S.
Evaluate
1 dx on I cos x
\{(2n
1) : n
T.
Evaluate
dx 9x 2
4
1 1
1
4x 2
1) : n 2
\ {n : n
on I
}
.
R\
(2n
1) 2
:n
2 2 , 3 3
dx on
tan 2 x )
dx on I
x2 dx on I 1 x2
\
(2n
1) 2
:n
}
\0
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7. e Tan
A.
1
1x
dx on I
2
x
(0,
1
sin(Tan
B.
)
1
x
x)
2
dx , x
C.
2x3 dx on R 1 x8
D.
x8 dx on 1 x 18
E.
cos x dx on 0, x
F.
x2 x4
G.
sec x log(sec x
H.
sin 3 x dx on
I.
cos 3 x dx on
J.
cos x cos 3x dx on
dx
K.
ex
1
L.
2
, x
log sec x ex dx on
tan x
M.
tan x )dx on 0,
Evaluate
x ex
S.
x
1
dx on I
2
\
,n
N.
x log x dx on 0,
Q.
1 2
, 2n
4 4 , 5 5
25x 2 dx on
16
1 2
2n
1 dx on 1, x log x[log(log x )]
O.
1 dx on 1
1
dx
Evaluate
x
2
2x
P.
cos x cos 2 x dx on
R.
ex
T.
ex
1
10
x log x dx on 0, x
x
2
x
3
2
dx on I
\
3
1 dx 1 cot x
U.
sin 2x
V.
2
W.
2
dx on I
\ x
a cos x
b sin x
ex sec x
sec x tan x dx on I
|a cos2 x
\ 2n
1
2
b sin 2 x
0
:n
8. Evaluate the following definite integrals 2
A.
Evaluate
5 0 sin 2
5 2 cos
x
3
x
5 cos 2
B.
dx x
2 0
2 cos d
sin x dx sin x cos x
6
2
C.
Evaluate
D.
1
1 x dx 0
E. 0
dx 3 2x
1
F.
xe
x2
dx
0
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5
G. 1
3
dx 2x 1
H. 2
2k
Evaluate lim
K.
4k
6k nk
n
2
2x dx 1 x2
.....
2n
I.
4
sec 4 d
J.
2
0
x dx
0
k
by using the method of finding definite integral as the
1
limit of a sum.
9. 2
A.
Find
sin 4 x dx
0 2
B.
Find
cos8 x dx
0
C.
Find the area under the curve f x
sin x in 0, 2
D.
Find the area under the curve f x
cos x in 0, 2
E.
Find the area bounded by the curves y
.
sin x and y
cos x between any two consecutive
points of intersection. F.
Find the area enclosed within the curve x
G.
Evaluate the following definite integrals. (i)
y 2
1 sin 5 x cos 4 x dx
0
2
H.
Find
2
(ii)
sin 6 x cos 4 x dx
0
sin 7 x dx
0
a
I.
a2
Evaluate
x 2 dx
0
x 2 , the X-axis and the lines x
J.
Find the area bounded by the parabola y
K.
Find the area bounded between the curves y 2
L.
Evaluate
x2 , y
1, x
2
x
tan 5 x cos8 x dx
0
10. A.
B.
Find the order and degree of
Find the order and degree of
d3y dx 3
d2 y dx 2
2
3
dy dx
dy dx
2
ex
4
3 6/5
6y
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C.
Find the order of the differential equation corresponding to Aex
y D.
Be3x
Ce5x ( A, B, C being parameters ) is a solution
Form the differential equation corresponding to y
b sin 3x , where A and B are
A cos 3x
parameters. E.
Form the differential equation corresponding to the family of circles of radius r given by (x
a)2
(y
b)2
r 2 , where a and b are parameters.
F.
Solve
dy dx
G.
Solve
dy dx
H.
Find the order and degree of the differential equation
I.
Find the order of the differential equation corresponding to y
y tan x
sin x
x 2 logx 1 sin y
y cos y
d2 y dx
2
p2 y . c x
c
2
, where c is an arbitrary
constant. 1/3
d2 y
1/2
dy dx
J.
Find the order and degree of x
K.
Find the general solution of x
L.
Find the general solution of
M.
Solve y(1
N.
Form the differential equations of the following family of curves where parameters are given in brackets. aex
(i) xy O.
x)dx
x(1
x
be
y
dy dx
y)dy
dy dx ex
y
0
0 y
0
; a , b (ii) y
Find the general solution of
dx
x
2
bx e kx ; a , b (iii) y
a
1 x2 dy
1 y 2 dx
a cos nx
b ; a, b
0
SHORT ANSWER QUESTIONS
11. A.
From the point A 0, 3 on the circle x 2 a point M such that AM
4x
3)2
(y
0 a chord AB is drawn and extended to
2 AB . Find the equation of the locus of M .
If the abscissae of points A, B are the roots of the equation x 2
B.
B are roots of y C.
2 py
q
2
x
y
2
0 and ordinates of A ,
y2
6x
4y
12
0 and passing
2,14 .
If a point 2
b2
0 then find the equation of a circle for which AB is a diameter.
Find the equation of a circle which is concentric with x 2 through
D.
2
2 ax
2x
P 4y
is moving such that the length of tangents drawn from 20
0 and x
equation of the locus of P is x 2
2
y y2
2
2x
2x
8y
12 y
1 8
P
to
0 are in the ratio 2:1 then show that the 0.
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E.
If 4 x
3y
0 is a tangent to the circle represented by x 2
7
y2
6x
4y
0 the find its
12
point of contact. F.
Find
the
x2 G.
y2
22x
common
4y
tangents
of
the
x2
circles
y2
4x
6y
y2
22x
4y
100
0
and
28
0
and
0.
100
x2
Find the transverse common tangents of the circles x2
H.
direct
y2
4x
10 y
0.
4
If a point P is moving such that the lengths of tangents drawn from P to the circles x2
y2
4x
6y
0 and x 2
12
y2
6x
18y
0 are in the ratio 2 : 3 then find the
26
equation of the locus of P . Find the equations of the tangents to the circle x 2
I.
x
y
to 3x K.
4x
6y
12
0 which are parallel to
0.
8
Find the equations of the tangents to the circle x 2
J.
y2
y
y2
2x
2y
0 which are perpendicular
3
0.
4
Find the equation of circles passing through 1, 1 , touching the lines 4 x 3x
4y
3y
5
0 and
0.
10
0 touches the circle x 2
L.
Show that x
M.
Tangents are drawn to the circle x 2
y
1
y2
y2
3x
7y
0 and find its point of contact.
14
16 from the point P 3,5 . Find the area of the
triangle formed by these tangents and the chord of contact of P . Show that the circles x 2
N.
y2
6x
2y
0, x 2
1
y2
2x
8y
0 touch each other
13
Find the point of contact and the equation of common tangent at their point of contact. Show that x 2
O.
y2
6x
9y
0, x 2
13
y2
2x
0 touch each other. Find the point of
16y
contact and the equation of common tangent at their point of contact.
12. A.
Find S
the
x
2
y
equation
2
3x
5y
show that f ' g C.
1
2
2
a D.
y2
b
of x
2
the
y
2
common
chord
3y
4
y2
2g'x
2 fy
0 and x 2
y2
2 ax
c
of
the
two
circles
0.
5x
2 gx
2 f 'y
0 touch each other then
fg ' .
Show that the circles x 2 1
length
0 and S '
4
If the two circles x 2
B.
and
0 and x 2
y2
2by
c
0 touch each other if
1 . c
Show that the circles x 2
y2
0 and x 2
2x
y2
6x
6y
2
0 touch each other. Find the
coordinates of the point of contact. Is the point of contact external or internal? E.
Show that the circles x 2
y2
4x
6y
0 and 5 x 2
12
y2
8x
14y
32
0 touch each
other and find their point of contact. F.
Show x
2
y
that 2
14x
four 6y
common 33
tangents
0 and x
2
y
2
can 30x
be 2y
drawn 1
for
the
circles
given
by
0 and find the internal and external
centres of similitude. FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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13. A.
Find the eccentricity, coordinates of foci, Length of latus rectum and equations of directrices of the following ellipse. (i) 9x 2
B.
16 y 2
36x
32 y
92
(ii) 3x 2
0
y2
6x
2y
5
0
Find the equation of the ellipse referred to its major and minor axes as the coordinate axes X, Y-respectively with latus rectum of length 4 and distance between foci 4 2 .
C.
If
are the eccentric angles of the extremities of a focal chord (other than the vertices) of the
1, 2
ellipse
E.
1, a
b2 1
(i) e cos D.
y2
x2 a2
2
2
b and e its eccentricity. Then show that 1
cos
2
e e
(ii)
2
1 1
cot
1
2
Find the equation of the ellipse with focus at 1, 1 , e
cot
2
2
2 / 3 and directrix as x
Prove that the equation of the chord joining the points ' ' and ' ' on the ellipse
x cos a
2
y sin b
cos
2
y
x2 a2
0.
2
y2 b2
1 is
2
F.
S and T are the foci of an ellipse and B is one end of the minor axis. If STB is an equilateral triangle, then fund the eccentricity of the ellipse.
G.
Find the equation of the ellipse in the standard form such that distance between foci is 8 and distance between directrices is 32.
H.
Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equations of directrices of the following ellipse. (i) 9x 2
I.
16 y 2
144
(ii) x 2
2y2
4x
12 y
14
0
A man running on a race course notices that the sum of the distances of the two flag posts from him is always 10m. and the distance between the flag posts is 8m. Find the equation of the race course traced by the man.
14. A.
Find the equation of tangent and normal to the ellipse 9x 2
16 y 2
144 at the end of the latus
rectum in the first quadrant. B.
If the normal at one end of a latusrectum of the of the ellipse end of the minor axis, then show that e 4
C.
2y
4
(iii) which makes an angle
y2
4
b2
1 passes through one
8 which are
(ii) perpendicular to x
0
y2
1 [ e is the eccentricity of the ellipse)
Find the equation of the tangents to the ellipse 2x 2 (i) parallel to x
D.
e2
x2 a2
y
2
0
with x-axis
A circle of radius 4, is concentric with the ellipse 3x 2 is inclined to the major axis at an angle
4
13y 2
78 . Prove that a common tangent
.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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E.
Show that the locus of the feet of the perpendiculars drawn from foci to any tangent of the ellipse is the auxiliary circle.
F.
If a tangent to the ellipse
y2
x2 a2
b2
1 a
a2
respectively then prove that
CM
b meets its major axis and minor axis at M and N
b2 2
CN
1 Where C is the centre of the ellipse.
2
y2
x2 If PN is the ordinate of a point P on the ellipse 2 a
G.
1 and the tangent at P meets the
b2
a2 where C is the centre of the ellipse.
X-axis at T then show that CN CT H.
Show that the points of intersection of the perpendicular tangents to an ellipse lie on a circle.
I.
Find the equation of tangent and normal to the ellipse x 2
J.
Find the condition for the line x cos
K.
Show that the foot of the perpendicular drawn from the centre on any tangent to the ellipse lies on the curve x 2
y2
2
a2 x 2
8y 2
33 at
1,2 .
p to be a tangent to the ellipse ...
y sin
b2 y 2 .
15. A.
Find the centre, eccentricity, foci, directrices and the length of the latus rectum of the following hyperbolas. (i) 4x 2
B.
9y 2
32
(ii) 4( y
0
3)2
9( x
2)2
1
Prove that the point of intersection of two perpendicular tangents to the hyperbola x2 a2
C.
y2 b
D.
1 lies on the circle x 2
2
y2
a2
b2 .
Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the following hyperbolas (i) x 2
4y 2
(ii) 9x 2
4
16 y 2
72 x
32 y
16
0
Show that the angle between the two asymptotes of a hyperbola 2 sec
E.
8x
1
(ii) perpendicular to the line x
2y
4y 2
b
2
1 is 2 Tan
1
b or a
Find the equation of tangents drawn to the hyperbola 2x 2
G.
If the line b 2 m2
lx
my
n
4 which are (i) parallel
0.
F.
H.
y2
e .
Find the equations of the tangents to the hyperbola x 2
a2 l 2
x2 a2
3y 2
0 is a tangent to the hyperbola
6 through (–2, 1)
x2 a2
y2 b2
1 then, show that
n2
Prove that the product of the perpendicular distances from any point on a hyperbola to its asymptotes is constant.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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I.
y2
x2 a2
Tangents to the hyperbola
b2
1 make angles
1, 2
with transverse axis of a hyperbola.
Show that the point of intersection of these tangents lies on the curve 2 xy tan
J.
tan
1
k( x 2
a2 ) when
k.
2
Show that the locus of feet of the perpendiculars drawn from foci to any tangent of the x2 a2
hyperbola
y2
1 is the auxiliary circle of the hyperbola.
b2
16. 2 ...... 1 n
n n
1 n
A.
Evaluate lim 1
B.
Evaluate lim
C.
Find the area of one of the curvilinear triangles bounded by y
D.
Find
n
n
1
n
a
x 2 a2
2
.....
n
n
n n
n
3 2
x2
sin x , y
cos x and X-axis.
dx
a 2
E.
Find
sin 2 x cos 4 x dx
2
F.
Find the area enclosed by the curves y
G.
Evaluate lim
H.
Evaluate lim
n
n
1 n
1 1
1 tan n 4n n
I.
i 1
J.
Evaluate lim 1
K.
Evaluate
n
2
0
4
2
tan
.....
2 4n
6x
x2
1 6n
......
tan
n 4n
i3
Evaluate lim n
n
3x and y
i4
n4
1 1 n2
22 ...... 1 n2
n2 n2
1 n
dx 5 cos x
1
L.
xTan 1 x dx
Evaluate 0
M.
Find the area of the region bounced by the parabolas y 2
4x and x 2
4y
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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17. Solve the following differential equations: A.
dy dx
ex
C.
dy dx
x tan( y
E.
(cos x )
G.
dy dx
y2
y
1
x2
x
1
I.
dy dx
4x y 1 x2
K.
x
M.
xdy
O.
dy dx
Q.
sin
S.
1
ex / y dx
U.
dy dx
sin( x
y
dy dx
2y3
x)
1
y sin x
dy dx
xy xy
tan x
0 1 1
x
2 2
y
x cos2
y
y dx x
y x
dy dx
1
y
x2 e
x
y
ex / y 1 y)
cos( x
x dy y
0
y)
B.
dy dx
x(2 log x 1) sin y y cos y
D.
(x2
y 2 )dx
F.
1
x2
dy dx
2 xy
H.
1
x2
dy dx
y
dy dx
J.
cos x.
L.
dy 2 3 x y dx
N.
x2
P.
xy 2
R.
dy dx
T.
x
V.
dy dx
y2
4x 2
xy dy dx
tan 2 x 1
dy dx
0
1x
eTan
sec2 x
y sin x
x dx
y
2 xydy
1 xy
yx 2
y dy
0
y 1
x 2y 1 2x 4y
LONG ANSWER QUESTIONS
18. A.
Find the equation of a circle which passes through 2, 3 and 4x
3y
1
4,5 and having the centre on
0.
B.
Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on 4 x 3y 24 0 .
C.
Find the equation of the circle whose centre lies on the X-axis and passing through
2,3 and
4,5 . D.
Find the equation of circle passing through the points 1,2 , 3, 4 , 5, 6
E.
Find the equation of the circle passing through (0, 0) and making intercepts 4, 3 on X-axis and Yaxis is respectively.
F.
Find the equation of the circle passing through (0, 0) and making intercept 6 units on X-axis and intercept 4 units on Y-axis.
G.
If 2,0 , 0,1 4,5 and 0,c are concylclic then find c .
H.
Find the equations of circles which touch 2 x
3y
1
0 at (1,1) and having radius
13 .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Show that the circles x 2
I.
y2
4x
6y
0 and x 2
12
y2
6x
18y
0 touch each other.
26
Also find the point of contact and common tangent at this point of contact. J.
Show that the tangent at x2
K.
y2
4x
If the polar of the points on the circle x 2
If
M.
1, 2
y2
4x
0 touches the circle
7
a2 with respect to the circle x 2
y2
y2
are the angles of inclination of tangents through a point P to the circle x 2 1
Prove
at
that
y2
the
2x
tangent
10 y
cot
b 2 touches
y2
a2 then
k.
2
of
3, 2
the
x2
circle
y2
touches
13
the
circle
0 and find its point of contact.
26
Find the equation of the circle which touches the circle x 2
N.
8y
c 2 then prove that a , b , c are in Geometrical progression.
find the locus of P when cot x2
y2
0 and also find its point of contact.
6y
the circle x 2 L.
of the circle x 2
1,2
y2
2x
4y
20
0 externally at
y2
2x
4y
11
0 and also find
5,5 with radius 5. Find the pair of tangents drawn from 1,3 to the circle x 2
O.
the angle between them.
19. A.
Find the equation of the circle which passes through (1, 1) and cuts orthogonally each of the circles x 2
B.
y2
8x
2y
0 and x 2
16
y2
4x
4y
1
0.
Find the equation of the circle which intersects the circle x 2
y2
6x
4y
3
0 orthogonally
and passes through the point (3, 0) and touches Y-axis. C.
Find the equations to all possible common tangents of the circles x2
D.
E.
2x
6y
0 and x 2
6
(i) x 2
y2
9 and x 2
(ii) x 2
y2
4x
y2
2y
16x
H.
1
2y
0 and x 2
4
49
y2
0
4x
2y
4
0
Find the equation of the circle which is orthogonal to each of the following three circles y2
2x
17 y
0 , x2
4
y2
7x
6y
11
0 and x 2
y2
x
22 y
3
0.
Find the equation of the circle passing through the origin, having its centre on the line x and intersecting the circle x 2
G.
y2
Find all common tangents of the following pairs of circles.
x2
F.
y2
y2
4x
2y
4
y2
10x
2x
3y
7.
If x
y
4y
21
0
4
0 orthogonally.
Find the equation of the circle which cuts the circles x2
y
x2
y2
4x
6y
11
0
and
orthogonally and has the diameter along the straight line
3 is the equation of the chord AB of the circle x 2
y2
2x
4y
0 , find the
8
equation of the circle having AB as diameter. I.
Show x2
y2
that 8x
the 6y
common 23
chord
of
the
circles
x2
y2
6x
4y
9
0
and
0 is the diameter of the second circle and also find its length.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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20. A.
B.
Show
that
the
xa1/3
yb1/3
common
a2 /3b2 /3
tangent
to
the
y2
parabola
C.
0 is
4 ax a
is
4by
0
Prove that area of the triangle formed by the tangents at x1 , y1 , x2 , y2 parabola y 2
x2
and
4 ax
1 y1 16 a
y2 y2
y3 y3
y1
and x3 , y 3
to the
sq. units.
Find the equation of the parabola whose axis is parallel to x -axis and which passes through the points ( 2,1) , (1, 2) and ( 1, 3) .
D.
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) .
E.
Prove
that
the
1 ( y1 8a
y 2 )( y 2
area
G.
H.
t12
in
the
parabola
y2
4 ax is
16 x which are parallel and perpendicular
0 , also find the coordinates of their points of contact.
5
y2
2 a2 and the parabola y 2
8ax
2a .
x
2
4 ax meets the parabola again in the point t 2 . Then prove that
0.
If a normal chord a point ' t ' on the parabola y 2 prove that t
4 ax subtends a right angle at vertex, then
2.
Find the coordinates of the vertex and focus, and the equations of the directrix and axes of the following parabolas (i) y 2
(ii) x 2
16 x
4y
3x 2
(iii)
9x
5y
2
0
(iv)
From an external point P, tangent are drawn to the parabola y 2 angles
1, 2
line y
bx .
with its axis, such that tan
Prove that the two parabolas y 2 of Tan
M.
y
The normal at a point t1 on y 2
t1t2
L.
inscribed
Show that the equation of common tangent to the circle x 2 are y
K.
triangle
y1 ) sq. units where y1 , y2 , y3 are the ordinates of its vertices.
y 3 )( y 3
respectively to the line 2 x
J.
the
Find the equations of tangents to the parabola y 2
F.
I.
of
3 a1 /3 b 1 /3
1
2 a 2 /3
b 2 /3
tan
1
4 ax and x 2
2
y2
x
4y
5
0
4 ax and these tangents make
is a constant b. Then show that P lies on the
4by intersect (other than the origin ) at an angle
.
Find the equation of the parabola whose focus is ( 2, 3) and directrix is the line 2 x
3y
4
0.
Also find the length of the latus rectum and the equation of the axis of the parabola. N.
Find the coordinates of the vertex and focus, the equation of the directrix and axis of the following parabolas. (i) y 2
4x
4y
3
0
(ii) x 2
2x
4y
3
0
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Show that the locus of point of intersection of perpendicular tangents to the parabola y 2
O.
the directrix x
0.
a
Show that the common tangents to the circle 2x 2
P.
4 ax is
at the focus of the parabola y 2
a2 and the parabola y 2
2y 2
4 ax intersect
4ax .
21. Evaluate A.
(3x
C.
6x
E.
x
2) 2 x 2 5
6
2x
3
3 x
2
x
1 dx
B.
2x2
x dx
D.
dx
4
cos x 3sin x 7 dx cos x sin x 1
H.
I.
9 cos x sin x dx 4sin x 5cos x
J.
K.
2 sin x 3sin x
L.
M.
dx on 1 x4
N.
O.
dx 5sin x
P.
dx 4 cos x 3sin x
R.
4
Q.
x
S.
x
1 x2
2x2
U.
x
3
x
3 x
1
1 2
sin x cos x cos x 3 cos x
W.
2
2
4 dx 5
dx
T.
dx
V.
2
x2
dx
5 2x
10
dx
dx 1
x
3
x2
2x
on
dx
F.
G.
3cos x 4 cos x
2x
x
1
2x
2 cos x 4 cos x
3x
cos x
1 x
a x x x
b x c 1
2
1
on I
\
1,
1 2
3sin x dx 5sin x
1 sin x
1
2
1,3
x
1
dx
dx
dx
1 dx 3cos 2 x
2
dx 1
x
3
2x
x2
on
3,1
x3 2x 3 dx x2 x 2 dx x
3
1
dx 3 cos x 4 sin x
X.
6
22. A.
If I n n
B.
cosn x dx , then show that I n
2 and deduce the value of
Obtain reduction formula for I n value of
1 cosn n
1
x sin x
n 1 In n
2,
n being a positive integer,
cos 4 xdx . cot n xdx , n being a positive integer, n
2 and deduce the
cot 4 xdx .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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C.
sin m x cosn x dx , then show that
If Im ,n
sin m
I m ,n
1
x cosn m n
1
x
m 1 Im m n
E.
Obtain reduction formula for
F.
Obtain reduction formula for
2,n ,
for a positive integer n and an integer m
sin n x for an integer n
tan n x I n
2 and deduce the value of
2. sin 4 x dx .
cot n xdx , n being a positive integer, n
2 and
tan 6 x dx .
deduce the value of G.
2 and
cosec5 x dx .
deduce the value of D.
cosecn x dx , n being a positive integer, n
Obtain the reduction formula for In
sec n x for an integer n
Obtain reduction formula for
2 and deduce the value of
sec 5 x dx .
23. A.
Show that the area enclosed between the curves y 2
B.
Show that the area of the region bounced by the circle x 2
y2
x2 a2
Find the area bounded between the curves y 2
D.
Find the area bounded between the curves 4x
y2
3 and y 2
20 5
x is 64
5 . 3
1 (ellipse) is ab . Also deduce the area of
b2
a2 .
C.
(i) y
12 x
x 2 and y
5
(ii) y 2
2x
4 ax , x 2
4x , y 2
4 4
4by a
0, b
0
x
Evaluate E. 0
4
x sin x dx 1 sin x
F. 0
4
H.
log 1
tan x dx
I.
0
0
b
K.
x a
a b
x dx
L. 0
sin x cos x dx 9 16 sin 2 x
x sin x dx 1 cos2 x
1
G. 0
2
J.
log 1 1
x x
2
dx
cos x dx 1 ex
2
x sin 3 x dx 1 cos2 x
2
M. 0
x sin x
cos x
dx
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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2
N. 0
sin 2 x dx cos x sin x
The circle x 2
O.
y2
8 is divided into two parts by parabola 2y
x 2 . Find the area of both the
parts. P.
Let AOB be the positive quadrant of the ellipse
y2
x2 a2
1 with OA
b2
a , OB
b . Then show
2 ab . 4
that the area bounded between the chord AB and the arc AB of the ellipse is
24. A.
Find the equation of a curve whose gradients is passes through the point 1,
4
Solve
dy dx
y2
y
1
2
x
1
D.
Solve
dy dx
3y 3x
F.
Solve
dy dx
H.
Solve x sec
I.
Give the solution of x sin 2
J.
Find the solution of the equation x( x
C.
Solve 2 x
7 3
E.
Solve
dy dx
x sin 2 y
x 3 cos 2 y
G.
Solve
dy dx
y .( ydx x
xdy )
x
7x 7y
9 when x
K.
Solve y 2
L.
Solve: y dx
M.
Solve 2 x
N.
Solve 1
x2
O.
Solve x
y dy
P.
Solve x 2
0
dy dx
y 2 dx
yco sec y dx x
y .( xdy x
ydx
2xy
x 2 dy
y x
x dy
x
y
x dy x cos 3
cos2
y , where x x
0, y
0 and which
y
1 dx
y tan x
x2
4x
2y
1 dy
sin x
y2
2x2
ydx )
xdy which passes through the point 1,
2)
0
dy dx
2( x
1)y
x3 (x
4
2) which satisfies the condition
3.
2xy dx
2y
y x
.
B.
that y
dy dx
dy dx
0 y dx y sin
y x
1
y
Tan 1 x
x
y
1 dx
2xy dy
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CENTRE
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN
MATHS-IIB 2017-18
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INTERMEDIATE PUBLIC EXAMINATION, MARCH 2017 Total No. of Questions - 24
Reg.
Total No. of Printed Pages - 2
No.
Part - III MATHEMATICS, Paper-II (B) (English Version) Time : 3 Hours]
[Max. Marks : 75
SECTION - A
10
2 = 20 M
I.
Very Short Answer Type questions:
1.
Obtain the parametric equation of the circle 4( x 2
2.
Find the value of k, if the points (4, 2) and ( k , 3) are conjugate points with respect to the circle x2
3.
y2
5x
8y
y2 )
9.
0.
6
Find the angle between the circles given by the equations x 2 x
2
y
2
4x
6y
Find the coordinates of the points on the parabola y 2
5.
If 3x
6.
Evaluate
7.
Evaluate
8.
Evaluate
k
0 is a tangent to the hyperbola x 2
1 cosh x
e x (1
sinh x
6y
0,
41
4y 2
8x whose focal distance is 10. 5 . Find the value of k.
dx on R.
x)
dx on I cos ( xe ) 2
12 x
0.
59
4.
4y
y2
x
R \{x R : cos(xex )
0}
2
sin x dx 2
3
9.
Evaluate 0
10.
x x2
16
dx
Find the order of the differential equation of the family of all the circles with their centres at the origin.
SECTION – B II.
5
4 = 20 M
Short Answer Type questions: (i) Attempt any five questions (ii) Each question carries four marks
11.
If a point P is moving such that the lengths of tangents drawn from P to the circles x2
y2
4x
6y
12
0 and x 2
y2
6x
18y
26
0 are in the ratio 2 : 3 then find the
equation of the locus of P . 12.
Find the equation and the length of the common chord of the following circles: x2
13.
y2
2x
2y
1
0 ; x2
y2
4x
3y
2
0
Find the equation of ellipse in the standard form, if it passes through the points ( 2, 2) and (3, 1) .
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Find the equation of the tangents to the ellipse 2x 2
14.
(i) parallel to x
2y
4
y2
(ii) perpendicular to x
0
8 which are
y
2
0
15.
If e , e 1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that
16.
Find the area of the region bounded by the parabola y 2
17.
Solve the following differential equation ( x
y
1)
dy dx
4x and x 2
1 e12
1.
4y .
1
SECTION – C III.
1 e2
5
7 = 35 M
Long Answer Type questions: (i) Attempt any five questions (ii) Each question carries seven marks
18.
If (2, 0) , (0,1) , (4, 5) and (0, c ) are concyclic the find c .
19.
Find the transverse common tangents of the circles x 2 x2
y2
4x
6y
4
y2
Derive the equation of a parabola in the standard form y 2
21.
Evaluate
22.
If I n
cos n xdx , then show that I n
0 and
4 ax with diagram.
1 cosn n
1
x sin x
n
1 n
In
2 and
for n
2 deduce the value
cos 4 x dx .
Show that 0
24.
28
9 cos x sin x dx . 4isnx 5 cos x
2
23.
10 y
0.
20.
of
4x
x sin x
cos x
dx
2 2
Solve the differential equation (x
1) .
log( 2
y)dy
(x
y
1)dx
BLUE PRINT (MATHS-IIB) S.No.
Chapter Name
Weightage Marks
Coordinate Geometry 1.
Circles
15 (2 + 2 + 4 + 7)
2.
System of Circles
3.
Parabola
9 (2 + 7)
4.
Ellipse
8 (4 + 4)
5.
Hyperbola
6 (2 + 4)
13 (2 + 4 + 7)
Calculus 6.
Indefinite Integration
18 (2 + 2 + 7 + 7)
7.
Definite Integration & Areas
15 (2 + 2 + 4 + 7)
8.
Differential Equations
13 (2 + 4 + 7)
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VERY SHORT ANSWER QUESTIONS
1. A.
Find the equation of circle with centre (1, 4) and radius 5.
B.
Find the centre and radius of the circle x 2
C.
Find the centre and radius of the circle 3x 2
D.
Find the equation of the circle whose centre is (-1, 2) and which passes through (5, 6)
E.
Find the equation of the circle passing through (2, 3) and concentric with the circle x2
y2
8x
12 y
If the circle x 2
F.
15
y2
y2
2x
4y
3y 2
6x
4
0
4y
4
0
0
ax
by
12
0 has the centre at (2, 3) then find a, b and the radius of the
circle. G.
Find the equation of the circle whose extremities of a diameter are (1, 2) and (4, 5)
H.
Find the other end of the diameter of the circle x 2
I.
Obtain the parametric equation of the circle represented by x 2
J.
Find the values of a , b if ax 2
bxy
3y 2
5x
y2
2y
8x
8y
0 if one end of it is (2, 3)
27 y2
6x
8y
96
0
0 represents a circle. Also find the radius
3
and centre of the circle. m2 x 2
y2
K.
Find the centre and radius of each of the circles
L.
Obtain the parametric equation of each of the following circles. (i) x 2
M.
y2
2
82
lies on the circle x 2
y2
(ii) x
4
Show that A 3, 1
3
2
y
4
1
2x
2cx
2mcy
0
0 . Also find the other end of the
4y
diameter through A.
2. A.
Locate the position of the point (2, 4) with respect to the circle x 2
y2
4x
6y
B.
Find the length of the tangent from (1, 3) to the circle x 2
4y
11
0.
C.
Find the equation of the tangent to x 2
D.
If the parametric values of two points A and B lying on the circle x 2
y2
6x
y2
6x
4y
y2
12
2x
11
0.
4y
12
0 are
to
the
circle
0 at ( 1,1)
30 and 60 respectively then find the equation of the chord joining A and B . E.
Find x2
the y2
area
22x
4y
of
the
triangle
formed
by
the
normal
If 4, k and 2, 3 are conjugate points with respect to the circle x 2
G.
Find the power of the point P with respect to the circle S
5, 6 and S
3, 4
0 with the coordinate axes.
25
F.
(i) P
at
x2
y2
8x
12 y
y2
17 then find k .
0 when
15
H.
If the length of the tangent from (2, 5) to the circle x 2
I.
Find the length of the chord formed by x 2
J.
Find the value of k if the points 4,2 and k , 3 are conjugate points with respect to the circle x2
y2
5x
8y
6
y2
y2
5x
4y
a2 on the line x cos
k
0 is y sin
37 then find k . p.
0.
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Find the equation of the normal to the circle x 2
K.
y2
4x
6y
0 and 3, 2 . Also find the
11
other point where the normal meets the circles. L.
Find the equation of the pair of tangents from 10, 4 to the circle x 2
M.
Find the equation of the normal at P of the circles S (i) P
3, 4 ,
S
x
2
y
2
x
y
Fin the polar of 1,2 with respect to x 2
O.
Find the pole of ax
by
c
P.
Find the pole of 3x
4y
45
Q.
Find the value of k if the point x2
y2
25 .
0 where P and S are given by
24
N.
y2
7.
0 with respect to x 2
0 c
y2
0 with respect to x 2
and
1, 3
y2
y2 6x
r2 . 8y
5
0
are conjugate with respect to the circle
2, k
35 .
R.
Find the angle between the tangents drawn from 3, 2 to the circle x 2
S.
Find the number of possible common tangents that exist for the following pairs of circles y2
2x
4y
If the angle between the circles x 2
y2
12 x
x2
y2
6x
6y
0 , x2
14
4
y2
6x
4y
2
0.
0
3. A.
6y
0 and x 2
41
y2
kx
6y
59
0 is 45
find k . B.
Find k if the following pairs of circles are orthogonal (i) x 2
C.
E.
y2
0, x 2
k
y2
2 ax
8
12x
6y
0, x 2
41
y2
0
4x
6y
59
0
3 . 4 Find the equation of the common tangent of the following circles at their point of contact
Show that angle between the circles x 2
x2
F.
2by
Find the angle between the circles given by the equations x2
D.
y2
y2
10x
2y
0 , x2
22
y2
y2
2x
a2 , x 2
8y
8
y2
ax
ay is
0
Find the equation of the radical axis of the following circles (i)
x2
(ii) x 2 G.
y2
y2
2x
2x
4y
1
0 , x2
4y
1
0 , x2
y2
y2
4x
4x
6y
y
5
0
0
Find the equation of the common chord of the following pair of circles. (i) x 2 (ii)
y2
x
a
4x 2
4y
y
b
3 2
0 , x2
c2 ,
y2
x b
5x 2
y
6y
a
4 2
0
c2 a
b
4. A.
Find the equation of the parabola whose vertex is 3, 2 and focus is 3,1 .
B.
Find the coordinates of the points on the parabola y 2
C.
Find the vertex and focus of 4 y 2
12 x
20 y
67
2x whose focal distance is
5 . 2
0.
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D.
Find the equation of the parabola whose focus is S(1, 7) and vertex is A(1, 2) .
E.
Find the coordinates of the points on the parabola y 2
F.
If (1/2, 2) is one extremity of a focal chord of the parabola y 2
8x whose focal distance is 10. 8x . Find the coordinate of the
other extremity. k is a tangent to the parabola y 2
G.
Find the value of k if the line 2 y
H.
Find the equation of the normal to the parabola y 2
I.
Find the vertex and focus of x 2
J.
Find the equations of axis and directrix of the parabola y 2
K.
Find the equation of the parabola whose focus is S(3, 5) and vertex is A(1, 3) .
L.
Find the equation of the parabola whose latus rectum is the line segment joining the points ( 3, 2) and ( 3,1) .
M.
Find the equations of the tangent and normal to the parabola y 2
5x
6x
6y
6
6x .
4x which is parallel to y
2x
0.
5
0. 6y
2x
0.
5
6x at the positive end of the
latus rectum. N.
Find the equation of the tangent and normal to the parabola x 2
O.
Show that the line 2 x
y
2
4x
0 is a tangent to the parabola y 2
8y
12
0 at 4,
3 . 2
16 x . Find the point of contact
also. Find the equation of tangent to the parabola y 2
P.
16 x inclined at an angle 60 with its axis and
also find the point of contact.
5. 1 e2
1 e12
A.
If e, e1 are the eccentricities of a hyperbola and its conjugate hyperbola prove that
B.
Find the equations of the hyperbola whose foci are ( 5,0) , the transverse axis is of length 8.
C.
If 3x
D.
If the eccentricity of a hyperbola is
4y
k
0 is a tangent to x 2
4y 2
5 find the value of k .
E.
5 , then find the eccentricity of its conjugate hyperbola. 4 Find the equation to the hyperbola whose foci are (4, 2) and (8, 2) and eccentricity is 2.
F.
If the lx
G.
Find the equations of the tangents to the hyperbola 3x 2
my
(i) parallel and
1
1 is a normal to the hyperbola
x2 a2
y2 b
1 then, show that
2
(ii) perpendicular to the line y
x
4y 2
a2 l2
b2 m2
( a2
b2 )2
12 which are
7
H.
Define rectangular hyperbola. What is its eccentricity?
I.
Find the product of lengths of lengths of the perpendiculars from any point on the hyperbola
J.
x2 y 2 1 to its asymptotes. 16 9 Find the equation of the hyperbola whose asymptotes are 3x
K.
Find the equation of the normal at
L.
If the angle between the asymptotes is 300 then find its eccentricity.
/ 3 to the hyperbola 3x 2
5 y and the vertices are ( 5,0) .
4y 2
12 .
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6. 3
1 x
x
dx , x
A.
Evaluate
B.
Find
C.
Evaluate
x5 dx on 1 x 12
D.
Find
x 1 . e x2
E.
Evaluate
sin 2x dx on
1
1
( ax
F.
1 x
(x
bx )2
5) x
dx , ( a
4
sec2 x cos ec 2 x dx on I
H.
1 cos 2x dx on I 1 cosh x
1 and b
)
)
1) on
0, b
\ {n : n
[2n , (2n
dx on
sinh x
(0,
dx on ( 4,
0, a
G.
I.
.
dx on where I
dx
ax b x
0
}
(2 n
1) ], n
.
cot 2 x dx on
J.
Evaluate
K.
Evaluate
x6 1 dx for x 1 x2
L.
Evaluate
cos3 x sin x dx on
M.
Evaluate
sin 4 x dx , x 1 cos6 x
N.
Evaluate
sin 2 x dx on
O.
Evaluate
P.
Evaluate
Q.
Evaluate
elog(1
R.
Evaluate
sin 2 x dx on I 1 cos 2 x
\ (2n
1) : n
S.
Evaluate
1 dx on I cos x
\{(2n
1) : n
T.
Evaluate
dx 9x 2
4
1 1
1
4x 2
1) : n 2
\ {n : n
on I
}
.
R\
(2n
1) 2
:n
2 2 , 3 3
dx on
tan 2 x )
dx on I
x2 dx on I 1 x2
\
(2n
1) 2
:n
}
\0
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7. e Tan
A.
1
1x
dx on I
2
x
(0,
1
sin(Tan
B.
)
1
x
x)
2
dx , x
C.
2x3 dx on R 1 x8
D.
x8 dx on 1 x 18
E.
cos x dx on 0, x
F.
x2 x4
G.
sec x log(sec x
H.
sin 3 x dx on
I.
cos 3 x dx on
J.
cos x cos 3x dx on
dx
K.
ex
1
L.
2
, x
log sec x ex dx on
tan x
M.
tan x )dx on 0,
Evaluate
x ex
S.
x
1
dx on I
2
\
,n
N.
x log x dx on 0,
Q.
1 2
, 2n
4 4 , 5 5
25x 2 dx on
16
1 2
2n
1 dx on 1, x log x[log(log x )]
O.
1 dx on 1
1
dx
Evaluate
x
2
2x
P.
cos x cos 2 x dx on
R.
ex
T.
ex
1
10
x log x dx on 0, x
x
2
x
3
2
dx on I
\
3
1 dx 1 cot x
U.
sin 2x
V.
2
W.
2
dx on I
\ x
a cos x
b sin x
ex sec x
sec x tan x dx on I
|a cos2 x
\ 2n
1
2
b sin 2 x
0
:n
8. Evaluate the following definite integrals 2
A.
Evaluate
5 0 sin 2
5 2 cos
x
3
x
5 cos 2
B.
dx x
2 0
2 cos d
sin x dx sin x cos x
6
2
C.
Evaluate
D.
1
1 x dx 0
E. 0
dx 3 2x
1
F.
xe
x2
dx
0
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5
G. 1
3
dx 2x 1
H. 2
2k
Evaluate lim
K.
4k
6k nk
n
2
2x dx 1 x2
.....
2n
I.
4
sec 4 d
J.
2
0
x dx
0
k
by using the method of finding definite integral as the
1
limit of a sum.
9. 2
A.
Find
sin 4 x dx
0 2
B.
Find
cos8 x dx
0
C.
Find the area under the curve f x
sin x in 0, 2
D.
Find the area under the curve f x
cos x in 0, 2
E.
Find the area bounded by the curves y
.
sin x and y
cos x between any two consecutive
points of intersection. F.
Find the area enclosed within the curve x
G.
Evaluate the following definite integrals. (i)
y 2
1 sin 5 x cos 4 x dx
0
2
H.
Find
2
(ii)
sin 6 x cos 4 x dx
0
sin 7 x dx
0
a
I.
a2
Evaluate
x 2 dx
0
x 2 , the X-axis and the lines x
J.
Find the area bounded by the parabola y
K.
Find the area bounded between the curves y 2
L.
Evaluate
x2 , y
1, x
2
x
tan 5 x cos8 x dx
0
10. A.
B.
Find the order and degree of
Find the order and degree of
d3y dx 3
d2 y dx 2
2
3
dy dx
dy dx
2
ex
4
3 6/5
6y
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C.
Find the order of the differential equation corresponding to Aex
y D.
Be3x
Ce5x ( A, B, C being parameters ) is a solution
Form the differential equation corresponding to y
b sin 3x , where A and B are
A cos 3x
parameters. E.
Form the differential equation corresponding to the family of circles of radius r given by (x
a)2
(y
b)2
r 2 , where a and b are parameters.
F.
Solve
dy dx
G.
Solve
dy dx
H.
Find the order and degree of the differential equation
I.
Find the order of the differential equation corresponding to y
y tan x
sin x
x 2 logx 1 sin y
y cos y
d2 y dx
2
p2 y . c x
c
2
, where c is an arbitrary
constant. 1/3
d2 y
1/2
dy dx
J.
Find the order and degree of x
K.
Find the general solution of x
L.
Find the general solution of
M.
Solve y(1
N.
Form the differential equations of the following family of curves where parameters are given in brackets. aex
(i) xy O.
x)dx
x(1
x
be
y
dy dx
y)dy
dy dx ex
y
0
0 y
0
; a , b (ii) y
Find the general solution of
dx
x
2
bx e kx ; a , b (iii) y
a
1 x2 dy
1 y 2 dx
a cos nx
b ; a, b
0
SHORT ANSWER QUESTIONS
11. A.
From the point A 0, 3 on the circle x 2 a point M such that AM
4x
3)2
(y
0 a chord AB is drawn and extended to
2 AB . Find the equation of the locus of M .
If the abscissae of points A, B are the roots of the equation x 2
B.
B are roots of y C.
2 py
q
2
x
y
2
0 and ordinates of A ,
y2
6x
4y
12
0 and passing
2,14 .
If a point 2
b2
0 then find the equation of a circle for which AB is a diameter.
Find the equation of a circle which is concentric with x 2 through
D.
2
2 ax
2x
P 4y
is moving such that the length of tangents drawn from 20
0 and x
equation of the locus of P is x 2
2
y y2
2
2x
2x
8y
12 y
1 8
P
to
0 are in the ratio 2:1 then show that the 0.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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E.
If 4 x
3y
0 is a tangent to the circle represented by x 2
7
y2
6x
4y
0 the find its
12
point of contact. F.
Find
the
x2 G.
y2
22x
common
4y
tangents
of
the
x2
circles
y2
4x
6y
y2
22x
4y
100
0
and
28
0
and
0.
100
x2
Find the transverse common tangents of the circles x2
H.
direct
y2
4x
10 y
0.
4
If a point P is moving such that the lengths of tangents drawn from P to the circles x2
y2
4x
6y
0 and x 2
12
y2
6x
18y
0 are in the ratio 2 : 3 then find the
26
equation of the locus of P . Find the equations of the tangents to the circle x 2
I.
x
y
to 3x K.
4x
6y
12
0 which are parallel to
0.
8
Find the equations of the tangents to the circle x 2
J.
y2
y
y2
2x
2y
0 which are perpendicular
3
0.
4
Find the equation of circles passing through 1, 1 , touching the lines 4 x 3x
4y
3y
5
0 and
0.
10
0 touches the circle x 2
L.
Show that x
M.
Tangents are drawn to the circle x 2
y
1
y2
y2
3x
7y
0 and find its point of contact.
14
16 from the point P 3,5 . Find the area of the
triangle formed by these tangents and the chord of contact of P . Show that the circles x 2
N.
y2
6x
2y
0, x 2
1
y2
2x
8y
0 touch each other
13
Find the point of contact and the equation of common tangent at their point of contact. Show that x 2
O.
y2
6x
9y
0, x 2
13
y2
2x
0 touch each other. Find the point of
16y
contact and the equation of common tangent at their point of contact.
12. A.
Find S
the
x
2
y
equation
2
3x
5y
show that f ' g C.
1
2
2
a D.
y2
b
of x
2
the
y
2
common
chord
3y
4
y2
2g'x
2 fy
0 and x 2
y2
2 ax
c
of
the
two
circles
0.
5x
2 gx
2 f 'y
0 touch each other then
fg ' .
Show that the circles x 2 1
length
0 and S '
4
If the two circles x 2
B.
and
0 and x 2
y2
2by
c
0 touch each other if
1 . c
Show that the circles x 2
y2
0 and x 2
2x
y2
6x
6y
2
0 touch each other. Find the
coordinates of the point of contact. Is the point of contact external or internal? E.
Show that the circles x 2
y2
4x
6y
0 and 5 x 2
12
y2
8x
14y
32
0 touch each
other and find their point of contact. F.
Show x
2
y
that 2
14x
four 6y
common 33
tangents
0 and x
2
y
2
can 30x
be 2y
drawn 1
for
the
circles
given
by
0 and find the internal and external
centres of similitude. FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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13. A.
Find the eccentricity, coordinates of foci, Length of latus rectum and equations of directrices of the following ellipse. (i) 9x 2
B.
16 y 2
36x
32 y
92
(ii) 3x 2
0
y2
6x
2y
5
0
Find the equation of the ellipse referred to its major and minor axes as the coordinate axes X, Y-respectively with latus rectum of length 4 and distance between foci 4 2 .
C.
If
are the eccentric angles of the extremities of a focal chord (other than the vertices) of the
1, 2
ellipse
E.
1, a
b2 1
(i) e cos D.
y2
x2 a2
2
2
b and e its eccentricity. Then show that 1
cos
2
e e
(ii)
2
1 1
cot
1
2
Find the equation of the ellipse with focus at 1, 1 , e
cot
2
2
2 / 3 and directrix as x
Prove that the equation of the chord joining the points ' ' and ' ' on the ellipse
x cos a
2
y sin b
cos
2
y
x2 a2
0.
2
y2 b2
1 is
2
F.
S and T are the foci of an ellipse and B is one end of the minor axis. If STB is an equilateral triangle, then fund the eccentricity of the ellipse.
G.
Find the equation of the ellipse in the standard form such that distance between foci is 8 and distance between directrices is 32.
H.
Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equations of directrices of the following ellipse. (i) 9x 2
I.
16 y 2
144
(ii) x 2
2y2
4x
12 y
14
0
A man running on a race course notices that the sum of the distances of the two flag posts from him is always 10m. and the distance between the flag posts is 8m. Find the equation of the race course traced by the man.
14. A.
Find the equation of tangent and normal to the ellipse 9x 2
16 y 2
144 at the end of the latus
rectum in the first quadrant. B.
If the normal at one end of a latusrectum of the of the ellipse end of the minor axis, then show that e 4
C.
2y
4
(iii) which makes an angle
y2
4
b2
1 passes through one
8 which are
(ii) perpendicular to x
0
y2
1 [ e is the eccentricity of the ellipse)
Find the equation of the tangents to the ellipse 2x 2 (i) parallel to x
D.
e2
x2 a2
y
2
0
with x-axis
A circle of radius 4, is concentric with the ellipse 3x 2 is inclined to the major axis at an angle
4
13y 2
78 . Prove that a common tangent
.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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E.
Show that the locus of the feet of the perpendiculars drawn from foci to any tangent of the ellipse is the auxiliary circle.
F.
If a tangent to the ellipse
y2
x2 a2
b2
1 a
a2
respectively then prove that
CM
b meets its major axis and minor axis at M and N
b2 2
CN
1 Where C is the centre of the ellipse.
2
y2
x2 If PN is the ordinate of a point P on the ellipse 2 a
G.
1 and the tangent at P meets the
b2
a2 where C is the centre of the ellipse.
X-axis at T then show that CN CT H.
Show that the points of intersection of the perpendicular tangents to an ellipse lie on a circle.
I.
Find the equation of tangent and normal to the ellipse x 2
J.
Find the condition for the line x cos
K.
Show that the foot of the perpendicular drawn from the centre on any tangent to the ellipse lies on the curve x 2
y2
2
a2 x 2
8y 2
33 at
1,2 .
p to be a tangent to the ellipse ...
y sin
b2 y 2 .
15. A.
Find the centre, eccentricity, foci, directrices and the length of the latus rectum of the following hyperbolas. (i) 4x 2
B.
9y 2
32
(ii) 4( y
0
3)2
9( x
2)2
1
Prove that the point of intersection of two perpendicular tangents to the hyperbola x2 a2
C.
y2 b
D.
1 lies on the circle x 2
2
y2
a2
b2 .
Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the following hyperbolas (i) x 2
4y 2
(ii) 9x 2
4
16 y 2
72 x
32 y
16
0
Show that the angle between the two asymptotes of a hyperbola 2 sec
E.
8x
1
(ii) perpendicular to the line x
2y
4y 2
b
2
1 is 2 Tan
1
b or a
Find the equation of tangents drawn to the hyperbola 2x 2
G.
If the line b 2 m2
lx
my
n
4 which are (i) parallel
0.
F.
H.
y2
e .
Find the equations of the tangents to the hyperbola x 2
a2 l 2
x2 a2
3y 2
0 is a tangent to the hyperbola
6 through (–2, 1)
x2 a2
y2 b2
1 then, show that
n2
Prove that the product of the perpendicular distances from any point on a hyperbola to its asymptotes is constant.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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I.
y2
x2 a2
Tangents to the hyperbola
b2
1 make angles
1, 2
with transverse axis of a hyperbola.
Show that the point of intersection of these tangents lies on the curve 2 xy tan
J.
tan
1
k( x 2
a2 ) when
k.
2
Show that the locus of feet of the perpendiculars drawn from foci to any tangent of the x2 a2
hyperbola
y2
1 is the auxiliary circle of the hyperbola.
b2
16. 2 ...... 1 n
n n
1 n
A.
Evaluate lim 1
B.
Evaluate lim
C.
Find the area of one of the curvilinear triangles bounded by y
D.
Find
n
n
1
n
a
x 2 a2
2
.....
n
n
n n
n
3 2
x2
sin x , y
cos x and X-axis.
dx
a 2
E.
Find
sin 2 x cos 4 x dx
2
F.
Find the area enclosed by the curves y
G.
Evaluate lim
H.
Evaluate lim
n
n
1 n
1 1
1 tan n 4n n
I.
i 1
J.
Evaluate lim 1
K.
Evaluate
n
2
0
4
2
tan
.....
2 4n
6x
x2
1 6n
......
tan
n 4n
i3
Evaluate lim n
n
3x and y
i4
n4
1 1 n2
22 ...... 1 n2
n2 n2
1 n
dx 5 cos x
1
L.
xTan 1 x dx
Evaluate 0
M.
Find the area of the region bounced by the parabolas y 2
4x and x 2
4y
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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17. Solve the following differential equations: A.
dy dx
ex
C.
dy dx
x tan( y
E.
(cos x )
G.
dy dx
y2
y
1
x2
x
1
I.
dy dx
4x y 1 x2
K.
x
M.
xdy
O.
dy dx
Q.
sin
S.
1
ex / y dx
U.
dy dx
sin( x
y
dy dx
2y3
x)
1
y sin x
dy dx
xy xy
tan x
0 1 1
x
2 2
y
x cos2
y
y dx x
y x
dy dx
1
y
x2 e
x
y
ex / y 1 y)
cos( x
x dy y
0
y)
B.
dy dx
x(2 log x 1) sin y y cos y
D.
(x2
y 2 )dx
F.
1
x2
dy dx
2 xy
H.
1
x2
dy dx
y
dy dx
J.
cos x.
L.
dy 2 3 x y dx
N.
x2
P.
xy 2
R.
dy dx
T.
x
V.
dy dx
y2
4x 2
xy dy dx
tan 2 x 1
dy dx
0
1x
eTan
sec2 x
y sin x
x dx
y
2 xydy
1 xy
yx 2
y dy
0
y 1
x 2y 1 2x 4y
LONG ANSWER QUESTIONS
18. A.
Find the equation of a circle which passes through 2, 3 and 4x
3y
1
4,5 and having the centre on
0.
B.
Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on 4 x 3y 24 0 .
C.
Find the equation of the circle whose centre lies on the X-axis and passing through
2,3 and
4,5 . D.
Find the equation of circle passing through the points 1,2 , 3, 4 , 5, 6
E.
Find the equation of the circle passing through (0, 0) and making intercepts 4, 3 on X-axis and Yaxis is respectively.
F.
Find the equation of the circle passing through (0, 0) and making intercept 6 units on X-axis and intercept 4 units on Y-axis.
G.
If 2,0 , 0,1 4,5 and 0,c are concylclic then find c .
H.
Find the equations of circles which touch 2 x
3y
1
0 at (1,1) and having radius
13 .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Show that the circles x 2
I.
y2
4x
6y
0 and x 2
12
y2
6x
18y
0 touch each other.
26
Also find the point of contact and common tangent at this point of contact. J.
Show that the tangent at x2
K.
y2
4x
If the polar of the points on the circle x 2
If
M.
1, 2
y2
4x
0 touches the circle
7
a2 with respect to the circle x 2
y2
y2
are the angles of inclination of tangents through a point P to the circle x 2 1
Prove
at
that
y2
the
2x
tangent
10 y
cot
b 2 touches
y2
a2 then
k.
2
of
3, 2
the
x2
circle
y2
touches
13
the
circle
0 and find its point of contact.
26
Find the equation of the circle which touches the circle x 2
N.
8y
c 2 then prove that a , b , c are in Geometrical progression.
find the locus of P when cot x2
y2
0 and also find its point of contact.
6y
the circle x 2 L.
of the circle x 2
1,2
y2
2x
4y
20
0 externally at
y2
2x
4y
11
0 and also find
5,5 with radius 5. Find the pair of tangents drawn from 1,3 to the circle x 2
O.
the angle between them.
19. A.
Find the equation of the circle which passes through (1, 1) and cuts orthogonally each of the circles x 2
B.
y2
8x
2y
0 and x 2
16
y2
4x
4y
1
0.
Find the equation of the circle which intersects the circle x 2
y2
6x
4y
3
0 orthogonally
and passes through the point (3, 0) and touches Y-axis. C.
Find the equations to all possible common tangents of the circles x2
D.
E.
2x
6y
0 and x 2
6
(i) x 2
y2
9 and x 2
(ii) x 2
y2
4x
y2
2y
16x
H.
1
2y
0 and x 2
4
49
y2
0
4x
2y
4
0
Find the equation of the circle which is orthogonal to each of the following three circles y2
2x
17 y
0 , x2
4
y2
7x
6y
11
0 and x 2
y2
x
22 y
3
0.
Find the equation of the circle passing through the origin, having its centre on the line x and intersecting the circle x 2
G.
y2
Find all common tangents of the following pairs of circles.
x2
F.
y2
y2
4x
2y
4
y2
10x
2x
3y
7.
If x
y
4y
21
0
4
0 orthogonally.
Find the equation of the circle which cuts the circles x2
y
x2
y2
4x
6y
11
0
and
orthogonally and has the diameter along the straight line
3 is the equation of the chord AB of the circle x 2
y2
2x
4y
0 , find the
8
equation of the circle having AB as diameter. I.
Show x2
y2
that 8x
the 6y
common 23
chord
of
the
circles
x2
y2
6x
4y
9
0
and
0 is the diameter of the second circle and also find its length.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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20. A.
B.
Show
that
the
xa1/3
yb1/3
common
a2 /3b2 /3
tangent
to
the
y2
parabola
C.
0 is
4 ax a
is
4by
0
Prove that area of the triangle formed by the tangents at x1 , y1 , x2 , y2 parabola y 2
x2
and
4 ax
1 y1 16 a
y2 y2
y3 y3
y1
and x3 , y 3
to the
sq. units.
Find the equation of the parabola whose axis is parallel to x -axis and which passes through the points ( 2,1) , (1, 2) and ( 1, 3) .
D.
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) .
E.
Prove
that
the
1 ( y1 8a
y 2 )( y 2
area
G.
H.
t12
in
the
parabola
y2
4 ax is
16 x which are parallel and perpendicular
0 , also find the coordinates of their points of contact.
5
y2
2 a2 and the parabola y 2
8ax
2a .
x
2
4 ax meets the parabola again in the point t 2 . Then prove that
0.
If a normal chord a point ' t ' on the parabola y 2 prove that t
4 ax subtends a right angle at vertex, then
2.
Find the coordinates of the vertex and focus, and the equations of the directrix and axes of the following parabolas (i) y 2
(ii) x 2
16 x
4y
3x 2
(iii)
9x
5y
2
0
(iv)
From an external point P, tangent are drawn to the parabola y 2 angles
1, 2
line y
bx .
with its axis, such that tan
Prove that the two parabolas y 2 of Tan
M.
y
The normal at a point t1 on y 2
t1t2
L.
inscribed
Show that the equation of common tangent to the circle x 2 are y
K.
triangle
y1 ) sq. units where y1 , y2 , y3 are the ordinates of its vertices.
y 3 )( y 3
respectively to the line 2 x
J.
the
Find the equations of tangents to the parabola y 2
F.
I.
of
3 a1 /3 b 1 /3
1
2 a 2 /3
b 2 /3
tan
1
4 ax and x 2
2
y2
x
4y
5
0
4 ax and these tangents make
is a constant b. Then show that P lies on the
4by intersect (other than the origin ) at an angle
.
Find the equation of the parabola whose focus is ( 2, 3) and directrix is the line 2 x
3y
4
0.
Also find the length of the latus rectum and the equation of the axis of the parabola. N.
Find the coordinates of the vertex and focus, the equation of the directrix and axis of the following parabolas. (i) y 2
4x
4y
3
0
(ii) x 2
2x
4y
3
0
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Show that the locus of point of intersection of perpendicular tangents to the parabola y 2
O.
the directrix x
0.
a
Show that the common tangents to the circle 2x 2
P.
4 ax is
at the focus of the parabola y 2
a2 and the parabola y 2
2y 2
4 ax intersect
4ax .
21. Evaluate A.
(3x
C.
6x
E.
x
2) 2 x 2 5
6
2x
3
3 x
2
x
1 dx
B.
2x2
x dx
D.
dx
4
cos x 3sin x 7 dx cos x sin x 1
H.
I.
9 cos x sin x dx 4sin x 5cos x
J.
K.
2 sin x 3sin x
L.
M.
dx on 1 x4
N.
O.
dx 5sin x
P.
dx 4 cos x 3sin x
R.
4
Q.
x
S.
x
1 x2
2x2
U.
x
3
x
3 x
1
1 2
sin x cos x cos x 3 cos x
W.
2
2
4 dx 5
dx
T.
dx
V.
2
x2
dx
5 2x
10
dx
dx 1
x
3
x2
2x
on
dx
F.
G.
3cos x 4 cos x
2x
x
1
2x
2 cos x 4 cos x
3x
cos x
1 x
a x x x
b x c 1
2
1
on I
\
1,
1 2
3sin x dx 5sin x
1 sin x
1
2
1,3
x
1
dx
dx
dx
1 dx 3cos 2 x
2
dx 1
x
3
2x
x2
on
3,1
x3 2x 3 dx x2 x 2 dx x
3
1
dx 3 cos x 4 sin x
X.
6
22. A.
If I n n
B.
cosn x dx , then show that I n
2 and deduce the value of
Obtain reduction formula for I n value of
1 cosn n
1
x sin x
n 1 In n
2,
n being a positive integer,
cos 4 xdx . cot n xdx , n being a positive integer, n
2 and deduce the
cot 4 xdx .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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C.
sin m x cosn x dx , then show that
If Im ,n
sin m
I m ,n
1
x cosn m n
1
x
m 1 Im m n
E.
Obtain reduction formula for
F.
Obtain reduction formula for
2,n ,
for a positive integer n and an integer m
sin n x for an integer n
tan n x I n
2 and deduce the value of
2. sin 4 x dx .
cot n xdx , n being a positive integer, n
2 and
tan 6 x dx .
deduce the value of G.
2 and
cosec5 x dx .
deduce the value of D.
cosecn x dx , n being a positive integer, n
Obtain the reduction formula for In
sec n x for an integer n
Obtain reduction formula for
2 and deduce the value of
sec 5 x dx .
23. A.
Show that the area enclosed between the curves y 2
B.
Show that the area of the region bounced by the circle x 2
y2
x2 a2
Find the area bounded between the curves y 2
D.
Find the area bounded between the curves 4x
y2
3 and y 2
20 5
x is 64
5 . 3
1 (ellipse) is ab . Also deduce the area of
b2
a2 .
C.
(i) y
12 x
x 2 and y
5
(ii) y 2
2x
4 ax , x 2
4x , y 2
4 4
4by a
0, b
0
x
Evaluate E. 0
4
x sin x dx 1 sin x
F. 0
4
H.
log 1
tan x dx
I.
0
0
b
K.
x a
a b
x dx
L. 0
sin x cos x dx 9 16 sin 2 x
x sin x dx 1 cos2 x
1
G. 0
2
J.
log 1 1
x x
2
dx
cos x dx 1 ex
2
x sin 3 x dx 1 cos2 x
2
M. 0
x sin x
cos x
dx
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
2
N. 0
sin 2 x dx cos x sin x
The circle x 2
O.
y2
8 is divided into two parts by parabola 2y
x 2 . Find the area of both the
parts. P.
Let AOB be the positive quadrant of the ellipse
y2
x2 a2
1 with OA
b2
a , OB
b . Then show
2 ab . 4
that the area bounded between the chord AB and the arc AB of the ellipse is
24. A.
Find the equation of a curve whose gradients is passes through the point 1,
4
Solve
dy dx
y2
y
1
2
x
1
D.
Solve
dy dx
3y 3x
F.
Solve
dy dx
H.
Solve x sec
I.
Give the solution of x sin 2
J.
Find the solution of the equation x( x
C.
Solve 2 x
7 3
E.
Solve
dy dx
x sin 2 y
x 3 cos 2 y
G.
Solve
dy dx
y .( ydx x
xdy )
x
7x 7y
9 when x
K.
Solve y 2
L.
Solve: y dx
M.
Solve 2 x
N.
Solve 1
x2
O.
Solve x
y dy
P.
Solve x 2
0
dy dx
y 2 dx
yco sec y dx x
y .( xdy x
ydx
2xy
x 2 dy
y x
x dy
x
y
x dy x cos 3
cos2
y , where x x
0, y
0 and which
y
1 dx
y tan x
x2
4x
2y
1 dy
sin x
y2
2x2
ydx )
xdy which passes through the point 1,
2)
0
dy dx
2( x
1)y
x3 (x
4
2) which satisfies the condition
3.
2xy dx
2y
y x
.
B.
that y
dy dx
dy dx
0 y dx y sin
y x
1
y
Tan 1 x
x
y
1 dx
2xy dy
wish you all the best FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
Regd. Off.: 29A, ICES House, Kalu Sarai, Sarvapriya Vihar, New Delhi - 110 016. Ph: 011 - 2651 5949, 2656 9493, Fax: 2651 3942
