2ndyear20yrs

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Compiled by: Faizan Ahmed math.pgseducation.com
24 YEARS
PAST
PAPERS
IN ACCORDANCE
WITH
THE CHAPTER
XII-Mathematics
FROM THE DESK OF: FAIZAN AHMED
SUBJECT SPECIALIST
SKYPE NAME: ncrfaizan

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CHAPTER # 01CHAPTER # 01CHAPTER # 01CHAPTER # 01
FUNCTIONS AND LIMITS
1992
Q. (a) (i) f : R IR is given by: =

−
(Q being the set of rational)
(1) Find f (ππππ)
(2) Find the range of ‘f’.
(3) Give reason why ‘f’ is not ‘ONTO’.
(4) Give reason why ‘f’ is not ‘ONE-TO-ONE’.
1993
Q. A function f from R to R is given by: =

| |
, ≠≠≠≠ є
, =
Find the graph of f and also draw its sketch in R2
.
1994
Q. Define even and odd functions and show that

is an odd function of x.
1995
Q. Find poq, qop and pq where p is defined by p (x) = x2
+ 1
∀ x є R and q is the cosine function.
1996
Q. A function of: is defined by =
– , ∀∀∀∀ − ∞∞∞∞,
+ , ∀∀∀∀ " , #$
, ∀∀∀∀ #, + ∞∞∞∞
Find (i) the image of zero, (ii) the value of f at 3,
(iii) f (√&), (iv) f (l) (v) the image of 5.
1997
Q. Define even and odd functions and show that

is an odd function of x.
1998
Q. If f : R R is given by: =

−
, (Q being the set of rational)
(i) Find f (√') (ii) Find the range of ‘f’. (iii) Give reason why ‘f’ is not ‘ONTO’.
(iv) Give reason why ‘f’ is not ‘ONE-TO-ONE’.
1999
Q. A function h(x) from R to R is given by: =

| |
, ≠≠≠≠ є
, =
Find the graph of h(x) and draw its sketch.
2000

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Q. (i) Define composite function.
(ii) If f(x) = tan (x+2) and g(x) = x2
+ 1, ∀∀∀∀x (, find the composite
functions, fog and gof.
2001
Q. If f: [-1, 5] R is given by f(x) = x2
for all x є [-1, 5], find f(2), f(− #
), image of zero and image of
5. Can you find the value of -2?
Does there exist a real number x such that f(x) = -1?
2002
Q. Define even and off functions and find whether

is an even
or an odd function of ‘x’.
2003
Q. f : R is given by:
f (x) = 0 when x є Q (Q being the set of rational numbers)
1 when x є R – Q
(i) Find f (√') (ii) Find the range of f.
(iii) Why is f not ONTO? (iv) Why is f not ONE-TO-ONE?
2004
Q. Define composite function. If f(x) = tan (x + 2) and g(x) = x2
+ 1 ∀∀∀∀ x є IR, find the
composite functions fog and gof.
OR
Define Even and Odd functions. Find whether
f(x) =

is even or odd function of x.
2005
Q. A function f : R R is given by =
,
, −
(Q being the set of rational numbers).
Find the following.
(i) f(ππππ) (ii) f(
##
)
) (iii) The range of the function
(iv) Why is f not ONTO?
2006
Q. A function f : N N is defined by f(x) = x + 1 (N being the set of all natural numbers). Then:
i. Find f(7) and f(11).
ii. State whether f(-3) can be found or not. If not, why not?
iii. State whether f is 1 -1 or not.
iv. Why is f ont onto?
2007
Q. A function ƒ : R R is Given By: =
− ,
, −
Find the following: (i) f *
##
)
+ (ii) f (ππππ) (iii) f,√#- (iv) The range of the function
2008
Q. If f : R R is given by: =
,
, −
, (Q being the set of rationals)
Find (i) f,√#- (ii) the range of f (x) (iii) f *'
+ (iv) f(2)
2009

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Q. Define even & odd functions. Find whether the following function is even, odd or neither:
( ) sin tanf x x x= −= −= −= −
2010
Mcqs: f(x) = sinx+cosx is a/an:
(a) Even function (b) odd function (c) neither even nor odd (d) modulus function
2011
Mcqs: A function f(x) = | |
, ≠ /0
(a) Even function (b) odd function (c)circular function (d) neither even nor odd
2012
Mcqs: (xv) f(x) = sinx+cosx is:
(a) Even function (b) odd function (c) neither even nor odd (d) modulus function
Q. Two polynomial functions f and g are defined by f(x) = x2
-3x+4 and g(x) = x + 1 ∀∀∀∀ x є R, Find fog
and gof and show that 12 ≠ 21 .
2013
Mcqs: (xiv) A function f(x) is said to be odd whenever:
(a) f(x)=0 (b) f(-x)=f(x) (c) f(-x)=-f(x) (d) f(-x)=1
2014
None
FUNCTIONS PORTION
1992
Q. A sequence is given by:
.&
#.4
,
&.'
4.5
,
'.)
5.6
, . . .
Where ‘ . ’ represents ordinary multiplication. Write down the General Term of the sequence
and find it limit.
1993
.#
&.4
,
&.4
'.5
,
'.5
).6
,
).6
7.
. . .
Where ‘ . ’ represents ordinary multiplication. Write down the General Term of the sequence
and find it limit.
1994
Q. Find the limit of the sequence:
.&
#.4
,
&.'
4.5
,
'.)
5.6
, . . .
1995
Q. Prove that 89: →∞ * + + =
1996
Q. Show that if m is an integer: 89: →∞ * + +
=
= =
1997
Q. In sequence,
.&
'.)
,
'.)
7.
,
7.
&. '
, . . . where ‘. ’ represents ordinary multiplication. Write down the
general term of the sequence and find its limit.
1998
Q. Discuss the Convergence OR Divergence of the following series:

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−
#
&
+
4
7
−
6
#)
+ …
1999
Q. Discuss convergence or divergence of the series: + #
+ &
+. . .
2000
.#
&.4
,
&.4
'.5
,
'.5
).6
, . . . where dot represents multiplication.
Write down the general term of the given sequence, also find the limit.
2001
Q. A sequence ?@ A is defined by @ = , @ = B + @ , ∀ n C. Show that the sequence is
monotonic increasing and bounded and further more is D/= @ = D then D#
− D − = .
2002
Q. A sequence is given by
.&
#.4
,
&.'
4.5
,
'.)
5.6
, . . . where (.) represents the ordinary multiplication. Write
down the general terms of sequence and find its limit.
2003
Q. Discuss whether the series
'
+
'# +
'& + . . . is convergent or divergent.
2004
Q. A sequence is given by
&
#
,
#
&
,
'
4
,
4
'
, . . . write down the general term of the given sequence. Also
find the limit.
2005
Q. Discuss the Convergence OR Divergence of the following series: −
#
&
+
4
7
−
6
#)
+ …
2006
.&
#.4
,
&.'
4.5
,
'.)
5.6
, . . .
2007
Q. write down the general term limit of the sequence:
&
#
,
#
&
,
'
4
,
4
'
, . . .
2008
Q. Find the nth term and the limit of the sequence:
.#
&.4
,
&.4
'.5
,
'.5
).6
, . . .
where ‘.’ represents multiplication.
2009
Q. Write the nth term of the sequence:
.#
&.4
,
&.4
'.5
,
'.5
).6
, . . . and calculate its limit.
2010
2011
Mcqs: 89: →∞ * + + =
(a) 0 (b) ∞ (c) e (d) 1
2012
None
2013
Mcqs: (xvi) 89: →∞ * + + =:
(a) (b) (c) (d) –

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Q. Find the nth term and limit of the sequence:
.&
#.4
,
&.'
4.5
,
'.)
5.6
, . . ., where ‘.’ Represents
multiplication.
2014
.#
&.4
,
&.4
'.5
,
'.5
).6
,
).6
7.
. . .
LIMIT OF FUNCTION PORTION
1992
Q. Determine any Two of the following limits:
(i) 89:Ө→G
#
H1IӨ JKLӨ
H10&Ө
(ii) 89: →
4 # M #
(iii) 89: →+∞ *
+
1993
Q. Determine any Two of the following limits.
(i) 89: →
H10'
H10)
(ii) 89: →

(iii) 89: →+∞
# # #
# 4
1994
Q. Evaluate any two of the following.
(i) 89: →+∞ *
+ (ii) 89: → *
#
#
−
+ (iii) 89:Ө→
G
#
H1IӨ JKLӨ
H10&Ө
1995
Q. Evaluate any Two of the following.
(i) 89: →&
# ' 5
# 6 '
(ii) 89: →+∞ *

+ (iii) 89:∆ →
L9 ∆ 0/
∆
1996
Q. Evaluate any TWO of the following.
(i) 89: →@
= @=
@
(ii) 89: →
I@ 0/

(iii) 89: →+∞
8 #
1997
(i) 89:O→
H10 HP H1IP
P
(ii) 89: →# Q
# & #
# 5 5
(iii) 89:R→∞
8 MS
S
1998
(i) 89: →∞
8 M
(ii) 89: →
4 # M #
(iii) 89: →
√
&√
1999
(i) 89:R→∞
8 MS
S
(ii) 89: → *
#
#
−

+ (iii) 89:αααα→
I@ αααα 0/ αααα
0/ &αααα
2000
Q. Evaluate any Two of the following limits.
(i) 89:∆ →
JKL ∆ H10
∆
(ii) 89: → *

−
&
&
+(iii) 89: →∞
8 M
2001
(i) 89:ΨΨΨΨ→
I@ ΨΨΨΨ 0/ ΨΨΨΨ
0/ &ΨΨΨΨ
(ii) 89: →
√ & #

(iii) 89: →−
B #

2002
Q. Determine any Two of the following limits.
(i) 89:T→
H1IT H10T
H10&T
(ii) 89: →
7 M 6
(iii) 89: →∞
8 M

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2003
(i) 89: →
H10U
H10V
(ii) 89:W→∞
B#I# I
&I '
(iii) 89:φφφφ→
I@ φφφφ 0/ φφφφ
φφφφ&
2004
Q. Evaluate any two of the following limits.
(i) 89: →
0/ 4
(ii) 89: → *
−
&
&
+ (iii) 89: →#
= #=
#
OR 89: →
=
m, n є ℜℜℜℜ
2005
(i) 89:∆ →
√ ∆ √
∆
(ii) 89: → *
#
#
−

0/ &αααα
2006
Q. Evaluate any two of the following:
(i) 89: →@
= @=
@
(ii) 89: → *

−
&
&
0/ &αααα
2007
(i) 89: →
√ – √
(ii) 89: →∞
8 M
(iii)
89: →
L9 ' L9 &
2008
Q. Evaluate any Two of the following:
(i) 89:αααα→
0/ &αααα
(ii) 89: →&
7 #
4 B # )
(iii) 89: →
=
m, n є ℜℜℜℜ
2009
(i)
0
lim
x
x a a
x→→→→
+ −+ −+ −+ −
(ii)
0
9 8
lim
e eθ θθ θθ θθ θ
θθθθ θθθθ
−−−−
→→→→
− −− −− −− −
(iii) 20
1 cos
lim
x
x
x→→→→
−−−−
(iv)
2 1
lim
x
x x→−∞→−∞→−∞→−∞
−−−−
2010
Mcqs: (xii) 89: →
0/ &
=
(a)
)
&
(b) ) (c)
&
)
(d) &
(xiii) 89: →#
# 4
#
=
(a) 6 (b) 4 (c) 1I X / X (d) 0
Q. Evaluate 30
tan sin
lim
sinx
x x
x→
−
2011
Mcqs: (xii) 89: →@
@
@
=
(a) 1 (b) @ (c) n (d) 0
(i) 89:αααα→
H10
# (ii) 89: → *

−
&
&
+(iii) 89: →
# Y
2012

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Mcqs: (xii) 89: →
0/
4
'
=
(a)
'
4
(b)
4
'
(c)
4
(d)
'
(a) 89: →#
# ' #
' # 5 4
(b) 89: →
& M #

(c) (i) 89:T→
JKL JT JKWT
T
2013
Mcqs: (i) 89: →4
# 5
4
=
(a) 4 (b) 6 (c) (d) ∞
(a) 89:T→G
#
JKWT JKLT
JKL&T
(b) 89: →
=
m, n є ℜℜℜℜ (c) 89: →

√ H10
OR
89: → * −
&
&+
2014
(i) 89: →@
= @=
@
(ii) 89: →
B # 5 4

(iii) 89:Z→
I@ 0/

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THE STRAIGHT LINE
1992
Q. The points L(3,3) , M(4,5), and N(2,4) are the mid-points of the sides of a triangle. Find its
vertices.
Q. Find the equation of the line which passes through the point (1, -5) and has the sum of its
intercepts equal to 5.
Q. Find the equation of the straight line which passes through the point (3, -4) and is such that the
portion of it between the axes is divided by the point in the ratio 2:3.
1993
Q. The vertices A, B, C of a triangle are (2, 1), (5, 2) and (3, 4) respectively. Find the coordinates of
the circum-centre and also the radius of the circum-circle of the triangle.
Q. The line segment joining P(-8, 10) and Q(6, -4) is cut by x and y-axes at A and B respectively; find
the ratio in which A and B divide PQ.
1994
Q. Find the coordinates of the in-centre of the triangle whose angular points are respectively (-36,
7) , (20, 7) and (0, -8).
1995
Q. The centroid of a triangle whose two vertices are (2, 4) and (3, -4) is found to be (3, 1); find the
third vertex.
Q. The line through (6, -4) and (-3, 2) is perpendicular to the line through (2, 1) and (0,y); find y.
1996
Q. Prove that if the diagonals of a parallelogram are perpendicular the figure is rhombus.
Q. If the points (a, b), (a`, b`) (a-a`, b-b`) are collinear, show that their join passes through the
origin and that ab` = a`b.
1997
Q. The points (3, 3), (5, y) and (-4, -6) are the three consecutive vertices of a rectangle. Find y and
its fourth vertex.
Q. Determine the equation of the line which passes through the points (-2, -4) and has the sum of
its intercepts equal to 3.
1998
Q. The straight line joining the points (1, -2), (-3, 4) is trisected, find the coordinates of the points
of trisection.
Q. Find the angles of the triangle whose vertices are A (-2, 1), B (4, -3) and C (6, 4).
1999
Q. The vertices A, B, and C of a triangle are (2, 1), (5, 2) and (3, 4) respectively; find the coordinates
of the circum-centre and also the radius of the circum-circle of the triangle.
2000
Q. For the triangle with vertices A (5, 1) , B(3, -5) and C(-3, 7). Find the equation of attitude from B.
2001
Q. Prove that the points whose co-ordinates are respectively (5, 1) , (1, -1) and (11, 4) lie on a
straight line. Find the intercepts made by this line on the axes.
Q. Prove that the diagonals of an isosceles trapezoid are equal.

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2002
Q. Determine the equation of the line which passes through the point (-4, -5) and has the sum of
its intercepts equal to ‘3’.
Q. Find the angles of the triangle whose vertices are A(-2, 1), B(4, -3) and C(6,4).
Q. Find the equation of the locus of a point whose distance from the point (2, -2) is equal to its
distance from the line x – y = 0.
2003
Q. Find the equation of a straight line passing through the point (a, b) such that the portion of the
straight line between the axes is bisected at the point.
2004
Q. The line through (6, -4) and (-3, 2) is perpendicular to the line through (2, 1) and (0, y); find y.
Also find the equations of both the lines.
Q. If A (2, 1), B(5, 2) and C(3, 4) are the vertices of the tri-angle, find the coordinates of the circum-
centre and the radius of the circum-circle of the triangle.
Q. The x-intercept of a line is the reciprocal of its y-intercept and passes through the point (2, -1);
find the equation of the lines.
2005
Q. Show that the line segment joining the mid-points of any two sides of a triangle is parallel to
the third side and equal to one-half of its length.
2006
Q. In what ratio does the point M(2,4) divide the join of L(7,9) and N(-1,1)?
Q. If the points (a,b), (a’ – a’.b – b’) are collinear, show that their join passes through
the origin and that ab’ = a’b.
2007
2008
Q. The vertices A,B and C of a triangle are (2,1), (5,2) and (3,4) respectively find the coordinates of
the circum – center and also the radius of the circum-circle of the triangle.
Q. Find the equation of the perpendicular bisector of the line segment joining the points A (15,14)
and B (-3, -4).
2009
Q. Prove that the diagonals of an isosceles trapezoid are equal.
Q. Find the equation of the line which passes through the point (–2, –4) and has the sum of its
2010
Q. If the line through (2, 5) and (–3, –2) is perpendicular to the line through (4, –1) and (x, 3),
find x.
Q. Find the equation of the line which passes through the point (–3, –4) and has the sum of its
Q. Find the equation of the locus of a moving point such that the slop of the line joining the point
to A(1, 3) is three times that of the slope of the line joining the point to B(3, 1).
2011
Mcqs: (ii) If a straight line is parallel to y-axis then its slope is:
(i) 1 (ii) 0 (iii) -1 (iv) ∞
(xvii) If a line is parallel to x-axis its equation is:

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(i) x=0 (ii) y=0 (iii) x=constant (iv) y=constant
Q. A in two-thirds the way from (1,10) to (-8,4) and B is the mid-point of (0,-7) and (6,-11). Find the
distance |[]]]]|.
Q. Find the equations of the straight line which passes through the point (3,4) and makes
intercepts on the axes such that the y-intercept is twice its x-intercept.
2012
Mcqs: (vi) Distance of the point (4,5) from the y-axis is:
(a) 5 units (b) 4 units (c) 9 units (d) 1 unit
(xix) The line 4x+5y+2=0 is perpendicular to the line:
(a) 5x+4y-2=0 (b) 5x-4y+3=0 (c) 4x+5y-2=0 (d) -5x-4y+2=0
Q. A straight line passes through the points A(-12,-13) and B(-2,-5). Find the point on the line
whose ordinate is -1.
Q. The vertices A,B and C of a triangle are (2,1), (5,2) and (3,4) respectively. Find the coordinates of
the circum-centre and radius of the circum-circle of the triangle ABC.
Q. Find the equation of a line which passes through the point (-1,2) and has sum of its intercepts
equal to 2.
2013
Mcqs: (iii) 3x-4y-15=0 is parallel to the line:
(a) 5x-3y-15=0(b) x-y+15=0 (c) 3x+y-15=0 (d) 6x-10y+15=0
(vii) Slope of Y-axis is:
(a) 0 (b) 1 (c) -1 (d) ∞
(xv) Point of concurrency of the medians of a triangle is called:
(a) In-centre (b) ortho-centre (c) centroid (d) circum-centre
Q. The line through (2,5) and (-3, -2) is perpendicular to the line through (4,-1) and (x,3); find .
Q. Find the value of k when the vertices of the triangle are the points (2,6), (6,3) and (4,k) and its
area is 15 Sq. units.
2014
Q. If the line through (2, 5) and (–3, –2) is perpendicular to the line through (4, –1) and (x, 3),
find x.

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THE GENERAL EQUATIONS OF
STRAIGHT LINES
1992
Q. Find the equation of the locus of a point whose distance from the point (2, -2) is equal to its
distance from the line − P = .
1993
Q. Find the combined equation of the pair of lines through the origin which are perpendicular to
the lines represented by 6x2
– 13xy + 6y2
= 0.
Q. The sides of a triangle are 4x+3y+7=0, 5x+12y+20=o, and 3x+4y+8=0. Find the equations of the
internal bisectors of the angles and show that they are concurrent.
1994
Q. Find the equation of a line parallel to x – axis and passing through the point of intersection 3x –
2y – 1 = 0 , and 2x + y + 1 = 0.
Q. Find the equation of the line perpendicular to x + y + 5 = 0, passing through the point of
intersection of x-2y+2=0 and 2x+y-1=0.
Q. Show that the equation 3x2
+ 7yx + 2y2
= 0, represents two distinct straight lines. Also find the
angle between them.
1995
Q. Find the equations of two straight lines passing through (3, -2) and inclined at 60o
to the line
√&x + y = 1.
Q. If ∆ denotes the area of a triangle and the coordinates of the points A,B,C,D are (6, 3), (-3, 5),
(4, -2) and (x, 3x) and
∆^_`
∆a_`
=
#
; find x.
1996
Q. A line whose y – intercept is 1 less than its x = intercept, forms with coordinate axes a triangle
of area 6 square units. What is its equation.
Q. Show that the line x2
– 4xy + y2
= 0 and x + y = 3 form an equilateral triangle.
1997
Q. D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle ABC. Prove that
∆ ABC = 4 ∆ DEF.
Q. Find the controid of the triangle, the equations of whose sides are 12x2
– 20xy + 7y2
= 0 and 2x –
3y + 4 = 0.
1998
Q. Find the equations of the straight lines through the intersection, of the lines 5x – 6y – 1=0,
3x + 2y + 5 = 0 and making an angles of 45o
with the line 5y – 3x = 11.
Q. Given that 3x–2y–5 = 0, 2x+3y+7=0 are the equations of two sides of a rectangle, and that
(-2, 1) is one of the vertices; calculate the area of the rectangle.
1999
Q. Find the equation of the line perpendicular to the line x – y + 5 = 0 and passing through the
intersection of the lines x – 2y + 7 = 0 and 2x + y – 1 = 0.
Q. The point (2, -5) is the vertex of a square, one of a square, one of whose sides lies on the line x –
2y – 7 = 0 calculate the area of the square.

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Q. Show that the equation of the line through the origin, making an angle of measure ∅∅∅∅ with the
line y = mx + b is
P
=
= I@ ∅∅∅∅
=I@ ∅∅∅∅
2000
Q. The point P (3, 2) is the foot of the perpendicular dropped from the origin to a straight line.
Write the equation of this line.
Q. A straight line forms a right triangle with the axes of coordinates. If the hypotenuse is 13 units
in length and the area of the triangle is 30 square units; find the equation of the straight line.
Q. What does equation x2
– y2
= 0 represent? Explain it, and if it is intersected by the line
y–2=0 at the points A and B and if O be the origin then find the area of triangle OAB.
2001
Q. A triangle is formed by the lines:
l1 ≡≡≡≡ 3x – 4y = 0 l2 ≡≡≡≡ 4x + 3y – 8 = 0 l3 ≡≡≡≡ 24x – 7y – 12 = 0
Find the equations of internal bisectors of angles of the triangle.
Q. Find the centroid and the area of the triangle; the equations of whose sides are
7x2
– 20xy + 12y2
= 0 and 2x – 3y + 4 = 0.
2002
Q. Find the equation of the straight line through the intersection of lines 5x – 6y – 1 = 0 and 3x + 2y
= -5 and perpendicular to the line 3x – 5y + 11 = 0.
Q. Find the equation of the locus of a point whose distance from the point (2,-2) is equal to its
distance from the line x-y=0.
2003
Q. Find the equations of the straight line passing through (1, -2) and making acute angles of ππππ/4
radians with the line 6x + 5y = 0. (Draw the figure)
Q. Determine the values of a and b for which the line (a + 2b – 3) x +
(2a – b + 1) y+6a+9=0 is parallel to the axis of x and has y-intercept = - 3.
Q. Show that the lines x2
– 4xy + y2
= 0 and x + y = 3 form and equilateral triangle; find the centroid
of the triangle.
2004
Q. The x-intercept of a line is k and y-intercept is the reciprocal of the x-intercept and passes
through the point (2,-1), find the equation of the line.
Q. Find the equation of the line passing through the intersection of the lines
3x – 4y + 1 = 0 and 5x + y – 1 = 0 and cutting off equal intercepts from the axes.
Q. The gradient of one of the lines of ax2
+ 2hxy + by2
= 0 is twice that of the other. Show that 8h2
–
9ab = 0. OR
Q. If A(2, 3), B(3, 5) are fixed points and a point P moves such that ∆ PAB = 8 sq. units, find the
equation of the locus of P.
2005
Q. The point A (-1, 3) is the foot of the perpendicular dropped from the origin to a straight line.
Find the equation of this line and also find the length of this perpendicular.
Q. Find the centroid of the triangle, the equations of whose sides are 12y2
– 20xy + 7x2
= 0 and 2x –
3y + 4 = 0.
Q. Find the equation of the straight line through the point of intersection of the lines
3x + 2y + 5 = 0 and 2x + 7y – 8 = 0, bisecting the join of (-1, -4) and (5, -6).
2006
Q. A line whose y-intercept is 1 less than its x-intercept forms with the coordinate axes a triangle
of area 6 square units. What is its equation?

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Q. What does the equation xy=0 represent? Also find the area if the triangle formed by
the lines x – 2 =0 and x2
– 7xy + 2y2
= 0.
2007
Q. Find the co-ordinates of the foot of the perpendicular from (-2,5) to a line x+3y +11= 0.
Q. Find the measures of the angle of the triangle, the equation of whose sides are x+y–5 = 0, x – y
+ 1 = 0 and y = 1 Also find its area.
Q. The gradient of one of the that lines of ax2
+ hxy + by2
= 0 is thrice that of the other, show that
8 h2
= 4ab.
2008
+2hxy+by2
=0 is twice that of the other;
Show that 8h2
= 9ab.
Q. Determine the values of a and b for which the line (a + 2b – 3) x + 2a–b +1) y+6a+9 = 0 is parallel
to the axis of X and has y – intercept -3 . Also write the equation of the line.
2009
Q. The point (2, – 5) is the vertex of a square one of whose sides lies on the line 2 7 0x y− − =− − =− − =− − = ;
calculate the area of the square.
Q. What does the equation 2 2
0x y− =− =− =− = represent? If the line 2 0y − =− =− =− = intersects 2 2
0x y− =− =− =− = at
points A and B and if ‘O’ be the origin, then find the area of the triangle OAB.
2010
Mcqs: The line 2x+3y+6=0 is perpendicular to the line:
(a) 2x+3y-8=0 (b) 2x-3y+7=0 (c) x-y+6 = 0 (d) 3x-2y+9=0
Q. Find the value of k when the vertices of the triangle are (2, 6), (6, 3) and (4, k) and its area is 17
square units
Q. The gradient of one of the lines 2 2
2 0ax hxy by+ + = is five times that of the other, show that
2
5 9h ab= .
Q. D, E, F are the mid-points of the sides BC, CA, AB respectively of the triangle ABC show that
4ABC DEF∆ = ∆ .
2011
Mcqs: The angle between the pair of lines 3x2
+8xy-3y2
=0 is:
(a) 900
(b) 450
(c) 00
(d) 1800
Q. The point (2,3) is the foot of perpendicular dropped from the origin to a straight line. Write its
equation.
Q. Find the distance between the parallel line 3x+4y+10=0, 6x+8y-9=0.
Q. Show that the lines x2
-4xy+y2
=0 and x+y=3 form an equilateral triangle. Also find the area of the
triangle.
2012
Mcqs: (v) The point of intersection of internal bisectors of the angles of triangle is called:
(a) Incentre (b) Centroid (c) Ortho-centre (d) circum-
centre
(vii) Two lines represented by ax2
+2hxy+by2
=0 are perpendicular to each other, if:
(a) a+b=0 (b) a−b=0 (c) a=0 (d) b=0
(xiii) If a line is perpendicular to y-axis then its equation is:
(a) x=0 (b) y=constant(c) x=constant (d) y=0
Q. Find the equation of a line through the intersection of the lines 7x-13y+46=0 and 19x+11y-41=0
and passing through the point (3,1) by using k-method.
Q. The point (-2,1) is a vertex of a rectangle whose two sides lie on the lines 3x-2y-5=0, 2x+3y+7=0.
Find area of the rectangle.

P a g e | 15
`
2013
Q. The gradient of one of the lines 2 2
2 0ax hxy by+ + = is five times that of the other, show that
' #
= 7@b.
2014
Q. Find the combined equation of the pair of lines through the origin which are perpendicular to
the lines represented by 6x2
– 13xy + 6y2
= 0.
+ 2hxy + by2
= 0 is twice that of the other. Show that 8h2
= 9ab .
Q. Find the distance between the parallel line ' − #P + = , ' − #P − 5 = .
Q. Find the equation of a line through the intersection of the lines # + &P + = , & − 4P −
' = and passing through the point (2,1).
OR Find the equation of the locus of the points which are equidistant from the point (0,3) and the
line P + & =

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DIFFERENTIABILITY
1992
Q. Find the derivative by the first principles at any point x in the domain D(f) of the function f: f
(x) = cot2
x
Q. Find
XP
X
for any Two of the following.
(i) x = a cos3
2 Ө , y = b sin3
2 Ө (ii) x3
+ y3
+ ax2
y + bxy2
= 0 (iii) P = #& #
+
1993
Q. Find the derivative by the first principles at any point x in the domain D(f) of the function f(x) =
cos2
x.
Q. Find
XP
X
(i) B # + P# = ln (x2
– y2
) (ii) x=a H10#
&c, y=b0/ #
&c (iii) P
.P = 1
1994
Q. Find the derivative by the First Principles at any point x in the domain D(f) of the function f(x) =
Sin2
x:
Q. Differentiate any Two of the following functions with respect to their independent variables.
(i) x = e t
cos 2t , y = e-2t
(ii) y = a cot-1
{m tan-1
(bx)}
(iii) y = (tanx)x
+ (x)tan x
1995
Q. Find the derivative by the first principles at any point x where, f(x) = 2x2
– x.
Q. Find
XP
X
(i) y =
0/ #
H10#
(ii) x = I
H10I; y = I
0/ I (iii) y = ln(secx + tanx)
1996
Q. Find the derivative, by first principles, at any point x ΣΣΣΣD(f) of f(x) = cosec x.
Q. Find the derivative of the function =
& #
& # + D √ + # + I@
Q. Find
XP
X
(i) y = xx
+ (ln x) sinx
(ii) P =
√4 √4
√4 √4
(iii) D P = 0/ P
1997
Q. Find the derivative by the first principles at any point x in the domain D(f) = R of the function
f(x) = sin 2x.
Q. Find
XP
X
(i) D P = 0/ P (ii) y = xcosecx
(iii) x=tant3
+sect3
, y= tant
+ 2sect
1998
Q. Find the derivative by the First Principle at any point x in the domain D(f) of the functions
= 0/ √
Q. Find
XP
X
(i) x=a(t-tsint), y=b(1-cost) at I =
G
#
(ii) P = D 0/
(iii) P =
I@ * #+

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`
1999
Q. Find the derivative by the first principles at any point x in the domain D(f) of the function f(x) =
sin x2
.
Q. Find
XP
X
(i) B # + P# = ln (x2
– y2
) (ii) P = L J I@ − H1I (iii) y = 0/
+ I@
2000
Q. Find the derivative by the first principles at x=a, in the domain D(f) of the function, where f(x) =
cot x2
.
Q. Find
XP
X
(i) xy
. yx
= 5 (ii) P =
' #
' # + D √ + # + H1I (iii) x=lnt+sint,
y=et
+cost
2001
Q. Find the derivative of the function f(x) = sin x2
at any point x in the domain of f by the first
principles.
Q. Find
XP
X
for the functional equation B # + P# = ln (x2
– y2
)
2002
Q. Find the derivative by the first principles at the point x = a in the domain D(f) of the function f(x)
= cos2
x.
Q. Find
XP
X
(i) 2x2
+ 3xy + 7y2
– 2x + 4y + 9 = 0
(ii) y = xx
– x cos x
(iii) x = a cosn
c, y = b sinn
c
2003
Q. Find the derivative, by first principles, at x = 1 in the domain of any one of the following
functions:
(i) f(x) = cot2
x (ii) f(x) =
#
&
Q. Find
XP
X
(i) B # + P# = ln (x2
– y2
) (ii) x=lnt+sint, y=et
+cost, also find
X#P
X #
(iii) P =
0 H *
#
# +
2004
Q. Find the derivative by using the definition at a point x of the function f(x) = sin x2
, a є ℜℜℜℜ.
Q. Find
XP
X
(i) ex
ln y = sin-1
y (ii) x = (t – sin t) , y = (1 – cos t) at I =
G
#
(iii)
P = B # − # + & &
2005
Q. Find the derivative of f(x) = cos2
x at any point of its domain of definition by using the first
principles.
Q. Find
XP
X
(i) x = a cos3
2 c, y = b sin3
2 c at c =
G
5
(ii) xB + P +y√ + = 3
(iii) y = I@ H10
2006
Q. Find the derivative of f(x) = sin 2x at any point of its domain by using the first principles.

P a g e | 18
`
Q. Find
XP
X
(i) y= tan-1
2
2
1
x
x−
(ii)
2
2
5 1
5
x
x
− 2 1
1 cotIn x x−
+ +
(iii) x = sint3
+cost3
, y = sint+cos-1
t
2007
Q. Find the derivative by the first principle at any point in the domain of any one of the following:
(i) f(x) = tan x (ii) f(x) = x2/3
Q. Find
XP
X
(i) P = I@ *
#
#
+ (ii) ex
lny = sin-1
y (iii) x = acos2
2 θθθθ , y = b sin2
3 θθθθ
2008
Q. Find the derivative by the first principle at the point ‘x’ in the D (f) of the function f (x) = sin2
x.
Q. Find
XP
X
(i) P =
& #
& # + D √ + # + I@ (ii) B # + P# = ln (x2
– y2
) (iii) x = sint3
+cost3
, y
= sint+cos-1
t
2009
Q. Find the derivative, by the first principles, at a point x a==== in the domain D(f) of the function
2
( ) cosf x x==== .
Q. Find
XP
X
(i) 2 2
2 3 7 2 4 9 0x xy y x y+ + + + + =+ + + + + =+ + + + + =+ + + + + = (ii) cos , sinn n
x a y bθ θθ θθ θθ θ= == == == =
(iii) y = + D 0/
2010
Mcqs: (xiv) If f(x) = tan9x, then f’(x) is:
(a) sec2
9x (b) 9sec2
x (c) 9sec2
9x (d) –sec2
9x
(xv) If f(x) = lnx3
, then f’(x) at x=-2 is:
(a)
#
&
(b) −
&
#
(c) −
#
&
(d)
Q. Find the derivative by the 1st
principles at x = a in the domain D(f) of f(x) = cosec x.
Q. Find
dy
dx
of any Two of the following:
a) sin cosx x
y x +
= b) 1
ln sinx
e y y−
=
c) ( ) ( )sin , 1 cos
2
x a y a at
θ
θ θ θ θ= − = − =
OR
Q. If ( ) cos sin ,y f x a x b x xε= = + ∀ , show that
2
2
0
d y
y
dx
+ = .
2011
Mcqs: (v) 89: →@
@
@
=:
(a) f’(x) (b) f’(a) (c) f’(0) (d) f’(1)
(vi)
XP
X
0/ #
+ H10#
/0:
(a) 1 (b) #0/ H10 (c) −#0/ H10 (d)
(xix) e = I@ # , I f
/0:
(a) # (b) 4 # (c) 4 # (d)
#
4 #
(xx) e P = H1I , I XP =∶
(a) −H10 H X (b) −H10 H#
(c) −H10 H#
X (d) −H1I#
X

P a g e | 19
`
principles at x = a in the domain D(f) of f(x) = cotx
Q. Find
dy
dx
(i) P = D √ + # + H1I (ii) P = D * + (iii) = @H10#
&c, P = b0/ #
&c
2012
Mcqs: (iii) e P = D 0/ , I
XP
X
=:
(a) 0/
(b) cosx (c) cotx (d) tanx
(xi) If y=8Kh@
, then dy =:
(a) D @X (b) D
X (c) D @
X (d) @ X
(xiv) If = I@ & , then f’(x) is:
(a)
7 #
(b)
7 #
(c)
&
7 #
(d)
&
& #
Q. Find
dy
dx
(i) P = D I@ M
(ii) P
. P = (iii) x = sint3
+cost3
, y = sint+cos-1
t
2013
Mcqs: (ii) e = 0/ 7 , I f
=:
(a) H107 (b) −H107 (c) 7H10 (d) 7H107
(xvii) Derivative of @
with respect to ‘x’ is:
(a) @
D @ (b) @
D (c)
@
D @
(d) @ @
(xi) If f(y)=8Kh@
P, for all P in ℝ , then
X
XP
8Kh@
P=:
(a)
P
D @XP (b)
PD
XP (c)
P
@P
XP (d)
P
D @
principles at x = a in the domain D(f) of f(x) = sin2x
Q. Find
dy
dx
(a) P = 0/ H10
(b) P = 0/ &
(c) P = B # + # + &
'
2014
principles at x = a in the domain D(f) of = H1I j( =
& &
−
Q. Find
dy
dx
(i) D P = L9 P (ii) B # + P# = D #
− P#
(iii) P
. P =

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`
APPLICATIONS OF
DIFFERENTIAL CALCULUS
1992
Q. Show that √ + k can be approximates as √ +
#√
k . Hence find the value of
√6. 7 .
Q. Determine the extreme values of the function.
f (x) = (x – 3)3
(x – 4)2
1993
Q. Calculate an approximate value of cos 46o
.
f(x) = (x – 2) (x – 3)
x2
1994
Q. If the radius of a sphere increases by 0.2%, show that the volume increases by about 0.6%.
f(x) = x3
– 9x2
+ 15x+3
1995
Q. Find the slope of the tangents to the curve y2
= 4x at its vertex and at the ends of the latus
rectum.
Q. From a square sheet of cardboard with side 12 units is made a topless box of maximum
volume by cutting equal squares at the corners and removing them and turning up the sides. Prove
that the length of the side of the square is 2 units.
1996
Q. Find a right-angled triangle of maximum area with hypotenuse of length h.
1997
Q. Find an appropriate value of cos 46o
.
Q. Find a right-angled triangle of maximum area with a hypotenuse of length ‘h’.
1998
Q. Calculate the approximate value of sin 44o
.
Q. Find the rectangle of maximum area inscribed inside the curve:
#
@#
+
P#
b#
=
1999
Q. If the radius of a sphere increase by 0.1%, show that the volume increases by about 0.3%.
Q. A rectangular reservoir with a square bottom and open top is to be lined inside with lead. Find
the dimensions of the reservoir to hold ½ a3
cubic meters such that the lead required is minimum.
2000
Q. Calculate an approximate value of 8Kh . . Given that 8Kh = . 4&4&.
Q. Using a tin sheet of length 48 cm and width 25cm. Make a topless box of maximum volume by
cutting equal squares of dimension x cm. at the corners and removing them and bending the tin so as
to form the sides of the box. Find the value of x for maximum volume. Also find the maximum volume
of box. Give your answer correct to three decimal places.

P a g e | 21
`
2001
#√
√&. 7 .
Q. Show that the rectangle of maximum area inscribed in a circle of radius ‘a’ is a square of area
2a2
.
2002
Q. Using differential, find the approximate value of cos 44o
.
Q. Determine the extreme values of the function f(x) = x3
– 9x2
+ 15x + 3.
2003
Q. Calculate the approximate value of tan 44o
.
Q. Find the right-angled triangle of the maximum area whose hypotenuse is of length “h”.
2004
Q. Calculate the approximate value of cos 47o
using differential.
Q. Find the extreme values of the given function using derivatives f(x) = x(x-1)(x-2), ∀ C.
2005
Q. Calculate the approximate value of tan 46o
using differential.
Q. Find all the stationary points and extreme values of the function ‘f’ such that f(x) =
&
&
− # #
+
& + , ∀ (
2006
#√
√&. 7 .
Q. Find the relative maximum and minimum values of the function f(x)=x3
-3x2
+2x+1.
2007
Q. Calculate the approximate value of cos 46o
by using differentials.
Q. Prove that the relative maximum value of
D
is . OR
Q. Find the right angle triangle of the maximum area whose hypotenous is of length “h”.
2008
Q. Using differentials calculate the approximate value of tan 44 .
Q. Determine the extreme values of the function f(x) = x3
– 9x2
+ 15x + 3.
2009
Q. Using differentials, find the approximate value of cos 44 .
Q. Determine the extreme values of the function (((( )))) 3 2
9 15 3f x x x x= − + += − + += − + += − + + .
2010
Mcqs:(xvi) If s=f(t), then
X#0
XI#
/0 ∶
(a) distance covered at time ‘t’ (b) speed at time ‘t’
(c) acceleration at time ‘t’ (d) velocity at time ‘t’
(xvii) The necessary condition for f(x) to have an extreme values is:
(a) f’(x)=1 (b) f’(x)=0 (c) f’(x)=0 (d) f’’(x)=0
Q. Using differentials, find the approximate value of cos44o
.
Q. Show that the maximum value of ( )
ln 1x
f x is
x e
= .

P a g e | 22
`
2011
#√
√&. 7 .
Q. Find the relative maximum and minimum values of the function f(x)= 0/
2012
Q. By using the differentials, calculate an approximate value of cos440
.
Q. Find the relative maximum and minimum values of the function f(x)=2 +
2013
Q. Using differentials, show that √ + k can be approximated to √ +
#√
k .
Hence find the value of √7. .
Q. Find the relative maximum and relative minimum values of the function
=
D
.
Q. Equation of a curve is given by x2
-2xy+y2
+2x-4=0, find the slope of the curve at the
point (2,2).
2014
#√
√&. 7 .
2014
#√
√&. 7 .
Q. Find the relative maximum and minimum values of the function = 0/
OR = &
– 7 #
+ ' + &.

P a g e | 23
`
ANTIDERIVATIVES
(INTEGRATION)
1992
(i) l + √ # + # + '
&
X (ii) l H1I'
#
X
G
G
#
(iii) l H10&
√0/
G
5 X
(i) l & √ # − 7 X (ii) l #
H10 X (iii) l
# #
&
X
Q. Find the area enclosed by the ellipse:
#
4
+
P#
7
= 1, x= -1 x=1
Q. Solve the following differential equation. P

XP
X
= x, when x=0 and y=0
1993
(i) l √# # + & #
X (ii) l H10
G
# X (iii) l I@ &
. 0 H
G
X
(i) l
I@
8Kh H10
X (ii) l #
X (iii) l D #
X
Q. Find the area enclosed by the parabola
ay = 3 (a2
– x2
) and the axis of x.
Q. Solve the differential equation
XP
X
= x + sin x, given that y = 3 when x = 0.
1994
(i) l
X
#−# ++'
&
#
(ii) l H104
X
G
# (iii) l
& X
4 #
&
#
(i) l
D .L9 m D #n
X (ii) l 0/ X (iii) l #
X
Q. Find the area bounded by the parabola y2
= 4x and the line y = x – 4.
Q. Solve the differential equation.
y (1 + x2
)
XP
X
= (1 + y2
)2
x
1995
(i) l
X
√ – √
(ii) l
X
B # 4
#√&
(iii) l 0/ &
X
G
#
Q. Prove that the area enclosed by the circle x2
+ y2
= 1 is ππππ sq. units.
Q. Solve the differential equation:
XP
X
=
0/ #P
H10#
1996

P a g e | 24
`
(i) l
0 H .I@
@ b0 H
X (ii) l
&
#
X (iii) l D X
XP
X
= B P − #P − & + 5 , y=12 when x=6
Q. Find the area above X-axis under the following curve between the given ordinates.
#
4
+
P#
7
= 1, x= -1 x=1
Q. Evaluate any TWO of the following:
(i) l
# X
4 #
&
#
#
(ii) l I@ &
0 H X
G
(iii) l # #
+ 4 X
1997
(i) l I@ 4
X
G
4 (ii) l
#
#+ − #

#
X (iii) l
I&
B4 I#
XI
(i) l
0/
# &H10 H10#
X (ii) l # &
X (iii) l
P &
P# #P '
XP
Q. Solve any One of the following differentiate equations:
(a)
X0
XI
= √0 + √&I + , s=3 when t=5
(b) # XP
X
= 3 4
P#
+ P#
when y(3)=1
Q. Find the area above the X-axis, Under the Curve
#
5
+
P#
#'
= 1, between the ordinates X = 1
and X = 2.
1998
(i) l I@
#
0 H5
#
X
G
# (ii) l
' X
B7 #
&√&
# (iii)
l H10# H10 X
G
#
(i) l I@ &
0 H&
X (ii) l 0/ & #
X (iii) l
&
#
X
Q. Solve the following differential equations:
#
(1+y)
XP
X
= −(1-x)P#
Q. Use ‘Integration by parts’ to evaluate l √7 − # X
Q. Find the area bounded by the parabolas y2
= 9x and x2
= 9y.
1999
(i) l H104
G
# X (ii) l
& X
B4 #
(iii) l I@ &
0 H
G
X
(i) l H10 5
X (ii) l 0/ &
0/ #
X (iii) l I@ X
y
XP
X
= x(y4
+2y2
+1) and y(-3) = 1
Q. Find the area above x-axis under the curve.
x2
+ y2
= 9 , between x = -2 and x = 1.
2000

P a g e | 25
`
(i) l
0 H I@
@ b0 H
X (ii) l I@ X (iii) l &
+
)
' '
X
X0
XI
= √0 + # √)I − ' , s=7 when t=3
Q. Find the area above x-axis under the curve.
x2
+ y2
= 9 , between x =
&
#
and x =
'
#
.
(i) l 0/ & 0/ # X (ii) l
X
B7 #
&
(iii) l
0/
H10 # H10
G
# X
2001
(i) l 0/ #
G
4 H10#
X (ii) l
P& X
B 5 P#
#
(iii)
l + √ # + # + #
#
X
(i) l &
√) + # X (ii) l
Xo
o#B@# o#
(iii) l
&
# # '
X
(i) l H10 X (ii) l #
D X (iii)
l
#
, #- & # X
Q. Find the are above the x – axis, between the ordinates x = #
and x =
&
#
, under the curve given by
P = √4 − #
2002
(i) l 0/ 4
X (ii) l L9 X (iii) l
& X
B7 #
&√&
#
# XP
X
=
BP#
+ P, P =
Q. Find the are above the x – axis, under the ellipse
#
5
+
P#
7
= 1between the ordinates x =1 and x
=3.
(i) l
X
0/
G
& (ii) l
& # #
#
X (iii) l #
0/ X
2003
(i) l
H10 H H1I
@ bH10 H
X (ii) l &
I@ X (iii) l
# X
4 #
&
#
(i) l D #
X (ii) l
I@
8 H10
X (iii) l @
0/ b X
Q. Find the area above the x-axis under the curve
#
4
+
P#
7
= 1, between bx = -1 and x = 1.
XP
X
= B P − #P − & + 5 , y=12 when x=6
2004

P a g e | 26
`
(i) l #
+ & + '
#
& * +
&
#
+ X
#
(ii) l I@ X
G
4 (iii) l D X
(i) l 0/ & H10' X (ii) l # & #
X (iii) l
0/
# &H10 H10# X
Q. Find the area above the X-axis under the circle x2
+ y2
= 9 between the ordinates x = 0.5 and x =
1.5.
XP
X
= sin2
y. cos2
x sin x.
OR
XP
X
= x + sin x , y = 3 when x = 0.
2005
(i) l
X
√ √
(ii) l
& '
√
X (iii) l 0/ '
X
(i) l 0/ # X (ii) l
X
B4 #
(iii) l
) #'
& 4
X
Q. Find the area above the x-axis, under the ellipse
#
4
+
P#
7
= 1 between the
ordinates x = -1 and x = 1.
XP
X
=
B H10#P
0/ #P
, P =
G
#
= &
2006
(i) l
X
√ √
(ii) l I@ &
0 H&
X (iii) l
B # 7
X
(i) 4
cos xdx∫ (ii) 3
2 x
x e xdx∫ (iii)
cos
(1 sin )(2 sin )
xdx
x x
∫
+ +
Q. Find the area above x-axis, under the curve y= tan x and between the ordinates
x=
6
and
π
x =
3
π
dy
dx
= , (9) 100xy y =
2007
Q. Evalute any two of the following:
(i) l 0/ '
& .H10&
&
G
5 X (ii) l
#
4 #
&
#
(iii)
l
H10# D
X
(i) l 5 ' &
X (ii) l
'0/
5 H10 H10# X (iii) l
X
√ √
Q. Find the area above the x-axis under the curve f(x) = tan2
x
Between =
G
5
@ X =
G
4
XP
X
=
B H10P
0/ P
p P & =
G
#
2008

P a g e | 27
`
(i) l H104
X
G
# (ii) l
0 H I@
@ b0 H
X (iii) l #
D X
(i) l
X
# 4 '
(ii) l I@ &
0 H X
G
(iii) l
# X
4 #
&
#
Q. Find the area above the X-axis, between the ordinates x = -2 and x = 1 under the curve
P = √7 − #
Q. Solve the differentiate equation:
y
XP
X
= x(y4
+2y2
+1) and y(-3) = 1
2009
(i) cosx x
e e dx∫∫∫∫ (ii) 1
0
tanx xdx
ππππ
−−−−
∫∫∫∫ (iii) 2
1
x
x
e dx
e++++∫∫∫∫
(i)
3
0
1 sin
dx
x
ππππ
−−−−∫∫∫∫ (ii)
(((( ))))
cos
sin 2 sin
x dx
x x++++∫∫∫∫ (iii)
1 sin
1 cos
x x
e dx
x
++++
++++∫∫∫∫
Q. Solve the differential equation (((( ))))2
2
1
, 1
dy
x y
dx y y
====
++++
.
Q. Find the area above the x-axis between the ordinates
4
x
ππππ
==== and
3
x
ππππ
==== under the curve
tany x==== .
2010
MCQS: (i) l I@
0 H#
X =:
(a) 0/
+ H (b) 0/ #
+ H (c) I@
+ H (d) 0 H#
+ H
(vii) An equation involving
XP
X
is called:
(a) polynomial equation (b) differential equation
(c) exponential equation (d) logarithmic function
(xviii) l U
X =: p U ≠ −
(a)
Uq
U
+ H (b)
UM
U
+ H (c)
Uq
U
+ H (d)
UM
U
+ H
(xx) l? A f
X =:
(a)
? A
+ H (b)
? A q
+ H (c)
? A M
+ H (d) D + H
a) lnx x dx∫ b) ( )
2
32 3 2
1
3 2x x x x y dx+ + +∫
c) sin 3 cos5x x dx∫ OR 2
2 3
2 2
x
dx
x x
−
+ +∫
Q. Evaluate of Two of the following:
i) 2
4x x dx+∫ ii)
( )
cos
sin 2 sin
x dx
x x+∫ iii)
tan
ln cos
x
dx
x∫
( ) ( )4 2
2 1 , 3 1
dy
y x y y y
dx
= + + − =
Q. Find the area above the x-axis between the ordinates
4
x
π
= under the curve tany x= .

P a g e | 28
`
2011
MCQS: (xii) l 0/ & X =:
(a) H10& + H (b)
H10&
&
+ H (c) (d) . ' + H
(xiv) If n=-1, then l? A f
X =:
(a)
? A q
+ H (b)
? A
+ H (c) D (d)
? A M
+ H
(xvi)l I@
0 H#
X =:
(a) 0 H#
+ H (b) 0 H
+ H (c) I@
+ H (d) I@ + H
(i) l I@ &G
0 H X (ii) l #
I@ X (iii) l
X
4 #
(i) l
#
, #-,& #-
X (ii) l
&X
B@# #
(iii) l
0 H I@
@ b0 H
X
XP
X
=
B H10P
0/ P
p P & =
G
#
Q. Find the area above the x-axis under the curve f(x)=3x4
-2x2
+1 and between the ordinates x=1
and x=2.
2012
MCQS: (xiv) l
r
X =:
(a)
? A q
+ H (b) + H (c) D + H (d) D f
(xvi)l 0/
H10 X =:
(a) H10
+ H (b) H10
0/ + H (c) 0/
0/ + H (d) 0/
+ H
(a) l 0/ 4P0/ #PXP (b) l
X
, # @#-
&
#
@
(c) l
XP
B4P P#
(a) l
X
# 4 '
(b)l
0/
H10
(c) l
H10 X
0/ # 0/
OR l I@ &
0 H X
G
# + #P
XP
X
= + & #
, P # =
Q. Find the area above the x-axis under the circle x2
+y2
=4 and between the ordinates = #
and
=
&
#
2013
MCQS: (v) l 0 H I@ X =:
(a) 0 H I@ + H (b) 0 H + I@ + H (c) 0 H + H (d) I@ + H
(ix) l X =:
(a) + H (b) + H (c) – + H (d) + H
(a) l
X
√ √
(b) l
X
7 # (c) l H104 H10# X
(a) l
H10 X
0/ # 0/
(b) l
0 H I@ X
@ b0 H
(c) l H104
G
# X
XP
X
= P#
0/
Q. Find the area under the curve P = & 4
− # &
+ , above the x-axis between = and = #
2014

P a g e | 29
`
(i) l *√ −
√
+ X (ii) l D X (iii) l
L9 D
& H10D #
X
# + #P
XP
X
= + & #
, P # = OR
Xo
Xs
= √os, o = ,s = 7
Q. Find the area under the curve P = −
'
# between the ordinates = #, = 4.
Q. Evaluate any TWO of the following:
(i) l
X
B4 #
(ii) l
# X
JKL# #
(iii) l
0/ X
H10 # H10

P a g e | 30
`
CIRCLE
1992
Q. Show that the four points (5, 7), (8, 1), (1, 3) a7nd (1, &
) are concyclic and find the equation of the circle
on which they lie.
Q. Prove that condition that the line: H10t + P0/ t = U may touch the circle x2
+y2
+2gx+2fy+c=0 is
B2H10t + 0/ t + U = B2# + # − H
Q. Prove that the conics @ #
+ bP#
= and @′ #
+ b′P#
= cut orthogonally if
@
−
@f
=
b
−
bf
1993
Q. Find the equation of the circle concentric with the circle x2
+ y2
+ 6x – 10y + 33 = 0, and touching
the line y = 0.
1994
Q. Find the equation of the circle which passes through the two points (a, 0) and (-a, 0) and whose
radius is √@# + b# .
Q. Prove that the curves 3x2
−y2
=12 and x2
+3y2
-24=0 intersect at right angles. Also find the point of
intersection.
1995
+ y2
+ 8x – 10y + 33 = 0 and touching
the x-axis.
Q. Find the equations of the tangents to the circle x2
+ y2
– 6x – 2y + 9 = 0 through the origin.
1996
Q. Show that four points (3,4) , (-1, -4) , (-1, 2) , (3, -6) are concyclic, and find the equation of the
circle on which they lie.
Q. Find the equation of the circle which is concentric with the circle x2
+y2
– 8x+12y+15 = 0 and
passes through the point (5, 4).
1997
Q. Find the equation of the circle passing through the points (-1, -1) and (3, 1) and with centre the
line x – y + 10 = 0.
1998
Q. Find the equation of the circle containing the point (6,0) and touching the line x=y at (4,4).
Q. Prove that the condition tangency of y = m x + b with the circle x2
+y2
+2gx+2fy + c = 0 is (g + fm)2
= b (b + 2f – 2mg) + c (l + m2
).
1999
Q. Find the equation of circle which touches x-axis and passes through the points (1, -2) and (3, -4).
2000
Q. Find the equation of the circle containing the points (-1, -1) and (3, 1) and with the center on
the line x – y + 10 = 0.
2001
+ y2
+ 6x – 10x + 33 = 0 and touching
the line y = 0.
Q. Prove that the two circles x2
+ y2
+ 2gx + c = 0 and x2
+ y2
+ 2fy + c = 0 touch each other if
# + 2# = H
2002

P a g e | 31
`
Q. Find the equation of the circle whose centre is at the point (2, 3) and it passes through the
centre of the circle x2
+ y2
+ 8x + 10y – 53 = 0.
Q. Find the equation of the circle concentric with x2
+ y2
+ 6x – 10y + 33 = 0 which touches the line
x = 0.
2003
Q. Find the equation of the circle which passes through the two points (a, 0) and (-a, 0) and whose
radius is √@# + b# .
2004
Q. Find the equation of the circle which passes through the point (-2,-4) and concentric with the
circle x2
+y2
-12y-23 = 0.
Q. Prove that the two circles x2
+ y2
+ 2gx + c = 0 and x2
+ y2
+ 2fy + c = 0 touch each other if
# +
2# =
H
2005
Q. Find the equation of the circle containing the points (-1, -2) and (6, -1) and touching the line y =
0.
2006
+ y2
– 4x – 6y – 23 = 0 and touching x-
axis.
Q. Prove that if wo circles x2
+ y2
+ 2gx + c = 0 and x2
+ y2
+ 2fy + c = 0 touch each other, then
2 2
1 1 1
f g c
+ = .
2007
Q. Find the equation of circle containing the point (-1, -1) and (3,1) and with the center on the
line x – y + 10 = 0.
2008
Q. Find the equation of the circle containing the points ( -1, -2) and (6, -1) touching X-axis.
2009
Q. Find the equation of the circle whose centre is at the point (2, 3) and it passes through the
centre of the circle 2 2
8 10 53 0x y x y+ + + − =+ + + − =+ + + − =+ + + − = .
Q. Prove that if two circles 2 2
2 0x y gx c+ + + =+ + + =+ + + =+ + + = and 2 2
2 0x y fy c+ + + =+ + + =+ + + =+ + + = touch each other, then
2 2
1 1 1
f g c
+ =+ =+ =+ = .
2010
Mcqs: (ii) The centre of the circle x2
+y2
-6x+8y-24=0 is:
(a) (3,-4) (b) (-3,4) (c) (4,3) (d) (3,4)
(iii) The length of the tangent from the point (-2,3) to the circle x2
+y2
+3=0.
(a) 3 (b) 4 (c) 5 (d) 6
(xix) The slope of the following tangent to the curve y=6x2
at (1,-1) is:
(a) -12 (b) 12 (c) 15 (d) 6
Q. Find the equation of the circle which is concentric with the circle 2 2
8 12 12 0x y x y+ − + − = and
passes through the pint (5, 4).
Q. Find the equation of the circle touching each of the axes in 4th
quadrant at a distance of 6 units
from the origin.

P a g e | 32
`
Q. Prove that two circles 2 2
2 0x y gx c+ + + = and 2 2
2 0x y fy c+ + + = touch each other if
2 2
1 1 1
f g c
+ = .
2011
Mcqs: (ii) The centre of the circle x2
+y2
+6x+10y+3=0 is:
(a) (-3,5) (b) (-3,-5) (c) (3,-5) (d) (3,5)
(xv) Which of the circles passes through origin?
(a) x2
+y2
+8x+7 =0 (b) x2
+y2
+9y+11=0
(c) x2
+y2
+8x+11y=0 (d) x2
+y2
+8x+11y+19=0
Q. Find the equation of the circle which passes through the origin and cuts off intercepts equal to 3
and 4 from the axes.
Q. Prove that the curves x2
+3y2
-24=0 and 3x2
+y2
=12 intersect at right angle at the point √5, √5 .
Q. Find the equation of circle containing the points (-1,-1) and (3,1) and with centre on the line
x−y+10=0.
2012
Mcqs: (xvii) The length of the tangent from the point (-2,3) to the circle x2
+y2
+3=0.
(a) 3 (b) 4 (c) 5 (d) 6
(x) The centre of the circle 2x2
+2y2
+8x=0 is:
(a) (0,0) (b) (-4,0) (c) (8,0) (d) (-2,0)
Q. Find the equation of the circle which is concentric with the circle x2
+y2
+6x-10y+33=0 and
touching the y-axis.
2013
Mcqs: (x) Centre of the circle x2
+y2
+6x-8y+3=0.
(a) (3,4) (b) (-3,-4) (c) (3,-4) (d) (-3,4)
Q. Find the equation of the circle touching each of the axes in 4th
quadrant at a distance of 5 units
from the origin.
+ y2
– 8x +12y 15 = 0 and passes
through the point (5,4).
Q. Find the equation of the circle containing the points (-1,-1) and (3,1) and with the centre on the
line − P + = .
2014
+ y2
-8x +12y -12 = 0, and passes
through the point (5,4).
Q. Find the equation of the circle passing through the focus of parabola #
+ 6P = and foci of
ellipse 5 #
+ #'P#
= 4 .
Q. Find the condition that conics @ #
+ bP#
= and @f #
+ bf
P#
= cut each other
orthogonally.

P a g e | 33
`
PARABOLA, ELLIPSE
AND HYPERBOLA
1992
Q. Find the centre, vertices, foci, eccentricity, and equation of directories of the ellipse:
25x2
+ 16y2
– 50x + 64y – 311 = 0
Q. Find the equation of the hyperbola with centre at origin and satisfying the following conditions.
HH Ip/H/IP =
&
'
, D@Io0 p HIo= =
#66
'
, Ip@ 0s p0 @ /0 /0 @D1 2 P − @ /0
Q. Find the equation of the tangents to the parabola x2
= 4y which are parallel and perpendicular to the line
y = 6x + 2.
1993
Q. Determine the vertex, axes, focus, latus rectum and the equation of the directrix of the
following parabola:
x2
+ 4x + 4y – 12 = 0
Q. Find the equation of the ellipse having the origin as its centre, one focus at the point (4, 0) and
the corresponding directrix x=6.
Q. Find the equation of the ellipse with centre at the origin satisfying the conditions =
#
&
and
directrix − & =
Q. Prove that the line lx + my + n = 0 and the ellipse
#
@# +
P#
b# = have just one point in common if
a2
l2
+ b2
m2
– n2
= 0.
1994
Q. Show that the equation ax2
+by2
+2gx+2fy+c=0, may represent a parabola if a≠≠≠≠0 and b=0. Find
the coordinates of the vertex.
Q. The length of the major axis of an ellipse is 25, and its foci are the points (+ 5, 0); find the
equation of the ellipse.
Q. Prove that a line parallel to an asymptote intersects the hyperbola in just one point.
Q. Prove that the curves 3x2
– y2
= 12 and x2
+ 3y2
– 24 = 0 intersect at right angles. Also find the
point of their intersection.
1995
Q. Determine the vertex, focus, and directrix of the parabola x2
+ 4x + 4y – 12 = 0.
Q. Find the distance between the vertices, foci and directrices of the ellipse 9x2
+ 13y2
= 117.
Q. Find the equation of hyperbola with center at the origin, length of the latus rectum = 64/3,
transverse axis along y-axis and eccentricity = 5/3.
Q. Find the slopes of the tangents at the ends of the latera recta of the hyperbola.
#
@# −
P#
b# =
1996
Q. The length of the major axis of an ellipse is 25 and its foci are the points (+ 5,0).
Find the equation of the ellipse.
Q. Find the equation of the hyperbola with centre at the origin and focus at the point (8,0) and the
directrix x=4.
#
@#
+
P#
b#
= have just one point in common if
a2
l2
+ b2
m2
– n2
= 0.
1997

P a g e | 34
`
Q. Prove that the line y = mx + c and the parabola y2
= 4 ax has just one point in common if c =
@
=
and the point of contact is *
@
=# ,
#@
=
+.
Q. The length of the major axis of an ellipse is 25 and its foci are the points (+ 5, 0), find the
equation of the ellipse.
Q. Find the equations of the tangents and normals at the ends of the Latus Rectum of the parabola
y2
= 4ax.
Q. Find the equation of the circle whose diameter is the major axis of the ellipse 16x2
+25y2
=400;
also find whether (4,-3) lies inside or outside the ellipse.
1998
Q. Find the equation of the circle whose diameter is the latus rectum of the parabola y2
=-16x.
Q. An ellipse is drawn to pass through the points (3, 12) (10, 10) and (3, -4) and to have the line x =
6 as an axis of symmetry; find the equation of the ellipse.
Q. Find the coordinates of vertices, foci, and equations of directrices and transverse axis of the
hyperbola 9x2
– 16y2
– 36x – 32y + 164 = 0.
1999
Q. Find the condition that the line x cos α + y sin αααα = p will touch the parabola y2
= 4ax.
Q. Find the equation of ellipse when e = 2/3, latus-rectum of length 20/3 and major axis along y-
axis.
Q. Find the condition that the conic ax2
+ by2
= 1 should cut the conic a’x2
+ b’y2
=1 orthogonally.
Q. Find the eccentricity, foci and directrices of the hyperbola 16 x2
– 9y2
= 144.
Q. Show that the eccentricities e1 and e2 of the two conjugate hyperbolas satisfy the relation e2
1 +
e2
2 = e2
1 e2
2.
2000
Q. Find the equation of a circle whose diameter is the latus rectum of the parabola x2
=36y.
Q. Find the coordinates of the center and the foci, the length of semi-transverse axis and the
eccentricity of the hyperbola.
9x2
– 16y2
+ 18x – 64y – 199 = 0
Q. If y = √'x+k , is a tangent to the ellipse
#
7
+
P#
4
= , what is the value of k.
Q. Find the equation of the ellipse whose center is at the origin, directrix x = 16 and length of latus
rectum 12.
Q. Find the equation of the tangent and normal to the hyperbola x2
– y2
= 64, at (10, 6).
2001
Q. Prove that the product of abscissa of the points where the straight line y = mx meets the circle
x2
+ y2
+ 2gx + 2fy + c = 0 is equal to
H
=#.
Q. If (x1, y2) , (x2 , y2) are the co-ordinates of the extremities of a focal chord of the parabola y2
=
4cx, prove that x1 x2 = c2
and y1 y2 = - 4c2
.
Q. Find the eccentricity, the semi-axes, the centre, the vertices and co-ordinates of the foci of the
ellipse:
4x2
– 32x + 25y2
– 300y + 864 = 0
Q. Find the equation of hyperbola with centre at the origin whose eccentricity is 3 and one of the
foci is (6, 0).
2002
Q. Find the equation of the ellipse when =
#
&
, latus rectum of length
#
&
and the major axis is
along x-axis.
Q. Find the vertex, focus and equation of the directrix of the parabola x2
– 4x + 5y – 11 = 0.

P a g e | 35
`
#
#'
+
P#
7
= have one point common if 25l2
+
9m2
– n2
= 0.
Q. Find the eccentricity, foci and directrices of the hyperbola 9y2
– 16x2
= 144.
2003
Q. Find the condition that the two conics ax2
+ by2
= 1 and a’x2
+ b’y2
= 1 intersect orthogonally.
Q. Find the coordinates of the vertices, foci and equation of directrices and principal axis of the
parabola y2
= x – 2y – 1.
Q. Find the equation of the ellipse with vertices at (0, +5) and passing through
the point *
4
'
, &+.
Q. Find the coordinates of the vertices, foci and equation of the directrices for the hyperbola 9x2
–
16y2
– 36x – 32y – 16 = 0.
Q. Show that the eccentricities e1 and e2 of two conjugate hyperbolas satisfy the relation e2
1 +
e2
2 = e2
1 e2
2.
2004
Q. Find the equation of the circle whose diameter is the latus rectum of the parabola y2
=-36x.
Q. Find the equation of the ellipse whose centre is at the origin, directrix x = 16 and length of latus
rectum is 12.
Q. Find the coordinates of the centre, foci, eccentricity and length of latus rectum of
hyperbola 16x2
– 36y2
+ 48x + 180y – 225 = 0.
Q. Find the equations of the tangent and normal to the hyperbola x2
– y2
= 49 at (8, 15).
2005
Q. Find the equation of the parabola whose focus is at (3, 4) and directrix is the line x+y–1=0.
Q. Find the equation of an ellipse whose centre is at the origin, equation of the directrix is y + 4 = 0
and the focus is at (0, -3).
Q. Find the eccentricity, the distance between focai, length of latus rectum and equations of
the directrices of the hyperbola
#
7
−
P#
5
= .
#
@# +
P#
b# = have just one point in
common if a2/2
+ b2
m2
= n2
.
2006
Q. Determine the focus, vertex and equation of directory of the parabola
x2
– 6x – 2y + 5 = 0
Q. Find the equation of the ellipse whose centre is at origin, vertices at (0,±5) and the length of
the tutus rectum is 3 units.
Q. Find the distance between the directories of the hyperbola 16x2
– 9y2
= 144
Q. if (x1,y1) , (x2 , y2) are the coordinates of the extremities of a focal chord of the parabola y2
= 4cx, prove that x1x2 = c2
and y1y2 = -4c2
.
2007
Q. Find the equation of ellipse with center at origin satisfying the condition =
#
&
and directrix x
– 3 = 0
Q. Find the distance between the directrices of the hyperbola 16x2
– 9y2
= 144 and also find the
equation of the directrices.
Q. Find the equation of the parabola whose focus is at (3,4) and directrix is the line x+y=1.
Q. Show that U
+
P
V
= touches the hyperbola
#
@#
−
P#
b#
= , if
@#
U#
−
b #
V#
=
2008
Q. Find the coordinates of the centres, foci, length of semi transverse axis and the eccentricity of
the hyperbola 16x2
− 36y2
+ 48x + 180y − 225 = 0.

P a g e | 36
`
Q. Find the length of, and the equation to the focal radii draw to c point (4√&, 4) of the ellipse 25x2
+ 16y2
= 1600.
Q. Find the condition that the conic ax2
+ by2
= 1 should cut a′x2
+ b′y2
= orthogonally.
Q. Find the equation of the tangents at the ends of the latus rectum of the parabola y2
= 4ax.
Q. Find the equation of the hyperbola with center at the origin whose eccentricity is 3 and one of
its foci is (6, 0).
2009
Q. Find the equation of the parabola whose focus is (3, 4) and directrix 1 0x y+ − =+ − =+ − =+ − = .
Q. Find the coordinates of the centre and the foci, the length of semi-transverse axis and the
eccentricity of the hyperbola 2 2
9 16 18 64 199 0x y x y− + − − =− + − − =− + − − =− + − − = .
Q. Show that the eccentricities e1 and e2 of two conjugates hyperbolas satisfy the relation
2 2 2 2
1 2 1 2e e e e+ =+ =+ =+ = .
Q. Find the equation of the ellipse whose
2
3
e ==== , latus rectum
20
3
==== and major axis is along Y-axis.
2010
Mcqs: (iv) If =
&
#
, then the conic is:
(a) parabola (b) hyperbola (c) ellipse (d) circle
(v) If b2
=a2
(1-e2
), the conic is:
(a) circle (b) parabola (c) ellipse (d) hyperbola
Q. Find the equation of the circle whose diameter is the latus rectum of the parabola 2
36y x= − .
Q. Find the eccentricity, foci and equations of directrices of 2 2
25 9 225x y+ = .
OR Q. Find the eccentricity of the hyperbola whose latus rectum is four times that of the transverse
axis.
Q. Show that the eccentricities 1e and 2e of two conjugate hyperbolas satisfy the relation
2 2 2 2
1 2 1 2e e e e+ = .
2011
Mcqs: (iv) If = , then the conic is:
(a) circle (b) ellipse (c) parabola (d) circle
(x) The vertices of hyperbola
#
5
−
P#
4
= are:
(a) ±#, (b) , ±# (c) , ±4 (d) ±4,
(xii) The distance between the foci of the ellipse
#
@# −
P#
b# = is:
(a) 2a (b) 2c (c) 2b (d) #
@
(xvii) The vertex of the parabola (x-1)2
=8(y+2) is:
(a) (1,-2) (b) (0,1) (c) (2,0) (d) (0,0)
Q. Determine the vertex, focus and equation of directrix of the parabola x2
+4x+4y-12=0.
Q. Find the eccentricity, foci and equations of directrices of the hyperbola 16x2
−9y2
=144.
OR Q. The length of Major axis of an ellipse is 20 and its foci are the points ±', . Find the equation
of the ellipse.
Q. Find the eccentricity, centre, vertices and foci of the ellipse given by equation:
4x2
−16x+25y2
+200y+316=0
2012
Mcqs: (iv) The distance between the foci of the ellipse
#
@# +
P#
b# = is:
(a) 2a (b) 2c (c) #
@
H
(d) 2b
Q. Find the equation of circle concentric with the circle x2
+y2
+6x−10y+33=0 and touching the y-
axis.
Q. Find the equation of parabola with focus (2,3) and directrix y−5=0.

P a g e | 37
`
Q. Find the equation of ellipse whose centre is at (0,0), =
#
&
, latus rectum of length
#
&
and major
axis is along x-axis.
OR Q. Find the eccentricity, foci and equations of directrices of hyperbola 9x2
−y2
+1=0.
2013
Q. Determine the vertex, focus, and directrix of the curve x2
+ 4x + 4y – 12 = 0.
Q. Find the equation of the hyperbola having focus (8,0) and directrix = 4.
OR Find the eccentricity, foci and equations of directrices of 25x2
+7y2
=225.
#
@# +
P#
b# = have just one point in common if
a2
l2
+ b2
m2
- n2
=0.
2014
Q. Find the equation of the parabola having focus (-5,3) and directrix P − ) =
Q. Find the centre, focus and eccentricity of the ellipse
& #
#'
+
P #
7
= .
Q. Find the equation of the hyperbola with focus 6, and directrix = 4
Q. Determine the focus, vertex and equation of directrix of P#
+ 4P + & − 7# =
Q. Show that the eccentricities e1 and e2 of the two conjugate hyperbolas satisfy the relation e2
1 +
e2
2 = e2
1 e2
2.
OR If P = √' + w is a tangent to the ellipse
#
7
+
P#
4
= . What is k?

P a g e | 38
`
VECTORS
1992
Q. Find the scalar area of the triangle ABC where A,B,C are the points (5, 1, -2),
(-2, 7, 3), (-4, -3, 1) by vector method.
Q. A,B,C are the points @x, bx, H] respectively ‘D’ divides [y]]]] in 4:1 and ‘E’ divides z]]]]] in 5:2. Find the position
vector of ‘E’.
Q. Find cosC in a triangle whose vertices are:
A (-5, -4) , B(-1, 3) , C(2, 03). (Use Vector Method).
1993
Q. A, B, C are the points @x, bx and #@x − b] respectively. D divides AC in 2:3 and E divides z]]]]] in 4:1;
find the position vector of E.
Q. Find cos ([]]]], [y]]]]) in a triangle whose vertices are A(-2, 0), B(4, 3), C(5, -1), (Use Vector Method).
Q. Prove that: "@x + b], b] + H], H] + @x$ = #"@x, b], H]$
1994
Q. The vertices of a quadrilateral are A:(1, 2, -1), B:(-4, 2, -2), C:(4, 1, -5), D:(2, -1, 3). At the point A,
forces of magnitude 2, 3, 2 act along the lines AB, AC and AD respectively; find their resultant.
Q. Determine @x unit vector perpendicular to each of the vectors a=2i–6j–3k and b=4i+3j+k. Also
calculate the sine of the angle between them.
Q. Evaluate the scalar triple product:
[2i + k , i, - i + 2j + k]
1995
Q. A, B, C are the points a, b and #@x − b] respectively. D divides [y]]]] in 2:3 and E divides z]]]]] in 4:1,
find the position vector of E.
Q. Find the unit vector perpendicular to the following pair of vectors: a = 3i + 5j – 4k and b = 4i – 3j
+ 5k.
Q. The vertices of a quadrilateral are A (1, 2, -1), B(-4, 2, -2), C (4, 1, -5) and D (2, -1, 3) At the point
A, the forces of magnitude 2, 2, 3 act along the lines []]]], [y]]]] and [z]]]] respectively; find the resultant.
1996
Q. Find cos ([]]]], [y]]]]) in a triangle whose vertices are A(-2, 0), B(4, 3) and C(5, -1).
Q. Find the unit vector perpendicular to both the vectors ox= 2i – 3k, sx = i + 2j – k.
Q. A particle acted on by forces 4i + j – 3k and 3i + j – k is displaced from the point (1, 2, 3) to the
point (5, 4, 1); find the work done on the particle.
1997
Q. If the resultant of two forces is equal in magnitude to one of them and is perpendicular to it in
direction, what is the relation between the two forces?
Q. Find sin (a, b) where a = 2i – 3j + k and b = i – 2k.
Q. In a parallelogram ABCD, mid-point of AB is X and & divides y]]]] in 1:2, show that if Z divides z{]]]]]
in 6:1, then it also divides [|]]]] in 3:4.
1998
Q. Find cos ( a , b ) where a = 4i – 2j + 4k, b = 3i – 6j – 2k.
Q. By using vecor method, find the lengths of the medians of the triangle formed by points (2, 4), (-
2, -2) and (4, -6).
Q. Find the volume of the parallelepiped whose three adjacent edges are represented by the
vectors.
a = i – 2j – 3k

P a g e | 39
`
b = 2i + j – k
c = i + 3j – 2k
1999
Q. Find the constant ‘a’ such that the vectors 2i – j + k, i+j–3k, 3i+aj+5k are coplanar.
Q. A particle acted on by the forces 4 i + j – 3 k , and 3 i + j – k is displaced from the point (1, 2, 3)
to the point (5, 4, 1); find the work done on the particle.
Q. Find the unit vector perpendicular to the vectors a = i –3j+4k , b = -3i +3k and also find sin (a,
b) for the vectors a and b.
2000
Q. Simplify the following and state the geometrical significance.
[ - a – b – c , 2b + 3c, – 4a + c ]
Q. Find the unit vector perpendicular to both the vectors, a = i + 2j + 2k and b = 3i – 2j + 4k. Also
calculate the sine of the angle between these two vectors.
Q. A particle is acted on by constant forces 4i + j – 3k and 3i + j – k and is displaced from the point i
+ 2j + 3k, to the point 5i + 4j + k , Find the work done by the forces on the particle.
2001
Q. Resolve the vectors @x = (2, 1, 0) , bx = (6, 8, -6) in the direction of vectors }]]]]=(1, -1, 2),
}#
]]]] = (2, 2, -1), }&
]]]] = (3, 7, -7).
Q. Forces of magnitude 5, 3, 1 act on a particle in the directions of the vectors (6, 2, 3),
(3, -2, 6) , (2, -3, -6) respectively. The particle is displaced from, the point (2, -1, -3) to the point (5, -1,
1); find the work done by the forces.
2002
Q. Find sin (@x, bx) where, @x = i – 3j + 4k , bx = -3i + 3k.
Q. Let the position vectors of the points A, B & C be a, b, c respectively and ‘D’ divides [y]]]] in 4:1
and the point ‘E’ divides z]]]]] in 5 : 2 ; find the position vector of the point ‘E’.
vectors, a = 2i – 3j + 4k , b = 3i – j + 2k, c = i + 2j – k.
2003
Q. Evaluate the scalar triple product of [a, b, c] where a = 2i– 3j, b=i+j–k and c = 3i – k.
Q. Find sin (a, b) where a = i – 3j + 4k , b = - 3i + 3k.
Q. Find the work done if a particle is acted upon by constant forces 4i + j – 3k and 3i + j – k, and is
displaced from the point i + 2j + 3k to the point 5i + 4j + k.
2004
Q. Resolve the vector a = (-1, 8, -13) in the direction of the vector P1 = (3, -2, 1), P2 = (-1, 1, -2) and
P3 = (2, 1, -3).
Q. Two points P and Q have the position vectors with respect to the origin O, given 3i + j + 2k and i
+ j – 2k respectively. Calculate the length PQ and show that the vectors OP and OQ are mutually
perpendicular.
Q. Find the volume of the parallelepiped with edges OA, OB, OC where A, B, C are the points (0, 1,
1), (-2, 1, 3), (2, -2, 0) respectively.
2005
Q. A particle, acted upon by constant forces, F1 = i – 2j – 3k , F2 = 2i + j – k, is displaced from the
point A (1, 2, 3) to the point B (5, 4, 1). Find the work done on the particle.
Q. Find the unit vector perpendicular to two given vectors a = 2i + 3j + 4k and b = i – j + k. Also find
sin (a, b).
Q. P, Q, R are the points p, q and 2p – q respectively. M divides PR in the ratio 2:3 and N divides
QM in the ratio 4:1. Find the position vector of N.
2006

P a g e | 40
`
Q. Show that the position vector of the mid-point of the line AB where A and B have position
vectors a and b respectively is
2
a b+
Q. Find a unit vector perpendicular to the vectors a = I – 3j + 2k and b = 3i + 2k.
Q. A particle acted upon by constant forces 3i + j – 3k and 3i + j – k is displaced from the point
(5,4,1) ; find the work done by the forces.
2007
Q. A particle acted upon the constant forces F1 = I – 2j - 3k and F2 = 2i + j - k is displaced from
the point (1,2,3), (5,4,1), find the work done on the particle.
Q. Find sin ( a , b), also find a unit vector perpendicular to both to the point and b, where
a = i – 3j + 4k and b = -3i + 3k.
Q. prove that: ~] + ], ] + 2x, 2x + ]• = #" ], ], 2x $
2008
Q. Find cos ([]]]] , [y]]]] ) in a triangle whose vertices are A (5, -1), B(-2, 0) and C (4.3) (6)
Q. A particle is acted on by the constant forces 4€̂ + ‚̂ − &wƒ and &€̂ + ‚̂ − wƒ and is displaced from
the point €̂ + #‚̂ + &wƒ to the point '€̂ + 4‚̂ + wƒ; Find the work done by the forces on the particle.
2009
Q. Find, (((( ))))sin ,u v and also find a unit vector perpendicular to u and v both, where
ˆ ˆ ˆ ˆ ˆ ˆ2 2 , 3 2 4u i j k v i j k= + + = − −= + + = − −= + + = − −= + + = − −
Q. Prove that 2a b b c c a a b c            + + + =+ + + =+ + + =+ + + =            
Q. Resolve the vectors (((( ))))2,1,0a ==== in the direction of the vectors.
(((( )))) (((( )))) (((( ))))1 2 31, 1,2 , 2,2, 1 , 3,7, 7p p p= − = − = −= − = − = −= − = − = −= − = − = −
2010
Mcqs: (vi) If @x. bxx = , then the angle between the vectors @x and bx:
(a) 0 (b)
G
#
(c)
G
&
(d) G
(vii) |@x| of a vector @x when @x = } }#
]]]]]]], where P1:(0,0,1) and P2:(-3,1,2).
(a) √ # (b) √ (c) √ (d) √ &
Q. Prove that ,2 3 , 2 5 , ,a b c a b c a b c   − − + + =    .
Q. Find the scalars x, y and z such that ( ) ( ) ( ) ( )ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ3 4 2 4 5 4 10i k y i j k z i k i j k− + − + + + − = + − .
Q. A particle acted upon by the forces ˆˆ ˆ4 3i j k+ − and ˆˆ ˆ3i j k+ − is displaced from the point
(1, 2, 3) to the point (5, 4, 1), find the work done.
2011
Mcqs: (xi) If @x and bx are any two vectors then ,@x − bx- @x + bx is equal to:
(a) a2
−b2
(b) 0 (c) @x × bx (d) # @x × bx
Q. Find the unit vector perpendicular to both the vectors @x = €̂ − &‚̂ + #wƒ and bx = −&€̂ + #wƒ and
find sin(a,b).
vectors @x = #€̂ − &‚̂ + 4wƒ and bx = &€̂ − ‚̂ + #wƒ.
2012
Mcqs: (xi) If @x = } }#
]]]]]]], where P1:(0,0,1) and P2:(0,4,4) then |@x| is:
(a) 4 (b) √' (c) 25 (d) 9
Q. Find the unit vector perpendicular to both the vectors @x = €̂ + ‚̂ and bx = ‚̂ + wƒ
Q. A particle acted upon by the forces ˆˆ ˆ4 3i j k+ − and ˆˆ ˆ3i j k+ − is displaced from the point
(1, 2, 3) to the point (5, 4, 1), find the work done.
2013

P a g e | 41
`
Q. Find sin(a , b) where @ = / – &… + 4w, b = −&/ + &w.
vectors @ = #/ + &… + 4w, b = / + #… − w, H = &/ − … + #w.
2014
Q. A particle acted upon by the forces 4€̂ + ‚̂ − &wƒ and &€̂ + ‚̂ − wƒ is displaced from the point (1, 2,
3) to the point (5, 4, 1), find the work done.
Q. Find the unit vector perpendicular to the following pair of vectors: €̂ + #‚̂ + #wƒ and &€̂ − #‚̂ −
4wƒ. Also find sine of the angle between them.
OR Simplify: ~@x, #b]]]] − &H]]]], −#@]]]] + bx + H]•

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