24 years past papers in accordance with the …pgseducation.com/assign-formulae/2ndyear20yrs.pdf ·...

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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

Q. A sequence is given by: �.#&.4 ,

&.4'.5 , '.5).6 , ).67.�� . . .

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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� − #& + 47 − 6#) +… 1999

Q. Discuss convergence or divergence of the series: �� + ��# + ��& +. . .

2000

Q. A sequence is given by: �.#&.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: � − #& + 47 − 6#) +… 2006

Q. Find the limit of the sequence: �.&#.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

Q. Find the limit of the sequence: �.#&.4 ,

&.4'.5 , '.5).6 , ).67.�� . . .

LIMIT OF FUNCTION PORTION

1992

Q. Determine any Two of the following limits:

(i) 89:Ө→G#H1IӨ�JKLӨH10&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�JKLӨH10&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�H1IPP (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/�αααα0/�&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/�ΨΨΨΨ0/�&ΨΨΨΨ (ii) 89:�→� √��&�#�� (iii) 89:�→−� B�#��

2002

Q. Determine any Two of the following limits.

(i) 89:T→� H1IT�H10TH10&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/�φφφφφφφφ&

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:�→� * #�#�� − ��+ (iii) 89:αααα→� I@�αααα�0/�αααα0/�&αααα

2006

Q. Evaluate any two of the following:

(i) 89:�→@ �=�@=��@� (ii) 89:�→� * �� − &��&+ (iii) 89:αααα→� I@�αααα�0/�αααα0/�&αααα

2007


(i) 89:�→� √��–√�� (ii) 89:�→∞ 8��M�� (iii) 89:�→� L9�'��L9�&��

2008

Q. Evaluate any Two of the following:

(i) 89:αααα→� I@�αααα�0/�αααα0/�&αααα (ii) 89:�→& 7��#4�B�#�) (iii) 89:�→� �=�� m, n є ℜℜℜℜ

2009


(i) 0

limx

x a a

x→→→→

+ −+ −+ −+ − (ii)

0

9 8lim

e eθ θθ θθ θθ θ

θθθθ θθθθ

−−−−

→→→→

− −− −− −− −

(iii) 20

1 coslimx

x

x→→→→

−−−− (iv)

2 1lim

x

x x→−∞→−∞→−∞→−∞

−−−−

2010

Mcqs: (xii) 89:�→� 0/�&�� =

(a) )& (b) ) (c)

&) (d) &

(xiii) 89:�→# �#�4��# =

(a)6 (b) 4 (c) �1IX��/��X (d) 0

Q. Evaluate 30

tan sinlim

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�JKWTT

2013

Mcqs: (i) 89:�→4 �#��5��4 =

(a) 4 (b) 6 (c)� (d) ∞


(a)89:T→G#JKWT�JKLTJKL&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 + 6y

2 = 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 + 2y

2 = 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 + y

2 = 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 + 7y

2 = 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 – y

2 = 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 + 12y

2 = 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 + y

2 = 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 + by

2 = 0 is twice that of the other. Show that 8h

2 –

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 + 7x

2 = 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?

P a g e | 14 `


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 + 2y

2 = 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 + by

2 = 0 is thrice that of the other, show that

8 h2 = 4ab.

2008

Q. The gradient of one of the lines of ax2+2hxy+by

2=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 20x 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 22 0ax hxy by+ + = is five times that of the other, show that 25 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-3y

2=0 is:

(a) 900 (b) 45

0 (c) 0

0 (d) 180

0

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+y

2=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+by

2=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 22 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 + 6y

2 = 0.

Q. The gradient of one of the lines of ax2 + 2hxy + by

2 = 0 is twice that of the other. Show that 8h

2

= 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 + & = �

P a g e | 16 `



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 XPX� for any Two of the following.

(i) x = a cos3 2 Ө , y = b sin3 2 Ө (ii) x3 + y3 + ax2y + 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.


(i) B�# +P# = ln (x2 – y

2) (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) =

Sin2x:

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.


(i) y = ��0/�#��H10#� (ii) x = ��IH10I; y = �I0/�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@��


(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.


(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/�√�


(i) x=a(t-tsint), y=b(1-cost) at I = G# (ii) P = �D��0/�� (iii) P =I@�� * ��#+

P a g e | 17 `


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.


(i) B�# +P# = ln (x2 – y

2) (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.


(i) xy . y

x = 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 XPX� for the functional equation B�# +P# = ln (x

2 – y

2)

2002

Q. Find the derivative by the first principles at the point x = a in the domain D(f) of the function f(x)

= cos2x.


(i) 2x2 + 3xy + 7y

2 – 2x + 4y + 9 = 0

(ii) y = xx – x

cos x (iii) x = a cos

n c, y = b sin

n 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) = cot2x (ii) f(x) = �#&


(i) B�# +P# = ln (x2 – y

2) (ii) x=lnt+sint, y=e

t+cost, also find

X#PX�# (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.


(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.


(i) x = a cos3 2c, y = b sin

3 2c at c = G5 (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 `



(i) y= tan-1

2

2

1

x

x− (ii)

2

2

5 1

5

x

x

− 2 11 cotIn x x−+ +

(iii) x = sint3 +cost

3, y = sint+cos

-1t

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


(i) P = I@�� * #��#+ (ii) ex lny = sin

-1 y (iii) x = acos

2 2 θθθθ , y = b sin

2 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.


(i) P = &�#��&�# + D�√� + �# + I@�� (ii) B�# +P# = ln (x2 – y

2) (iii) x = sint

3 +cost

3, 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==== .


(i) 2 22 3 7 2 4 9 0x xy y x y+ + + + + =+ + + + + =+ + + + + =+ + + + + = (ii) cos , sin

n nx a y bθ θθ θθ θθ θ= == == == =

(iii) y = �� + �D��0/��

2010

Mcqs: (xiv) If f(x) = tan9x, then f’(x) is:

(a) sec29x (b) 9sec

2x (c) 9sec

29x (d) –sec

29x

(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 xy x += b) 1ln sinxe y y−=

c) ( ) ( )sin , 1 cos2

x a y a atθ

θ θ θ θ= − = − =

OR

Q. If ( ) cos sin ,y f x a x b x xε= = + ∀ � , show that 2

20

d yy

dx+ = .

2011

Mcqs: (v) 89:�→@ ��@��@ =:

(a) f’(x) (b) f’(a) (c) f’(0) (d) f’(1)

(vi) XPX� �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


(i) P = D�√� + �# + H1I�� (ii) P = D� * ��+ (iii) � = @H10#&c, P = b0/�#&c

2012

Mcqs: (iii) e�P = D�0/��, I�� XPX� =:

(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


(i) P = �D��I@�M�� (ii) �P. P� = � (iii) x = sint3 +cost

3, y = sint+cos

-1t

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

XXP 8Kh@ P=:

(a) �P D�@XP (b)

�PD��XP (c)�P@PXP (d)

�P D�@


principles at x = a in the domain D(f) of f(x) = sin2x

Q. Find dy


(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


(i) ��D�P = L9�� P (ii) B�# + P# = D��# − P#� (iii) �P. P� = ��

P a g e | 20 `



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 – 9x

2 + 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

Q. Show that √� + k� can be approximates as √�+ �#√�k�. Hence find the value of

√&. 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 – 9x

2 + 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)=x

3-3x

2+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 tan44�.

Q. Determine the extreme values of the function f(x) = x3 – 9x

2 + 15x + 3.

2009

Q. Using differentials, find the approximate value of cos 44�.

Q. Determine the extreme values of the function (((( )))) 3 29 15 3f x x x x= − + += − + += − + += − + + .

2010

Mcqs: (xvi) If s=f(t), then X#0XI# /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 cos44

o.

Q. Show that the maximum value of ( ) ln 1xf 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+y

2+2x-4=0, find the slope of the curve at the

point (2,2).

2014


√&. 7. Q. Using differentials, find the approximate value of cos 44�.

2014


√&. 7. Q. Using differentials, find the approximate value of cos 44�.

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�GG# (iii) l H10&�√0/��G5� X�


(i) l�& √�# − 7X� (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�� XPX� = 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 – x

2) and the axis of x.

Q. Solve the differential equation XPX� = 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) XPX� = (1 + y

2)

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 + y

2 = 1 is ππππ sq. units.

Q. Solve the differential equation: XPX� = 0/�#PH10#�

1996


P a g e | 24 `


(i) l 0�H�.I@��@�b0�H� X� (ii) l �&��# X� (iii) l�D��X�

Q. Solve the differential equation:

XPX� = 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�G4� (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) X0XI = √0 + � √&I + � , s=3 when t=5

(b) �# XPX� = 3�4P# + 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) XPX� = −(1-x)P#

Q. Use ‘Integration by parts’ to evaluate l√7 − ��# X�

Q. Find the area bounded by the parabolas y2 = 9x and x

2 = 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�


yXPX� = x(y

4+2y

2+1) and y(-3) = 1

Q. Find the area above x-axis under the curve.

x2 + y

2 = 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�


X0XI = √0 + # √)I − ' , s=7 when t=3

Q. Find the area above x-axis under the curve.

x2 + y

2 = 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/�#�G4� H10#�X� (ii) l P&X�B�5�P##� (iii)

l �� + ��√�# + #� + ##� X� Q. Evaluate any Two of the following:

(i) l�& √) + �#X� (ii) l Xoo#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��#&√&#�


�# XPX� = �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. Q. Evaluate any Two of the following:

(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.


XPX� = B�P − #P − &� + 5 , y=12 when x=6

2004


P a g e | 26 `


(i) l ��# + &� + '��#& *� + &#+X�#� (ii) l I@��X�G4� (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 + y

2 = 9 between the ordinates x = 0.5 and x =

1.5.

Q. Solve the differential equation XPX� = sin

2y. cos

2 x sin x.

OR XPX� = x + sin x , y = 3 when x = 0.

2005


(i) l X�√��√� (ii) l &��'√��X� (iii) l0/�'�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.


XPX� = B��H10#P0/�#P , P = G# �� = &

2006


(i) l X�√��√� (ii) l I@�&� 0�H&�X� (iii) l B�#�7� X�


(i) 4cos xdx∫ (ii)

32 xx 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= 6and

π x =3

π

Q. Solve the differential equation dy

dx = , (9) 100xy y =

2007

Q. Evalute any two of the following:

(i) l 0/�'&�. H10&&�G5� X� (ii) l �#�4��#�&#

�� (iii)

l H10#�D�� X�


(i) l5�' ��&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 � = G5 @�X� = G4


XPX� = B��H10P0/�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:

yXPX� = x(y

4+2y

2+1) and y(-3) = 1

2009


(i) cosx xe e dx∫∫∫∫ (ii) 1

0

tanx xdx

ππππ−−−−∫∫∫∫ (iii)

21

x

x

e dx

e++++∫∫∫∫


(i) 3

01 sin

dx

x

ππππ

−−−−∫∫∫∫ (ii) (((( ))))

cos

sin 2 sin

x dx

x x++++∫∫∫∫ (iii) 1 sin

1 cos

x xe dx

x

++++++++∫∫∫∫

Q. Solve the differential equation (((( ))))2

2

1, 1

dyx ydx y y

====++++

.

Q. Find the area above the x-axis between the ordinates 4

xππππ

==== and 3

xππππ

==== 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 XPX� is called:

(a) polynomial equation (b) differential equation

(c) exponential equation (d) logarithmic function

(xviii) l�UX� =:��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

xdx

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

xdx

x∫


( ) ( )4 22 1 , 3 1dyy x y y ydx

= + + − =

Q. Find the area above the x-axis between the ordinates 4

xπ

= under the curve tany x= .

P a g e | 28 `


2011

MCQS: (xii) l0/�&�� 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�


XPX� = B��H10P0/�P ��p�P�&� = G#

Q. Find the area above the x-axis under the curve f(x)=3x4-2x

2+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 XPB4P�P#


(a) l X��#�4��' (b)l�� 0/��H10� (c) l H10�X�0/��#�0/�� OR l I@�&�0�H�X�G�


# + #P XPX� = � + &�#, P�#� = �

Q. Find the area above the x-axis under the circle x2+y

2=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) lH104�H10#�X�


(a) l H10�X�0/��#�0/�� (b) l 0�H�I@��X�@�b0�H� (c) l H104�G#� X�

Q. Solve the differential equation: XPX� = 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 XPX� = � + &�#, P�#� = � OR XoXs = √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+y

2+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 + y

2 + 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 3x

2−y2=12 and x

2+3y

2-24=0 intersect at right angles. Also find the point of

intersection.

1995


2 + 8x – 10y + 33 = 0 and touching

the x-axis.

Q. Find the equations of the tangents to the circle x2 + y

2 – 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+y

2– 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+y

2+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


2 + 6x – 10x + 33 = 0 and touching

the line y = 0.

Q. Prove that the two circles x2 + y

2 + 2gx + c = 0 and x

2 + y

2 + 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 + y

2 + 8x + 10y – 53 = 0.

Q. Find the equation of the circle concentric with x2 + y

2 + 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+y

2-12y-23 = 0.

Q. Prove that the two circles x2 + y

2 + 2gx + c = 0 and x

2 + y

2 + 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


2 – 4x – 6y – 23 = 0 and touching x-

axis.

Q. Prove that if wo circles x2 + y

2 + 2gx + c = 0 and x

2 + y

2 + 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 28 10 53 0x y x y+ + + − =+ + + − =+ + + − =+ + + − = .

Q. Prove that if two circles 2 22 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+y

2-6x+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+y

2 +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 28 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 22 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+y

2+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+y

2+8x+7 =0 (b) x

2+y

2+9y+11=0

(c) x2+y

2+8x+11y=0 (d) x

2+y

2+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+3y

2-24=0 and 3x

2+y

2=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+y

2 +3=0.

(a) 3 (b) 4 (c) 5 (d) 6

(x) The centre of the circle 2x2+2y

2+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+y

2+6x-10y+33=0 and

touching the y-axis.

2013

Mcqs: (x) Centre of the circle x2+y

2 +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.


2 – 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


2 -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�# + bfP# = � 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@Io0p�HIo= = #66' , Ip@�0s�p0�@�/0/0@D1�2P − @�/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 l

2 + b

2 m

2 – n

2 = 0.

1994

Q. Show that the equation ax2+by

2+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 – y

2 = 12 and x

2 + 3y

2 – 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 + 13y

2 = 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.


a2 l

2 + b

2 m

2 – n

2 = 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+25y

2=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 – 16y

2 – 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 + by

2 = 1 should cut the conic a’x

2 + b’y

2 =1 orthogonally.

Q. Find the eccentricity, foci and directrices of the hyperbola 16 x2 – 9y

2 = 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 – 16y

2 + 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 – y

2 = 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 + y

2 + 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 = - 4c

2.

Q. Find the eccentricity, the semi-axes, the centre, the vertices and co-ordinates of the foci of the

ellipse:

4x2 – 32x + 25y

2 – 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 `


Q. Prove that the line lx + my + n = 0 and the ellipse �##' + P#7 = � have one point common if 25l

2 +

9m2 – n

2 = 0.

Q. Find the eccentricity, foci and directrices of the hyperbola 9y2 – 16x

2 = 144.

2003

Q. Find the condition that the two conics ax2 + by

2 = 1 and a’x

2 + b’y

2 = 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 – 36y

2 + 48x + 180y – 225 = 0.

Q. Find the equations of the tangent and normal to the hyperbola x2 – y

2 = 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 = � .

Q. Prove that the line lx + my + n = 0 and the ellipse �#@# + P#b# = � have just one point in

common if a2/2

+ b2 m

2 = n

2.

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 – 9y

2 = 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 = -4c

2.

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 16x

2 – 9y

2 = 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 + PV = � 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 − 36y

2 + 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 + by

2 = 1 should cut a′x

2 + b′y

2 = 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

3e ==== , 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 b

2=a

2(1-e

2), 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 236y 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 16x2−9y

2=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+25y

2+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+y

2+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−y

2+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+7y

2=225.


a2l2 + b

2 m

2 - n

2=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. Prove that: "@x + b],b] + H], H] + @x$ = #"@x, b], H]$ 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

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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. Prove that: "@x + b],b] + H], H] + @x$ = #"@x, b], H]$ 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�̂ + �̂ − &w� and &�̂ + �̂ − w� and is displaced from

the point �̂ + #�̂ + &w� to the point '�̂ + 4�̂ + w�; 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−b

2 (b) 0 (c) @x × bx (d) #�@x × bx�

Q. Find the unit vector perpendicular to both the vectors @x = �̂ − &�̂ + #w� and bx = −&�̂ + #w� and

find sin(a,b).


vectors @x = #�̂ − &�̂ + 4w� and bx = &�̂ − �̂ + #w�.

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 = �̂ + w�

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�̂ + �̂ − &w� and &�̂ + �̂ − w� 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: �̂ + #�̂ + #w� and &�̂ − #�̂ −4w�. Also find sine of the angle between them.

OR Simplify: ~@x, #b]]]] −&H]]]], −#@]]]] + bx + H]�

24 years past papers in accordance with the …pgseducation.com/assign-formulae/2ndyear20yrs.pdf ·...

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