Page 1 of 25
01. If A,B,C A,B,C and D denotes the interior interior angles angles of a quadrilateral quadrilateral then then
tan A tan A cot A (C) cot A tan A tan A
tan A ( tan A)( cot A) (D) tan A tan A cot A
(A)
(B)
02. For every X R the value of the expression y
03.
x 2 8
x cos x cos 2 x is never
less than (A) -1 (B) 0 (C) 1 (D) 2 One side of a rectangular piece of paper is 6cm, the adjacent sides being longer than 6 cms. One corner of the paper is folded so that it sets on the opposite longer side. If the length of the crease is l cms and it makes an angle with the long side as shown, then l is (A)
3 sin cos cos 2
(B)
6 sin 2 cos
l
6
(C) 04.
3 sin cos
06.
07.
(D)
3 sin 3
The average of the numbers n sin n for n=2, 4, 6,…180
(A)1 05.
(B) cot1
(C) tan1
1
(D)
2
If u, v, w are real distinct numbers such that u v w 3uvw then the quadratic equation ux 2 vx w 0 has (A) real roots (B) roots lying on either side of unity (C) both roots are negative (D) non-real roots Let P(x) be polynomial polynomial of degree degree 4 with with leading leading coefficient coefficient 1. Given that P(1)=1, P(2)=3, P(3)=5 and P(4)=7. The value of P(5) will be (A) 9 (B) 5 (C) 24 (D) 33 Let an 16,4,1..... be a geometric sequence. Define P n as the product of the 3
3
3
first n terms. The value of
n
P n
n 1
08.
(A) 8 (B) 16 (C) 32 (D) 64 Let s1 , s2 , s3 ... ... and t1 , t2 , t 3 ...... are two arithmetic arithmetic sequences such that s1
t1 0; s2 2 t 2 and
10
value of s t . Then the value t i 1
09.
s2 s1
15
i
i 1
i
2
t 1
is
(A) 8/3 (B) 3/2 (C) 19/8 (D) 2 Starting with a unit square, a sequence of square is generated. Each Each square in the sequence has half the side length of its predecessor and two of its sides bisected by its predecessor’s side as shown. This process is repeated
Page 2 of 25
indefinitely. indefinitely. The total area enclosed by all the squares in limiting situation, is 5
(A) sq.units
(B)
4
79 64
sq.units
1 1
(C) 10.
11.
75 64
sq.units
(D)
1 12
sq.units
Let triangle ABC be an an isosceles isosceles triangle triangle with AB=AC. Suppose Suppose that the angle bisector of its angle B meets the side AC at a point D and that BC=BD+AD. Measure of the angle A in degrees, is (A) 80 (B) 100 (C) 110 (D) 130 In the figure figure ∆ABC ∆ABC is a right triangle at C. A semicircle semicircle with with centre centre O is a tangent to the side AC and BC. If the area of the triangle is∆, then the radius of the semicircle is 2 (A) 2 (B) 2 c 2 c 2 C
A O
(C) 12.
13.
14.
2 c 2 4
(D)
B
2 c2
The true set set of real values of of a such that the point point M (a, sin a) lies lies inside inside the triangle formed by the lines x-2y+2=0, x+y=0, and x y 0 , is (A) (0, (B) , (0, ) 3 2 2 (C) o, , (D) , , 2 6 3 2 2 3 Let ABC be a fixed fixed triangle triangle and and P be variable variable point in the plane of triangle triangle ABC. suppose a, b, c are lengths of sides BC, CA, AB opposite to angles A, B, C respectively. If a( PA) 2 b( PB) 2 c( PC ) 2 is minimum, then the point P with respect to ABC, is (A) centroid (B) circumcentre (C) orthocentre (D) incentre
If the points of intersection of lines L1 : y m1 x k 0 and L2 : y m2 x k 0 ( m1 m2 ) lies inside a triangle formed by the lines 2x+3y=1, x+2y=3 and 5x-6y-1=0, then true set of values of k are 1 3 1 3 3 (A) , (B) ,1 (C) 0, (D) , 0 3 2 2 2 2
Page 3 of 25
15.
16.
17.
Given A(0,0) and B(x,y) with x (0,1) and y>0. Let the slope of the line AB equals m1 . Point C lies on the line x=1 such that the slope of BC equals m2 where 0 m2 m1 . If the area of the triangle ABC can be expressed as (m1 m2 ) f(x), then the largest possible value of f(x) is (A)1 (B) 1/2 (C) 1/4 (D) 1/8 Locus of a point P(x, y) satisfying the equation x y 2 24 y 144 13 x 2 y 2 10 x 25 is (A) a finite line segment (B) an infinite line segment (C) part of a circle with finite radius (D) pair of straight lines In a triangle ABC with fixed base BC, the vertex A moves such that cos C cos B cos 2
18.
A 2
. If a, b, and c denote the lengths of sides of the triangle
opposite to angles A, B and C, respectively, then which one of the following is correct? (A) b-c=2a (B) 2b-2c=a (C) Locus of point A is circle (D) Locus of point A is pair of straight line The locus of the point of intersection of the tangent to the circle x 2 y 2 a 2 , which include an angle of 45 is the curve ( x 2 y 2 ) 2 a 2 ( x 2 y 2 a 2 ) . The value of is (A) 2 (B) 4 (C) 8 (D) 16 Equation of the circle which bisects the circumference of the circle x 2 y 2 2 y 3 0 and touches the curve y tan(tan 1 x) at the origin is (A) 2( x 2 y 2 ) 5 x 5 y 0 (B) x 2 y 2 5 x 5 y 0 (C) x 2 y 2 5 x 5 y 0 (D) none Let a and b represent the length of a right triangle’s legs. If d is the diameter of a circle inscribed into the triangle, and D is the diameter of a circle superscribed on the triangle, then d+D equals
19.
20.
(A) a+b
(B) 2(a+b) (0,a) A
(C)
21.
1
(a b )
(D) a 2 b 2
O
B
x (b,0)
2 Let pn denotes the number of ways in which three people can be selected
out of ‘n’ people sitting in a row, if no two of them are consecutive. If, Pn 1 P n 15 then the value of ‘n’ is (A) 7 (B)8 (C) 9 (D) 10
Page 4 of 25 8
22. 23.
24.
1 1 The constant term in the expansion of x 2 2 y is x y (A) 4900 (B) 4950 (C) 5050 (D) 5151 1 4 2 The range of the function, f(x)= cot log 0.5 ( x 2 x 3) is 3 3 3 (A) (0, ) (B) 0, (C) , s (D) , 4 4 2 4 1 2 The set of values of x, satisfying the equation tan (sin x) 1 is
(B)
(A) [-1, 1]
25.
2 2
,
2 2
2 2 2 2 , (C) (1,1) , (D) [1,1] 2 2 2 2 Which of the following functions defined below is NOT differentiable at the indicated point? f ( x)
x 2 if
x
(A)
2
1 x 0
if 0 x 1
x 2 sin
g ( x)
at x 0
1 x
0
if
x0
if
x 0
at x 0
(B) h( x )
sin 2 x if x 0 2 x if x 0
at x 0
(C) x if 0 x 1
k ( x)
at x 1
2 x if 1 x 2
26.
(D) Which one of the following functions is discontinuous for atleast one real value of x? (A) f ( x) 1 sgn x
(B) g ( x)
e x 1 e x
3
5
22 x 1 7 (C) h( x) 3 x 2 5 27.
f ( x) x 2 sin
Let
g ( x) x tan 1
28.
1 x
(D) k ( x) 3 2 sin x 1
; x cot 1 x, x 2tan1
x 0 f (0) g (0) 0
for
x 1 x
; x cot 1 x 2 / x 0 where
g(x)
is
and an
even
function, then (A) f is derivable at x=0 but g is not (B) g is derivable at x=0 but f is not (C) both f and g are derivable at x=0 (D) nither f nor g is derivable at x=0 If f x y f x f y x y xy 2 , x, y R and f ` (0) 0 , then (A) f need to be differentiable at every non zero x
Page 5 of 25
29.
30.
(B) f is differential for all x R (C) f is twice differentiable at x = 0 (D) none The number of values of a for the which the function f ( x) ( x 1) | x a | is differentiable x R s is, (A)0 (B) 1 (C) 2 (D) more than 2 f : (1, ) (0, ) Let be a continuous decreasing function with Lim x
f (4 x)
f (6 x)
1 .Then Lim 1 is equal to f (8x) f (8x) x
(A) 31.
8
(B)
6
(B) 1
(C)
(C)
6
(D) 1
8
1
2
(D)
1 4
Suppose the function f(x)-f(2x) has the derivate 5 at x=1 and derivative 7 at x=2. The derivative of the function f(x)-f(4x) at x=1 has the value to equal to (A) 19 (B) 9 (C) 17 (D) 14 /2
33.
4
Let f :( 3,3) R be a differentiable function with f(0)=-2 and f`(0)= -1.and g ( x) ( f (3 f ( x) 6)) 3 . Then g`(0) is equal to (A) 0
32.
4
2 2 x dx and v Let u cos sin 3 0 u and v is (A) 2u=v (B) 2u=3v
/2
cos 3 sin x dx , then the relation between 0
(C) u=v
(D) u=2v
/3
34.
The value of the definite integral
ln(1
3 tan x) dx equals
0
(A) 35.
3
ln 2
(B)
3
(C)
2 6
ln 2
(D)
2
ln 2
Let f be a one-to-one continuous function such that f(2)=3 and f(5)=7. 5
Given f ( x) dx 17 , then the value of the definite integral 2
(A) 10
(B) 11
(C) 12
7
f
1
( x) dx equals
3
(D) 13
/2
36.
The value of the definite integral
tan x dx is
0
(A) 2
(B)
2
(C) 2 2
(D)
2 2
6
y
i
6
37.
If
(sin i 1
1
xi
cos
1
e x yi ) 9 ,then x ln(1 x ) dx is equal to 1 e 2 x x i 1
2
6
i
i 1
(A) 0 38.
(B) e6 e 6
(C) ln
37
(D) e6 e6
12 At the point P(a, a ) on the graph of y x n ( n N ) in the first quadrant a n
normal is drawn. The normal intersects the y-axis at the point (0,b). If
Page 6 of 25
Lim b a 0
39.
2
, then n equals
(A) 1 (B) 3 (C) 2 (D) 4 Let g :[1,6] [0, ) be a real valued differentiable function satisfying g `( x)
40.
2 x g ( x)
and g(1)=0, then the maximum value of g cannot exceed
(A) ln 2 (B) ln 6 (C) 6 ln 2 (D) 2 ln 6 3 2 Let f ( x) x 3 x 2 x . If the equation f(x)=k has exactly one positive and one negative solution then the value of k equals 2 3
(A)
(B)
9
41.
9
3 1
2 2
(B) sin
44.
45.
46.
2 3 3
(D)
1 3 3
5 1
(C) sin
2
5 1
(D) sin
2
5 1 4
Two sides of a triangle are to have lengths ‘a’ cm & ‘b’ cm. If the triangle is to have the maximum area, then the length of the median from the vertex containing the sides ‘a’ and ‘b’ is a 2 b2 1 2a b a 2b 2 2 a b (A) (B) (C) (D) 2
43.
(C)
P and Q are two points on a circle of centre C and radius α, the angle PCQ being 2θ then the radius of the circle inscribed in the triangle CPQ is maximum when (A) sin
42.
2
3
2
3
If f be a continuous on [0,1], differentiable in (0,1) such that f(1)=0, then their exists some c (0,1) such that (A) c f` (C)-f(C)=0 (B) f`(C)+c f(C)=0 (C) f` (C)-c f (C)=0 (D) c f`(C)+f(C)=0 Let A and B be 3 3 symmetric matrices such that X=AB+BA and Y=AB-BA, then ( XY )T is equal to (A) XY (B) YX (C) –XY (D) –YX
3 Let A 2 1 2 3 1 (A) 2 2 1 0
2 3 2
1
1 0
1
B
and C AB AT , then AT C 3 A is equal to 1
(B)
1 3 2
0
1
1 (C) 0
3 2 3
1 3 (D) 0 1
Given planes P1 : cy bz x P2 : az cx y P3 : bx ay z
P 1 , P 2 , and P 3 pass through one line, if
(A) a 2 b 2 c 2 ab bc ca 2 2 2 (C) a b c 1
(B) a 2 b 2 c 2 2abc 1 (D) a 2 b 2 c 2 2ab 2bc 2ca 2abc 1
Page 7 of 25
47.
Suppose families always have one, two or three children, with 1 1
1
4 2
4
probabilities , and respectively. Assume everyone eventually gets married and has children, the probability of a couple having exactly four grandchildren is (A) 48.
49.
(B)
37 128
(C)
25 128
(D)
20 128
1 105
(B)
2 105
(C)
3 105
(D)
4 105
A jar contains 2 yellow candies, 4 red candies and 6 blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the 2 yellow candies will be eaten before any of the red candies are eaten, is (A)
51.
128
Shalu bought two cages of birds: cage I contains 5 parrots and 1 owl, and cage II contains 6 parrots, as shown one day shalu forgot to lock the both cages and two birds flew from cage I to cage II. Then two birds flew back from cage II to cage I. Assume that all birds have equal chance to flying, the probability that the owl is still in cage I, is (A) 1/6 (B) 1/3 (C) 2/3 (D) 3/4 There are 8 students from 4 schools A,B,C,D, 2 students from each school. Let these 8 students enter in 4 rooms R1 , R2 , R3 , R4 , so that each room will have 2 students. The probability that each room have students from the same school, is (A)
50.
27
1 18
(B)
1 15
(C)
1 24
(D)
1 36
A butterfly randomly lands on one of the six squares of the T-shaped figure shown and then randomly moves to an adjacent square. The probability that the butterfly ends up on the R square is G Y
W
R
B
O
52.
53.
(A) 1/4 (B) 1/3 (C) 1/5 (D) 2/5 A function y=f(x) satisfies the differential equation with initial condition y(0)=0.The f ( x).sin 2 x cos x (1 sin 2 x) f `( x) 0 value of f ( / 6) is equal to (A)1/5 (B) 3/5 (C) 4/5 (D) 2/5 Water is drained from a vertical cylindrical tank by opening a value at the base of the tank. It is known that the rate at which the water level drops is proportional to square root of water depth y, where the constant of proportionality k > 0 depends on the acceleration due to gravity and the geometry of the hole. If it is measured in minutes and k
1 15
then the times to drain the tank if the water is 4 meter deep to start
Page 8 of 25
with is (A) 30 min
(B) 45 min
(C) 60 min
(D) 80 min x
54.
( x t 1) g (t )dt x
Let g be a differentiable function satisfying
4
x2 for
0
1
all x 0 . The value of
12
g `( x) g ( x) 10 dx is equal to 0
(A) 55.
6
(B)
3
(C)
4
(D)
and
f(x)
[ f ( x) g( x)]dx 10 0
57.
59.
4
(B) x 2 ( y 3)
(C) x 2
3
and [ g ( x) f ( x)]dx 5 , the 0
4 3
( y 3)
4
(D) x 2 ( y 3) 3
Write the correct order sequence of the statement in respect of the statement given below. F stands for false and T stands for true. If a variable circle is described to pass through the point (1,0) and tangent to the curve y tan(tan 1 x) . The locus of the centre of the circle is a parabola whose I. length of the latus rectum is 2 2 II. Axis of symmetry has the equation x+y=1 III. Vertex has the co-ordinates (3/4, 1/4) IV. Directrix is x-y=0 (A) T F F F (B) F T T F (C) F T T T (D) F T F T 2 If a normal to a parabola y 4ax makes an angle with its axis, then it will cut the curve again at an angle
1
(A) tan 1 (2tan ) 60.
4
area between two curves for 0
58.
2
Suppose y=f(x) and y=g(x) are two continuous functions whose graphs intersect at the three points (0,4), (2,2) and (4,0) with f(x)> g(x) for 0
56.
1
(B) tan 1 tan (C) Cot 1 tan (D) none 2 2 The area of the rectangle formed by the perpendicular from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is / 4 is (A)
(a 2 b 2 ) ab
a 2 b2
b2 ) (B) 2 2 (a b ) ab (a
2
b2 ) (C) ab(a 2 b 2 ) (a
2
b2 ) (D) 2 2 (a b ) ab (a
2
Page 9 of 25
61.
For each positive integer n, consider the point P with abscissa n on the curve` y 2 x 2 1 . If d n represents the shortest distance from the point P to the line y=x then Lim(n.d n ) has the value equal to n 8
1
(A) 62.
63.
2 2
| z2 z3 |
1
(C)
2
(D) 0
2
4 | z3 z1 |
5 | z1 z 2 |
, then the value of
9 z2 z 3
16 z3 z1
25
,
z1 z 2
equals
(A) 0 (B) 3 (C) 4 (D) 5 If P and Q are represented by the complex numbers z 1 and z 2 such that 1 z1
(A) 65.
1
(B)
Number of values of z (real or complex) simultaneously satisfying the system of equations 1 z z 2 z 3 ...... z 17 0 and 1 z z 2 z 3 ...... z 13 0 is (A) 1 (B) 2 (C) 3 (D) 4 z1 , z2 , z 3 are If 3 distinct complex numbers such that 3
64.
1 z2
1 z1
1 z 2
then the circumcentre of OPQ (where O is the origin) is
z1 z 2
z1 z 2
(B)
2
(C)
2
z1 z 2
(D) z1 z 2
3
On the Argand plane point ‘A’ denotes a complex number z 1 . A triangle OBQ is made directly similar to the triangle OAM, where OM=1 as shown in the figure. If the point B denotes the complex number z 2 , then the complex number corresponding to the point ‘Q” is B(z2)
Q(z) O
(A) z1 , z 2 66.
67.
68.
z 1
(B)
1
(C)
z 2
A(z1)
1M
z 2 z 1
(D)
z1 z 2 z 2
An integer m is said to be related to another integer n if m is a multiple of n. Then the relation is (A) reflexive and symmetric (B) reflexive and transitive (C) symmetric and transitive (D) an equivalence relation If A={1,2,3,4}, then which of the following is a function from A to itself? (A) f1 {( x, y) : y x 1} (B) f 2 {( x, y) : x y 4} (C) f3 {( x, y ) : y x } (D) f 4 {( x, y ) : x y 5} For all n N,
n5 5
n3 3
7 15n
(A) an integer (C) a positive fraction
is (B) a natural number (D) a rational number
Page 10 of 25
69.
AB is a vertical pole. The end A is on the level ground C is the middle point of AB.P is a point on the level ground. The portion BC students angle β at P. If AP=nAB, then tan β = (A)
70.
71.
n 2n
74.
75.
n n
2
(C)
1
n n
2
1
(D) none of these
(B)
2
(C) 5
3
(D) data insufficient
The arithmetic mean of a set of observation is x . If each observation is divided by α and then is increased by 10, then the mean of the new series is (A)
73.
1
(B)
The mean height of 25 male workers in a factory is 61 cm, and the mean height of 35 female workers in the same factory is 58 cm. The combined mean height of 60 workers in the factory is (A) 59.25 (B) 59.5 (C) 59.75 (D) 58.75 If the standard deviation of n observation x1 , x2 ,..... xn is 4 and another set of n observations y1 , y2 ,..... yn is 3 the standard deviation of n observations x1 y1 , x2 y2 ......xn yn is (A) 1
72.
2
x
(B)
x 10
(C)
x 10
(D) x 10
Which of the following is the inverse of the proposition “if a number is a prime, then it is odd”? (A) If a number is not a prime, then it is odd. (B) If a number is not odd, then it is odd (C) If a number is prime, then it is prime (D) If a number is not odd, then it is not a prime ( p q) p p is (A) a tautology (B) a contradiction (C) neither a tautology nor a contradiction (D) cannot come to any conclusion Which of the following is always true? (A) ( p q) q p (B) ( p q) p q (C) ( p q ) p q (D) ( p q) p q The graph y px2 qx r , x R of is plotted in adjacent diagram. Given AM=2 and CM=1 Y
C A B
O (0,-8)
M
X
Page 11 of 25
76.
77.
Which of the following statement(S) is/(are) correct? (A) The value of (4p-r) is equal to 7. (B) The value of (4p-r) is equal to 5. (C) The sum of roots of equation px 2 qx r 0 is equal to 10. (D) The sum of roots of equation px 2 qx r 0 is equal to 12. Which of the following statement (S) is/(are) incorrect? (A) The value of Lim px2 qx r is not equal to zero x 8
(B) The inequality px 2 qx r 0 is true for all x (6, ) 32
(C) Harmonic mean of roots of the equation px 2 qx r 0 is
78.
(D) The value of q is equal to -3. In a triangle ABC, let tan A=1, tan B=2, tan C=3. And C=3 Area of the triangle ABC is equal to (A)
79.
3 2 2
(B) 3
(C) 2 3
(D) 3 2
The radius of the circle circumscribing the triangle ABC, is equal to (A)
80.
3
10 2
(B) 5
(C) 10
(D)
5 2
Let denote the area of the triangle ABC and p be the area of its pedal triangle. If k p then k is equal to (A) 10 (B) 2 5 (C)5 (D) 2 10 A line L1 with gradient 1 intersects a line L2 with gradient The line L2 intersects the line L3 with the gradient
1 3
1 2
at the point A.
at B and L3 intersects
L1 at the point C, so that a triangle ABC is formed. The abscissa of the
points A, B and C are 9,7 and 6 respectively. Let p be the x-coordinate of the circumcentre and h be the x co-ordinate of the orthocenter 81.
The value of (sin 2A+sin 2B+sin 2C) is equal to (A)
82.
4 25
(B)
1 25
The value |p+h| is equal to (A) 8 (B) 10
(C)
2 25
(C) 12
(D)
3 25
(D) 14
A path of length n is a sequence points ( x1, y1 ), (x2 , y2 )....., (xn , yn ) with integer coordinates such that for all i between 1 and n-1 both exclusive, either X i 1 xi 1 and yi 1 yi (in which case we say the ith step is rightward) or X i1 xi 1 and yi 1 yi 1 (in which case we say that the ith step is upward) This path is said to start at ( x1 , y1 ) and end at ( xn , yn ) . Let (a, b), for
Page 12 of 25
a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a, b) 10
83.
The value of p (i,10 i) is i 0
84.
85.
(A) 1024 (B) 512 (C) 256 (D) 128 Number of ordered pairs (i, j) where i ≠ j for which P(i,100-i)=P(j, 100-j) is (A) 50 (B) 99 (C)( 100 (D) 101 The sum P 43, 4
5
P(49 j, 3) is equal to j 1
(A) P(4,18) (B) P(3,49)
(C) P(4,47)
(D) P(5,47)
Consider a function, f ( x) ln 3/ 2 (sin x cos x) for x o, 3 2 2
86.
The value of Lim x
(A) is
2 3
2
f ( x)
is
3/ 2
2 x
(B) is 1
(C) is
2 3
(2) 3/ 4
(D) non existent
3 f ( x) 2 e
87
3 2 f x 2
2
The functioning g(x) is defined as g(x)= (A) is continuous at x
2
2
x 0,
2/3
, 2 .Then g(x)
; x
2
(C) g(x) is discontinuous at x
2
2
and jump of discontinuity is equal to
(D) has a non-removable discontinuity at x
2
2
The range of g(x) is (A) (0, 1]
89.
2/ 3
for g 0
(B) has a removable discontinuity at x
88.
1 1 ,1 1 (C) ,1 (D) 0, ln 2 2 2 2 be a differentiable function and satisfies 3 f(x +
1
(B) 0, ln 2 2
Suppose f : R R
1 and y)=f(x).f(y) for all x, y R with f(1)=6. If U Lim n f 1 f (1) n
n
3
V
f ( x)dx then find 0
90.
The range of f(x), is (A) (0, ) (B) ( 6, )
(C) (12, )
(D) [0, )
Page 13 of 25
91. 92.
The value of U, is (A)2 ln 2 (B) ln 2 The value of the product UV, is (A) 96 (B) 112
(C) 6 ln 2
(D) 8 ln 2
(C) 120
(D) 126
Let f(x) be a monic polynomial of degree 3 having extremum at x
93.
1 3
,1 and f(2)=0
Which of the following statement is correct? 1 (A) f(x) is increasing function on ,1 3 (B) f`(x) has a local maximum at x
1 3
(C) f(x) is negative on (, 2) and positive on (2, ) 1 (D) f`(x) is decreasing on , 3
94.
Number of distinct real roots of the equation f(f(x))=0, is (A) 1 (B) 3 (C) 5 (D) 9 2
95.
The value of definite integral f ( x) f ( x) dx , is equal to 0
(A) 0
96.
97.
98.
(C) -4
(D) -1
Suppose you have 10 keys and you wish to open a door and try the keys one at a time, randomly until you open the door. Only one of the keys will open the door If you eliminate each unsuccessful trials then the probability that the door gets opened on the 5th, trial, is given by 4 1 1 1 9 1 10 (A) (B) (C) (D) C 5 . . 10 5 2 10 10 If you do not eliminate the each unsuccessful trial from the remaining keys then the probability that the door gets opened on the 5 th trial, is given by 4 4 1 1 9 1 9 1 10 (A) (B) (C) C 5 . . (D) . 2 10 10 10 10 10 2 Tangents are drawn to the parabola y 4 x from the point P(6,5) to touch the parabola at Q and R. C 1 is a circle which touches the parabola at Q and C 2 is a circle which touches the parabola at R. Both the circles C 1 and C 2 pass through the focus of the parabola Area of the ∆PQR equals (A)
99.
(B) 4
1 2
(B) 1
1
(C) 2
(D)
(C) 10 2
(D) 210
4
Radius of the circle C 2 is (A) 5 5
(B) 5 10
Page 14 of 25
100. The common chord of the circles C 1 and C 2 passes through the (A) incentre of the ∆PQR. (B) circumcentre of the ∆PQR (C) centroid of the ∆PQR (D) othocentre of the ∆PQR 101. Statement 1: If x1 , x2 , x3 are non-zero real numbers such that 3( x12 x2 2 x3 2 1) 2( x1 x 2 x3 x1 x2 x2 x3 x 3x1) then x1 , x2 , x3 are in A.P as well as in G.P Statement 2: A sequence is in A.P as well as in G.P. If all the terms in the sequence are equal and non-zero. (A) Statement 1is true, statement 2is true and statement 2 is correct explanation for statement 1. (B) Statement 1 is true, statement 2 is true and statement 2 is NOT the correct explanation for statement 1. (C) Statement 1 is true, statement 2 is false (D) Statement 1 is false, statement 2 is true. 102. Statement 1: There lies exactly 3 points on the curve 8 x3 y 3 6 xy 1 , which form an equilateral triangle. Statement 2: The locus of all point P(x, y) satisfying 8 x3 y 3 6 xy 1 consists of union of a straight line and a point not on the line (A) Statement 1is true, statement 2 is true and statement 2 is correct explanation for statement 1. (B) Statement 1 is true, statement 2 is true and statement 2 is NOT the correct explanation for statement 1. (C) Statement 1 is true, statement 2 is false (D) Statement 1 is false, statement 2 is true. 103. Consider the curves C1 : x
2
y 2 3
a 2 and C2 : xy 3 C
Statement 1: C 1 and C 2 are orthogonal curves Statement 2: C 1 and C 2 intersect at right angles everywhere wherever they intersect. (A) Statement 1is true, statement 2 is true statement 2 is correct explanation for statement 1. (B) Statement 1 is true, statement 2 is true and statement 2 is NOT the correct explanation for statement 1. (C) Statement 1 is true, statement 2 is false (D) Statement 1 is false, statement 2 is true. 104. Let n denotes the interior angles (in degree) of a regular polygon of side n, then which of the following is (are) correct? (A) The value of tan 16 cot 16 is equal to 2 2
(B) The value of cos ec 24 is equal to 6 2
Page 15 of 25
(C) The value of Cot ( 8 ) sec( 6 ) is equal to 1.
5
( 5 ) cos( 10 ) is equal to (D) The value of Cos
105. Let E cos 2
7
cos 2
7
cos 2
3 7
. Then
2
which of the following alternative (S)
is/are incorrect? (A)
1 2
E
3
(B)
4
3 4
E 1
(C) 1 E
3
(D)
2
3 2
E
7
4 106. For which of the following graphs of quadratic expression y ax 2 bx c,
then product abc is positive? y
y (0,0)
(A)
(0,0)
x
(B)
y
y
(0,0)
(C)
x
(0,0) x
x
(D)
107. Which of the following statement (S) is (are) correct? (A) Sum of the reciprocal of all the n harmonic means inserted between a and b is equal to n times the harmonic mean between two given numbers a and b. (B) Sum of the cubes of first n natural number is equal to square of the sum of the first n natural numbers. 2n
(C) If a, A1 , A2 , A3 ,.... A2n , be are in A.P then
A n(a b) i 1
i
(D) If the first term of the geometric progression g1 , g 2 , g 3...... is unity, then the value of the common ratio of the progression such that (4 g 0 2 5 g 3 ) is minimum equals
2 5
108. If the roots of the equation, x3 px 2 qx 1 0 form an increasing G.P. where p and q are real, then (A) p + q=0 (B) p (-3, ∞) (C) one of the roots is unity (D) one root is smaller than 1 and one root is greater than 1. 109. Difference between the sum of the squares of the first fifty even natural numbers and the sum of the square of the first fifty odd natural numbers
Page 16 of 25
is equal to (A) The value of the expression y
x 2
1 x 1 ( x 1) 1 ( x 2)( x 4) where
x=100 (B) f(100) where f(1) =1 and f(x)=x+f(x-1) (C) the sum of all such friendly natural numbers which lie in the interval (0,101), where ‘m’ defines a friendly natural number satisfying the inequality mx 2 4 x 3m 1 0 , for every x R (D) Sum of the reciprocals of all the 100 harmonic means if these are inserted between 1 and 1/100 110. In a ∆ABC, a semicircle is inscribed, whose diameter lies on the side c. If x is the length of the angle bisector through angle C then the radius of the semicircle is (A)
abc
(B)
4 R 2 (sin A sin B )
(C) x sin
C
(D)
2
4 5 sgn | x 3 |,
111. Let f(x) =
x
2
if | x | 3 if | x | 3
x 2 s( s a)( s b)( s c) s
and g ( x) 2 tan 1( e x )
2
for all x R, then
which of the following is (are) correct? (A) fog(x) is an even function (B) gof (x) is an even function (C) gog (x) is an odd function (D) fof (x) is an odd function 112. Which of the following pair(S) of function is (are) identical ? x
(A) f(x)=sin (tan 1 x), g ( x)
(B) f ( x) sgn(cot
1
1 x 2
x ), g (x ) sec x tan x, where sgn x denotes signum function 2
2
of x.
x 2 1 x 2 1
ln cos1
x2 1 , g ( x) cos 2 (C) f(x) = e 0 x 1 2 x , g ( x) 2 tan 1 x (D) f(x) = sin 1 2 1 x 113. Let {an },{bn },{cn } be sequences such that (i) an bn cn 2n 1 (ii) anbn bn cn cn an 2n 1 (iii) an bn cn 1 and (iv) an bn cn then which of the following is/are correct ? (A) lim n
an n
1
1
(B) lim n
2
an n
2
(C) lim
n
an n
0
(D) lim
n
an n
2
/ 4
114. Let I n
(tan x) dx and n
let J n (1) n I 2 n for n=0,1,2 then which of the
0
following hold(S) goods?
Page 17 of 25
(A) I n I n 2
1
(B) J n J n 1
n 1 1
(C) If u tan then I n
un
(1) n 2n 1
for n 1
du (D) lim J n 0
1 u2 0
x
115. A function f is defined by f ( x) cos t cos ( x t) dt, 0 x 2 then which of the 0
following hold(S) good? (A) f(x) is continuous but not differentiable in (0,2 ) (B) Maximum value of f is (C) There exists atleast one c (0, 2) s.t.f`(c)=0 (D) Minimum value of f is
2
116. Consider a plane P passing through A(,3, ) B(-1,3,2) and C(7,5,10) and a straight line L with positive direction cosines passing through A, bisecting BC and makes equal angles with the coordinate axes. Let L1 be a line parallel to L and passing through origin. Which of the following is (are) correct? (A) The value of ( +) is equal to 5 (B) Equation of straight line L1 is
x 1 1
y 1 1
z 1 1
(C) Equation of the plane perpendicular to the plane P and containing line L1 is x-2y+z=0. (D) Area of triangle ABC is equal to 3 2 117. Which of the following statement (S) is/are correct? (A) 3 coins are tossed once. Two of them atleast must land the same way. No mater whether they land heads or tails, the third coin is equally likely to land either the same way or oppositely. So, the chance that all the three coins land the same way is ½. (B) Let 0
Page 18 of 25
29 96
then the possible values (S) of p can be
(A)
1 8
(B)
3
(C)
8
5
(D)
8
7 8
119. PQ is double ordinate of the parabola y 8 x . If the normal at P intersect the line passing through Q and parallel to axis of x at G, then the locus of G is a parabola with (A)vertex at (8,0) (B) focus at (10,0) (C) length of latus rectum equals 8 (D) equation of directrix is x=6 2 120. If from the vertex of a parabola y 4 x a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be made, then the locus of the further end of the rectangle is (A) an equal parabola (B) a parabola with focus at (9,0) (C) a parabola with directrix as x-7=0 (D) a parabola having tangent at its vertex x=8 121. Let p and q be non-zero real numbers. Then the equation ( px 2 qy 2 r )(4 x 2 4 y 2 8 x 4) 0 represents (A) two straight lines and a circle, when r=0 and p, q are of the opposite sign. (B) two circles, when p=q and r is of sign opposite to that of p. (C) a hyperbola and a circle, when p and q are of opposite sign and r≠ 0. (D) a circle and an ellipse, when p and q are unequal but of same sign and r is of sign opposite to that of p. Column II 122. Column I (A)If the sum of all the solution of (P) 4950 2 for the equation tan 33 x cos 2x 1 lying in the interval [0,314] is k , then the value of k, is (B) sum of all natural numbers (Q) 5048 2
satisfying the inequality
x 2 5 x 4 x 2 4
1
and not greater than 100, is (C) sum of all the possible natural values of b in [1,100] satisfying the
0 is
(S) 5050
(D) The sum of all such friendly natural numbers m which lie in the interval (0,101), where ‘m’ satisfies the inequality mx 2 4 x 3m 1 0, x R
(T) 5051
Column I
Column II
(A) Consider the triangle pictured shown. If 0 / 2 then the number
(P) 20
inequality
123.
e b (b 1)(b 2)
(R) 5049
(b 1)(b 2 2b 2)
Page 19 of 25
of integral value of c is A c
17
C
B
19
(B) In an acute angled triangle ABC, point D, E and F are the feet of the perpendiculars from A, B and C onto BC, AC and AB respectively. H is the intersection of AD and BE. If sin A=3/5 and BC=39, the length of AH is (C) In ∆ABC, a circle with centre Q is inscribed. If AB=AC=40 and BC=64 then the distance from A to Q equals a/b, then the value of (a+b) equals (D) A triangle has base 10 cm long and the base angles of 50 and 70 . If the perimeter of the triangle x y cos z is where z ε (0,90) then the value of x+y+z equals
(Q) 23
(R) 40
(S) 43
(T) 52 124.
Column I
Column II
(A) Let f be a real valued differentiable function on R such that f `(1)=6 and f ` (2)=2 Then is f (3 cos h 4 sin h 2) f (1) is equal to Lim h 0 f (3e h 5sec h 4) f (2) (B) For a>0, let f: [-4a, 4a] R be an even function such that f ( x) f (4 a x) x [2 a, 4 a] and
(P) 4
Lim h0
Lim h0
f (2a h) f (2 a)
4
h f ( h 2 a) f ( 2 a) 2h
G`(0) F `(0)
(R) 3
then is equal to
(C) Suppose f is a differentiable function on R. Let F ( x) f ( e x ) and G ( x) e f ( x ) .If f`(1)= e3 and f(0)=f `(0) =3 then
(Q) 5
(S) 2
is equal to
(D) Let f(x)=max. (cos x, x, 2x-1) where x 0 Then the number of
(T) 1
Page 20 of 25
points of non-differentiability of f(x) is equal to 125.
Column I
Column II
ln(1 x3 ) sin 1 , (A) f(x)= x 0,
if x 0
(P) continuous everywhere but
if x 0
not differentiable at x = 0
ln 2 (1 x) sin 1 , if x 0 (B) g(x) = x 0, if x 0
(Q) differentiable at x=0 but derivative is discontinuous at x=0
sin x ln 1 , (C) u (x)= 2 0,
if x 0
(R) differentiable and has
if x 0
continuous derivative (D) v (x) = Lim t 0
2 x
tan
1
2 t 2
(S) continuous and differentiable at x=0
126.
Column I
Column II
(A) Number of points of nondifferentiability of the function 3 f ( x) x( x 1) e x 1 . x 3 on
(P) 0
x ( , ) is equal to
(B) Number of values of α , where α (0,2) for which the system of
(Q) 3
x (sin ) y (cos ) z 0
equations x (cos ) y (sin ) z 0
x (sin ) y (cos ) z 0 has a solution ( x0 , y0 , z 0 ) with x0 2 y0 2 z 0 2 0, is (C) if the angles B and C of a triangle ABC are
6
and
4
respectively, and the included p q side is 3 1 , then area of the 2
triangle is (p, q, ε N). The value of (p+q) is
(R) 4
Page 21 of 25
1 (D) If f ( x) a cos ( x) b, f ` 2
(S) 5
3 2
2
f ( x)dx 1 then the value
and
3 2
of
12
(sin 1 a cos 1 b) is equal to
(T) 6 127.
Column I
Column II g ( x )
(A) If f ( x)
dt
1 t 3
0
where
(P) 3
cos( x )
g ( x )
(1 sin t 2 ) dt then the
0
value of f’ ( / 2) (B) If f(x) is a non-zero differentiable x
function such that f (t ) dt f x
(Q) 2
2
0
for all x, then f(2) equals a
(C) If (2 x x 2 ) dx , (a
(R) 1
b
then (a+b) is equal to (D) If Lim x 0
sin 2 x x
3
a
b x
2
0 then
(S) -1
(3a+b) has the value equal to 128.
Column I
Column II
(A) The least value of ‘a’ for which
(P) 20
the equation,
4 sin x
1
a has
1 sin x
atleast one solution on the interval (0, / 2) is (B) A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the depth of the liquid in it is x cm, the volume of the liquid in it is x 2 (15 x) cu.cm. The length EF is (C) If Rolle ’s Theorem is applicable to the function f ( x)
ln x x
(Q) 13
(R) 10
( x 0) over the
interval [a, b] where a, b I, then the value of (a 2 b 2 ) is equal to
(S) 9
Page 22 of 25
129.
Column I
Column II
P is point in the plane of the triangle ABC. Pv’s of A,B and C are a, band c respectively with respect to P as the origin. If b c . b c 0 and c a . c a 0,
(P) centroid
then w.r.t the triangle ABC, P is its (B) If a, b, c are the position vectors of (Q) orthoicentre the three non collinear points A, B and C respectively such that the vector V PA PB PC is a null vector then w.r.t the ∆ABC, p is its (C) If P is a point inside the ∆ABC such (R) Incentre that the vector R ( BC )( PA) (CA)(PB ) ( AB )(PC ) is a null vector then w.r.t the ∆ABC, P is its (D) If P is a point in the plane of the (S) Circumcentre triangle ABC such that the scalar product PA.CB and PB. AC vanishes, then w.r.t the ∆ABC, P is its
130.
Column I
Column II
(A) centre of the parallel piped whose (P) a b c . 3 coterminous edges OAOB and OC have position vectors a, b and c respectively where O is the origin is
(B) OABC is a tetrahedron where ‘O’ is the
(Q)
abc 2
origin. Positions vectors of its angular points A,B and C are a, b and c respectively. Segments joining each vertex with the centroid of the opposite face are concurrent at a point P whose p.v.’s are
(C) let ABC be a triangle the position
(R)
abc 3
vectors of its angular points are a, b and c
respectively. If | a b || b c || c a | then p.v of the orthocenter of the triangle is
(D) Let a, b, c be 3 mutually perpendicular vectors of the same magnitude. If an unknown vectors x satisfies the equation.
a
xb
a b x c b c x a c 0
Then x is given by
(S)
abc 4
Page 23 of 25
(e) ABC is a triangle whose centroid is G, orthocentre is H and circumcentre is the origin. If position vectors of A, B, C ,G and H are a, b, c, g and h respectively,
then h in terms of a, b and c is equal to 131. Consider the parabola y 2 12 x Column I
Column II
(A) Tangent and normal at the extremities of the latus rectum intersect the x axis at T and G is respectively. The coordinates of the middle point of T and G are (B) Variable chords of the parabola passing through a fixed point K on the axis, such that sum of the squares of the reciprocals of the two parts of the chords through K, is a constant. The coordinate of the point K are (C) All variable chords of the parabola subtending a right angle at the origin are concurrent at the point (D) AB and CD are the chords of the parabola which intersect at a point E on the axis. The radical axis of the two circles described on AB and CD as diameter always passes through
(P) (0,0)
(Q) (3,0)
(R) (6,0)
(S) (12,0)
132. Match the properties given in Column I with the corresponding curves given in the column II Column I
Column II
(A) The curve such that product of the distances of any of its tangent from two given points is constant, can be (B) A curve for which the length of the subnormal at any of its point is equal to 2 and the curve passes through (1,2) can be
(P) Circle
(Q) Parabola
Page 24 of 25
(C) A curve for passes through (1,4) and is such that the segment joining any point P on the curve and the point of intersection of the normal at P with the x-axis is bisected by the y-axis. The curve can be (D) A curve passes through (1,2) is such that the length of the normal at any of its point is equal to 2. The curve can be
(R) Ellipse
(S) Hyperbola
1)
b
2)
a
3)
a
4)
b
5)
a
6)
d
7)
c
8)
c
9)
a
10)
b
11)
c
12)
a
13)
d
14)
a
15)
d
16)
a
17)
b
18)
c
19)
c
20)
a
21)
b
22)
a
23)
c
24)
c
25)
d
26)
a
27)
a
28)
b
29)
b
30)
d
31)
c
32)
a
33)
a
34)
a
35)
c
36)
b
37)
a
38)
c
39)
d
40)
a
41)
b
42)
a
43)
d
44)
d
45)
d
46)
b
47)
a
48)
d
49)
a
50)
b
51)
a
52)
d
53)
c
54)
c
55)
c
56)
a
57)
b
58)
c
59)
b
60)
a
61)
a
62)
a
63)
a
64)
b
65)
c
66)
b
67)
d
68)
d
69)
a
70)
a
77)
a b c d
78)
b
79)
a
80)
c
76)
a d
a
86)
a
87)
d
88)
d
89)
a
90)
c
95)
a
96)
d
97)
a
98)
b
99)
c
100)
a
105)
b d
106)
b c
107)
a c d
108)
109)
a c
110)
115)
b c d
116)
b c d
a b d a b c d
119)
a b c d
a b c a b c d
71)
d
72)
c
73)
d
74)
c
75)
c
81)
c
82)
b
83)
a
84)
c
85)
91)
d
92)
c
93)
a
94)
b
103)
b c
104)
a b d
113)
a b c d
101)
a
111)
a c
121)
a b c d
102)
112)
a
b c
114)
122. (A) P, (B) Q, (C) S, (D) R.
c d
118) 117)
c d
120)
Page 25 of 25
123. (A) Q, (B) T, (C) S, (D) R 124. (A)P, (B) S, (C) R, (D) S 125. (A) R,S (B),Q,S (C)P, (D) 126. (A)P (B) Q (C) R(D) T 127. (A) S (B) R(C) R(D) Q 128. (A) S (B) R(C) P(D) Q 129. (A)S (B) P(C) R(D) Q 130. (A)Q (B) S(C) R(D) Q (E) P 131. (A) Q (B) R(C) S(D) P 132. (A)R,S (B) Q(C) R(D) P
********
