MATHEMATICS
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Q.1 to Q.30 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
Q.1
If the integral
[30 × 4 = 120]
1 cos 8x
cot 2x tan 2x dx = A cos 8x + k, where k is an arbitrary constant,
then A is equal to (A) Q.2
Q.3
1
(B)
8
1
(C)
16
1
(D)
16
1 8
There exists a natural number N whichis 50times its own logarithm to the base 10, then the sum of the digitsinNis (A) 1 (B) 5 (C) 9 (D) none
Let S =
a11 a12 a13 a 21 a 22 a 23 : a ij {1, 0, 1} a 31 a 32 a 33
then the number of symmetric matrices with trace equals zero, is (A)729 (B) 189 (C) 162
(D) 27
Q.4
Consider the system of equations x + ky = 0, y + kz = 0 and z + kx = 0. The set of all real values of k for whichthe system has a unique solution, is (A) R – {– 1} (B) R – {1} (C) {– 1} (D) {– 1, 1}
Q.5
The value of
log8 17 log9 23
(A) –1
log 2
17 2 is equal to log3 23
(B) 0
(C) 1
(D) –2
Q.6
The sum of the series (B)2 + 2 (D)2 + 3 ( 6 )2 + ………… upto 10 terms is equal to (A) 11300 (B) 12100 (C) 12300 (D) 11200
Q.7
Number of values of x [0, ] satisfying cos25x – cos2x + sin 4x ·sin 6x = 0, is (A) 2 (B) 3 (C) 5
(D) Infinitelymany
Q.8
The number of solutions of the equation sin 2 – 2cos +4sin = 4 in [0, 5] is equal to (A) 3 (B) 4 (C) 5 (D) 6
Q.9
The value of log(333.....3) 1111 .... 1 222 ...... 2
50 times
(A) 1
Q.10
100 times
(B) 2
, is
50 times
(C) 0
Let f be a composite function of x defined by f(u) =
Then the number of points x where f is discontinuous is (A) 4 (B) 3 (C) 2 XIII (VX)
(D) Not defined
1 u 3 6u 2 11u 6 (D) 1
, where u(x) =
1 x
.
Q.11
Q.12
Let R = {(x, y) : x, y N and x 2 – 4xy + 3y2 = 0} where N is the set of all natural numbers. Then the relation R is (A) reflexiveand symmetric (B) reflexive andtransitive (C) symmetric and transitive (D) reflexive but neither symmetric nor transitive
If x =
4
1 (a + b) is (A) 8
2
and y =
2 22
where is a real parameter then x2 – xy + y2 lies between [a, b] then
1
2
(B) 10
(C) 13
(D) 25
Q.13
The number of solutions of the equation sin –1x = 2 t a n –1x is equal to (A) 4 (B) 3 (C) 2 (D) 1
Q.14
In a triangle PQR, R =
then (A) a + b = c
2
. If tan
P 2
Q
and tan
are the roots of the equation ax2 + b x + c = 0 ( a 0),
2
(B) b + c = a
(C) a + c = b
(D) b = c
Q.15
The zeroes ofthe quadratic polynomial f (x) = 2x2 – 3x + k 2 – 3k + 4 lie on either side of the origin then the set of the values of k is (A) (B) (– 4, 1) (C) (4, – 1) (D) (– , – 4) (1, )
Q.16
If the three lines x – 3y= p, ax + 2y = q and ax + y = r form a right-angled triangle, then (A) a2 – 9a + 18 = 0 (B) a2 – 6a – 18 = 0 (C) a2 – 9a +12 = 0 (D) a2 – 6a – 12 = 0
Q.17
Given a right triangleABC with hypotenuseAC and A 50 . Points K and Llie onthe cathetus BC CK are such that KAC LAB = 10°. The ratio is equal to : LB 1 (A) (B) cos 10° (C) 2 (D) 2 cos 10° 2
Q.18
The sum of the infinite series (A)
1 3
(B)
1
1 9
1 18
1
1
30
45
4
1
..... , is
63 1 (C) 2
2
(D)
3
Q.19
Let S be the sum of the first n terms of the arithmetic sequence 8, 12, 16, .....……., and T be the sum of first n terms arithmetic sequence 17, 19, 21, ….......….. . If S – T = 0, then the value of n is equal to (A) 8 (B) 10 (C) 18 (D) 22
Q.20
Let n denotes the number of skew-symmetric matrices out of the matrices given below.
0 1 1 0 5 3
5 3 , S = R= 0 The value of n is equal to (A)
tan 2 10 · sin 2 10 tan 2 10 sin 2 10
(C) 1 + cos 2x + 2sin2x
XIII (VX)
1 2 3 2 2 1 3 1 4
,
T=
0 2 2 0
,
3
(B) (D)
cos 9 cos 27 cos 9
U =
1 5
3
sin 9 sin 27 sin 9
8 sin 40 · sin 50 · tan 10 cos 80
5 1
Q.21
In a triangleABC, R(b + c) = a bc where R is the circumradius of the triangle.Then the triangle is (A) Isosceles but not right (B) right but not i sosceles (C) right isosceles (D) equilateral
Q.22
x Let f : R R, be defined as f(x) = e + cos x , then f is (A) one-one and onto (B) one-one and into (C) many-one and onto (D) many-one and into
Q.23
LetABCDbe a square ofside length 2. Let E bethe middle point of the segment CD. The radius 'r' of the circle inscribed in ABE, is
2
5 1
(A)
5 1
(B)
2
3
(C)
2
5 1
(D)
4
4
Q.24
Consider the quadratic function f(x) = ax2 + bx + c where a, b, c R and a 0, such that f(x) = f(2 – x) for all real number x. The sum of the roots of f (x) is (A) 1 (B) 2 (C) 3 (D) 4
Q.25
Let f(x ) be a differentiable function such that f(x) + 2 f(– x) = sin x for all x R.
The value of f ' is equal to 4 1
(A)
(B)
2
1
2
(C)
2
(D) – 2
log x log 2 2 log2 2 2 log2 2
Q.26 The value of x satisfying the equation (A) 5
Q.27
Q.28
y x 2y
(B)
–1
The sum of series cot
dx 2
(C)
( x 2 y) 3
–1
+ cot
(B) 4
2 ( x 2 y) 3
33 129 + cot –1 4 8
(B) cot –1 3
(D) 32
is equal to
2xy 2 y
9 2
= 5, is
(C) 25
3 2
(D)
2x 2 y2 ( x 2 y) 3
+ ............. is equal to
(C) cot –1(–1)
2 Let f(x) = cos –1 2x 1 x then f '
(A) – 4
Q.30
d2y
For the curve x y + y = 1,
(A) cot –1(B)
Q.29
log 2 2
(B) 16 2
(A)
2
(D) cot –1(A)
equals
(C)
1 4
(D)
1 4
If a and b are positive real numbers such that a + b = 6, then the minimum value of is equal to (A)
2
3 XIII (VX)
(B)
1 3
(C) 1
(D)
3 2
4 1 a b
[SINGLE CORRECT CHOICE TYPE] Q.1 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
Q.1
[10 × 3 = 30]
Let f(x) = x | x | – 4x – 1 for all x R, then f (x) is
(A) continuous and derivable for all x R. (B) continuous for all x R but non-derivable only at x = 0. (C) neither continuous nor derivable at x = 0. (D) continuous for all x R but non-derivable at two points. Q.2
Q.3
Number of values of x [0, ] where f (x) = [4sin x – 7] is non-derivable is [Note: [k] denotes the greatest integer less than or equal to k.] (A) 7 (B) 8 (C) 9 (D) 10
The value of Lim
x –
2x 1 sin x – x 2 1
(A) –2 Q.4
Q.5
3sin x
(B) 0
(C) 2
Let
2 tan x · f(x) = k ,
1
cot x
(A) e
2
sin x cos x
if x
,
if x
4
4
4
, then the value of k is
1
1
1
1
(B) e 2
(C) e 2
(D) e
2
Whichone ofthefollowingfunction is non-differentiable for atleast one real value of x?
x 2 sin 1 , (A) f(x) = x 0,
x0
(B) g(x) = cos | x | + sgn (x) + sgn (– x)
x0
(C) h(x) = x 2 2 x 1 [Note : sgn k denotes signum function of k.]
Q.7
(D) does not exist
Let P(x) = x10 + a2x8 + a3 x6 + a4x4 + a2x2 be a polynomial with real coefficients. If P(1) = 1 and P(2) = – 5, then the minimum number of distinct real zeroes of P(x) is (A) 5 (B) 6 (C) 7 (D) 8
If f (x) is continuous at x =
Q.6
is equal to
(D) k(x) = sgn (x2 + 3x + 4)
Let f be a differentiable function such that Lim
1 (A) XIII (VX)
f (π h)
h 0
(B)
1 2
(C)
3
h 1 3
f (π ) = ,thenthevalueof f'( ) is 3
3
(D)
1 6
1
Q.8 Thevalueof Lim log a x
x a
x a
l n a
1
a
a
a l n a
l n a
(A) e
Q.9
, where 0 < a 1, is equal to
(B) eal n a
Let f (x) =
(C) e
sgn ( x 2 x 1) cot 1 ( x 2 )
. If f (x) is continuous for all x R, then number of integers in the
range of , is (A) 0 (B) 4 (C) 5 [Note : sgn k denotes signum function of k.]
Q.10
Let f(x) =
(D) e
(D) 6
x 2 x2
[cos x ], x 2,
Number of points where f(x) is discontinuous in (– , ) is [Note: [k] denotes greatest integer less than or equal to k.] (A) 3 (B) 4 (C) 5
(D) 6
[PARAGRAPH TYPE] Q.11 to Q.13 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Paragraph for question nos. 11 to 13
Lim n
n
2
n 1
n
2
n 1 x ,
Let
f (x) =
and
g(x) = | x | + | x2 – 1 |, for all x R.
0,
x0 x0
Q.11
Which one of the following statement is correct? (A) f (x) is continuous at x = 0. (B) f (x) is non-differentiable at x = 0. (C) f (x) has non-removable type of discontinuity at x = 0. (D) f (x) has removable type of discontinuity at x = 0.
Q.12
Number of points where g(x) is non-derivable, is (A) 0 (B) 1 (C) 2
(D) 3
Number of points of non-differentiability of g f ( x ) , is (A) 0 (B) 1 (C) 2
(D) 3
Q.13
XIII (VX)
[3 × 3 = 9]
[REASONING TYPE] Q.14 to Q.16 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
Q.14
Let f : R R be a continuous function defined by f (x) =
Statement-1 : f(c) =
1 3 2
Statement-2 : 0 < f (x)
[3 × 3 = 9]
1 e
5x
4e – 5 x
, for some c R. 1
, for all x R. 4 (A) Statement-1 is true, statement-2 is false. (B) Statement-1 is false, statement-2 is true. (C) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (D)Statement-1 is true, statement-2 is true andstatement-2 is NOT thecorrect explanationforstatement-1. Q.15
Consider the function f(x) = [x] + |1–x| , –1 x 3, where [x] is the greatest integer less than or equal to x. Statement-1 : f (x) is discontinuous at x = 1. Statement-2 : f (x) is non-differentiable at x = 1. (A) Statement-1 is true, statement-2 is false. (B) Statement-1 is false, statement-2 is true. (C) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (D)Statement-1 is true, statement-2 is true andstatement-2 is NOT thecorrect explanationforstatement-1.
Q.16
Let f be a real-valued function defined on R such that f (x) = Lim n
x x
2n
1
.
Statement-1 : f (x) is non-differentiable at three points. Statement-2 : f (x) is an odd function. (A) Statement-1 is true, statement-2 is false. (B) Statement-1 is false, statement-2 is true. (C) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1. (D)Statement-1 is true, statement-2 is true andstatement-2 is NOT thecorrect explanationforstatement-1.
[MULTIPLE CORRECT CHOICE TYPE] Q.17 to Q.21 has four choices (A), (B), (C), (D) out of which ONE OR MORE maybe correct.[5×4=20]
Q.17
Let f(x) =
l n (1 2 x ) , x 2 cos x, e2 x 1 , x e 2 1,
1
x0
2 x0
0 x 1 x 1
then (A) f(x) is continuous at x = 0.
(B) f(x) is not differentiable at x = 0.
(C) f(x) is continuous at x = 1.
(D) Lim [f ( x )] = 1. x 0
[Note: [k] denotes greatest integer less than or equal to k.]
XIII (VX)
Q.18
max . {e x , e x , 2}, Let f(x) = x x min .{e , e , 2},
x0 x0
.
Which of thefollowingstatement(s) is/are correct? (A) f(x) is discontinuous at x = 0. (B)f(x) is non-derivable at exactly twopoints. (C) f(x) has non-removable type of discontinuity at x = 0 with jump of discontinuity equal to 2. 1
(D) f (x) is continuous but non-derivable at x = l n
Q.19
2x 1 x Let f (x) = sin –1
2
, then
2
1
(A) f (x) is continuous and differentiable at x =
2
(B) f (x) is continuous and differentiable at x =
4
(C) f (x) is continuous and differentiable at x =
If f (x) =
.
.
6
.
(D) f (x) is continuous but non-differentiable at x =
Q.20
.
1 2
.
a cos x bx sin x ce x 2x , x0 2 x 0, x0
is differentiable at x = 0, then (A) a + b + c = 2 Q.21
(B) a + b = – 4
(C) f '(0) =
1 3
(D) a – c = 4
Which of the followingstatements is(are) correct? (A) If
sin x x x
f ( x )
x2 x 1 x
2
1
for all x > 5, then Lim f ( x ) = 1. x
(B) If f is continuous on [–1, 1] such that f (–1) = 2 and f (1) = 10 then there exists a number c such that | c | < 1 and f (c) = e2, where 'e' is napier constant. (C) If f(x) and g(x) both are discontinuous at x = c, then the product function f(x) · g(x) must be discontinuous at x = c. (D) If f(x) and g(x) both are continuous at x = c, then gof(x) must be continuous at x = c.
XIII (VX)
[MATRIX TYPE]
[3 + 3 + 3 + 3 = 12]
Q.1 has four statements (A, B,C, D) given in Column-I and five statements (P, Q, R, S,T) given in Column-II. Any givenstatement in Column-I canhave correct matchingwith one or more statement(s) given in Column-II.
Q.1
Column-I
(A)
Let
Column-II
sin x, x , f ( x ) 3 cos x, 1,
x0 0 x 2
(B)
If Lim
(1 x x 2 ........ x n )
x 0
1
(Q)
2
(R)
3
(S)
4
(T)
5
2 x 3 x 3
then number of points where f(x) is discontinuous in (– , ), is equal to [Note : [k] denote greatest integer less than or equal to k. ] l n
(P)
nx then the value of n, is equal to
exists and is equal to
(C)
Let g ( x) = |4x3 – x| cos (x) then number of points where g (x)is non-differentiable in (– , ), is equal to
(D)
Let f be a differentiable function such that f '(2) =
1 5
1 4
f (2 3h 4 ) f ( 2 5h 4 ) is equal to then Lim 4 h 0 h
[INTEGER TYPE] Q.1 to Q.5 are "Integer Type" questions. (The answer to each of the questions are Single digits)[5×6=30]
Q.1
Let a, b, c R. If f(x) =
sin (ax 2 bx c) where (a b c) , 2 x 1 1 , a sgn (x 1)cos(2 x 2) bx 2 ,
a 2 b 2 . is continuous at x = 1, then find the value of 5 [Note : sgn k denotes signum function of k.]
XIII (VX)
if x 1 if x 1 if 1 x 2
Q.2
Let f (x) =
of
cos 1 1 {x}2 ·cos 11 {x}2 2 2 {x} {x}3
. If f(0+)=p and f(0 – ) = q, then find the value
p . q
[Note : {k} denote the fractional part of k.]
Q.3
If number of points of discontinuityof the function f (x) = [2 + 10sin x], in x 0, is same as 2 number of points of non-differentiabilityof the function g(x) = x 1x 2 ( x 1)(x 2).......( x 2m) , (m N) in x (– , ) then findthe value of m. [Note : [k] denotes largest integer less than or equal to k.]
Q.4
Let
cot x , 0 | x | 1 2 x x f (x) = . 1 , x0 3
If f (x) is continuous at x = 0, then find the value of (2 2).
Q.5
– 1
If and ( < ) are the roots of the equation Lim cos t
then find the value of (8 + 2 – .
XIII (VX)
– 1 sin tan
= 2 tx – 3tx t – 1 – x 6 tx
[SINGLE CORRECT CHOICE TYPE] Q.1 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
Q.1
The value of Lim x 0
e( x
(x
(A) 1
Q.2
Let
x
1)
2 x
)
xx 1
(B)
x2 , f ( x ) k ( x 2 4) 2 x ,
[10 × 2 = 20]
is equal to
2
1
(C)
8
3
(D)
2
1 4
xZ xZ
where Z is the set of all integers then f(x) is continuous at x = 2 for (A) k = 1 only (B) every real k (C) every real k except k = –1 (D) k = – 1 only Q.3
If f(x), g(x) and h(x) are three polynomials of degree 2 and
f (x )
g( x )
h ( x)
( x) f ' ( x) g' ( x ) h ' ( x ) f " (x )
g" ( x )
h" ( x )
then (x) is a polynomial of degree (dashes denote the differentiation) (A) 2 (B) 3 (C) 0 (D) atmost 3 Q.4
If the roots of equation x 2 + qx + p = 0 are each 1 less than the roots of the equation x2 + px + q = 0, then (p + q) is equal to (A) – 2 (B) – 4 (C) – 5 (D)– 6
Q.5
In the interval 0,
, the equation 2
(A) no solution (C) exactly two solutions
Q.6
Q.7
(B) exactly one solution (D) more than two solutions
If yx – xy =1, then the value of (A) 2 (1 – l n 2)
cos2 x – cos x – x = 0 has
dy
dx (B) 2 + l n 2
at x = 1 is equal to (C) 2 (1 + l n 2)
(D) 2 – l n 2
The first termof aninfinitelydecreasingG.P. is unityand its sum is S.The sum of the squares ofthe terms of the progression is : (A)
S
2S 1 XIII (VX)
(B)
S2 2S 1
(C)
S 2S
(D) S2
Q.8
Three angles A, B, C (taken in that order) of triangle ABC are in arithmetic progression. If a2 + b2 – c2 = 0 and c = 2 3 , then the radius of circle inscribed in triangle ABC is equal to [Note : All symbols used have usual meaningin triangleABC.] (A)
Q.9
3 3
(B)
2
Let
A=
tan 3 cot 2013 2
3 3
(C)
2
(D)
4
3 3 4
2
3 cos ( 2012 ) sec
3 3
P is a 2 × 2 matrix such that PPT = I,
and
where I is an identity matrix of order 2. If Q = P A PT and R = [r i j]2 × 2 = PT Q8 P, then (A) r 11 = 81 Q.10
(B) r11 = 27 3
(C) r 11 = 4 3
(D) r 11 = – 3
The equation log3(3–x)–log3(3+x)=log3(1–x)–log3(2x + 1) has (A) two real solutions (B) one prime solution (C) noreal solution (D) none
[MULTIPLE CORRECT CHOICE TYPE] Q.11 to Q.15 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [5×4=20]
Q.11
Let f be a biquadratic function of x given by f(x) = Ax4 + Bx3 + Cx2 + D x + E 1
where A, B,C, D, E
R
(A) A + 4 B = 0
and A 0. If
(B) A – 3 B = 0
f ( x ) x Lim x 0 2 x 3
e 3 , then
(C) f (1) = 8
(D) f ' (1) = – 30
Q.12
Identify the statement(s) whichare always True? (A)Asummable infinite geometric progression with non zero common ratio less than unityin absolute value is a decreasingprogression. (B)An infinitelydecreasing geometric progressionhavingthe property that itssum is twicethesumof its first n terms (n > 2) has a unique common ratio. (C) tan 1 is greater than tan 2. (D) The expression y = cos2x + c o s2(x + ) – 2 c o s cos x cos (x + ) is independent of x.
Q.13
Let f : R R be defined as (A) g ' (2) =
Q.14
3
(B) g ' (2) =
7
4
f (x) = (2x – 3)3 + 7 3
3
x + cos x
(C) g " (2) = 0
and g = f –1 , then (D) g " (2) =
27 343
Let Tr be the r th term of a sequence, for r = 1, 2, 3, 4, .............. . If 3 Tr+1 = Tr and T7 = then 5
T
(A)
5
1
=9
(C)
(B)
r 1
r 1
T ·T = 8 r
r 1
XIII (VX)
r 1
T r 1
27
1
= 11 1
r 1
(D)
T ·T = 92 r
r 1
r 1
1 243
,
Q.15
If the graph of quadratic expression f(x) = x2 + ax + b cuts positive x-axis at two points P (3, 0) and Q such that (A) b + 3a = 0
1 OP
1 OQ
2 OR
= 0, where O is origin and R is (4, 0), then
(B) b + 2a = 0
(C) 2a + 3b = 42
(D) 2a + 3b = 36
[INTEGER TYPE] Q.1 to Q.5 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)[5×4=20]
Q.1
Let M be a 3 × 3 non-singular matrix of real entries with det.(M) = 2. If M 1 adj (adj M ) = I3, where I3 is an identity matrix of order 3 then find the value of .
Q.2
If x, y R satisfy the equation x2 + y2 – 4x – 2y + 5 = 0, then compute the value of the sum 99
(x r y) . r 0
Q.3
The incircle of a triangle ABC touches AB at point P and has radius equal to 21. If AP = 23 and PB = 27, then findthe sum of the digits in the perimeter ofthe triangleABC.
Q.4
Let a polynomial P(x) when divided by x – 1, x – 2, x – 3 leaves the remainder 4, 5, 6 respectively. If P(x) is divided by (x – 1) (x – 2) (x – 3) and remainder is R(x), then find the value of R(100).
Q.5
Find the number of solutions of the equations 2 sin2 + sin 2 2 = 2 and sin 2 + cos 2 = tan in [0, 4] satisfying the inequality 2 cos2 +sin 2.
XIII (VX)
[MULTIPLE CORRECT CHOICE TYPE] Q.1 to Q.8 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [ 8 × 3 = 2 4 ]
Q.1
The value(s) of 'p' for which the equation ax2 p x + a b = 0 and x 2 a x b x + a b = 0 may have a common root, given a, b are non zero real numbers, is (A) a + b2 (B) a2 + b (C) a(1 + b) (D) b(1 + a)
Q.2
An arithmetic progressionhas the followingproperty: For aneven number ofterms, the ratioofthesum of first half of the terms to the sum of second half is always equal to a constant 'k'. Let the first term of arithmetic progression is 1. Then which of the following statement(s) is(are) correct? (A)
Absolute difference ofall possible values of k is
(B)
The sum of all possible values of k is
(D)
(B)
3
Let N =
45 sin r º r 1
(A) sin –1 N = (C) tan Q.5
.
4
Let , and satisfies 0 < < < < 2. If cos (x + ) + cos (x + ) + cos (x + ) = 0 for all x R. Then ( – ) can be equal to (A)
Q.4
3
. 3 If the number of terms of arithmetic progression is 20, then the sum of all terms of all possible arithmetic progressions is 420. The number of possible non-zero values of common difference of arithmetic progressions is 1.
(C)
Q.3
2
1
2
(C)
3
(D)
3
5 3
89 sec k º , then k 46
(B) cot –1(log2 N) = – tan –12
6
2
4
log N 2
(D) sin –1(sin N) = – N
= tan –1 2
Let f be a constant function with domain R and g be a certain function with domain R. f 2 Two ordered pairs in f are (4, a – 5) and (2, 4a – 9) for some real number a. Also domain of is g R – {7}. Then 1
(A) a = 2
100
(B) f (10)
=1
(C) (100)
g(7)
=1
(D)
f ( x) dx = 1 0
Q.6
For 3 × 3 matricesA and B, which of the following statement(s) is(are) correct? (A) AB is skew symmetric ifA is symmetric and B is skew symmetric. (B) (adjA)T =adjAT for all invertible matrixA. (C) AB+ BAis symmetric for all symmetric matricesA and B. (D) (adjA) –1 =adj(A –1) for all invertible matrix A. XIII (VX)
Q.7
The expression cos2( + ) + cos2( – ) – cos 2 · cos 2, is (A) independent of independent of (D) dependent on and . C) independent of and
Q.8
The lengths of two sides of a triangle are log2 4 and log4 2. If the length of the third side is log3 x, then a possible value of x can be (A) 5 (B) 7 (C) 9 (D) 11
[PARAGRAPH TYPE] Q.9 to Q.16 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[8 × 3 = 24]
Paragraph for question nos. 9 & 10 Consider two quadratic polynomials in x as f (x) = x2 – x + m and g(x) = x 2 – x + 3m, where m is non-zero real number. Let be the roots of equation f (x) = 0 and be roots of equation g (x) = 0.
Q.9 Q.10
If = 0, then (m + ) is equal to (A) – 2 (B) 3
(C) – 3
(D) 2
If min. f ( x ) > min. g( x ) , then the true set of real values of m is (A) (0 , )
(B) (– ) – {0}
(C) (– , 0) 0,
1
3 , 4 4
(D) (– , 0)
Paragraph for question nos. 11 & 12 Let f be a monic biquadratic polynomial satisfying f (–x) = f (x) for all x R and having minimum value – 4 at x = ± 2.
Q.11
The number of integral values of k for which the equation f (x) = k has four distinct real solutions, is (A) 2 (B) 7 (C) 15 (D) 21 n
Q.12
n
(A) 2 – tan –1 4
Q.13
Q.14
8 r tan f ( r ) 5
The value of Lim
1
is equal to
r 1
(B) – tan –1 4
(C)
3 2
– tan –14
(D)
2
– tan –14
Paragraph for question nos. 13 & 14 Let Abe a non-singular matrix of order 3 such that det.(A) = 5 and B is also a non-singular matrix satisfying A –1 B2 + A B = O. The value of det.(A6 – 2A4B + A2B2) is equal to (A) 0 (B) 56 (C) 23 · 56 (D) 106
A2 det (A2) – Adj (Adj. B) is equal to where Adj. (P) denotes the adjoint matrix of matrix P. (A)nullmatrix (B) 25 A2 – 5 B (C) 50 A2
XIII (VX)
(D) 20 A2
Paragraph for question no. 15 & 16 Let f(x)=x – ax + b where 'b' is an even positive integer. If roots of the equation f (x) = 0 are two distinct prime numbers and 2a + 3b = 44. If b – a, a and b +1 are the first three terms of a sequence then 15th term is (A) 3 · 214 (B) 59 (C) 63 (D) 5 · 219 2
Q.15
Q.16
If
2
3
a a S = b b b a
then [S] is equal to
[Note : [y] denotes greatest integer less than or equal to y.] (A) 2 (B) 3 (C) 5
(D) 9
[MATCHING LIST TYPE] Q.17 to Q.20 are MatchingListtype questions. Each question has matching lists.The codes for the lists have [4 × 3 = 12] choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
Q.17 (P)
List - I If 2x3 – 4x2 + 6 is written in the form p(x – 1)3 + q(x – 1)2 + r(x – 1) + s, then the value of (p + q + r + s), is
(Q)
Suppose that a, b and c are positive integers such that a log1443 + b l o g1442 = c, then the value of
(R)
(S)
Let H.M. oftwo number is
c
is
2
. An even number of H.M. a re inserted 7 between two given number such that sum of their reciprocal exceeded their number by 20, then number of means inserted, is Number of integral values of m so that the equation s i n x – 3 cos x =
Codes : P (A) 3 (B) 3 (C) 3 (D) 3
XIII (VX)
a b
Q 4 3 3 4
R 4 4 4 4
4m 6 4m S 3 4 2 2
has a solution for some x
[0, 2], is
List - II (1) 2
(2)
4
(3)
6
(4)
8
Q.18
Let in ABC, tanA + tan B =
2 3
and tan B + tan C =
2 3
.
[Note : All symbols used have usual meaningin triangle ABC.] List - I List - II
The value of tan A is
(1)
(Q)
The value of tan B is
(2)
(R)
If c = 4 3 , then circumradius of ABC is
(3)
4 3
(4)
4
If c = 4 3 , then area of ABC in s quare units is Codes : P Q R S (A) 2 1 3 4 (B) 4 3 2 2 (C) 3 2 1 4 (D) 2 2 4 3 (S)
Q.19
1
(P)
Consider f (x) = x 2 – x + k – 2, k R and g(x) = x2 – x + 1. List-I (P)
If the complete set of values of k for which y = f | x | is
3 1 3
List-II
(1)
0
(2)
1
(3)
2
(4)
3
non-derivable at 5 distinct points is (a, b) then 8(b – a) equals (Q)
If k 3 then number of tangents drawn to the curve y=
(R)
f ( x ) g( x )
which are parallel to x-axis, is
If the complete set of values of k for which y =
f (x ) g( x ) ,
where [p] denotes greatest integer less than or equal to p, is
continuous for all x R is (l , m) then m (S)
If f (x )
g( x )
equals
4
= f ( x ) g(x ) is true for all real values
of x then smallest integral value of k is Codes : P Q R S (A) 1 2 3 4 (B) 3 2 1 4 (C) 4 3 2 1 (D) 2 4 1 3
XIII (VX)
5l
Q.20
List -I contains the function and List-II contains their derivatives at x = 0. Select the correct answer using the codes given below the list. List-I List-II (P)
f(x) = cos 1
2x 1 x 2
(1)
2
(Q)
g(x) = cos –1(2x2 – 1)
(2)
3
(3)
–2
(4)
non-existent
(R)
(S)
h(x) = sin
k(x) = tan
Codes : P (A) 3 (B) 3 (C) 4 (D) 2
XIII (VX)
1 1 x
Q 4 4 3 4
1 x2 2
1 3x x
1 3x 2
R 4 4 2 4
3
S 1 2 1 2
ANSWER KEY
Q.1 Q.6 Q.11 Q.16 Q.21 Q.26
C B D A C C
Q.2 Q.7 Q.12 Q.17 Q.22 Q.27
A D A C D C
Q.3 Q.8 Q.13 Q.18 Q.23 Q.28
B A B A A A
Q.4 Q.9 Q.14 Q.19 Q.24 Q.29
A A A B B B
Q.5 Q.10 Q.15 Q.20 Q.25 Q.30
B A A C B D
Q.1 Q.6 Q.11 Q.16 Q.21
A C A B AB
Q.2 Q.7 Q.12 Q.17
A C D ABC
Q.3 Q.8 Q.13 Q.18
C C D ABD
Q.4 Q.9 Q.14 Q.19
A B C BCD
Q.5 Q.10 Q.15 Q.20
D B D BC
Q.1
(A) R; (B) T; (C) P; (D) Q
Q.1
5
Q.2
4
Q.3
6
Q.4
2
Q.5
9
Q.1 Q.6 Q.11
B A BD
Q.2 Q.7 Q.12
D B BCD
Q.3 Q.8 Q.13
C B AC
Q.4 Q.9 Q.14
B A BC
Q.5 Q.10 Q.15
B D BD
Q.1
2
Q.2
5150
Q.3
4
Q.4
103
Q.5
4
Q.1 Q.6 Q.11 Q.16
BC BCD C A
Q.2 Q.7 Q.12 Q.17
ABCD ABC C C
Q.3 Q.8 Q.13 Q.18
BC BCD D D
Q.4 Q.9 Q.14 Q.19
BC C A B
Q.5 Q.10 Q.15 Q.20
ABC D B B
XIII (VX)
