Single Correct Answer Type
(This section contains 8 single correct choice type questions. Each question has 4 choices a, b, c, d for its answer, out of which ONLY ONE is correct.)
1) Uf the curves x² - y² = 4 and xy =√5 intersect at points A and B, then the possible number of point(s) on the curve x² - y² = 4 such that triangle ABC is equilateral is
a) 0 b) 1 c) 2 d) 4
2) If A and B are two events such that P(A∩B)= 0.3 and P(A'∩ B')= 0.6, then the value of P(A∩ B') or (A' ∩B) is..
a) 0.9 b) 0.7 c) 0.3 d) 0.1
3) The value of lim ₓ→∞{²ˣ₀ ∫ ₓ ₑx²/ₑ4x²)dx equals to
a) 1/2 b) 1 c) 2 d) ∞
4) The number of values of k for which the equation x³ - 3x + k = 0 has two distinct roots lying in the interval (0,1) are
a) three b) two c) Infinity many d) no value of k satisfies the requirement
5) The domain of the function f(x)= cos⁻¹(cos⁻¹x) is
a) [-1,1] b) [0,π] c) [0, cos 1] d) [cos 1, 1]
6) If dy/dx = f(x) + ¹₀∫ f(x) dx, then the equation of the curve y= f(x) passing through (0,1) is..
a) f(x)= (2eˣ - e+1)/(3- e)
b) f(x)= (3eˣ - 2e+1)/{2(2-e)}
c) f(x)= (eˣ - 2e +1)/(e+1)
d) none of these
7) If the complex number z satisfies the equation z z'= z² + z'² then the maximum value of arf z - arg z' is (z' is conjugate)
a) 5π/3 b) 4π/3 c) 2π/3 d) π/3
8) If a, b, c are in AP and f(x)=
x+ a x²+1 1
x+b 2x² -1 1
x+ c 3x² -2 1 then f(1) is
a) -1 b) 0 c) 1 d) none
SECTION - II
Multiple Correct Answer Type
(this section contains 4 multiple correct choice type questions. Each question has 4 choices: a, b, c, d for its answer, out of which ONE or MORE is/are correct)
9) Let P is any point on the curve S = 0 such that tangent from P to x² + y² - 2x - 4y - 4 = 0 makes 60° with each other and from point Q perpendicular tangents are drawn to S, then
a) locus of P is circle of radius 5.
b) locus of P is a circle of radius 6.
c) locus of Q is a circle of radius 5√2.
d) Locus of Q is a circle of radius 6√2.
10) A function f: R--> R⁺ satisfies f(x +y)= f(x). f(y), x and y belongs to R, f(0)= 1; f'(0)= 2 then
a) ˡⁿ₀∫ [f(x)e⁻ˣ] dx = ln 4.5 where [.] denotes greatest integer function.
b) lim ₓ→₀[f(x)] does not exist (where [.] denotes greatest integer function).
c) f⁻¹(x) = ln √x, x > 0
d) f(x)= ₑx²- 4x has infinite solution in (0,6).
11) Consider the two lines r= i + j + M(i+ k) and s= j+ k+ L(i+ j), if PQ is the line of shortest distance, then
a) their shortest distance is √26.
b) their point of intersection is j + k.
c) their shortest distance is zero.
d) the length of projection of OP on i+ j+ k is 2/√3.
12) Consider n= 21⁵²,then
a) number of even divisors of n is 704.
b) number of divisors of n is 2809.
c) last two digits of n are 41.
d) number of divisors of n which is multiple of 9 is 2705.
SECTION- III
Comprehensive Type
(this section contains 2 groups of questions. Each group has 3 multiple choice questions based on a paragraph. Each question has 4 choices: a, b, c, d for its answer, out of which ONLY ONE is correct.
paragraph for question number 13 to 15
Consider an ellipse S= x²/a² + y²/{a²(1- e²)} - 1 = 0.
Let M= 0 be a parabola on the right of y axis confocal with S = 0 having vertex at the centre of S= 0.
Let P be the point of intersection of the parabola and the directix of ellipse in the first quadrant and L= 0 is the directrix of the parabola.
13) The ordinate of the point from the which the pair of tangents drawn to both the parabola and the ellipse, are separately at right angles is
a) a√(1- e²) b) √2 a √(1- e²)
c) 2a √(1- e²) d) a √(2- e²)
14) Pair of tangents are drawn from any point on L= 0 to S = 0 and M= 0. Then the locus of point of intersection of their chord of contact is
a) x= ae b) x= a/e c) x= {a√(1 - e²)/e} d) x = ae √(1- e²)
15) If the tangent at point Q on S = 0 and the line joining the points P to the locus of M= 0 intersect at the auxiliary circle of S= 0, then eccentric angle of the point Q is (Q lies in first quadrant)
a) tan⁻¹{e/√(1+ e²)}
b) tan⁻¹{2e/√(1- e²)}
c) tan⁻¹{√2 e/√(1+ e²)}
d) tan⁻¹{e/√(1- e²)}
Paragraph for question number 16 to 18
Let m, n, q be the 3 real numbers such that m²+ n² + q² - q = 0 and z= (m+ ni)/(1- q)
16) |z|² equals
a) q b) 1 - q. c) q/(1- q) d) (1- q)/q
17) m equals
a) (z + z')/2(1+ |z|²)
b) (z'+ z)i/2(1+ |z|²)
c) (z - z')i/2(1+ |z|²)
d) none
18) n equals
a) (z- z')/2(1+ |z|²)
b) (z'- z)i/2(1+ |z|²)
c) z/2(1+ |z|²)
d) 2z/(1+ |z|²)
SECTION - IV
Matrix Match Type
This section contains 2
questions. Each question contains statements given in two columns which have to be matched. The statements in Column- I are labelled A, B, C, D, while the statements in column-II are labelled 1, 2, 3, 4, 5. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column- II.
19) Column - I
A) If f(a)= 4 and a< 60, then number of possible values of a are
B) if f(a)=0, then a can be
C) if f(a)< 0, then a can't be
D) if f(a)= -(2k +1), where k belongs to natural number, then a can be
Column - II
1) 4
2) 14
3) 25
4) 6
5) 10
20) Column - I
A) if maximum and minimum values of 5 cos k + 3 cos k (k+ π/3) + 4 for all real values of k are m and n respectively, then
B) if minimum and maximum values of 4+ sin(π/4+ k) + 2 cos(√)4 - k) for all real values of k are n and m respectively, then
C) If number of solution of log|sinx| = x² - 16x, when x belongs to (0,π) is n and when [0,5π] is m, then
D) if number of solutions of 1+ [sin x] = cos x , +where [.] denotes the greatest integer function) when x belongs to (0, 10π) is m
Column - II
1) m+ n= 12
2) m- n= 6
3) m+ n= 8
4) m- n= 8