1. Let L₁ be a straight line passing through the origin and L₂ be the straight line x + y = 1. If the intercepts made by the circle x² + y² – x + 3y = 0 on L₁ and L₂ are equal, then which of the following equation can represent L₁ ?
(a) x + 7y = 0
(b) x – y = 0
(c) x – 7y = 0
(d) Both (a) and (b)

2. 

then incorrect about the matrix (PQ)^–1 is
(a) nilpotent
(b) idempotent
(c) involutory
(d) symmetric

3. The equation sin x + x cos x = 0 has at least one root in
(a) (−π/2,0)
(b) (0, p)
(c) (π, 3π/2)
(d) (0,π/2)

4. The area enclosed by the parabola y² = 12x and its latus rectum is
(a) 36
(b) 24
(c) 18
(d) 12

5. Number of permutations 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time are such that the digit. 1 appearing somewhere to the left of 2, 3 appearing to the left of 4 and 5 somewhere to the left of 6, is
(e.g., 815723946 would be one such permutation)
(a) 9 . 7!
(b) 8!
(c) 5! . 4!
(d) 8! . 4!

6. If the function f : [0, 16] → R is differentiable. If 0 < α < 1 and 1 < β < 2, then ∫¹⁶₀ f (t) dt is equal to
(a) 4 [a³f (a⁴) – b³ f (b⁴)]
(b) 4 [a³f (a⁴) + b³ f (b⁴)]
(c) 4 [a⁴f (a³) + b⁴f (b³ )]
(d) 4 [a²f (a²) + b² f (b²)]

7. Three distinct points P (3u², 2u³), Q (3v², 2v³) and R (3w², 2w³) are collinear then
(a) uv + vw + wu = 0
(b) uv + vw + wu = 3
(c) uv + vw + wu = 2
(d) uv + vw + wu = 1

8. Let ‘a’ denote the root of equation
cos (cos^–1 x) + sin^–1 sin (1 + x²/2) = 2 sec^–1 (sec x)
then possible values of [ | 10a | ] where [ . ] denotes the greatest integer function will be
(a) 1
(b) 5
(c) 10
(d) Both (a) and (c)

9. The two of the straight lines represented by the equation ax³ + bx² y + cxy² + dy³ = 0 will be at right angle if
(a) a² + c² = 0
(b) a² + ac + bd + d² = 0
(c) a²c² + bd + d² = 0
(d) None of these

10. If x² – 2x cos θ + 1 = 0, then the value of x²ⁿ – 2xⁿ cos nθ + 1, n ∈ N is equal to
(a) cos 2nq
(b) sin 2nq
(c) 0
(d) some real number greater than 0

11. Evaluate ∫8/(x + 2)(x² + 4) dx
(a) log | x + 2 | – 1/2 log (x² + 4) + tan^–1 x/2 + C
(b) log | x + 2 | – 1/2 log (x² + 4) + sin^–1 x/2 + C 
(c) log | x + 2 | – 1/2 log (x² + 4) + cos^–1 x/2 + C
(d) log | x + 2 | – 1/2 log (x³+ 4) + tan^–1 x/2 + C 

12. Vertices of a parallelogram taken in order are A (2, –1, 4), B (1, 0, –1), C (1, 2, 3) and D. Distance of the point P (8, 2, –12) from the plane of the parallelogram is 
(a) 4√6/9
(b) 32√6/9
(c) 16√6/9
(d) None of these

13. Given A̅ = 2î + 3ĵ + 6k̂, B̅ = î + ĵ – 2k̂ and C̅ = î + 2ĵ + k̂. Compute the value of | A̅×[A̅×(A̅×B̅)].C̅ |., 
(a) 343
(b) 512
(c) 221
(d) 243

14. The value of the definite integral

(a) 2(√2 +1)
(b) 2(√2 –1)
(c) √2 +1
(d) √2 –1

15. Area of triangle formed by common tangents to the circle x² + y2 – 6x = 0 and x² + y² + 2x = 0 is
(a) 3√3
(b) 2√3
(c) 9√3
(d) 6√3

16. The locus of the centres of the circles which cut the circles x² + y² + 4x – 6y + 9 = 0 and x² + y² – 5x + 4y – 2 = 0 orthogonally is
(a) 9x + 10y – 7 = 0
(b) x – y + 2 = 0
(c) 9x – 10y + 11 = 0
(d) 9x + 10y + 7 = 0

17. The sum to infinity of the series

(a) 3
(b) 1
(c) 2
(d) 3/2

18. The straight line ĵoining any point P on the parabola y² = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is
(a) x² + 2y² – ax = 0
(b) 2x² + y² – 2ax = 0
(c) 2x² + 2y² – ay = 0
(d) 2x² + y² – 2ay = 0

19. A box contains 6 red, 5 blue and 4 white marbles. Four marbles are chosen at random without replacement. The probability that there is atleast one marble of each colour among the four chosen, is
(a) 48/91
(b) 44/91
(c) 88/91
(d) 24/91

20. Statement-1 : If a, b, c are non real complex and α, β are the roots of the equation ax² + bx + c = 0 then Im (αβ) ≠ 0. because
Statement-2 : A quadratic equation with non real complex coefficient do not have root which are conĵugate of each other.
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is false.

21. Statement-1 : The line x/b + y/b = 1 touches the curve y = be^–x/a at some point x = x₀. because
Statement-2 : dy/dx exists at x = x₀.
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is false.

22. Let C be a circle with centre O and HK is the chord of contact of pair of the tangents from points A. OA intersects the circle C at P and Q and B is the midpoint of HK, then 
Statement-1 : AB is the harmonic mean of AP and A because
Statement-2 : AK is the Geometric mean of AB and AO, OA is the arithmetic mean of AP and A
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is false.

23. Statement-1 : The statement (p ∨ q) ∧ ~ p and ~ p ∧ q are logically equivalent.
Statement-2 : The end columns of the truth table of both statements are identical.
(a) Statement-1 is false, Statement-2 is true. 
(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is false.

24. Statement-1 : Period of f(x) = sin 4π {x} + tan π [x], where, [x] & {x} denote the G.I.F. & fractional part respectively is 1.
Statement-2: A function f(x) is said to be periodic if there exist a positive number T independent of x such that f(T + x) = f(x). The smallest such positive value of T is called the period or fundamental period.
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(d) Statement-1 is true, Statement-2 is false.

25. The value of the expression

where ω is an imaginary cube root of unity, is
(a) n (n² – 2)/3
(b) n (n² + 2)/3
(c) n (n² – 1)/3
(d) None of these

26. If s, s’ are the length of the perpendicular on a tangent from the foci, a, a’ are those from the vertices is that from the centre and e is the eccentricity of the ellipse, x²/a² + y²/b² = 1 then ss’ – c²/ aa’ – c² =
(a) e
(b) 1/e
(c) 1/e²
(d) e²

27. One percent of the population suffers from a certain disease. There is blood test for this disease, and it is 99% accurate, in other words, the probability that it gives the correct answer is 0.99, regardless of whether the person is sick or healthy. A person takes the blood test, and the result says 
that he has the disease. The probability that he actually has the disease, is
(a) 0.99%
(b) 25%
(c) 50%
(d) 75%

28. Set of values of m for which two points P and Q lie on the line y = mx + 8 so that ∠APB = ∠AQB = π/2 where A ≡ (– 4, 0), B ≡(4, 0) is
(a) (–∞, – √3)∪( √3,∞) –{–2,2}
(b) [– √3, – √3]–{–2, 2}
(c) (–∞, –1)∪(1,∞) –{–2, 2}
(d) {– √3, √3}

29. The trace T_r(A) of a 3 × 3 matrix A = (a_iĵ ) is defined by the relation Tr(A) = a₁₁ + a₂₂ + a₃₃ (i.e., T_r(A) is sum of the main diagonal elements). Which of the following statements cannot hold ?
(a) T_r(kA) = kT_r(A) (k is a scalar)
(b) T_r(A + B) = T_r(A) + T_r(B)
(c) T_r(I₃) = 3
(d) T_r(A²) = T_r(A)²

30.

(a) 1/2
(b) 1
(c) 4/3
(d) 3/2