1. If |z – i| ≤ 2 and z₀ = 5 + 3i, then the maximum value of |iz + z₀| is
(a) 2 + √31
(b) 7
(c) √31 − 2
(d) None of these 

2. The continued product of four roots of

(a) 0
(b) 1
(c) i
(d) –1 

3. If capital letters denote the cofactors of the corresponding small letters in

(a) Δ²
(b) 2Δ
(c) 0
(d) Δ 

4.

(a) – abc
(b) 0
(c) abc
(d) None of these 

5. 

Then x + y is equal to
(a) –1
(b) 1
(c) 6
(d) – 6

6. The position vectors of three points are 2a̅  − b̅ + 3c̅, a̅ − 2b̅ + mc̅ ,  and na̅ − 5b̅  where a̅ , b̅ , c̅ are non-coplanar vectors and m, n are scalars. The three points are collinear if and only if
(a) m = −2 n = 9/4
(b) m =9/4, n = − 2
(c) m = 2 n = 9/4 
(d) m = −9/4, n = 2 

7. If the vectors   a̅ = î + ĵ + k̂ ,b̅ = 4î + 3ĵ + 4k̂ c̅= î + αĵ + βk̂  are linearly dependent and |c̅| = √3, then
(a) α = ±1, β = 1
(b) α = 1, β = –1
(c) α = 1, β = ±1
(d) α = –1, β = ±1 

8. On each evening a boy either watches Doordarshan channel or Ten Sports. The probability that he watches Ten Sports is 4/5. If he watches Doordarshan, there is a chance of 3/4 that he will fall asleep, while it is 1/4 when he watches Ten Sports. On one day, the boy is found to be asleep. The probability that the boy watched Doordarshan is
(a) 5/7
(b) 2/7
(c) 3/7
(d) 4/7   

9. Three groups A, B and C are competing for the positions on the Board of Directors of a company. The probabilities of their winning are 0.5, 0.3 and 0.2, respectively. If the group A wins, the probability of introducing a new product is 0.7 and other corresponding probabilites for groups B and C are, respectively, 0.6 and 0.5. The probability that new product will be introduced is 
(a) 0.43
(b) 0.53
(c) 0.63
(d) 0.73   

10. The range of random variable X is {1, 2, 3} and P(X = 1) = 3λ³, P(X = 2) = 4λ – 10λ²,
P(X = 3) = 5λ – 1 where l is constant. Then P(2 ≤ X ≤ 3) is equal to
(a) 8/9
(b) 2/3
(c) 4/9
(d) 1/3 

11. In a book of 750 pages, there are 500 typographical errors. Assuming Poisson law for the number of errors per page, find the probability that a random sample of 5 pages will contain no error.
(a) e^–10
(b) e^–2/3
(c) e¹⁰
(d) e^–10/3 

12. The area enclosed within the curve |x| + |y| = 1 is
(a) 2
(b) 4
(c) 6
(d) none of these 

13. The radius of the circle having centre at (2, 1), whose one of the chord is a diameter of the circle x² + y² – 2x – 6y + 6 = 0 is 
(a) 1
(b) 2
(c) 3
(d) √3 

14. The equation of the parobola whose focus is the point (0, 0) and the tangent at the vertex is x – y + 1 = 0 is
(a) x² + y² + 2xy – 4x – 4y + 4 = 0
(b) x² + y² – 2xy – 4x + 4y – 4 = 0
(c) x² + y² – 2xy + 4x – 4y – 4 = 0
(d) x² + y² + 2xy – 4x + 4y – 4 = 0 

15. The angle between pair of tangents drawn to the ellipse 3x² + 2y² = 5 from the point (1, 2) is

16. 

which of the following remains constant with change in ‘α’?
(a) abscissa of vertices
(b) abscissa of foci
(c) eccentricity
(d) directrix 

17. Find the maximum value of f(x) = |x log_e x| in (0, 1).
(a) e
(b) –e
(c) –1/e
(d) none of these 

18. Semi-vertical angle of a cone is 45°. The height of cone is 30.05 cm. Find the approximate volume.
(a) 1325π c.c
(b) 9045π c.c
(c) 2020π c.c
(d) 330π c.c 

19.

(a) 0
(b) π(π + 1)
(c) π²(π + 1)
(d) none of these 

20. 

21.

22. The function f(x) = (x – 1)^m (x – 2)^m+1 has
(a) inflection at x = 1, 2
(b) inflection at x = 1 or x = 2
(c) no point of inflection
(d) two points of extrema

23. Let Φ(x) = ∫^x₀g(t) dt, where g is such that
–1/2 ≤ g(t) ≤ 0 for t ∈ [0, 1]
–1/2 ≤ g(t) ≤ 1 for t ∈ [1, 3]
           g(t) ≤ 1 for t ∈ [3, 4]
Then Φ(4) satisfies the inequality
(a) 1/2 ≤ Φ(4) ≤ 3
(b) 0 ≤ Φ(4) ≤ 2
(c) Φ(4) ≤ 3
(d) none of these 

24. ∫₀^[x] ,[x]dx , is equal to
(a) [x]² /2
(b) [x]([x]+ 1)/2
(c) [x]([x]− 1)/2
(d) ([x]− 1)([x]+ 1)/2 

25. Find the shortest distance between the curves
y² = x³ and 9x² + 9y² – 30y + 16 = 0.
(a) 0.5
(b) 0.2
(c) 1
(d) 3 

26. The false statement in the following is
(a) (p ⇒ q) ⇔ (~ q ⇒ ~ p) is a contradiction
(b) ~ (~ p) ⇒ p is a tautotogy
(c) p ∨ (~ p) is a tautology
(d) p ∧ (~ p) is a contradiction   

27. The inverse of the proposition (p ∧ ~ q) ⇒ r is
(a) ~ r ⇒ ~ p ∨ q
(b) r ⇒ p ∧ ~ q
(c) ~ p ∨ q ⇒ ~ r
(d) None of these 

28. Which venn diagram represents the truth of the statement ‘No policeman is a thief’?

29. A line is uniquely determined if
(i) it passes through a given point and has given direction
(ii) it passes through two given points 
(iii) it passes through a point and is parallel to a line.
(iv) it passes through two given points and is perpendicular to the plane
(a) only (i)
(b) (i) and (ii)
(c) (i), (ii) and (iii)
(d) all of these 

30. If a, b, c are real numbers and planes 
ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 meet in a line then
(a) a² + b² + c² = 0
(b)  a² + b² + c² = ab + bc + ca
(c) 3abc =  a² + b² + c²
(d) 3( a² + b² + c²) = 2(ab + bc + ca)

31. The equation of the plane containing the

32. If π/2 < θ < 3π/2, the modulus and argument form of (1 + cos 2θ) + i sin 2θ is
(a) 2cos θ [cos θ + i sin θ]
(b) – 2cos θ [cos (π + θ) + i sin (π + θ)]
(c) 2cos θ [cos (– θ) + i sin (– θ)]
(d) – 2cos θ [cos (π – θ) + i sin (π – θ)]   

33. The solution of the equation (x – h)² + (y – k)² = a²,where h and k are parameters’ is 

34. The solution of the differential equation
(1 + y²)dx = (tan^–1 y – x)dy is
(a) x = tan^–1 y + 1 + Ce^{– tan–1} y
(b) y = tan^–1 y + 1 + Ce^{– tan–1} y
(c) x = tan^–1 y + Ce^{– tan–1} y
(d) None of these   

35. The equation of the curve passing through the point (1,π/4) and having slope of tangent at any point (x, y) as y/x − cos² y/x’ is 

36. The area of the region bounded by the curves
y = 2^x, y = 2x – x² and x = 2, is 

37. 

38.  

(a) log 4
(b) – log 4
(c) 1 – log 4
(d) None of these   

39. ∫^∞₀e^−x .x³ dx is equal to
(a) 0
(b) 2
(c) 6
(d) 9 

40. The value of c from the Lagrange’s mean value theorem for which f(x) = √25 − x² in  [1, 5], is
(a) 5
(b) 1
(c) √15
(d) None of these