2. SECTION – A Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them is correct. Indicate you choice of the correct answer for each part in your answer-book by writing the letter (a), (b), (c) or (d) whichever is appropriate
3. Problem 01 Let a, b, c be such that b(a + c) ≠ 0 . If then the value of ‘n’ is zero any even integer any odd integer any integer
4. Problem 02 If the mean deviation of number 1, 1 + d, 1 + 2d, ….. , 1 + 100d from their mean is 255, then the d is equal to 10.0 20.0 10.1 20.2
5. Problem 03 If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is greater than 4ab less than 4ab greater than – 4ab less than – 4ab
6. Problem 04 Let A and B denote the statements A: cosα + cosβ + cosγ = 0 B: sinα + sinβ + sinγ = 0 If cos(β − γ )+cos(γ − α)+cos(α − β)=3/2, then A is true and B is false A is false and B is true both A and B are true both A and B are false
7. Problem 05 The lines p(p2 +1)x − y + q = 0 and (p2 + 1)2 x + (p2 +1)y + 2q 0 are perpendicular to a common line for no value of p exactly one value of p exactly two values of p more than two values of p
8. Problem 06 If A, B and C are three sets such that A ∩B = A ∩C and A ∪B = A ∪C , then A = B A = C B = C A ∩B = φ
9. Problem 07 If are non-coplanar vectors and p, q are real numbers, then the equality holds for exactly one value of (p, q) exactly two values of (p, q) more than two but not all values of (p , q) all values of (p, q)
10. Problem 08 Let the line lies in the plane x + 3y − αz + β = 0 . Then (α, β) equals (6, - 17) ( - 6, 7) (5, - 15) ( - 5, 15)
11. Problem 09 From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is less than 500 at least 500 but less than 750 at least 750 but less than 1000 at least 1000
12. Problem 10 dx,[•] denotes the greatest integer function, is equal to π/2 1 -1 -π/2
13. Problem 11 For real x, let f (x) = x3 + 5x + 1, then f is one-one but not onto R f is onto R but not one-one f is one-one and onto R f is neither one-one nor onto R
14. Problem 12 In a binomial distribution B (n, p= ¼), if the probability of at least one success is greater than or equal to 9/10 , then n is greater than a. b. c. d.
15. Problem 13 If P and Q are the points of intersection of the circles x2 + y2 + 3x + 7y + 2p − 5 = 0 and x2 + y2 + 2x + 2y − p2 = 0 , then there is a circle passing through P, Q and (1, 1) for all values of p all except one value of p all except two values of p exactly one value of p
16. Problem 14 The projections of a vector on the three coordinate axis are 6, - 3, 2 respectively. The direction cosines of the vector are a.6, − 3, 2 b. c. d.
17. Problem 15 If , then the maximum value of is equal to √3 +1 √ 5 +1 2 2 + √ 2
18. Problem 16 Three distinct points A, B and C are given in the 2 – dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point ( - 1, 0) is equal to 1/3. Then the circumventer of the triangle ABC is at the point (0, 0 (5/4 , 0) (5/2, 0) (5/3, 0)
19. Problem 17 The remainder left out when 82n –(62) 2n+1 − is divided by 9 is 0 2 7 8
20. Problem 18 The ellipse x2 + 4y2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is x2 +16y2 = 16 x2 +12y2 = 16 4x2 + 48y2 = 48 4x2 + 64y2 = 48
21. Problem 19 The sum to the infinity of the series is 2 3 4 6
22. Problem 20 The differential equation which represents the family of curves y = c1 ec2x , where c1 and c2 are arbitrary constants is y ' = y2 y " = y ' y yy" = y' yy" =(y)”2
23. Problem 21 One ticket is selected at random from 50 tickets numbered 00, 01, 02, …., 49. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals a.1/14 b.1/7 c.5/14 d.1/50
24. Problem 22 Let y be an implicit function of x defined by x2x − 2xx cot y −1= 0 . Then y '(1) equals – 1 1 log 2 – log 2
25. Problem 23 The area of the region bounded by the parabola (y − 2)2 = x −1, the tangent to the parabola at the point (2, 3) and the x-axis is 3 6 9 12
26. Problem 24 Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0 . If P(−1) < P(1) , then in the interval [−1, 1] P(−1) is the minimum and P(1) is the maximum of P P(−1) is not minimum but P(1) is the maximum of P P(−1) is the minimum and P(1) is not the maximum of P neither P(−1) is the minimum nor P(1) is the maximum of P
27. Problem 25 The shortest distance between the line y − x = 1 and the curve x = y2 is 3√2/8 2 √ 3/8 3 √ 2/5 √ 3/4
28. Problem 26 Let f(x) = (x +1)2 −1, x ≥ −1 Statement-1 : The set {x : f (x) = f −1 (x)} = {0, −1} Statement-2 : f is a bisection. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
29. Problem 27 Let f (x) = x x and g(x) = sinx . Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at x = 0. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
30. Problem 28 Statement-1 : The variance of first n even natural numbers is n2 -1/4 Statement-2 : The sum of first n natural numbers is n(n+1)/2 and the sum of squares of first n natural numbers is Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
31. Problem 29 Statement-1 : ~ (p ↔~ q) is equivalent to p ↔ q. Statement-2 : ~ (p ↔~ q) is a tautology. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
32. Problem 30 Let A be a 2 x 2 matrix Statement-1 : adj(adj A) = A Statement-2 : |adjA |= |A | Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 Statement-1 is true, Statement-2 is false Statement-1 is false, Statement-2 is true
