1. Straight Objective Type 1. If a ∈ Z and the equation (x - a) (x - 10) + 1 = 0 has integral roots, then the values of a are
1) 10, 8 2) 12, 10 3) 12, 8 4) 10, 14
2. If the sum of the squares of the roots of the equation
x2 − (sinα - 2)x - (1 + sinα) = 0 is least then α =
3. If a < 0 and b2 − 4ac < 0 then the graph of y = ax2 + bx + c
1) lies entirely below the x-axis
2) cuts the x-axis 3) touches the x-axis
4) lies entirely above the x-axis
4. If the sum of two roots of x5 + lx + m = 0 is zero then the value of m =
1) 1 2) -1 3) 0 4) -3
5. At least one root of x3 − 7x + 7 = 0 lies between
1) 0, 1 2) 1, 2 3) - 4, -3 4) - 1, 0
6. The greatest value of term independent of x
(where θ ∈ R) is
7. The number of zeros at the end of (101)11- 1 is.....
1) 2 2) 4 3) 6 4) 8
1) 5 2) 7 3) 9 4) 11
10. The number of rational terms in
1) 2 2) 4 3) 6 4) 8
12. If |Z − 1| + |Z − 3| ≤ 8, then maximum value of |Z − 4| is where Z is a complex number.
1) 8 2) 9 3) 10 4) 2
13. If |Z1| = 2, |Z2| = 3, |Z3| = 4 and |Z1 + Z2 + Z3| = 5 then |4 Z2 Z3 + 9 Z3 Z1 + 16 Z1 Z2| = ?
1) 14 2) 48 3) 96 4) 120
14. If Z is a complex number and P is a point satisfying arg Z = π/4 Such that sum of its 4 distances from two given point (0 1) and (0 2) is minimum, then P must be k/3 (1 + i), the numerical value of k should be
1) 1 2) 2 3) 3 4) 4
15. If z is a non zero complex number then
1) 0 2) 50 3) 50π 4) not defined
16. The coefficient of x8 in
17. The remainder when 2323 is divisible by 53 is
1) 7 2) 1 3) 30 4) 0
18. If ax2 + bx + 6 = 0 does not have two distinct real roots, where a, b ∈ R then the least value of 3a + b is ....
1) -1 2) -2 3) -3 4) -4
19. The equations x2 + 3x + 5 = 0 and ax2 + bx + c = 0 have a common root, if a, b, c ∈ N then the least possible value of a + b + c is equal to
1) 1 2) 6 3) 9 4) 12
20. Let f(x) = (1 + b2) x2 + 2bx + 1 and m(b) be the minimum value of f(x). As b varies, the range of m (b) is
1) [-1 1] 3) (1 2] 4) (0 1]
21. The set of values of x for which the inequality [x]2 - 5[x] + 6 ≤ 0 holds good (where [.] denote the greatest integral function)
1) 1 ≤ x < 2 2) 2 ≤ x < 3 3) 1 ≤ x < 3 4) 2 ≤ x < 4
22. If x3 + ax + 1 = 0 and x4 + ax2 + 1 = 0 have a common root then a =
1) -2 2) 2 3) 3 4) -3
23. The number of real roots of the equation
1) 2 2) 3 3) 6 4) 0
Numerical Value Type
24.
25.
26. If α is a positive root of x4 + x3 - 4x2 + x + 1 = 0 then α + 1/α = .......
27. Consider the equation x2 + 2x − n = 0 (n ∈ N) and n ∈ [5 100] Total number of different values of ‘n’ so that the given equation has integral roots is .....
28. If α is a complex number such that 2α + 1 = z
where z = √-3 and
= - kz then k = ........
29. The fractional part of
30. The number real roots of (6 - x)4 + (8 - x)4 = 16 is .....
31. If both the roots of the equation x2 - 2ax + a2 - 1 = 0 lie in the interval (-3 4) then sum of integral parts of ‘a’ is ......
Answers: 1-3; 2-4; 3-1; 4-3; 5-3; 6-4; 7-1; 8-2; 9-3; 10-1; 11-1; 12-2; 13-4; 14-2; 15-3; 16-4; 17-3; 18-2; 19;-3 20-4; 21-4; 22-1; 23-4; 24-1.04; 25-9; 26-2; 27-8; 28-3; 29-0.06; 30-2; 31-0.