1. If x is real and 3 + 5x - 2x2 > 0, then x lies in the set
Ans: 
Solution:
3 + 5x - 2x2 = 0 -2x2 + 5x + 3 > 0
Here a = -2 < 0, 3 + 5x - 2x2 > 0
2x2 - 5x - 3 = 0
2x2 - 6x + x - 3 = 0
2x(x - 3) + 1 (x - 3) = 0
(x - 3) (2x + 1) = 0
x = 3, x = -
... - 
< x < 3
2. If α, β, γ are the roots of x3 - 2x2 + 3x - 4 = 0, then α2β2 + β2γ2 + γ2α2 =
Ans: −7
Solution:
α2β2 + β2γ2 + γ2α2
= (αβ + βγ + γα)2 - 2 αβγ (α + β + γ)
= 32 - 2 × 4 (2) = 9 - 16 = -7
3. If α, β, γ are the roots of the equation x3 - 10x2 + 7x + 8 = 0, then match the following:
Ans: 1-d, 2-c, 3-b, 4-a
Solution:
Here α + β + γ = 10
α2 + β2 + γ2
= (α + β + γ)2 - 2 (αβ + βγ + γα)
= 102 - 2 × 7 = 100 - 14 = 86
Ans: 2
Solution:
(x2 + 2)
= (Ax + B) (x2 + 3) + (Cx + D) (x2 + 1)
= (A +C)x3 + (B + D)x2 + [3 A + C] x + 3B + D
Coefficient x3 equation on both sides
A + C = 0
Coefficient x2 equating B + D = 1 and
3 B + D = 2 constant term
... A + C + 3B + D = 2
5. The remainder when x64 + x27 + 1 is divided by (x + 1) is
Ans: 1
Solution:
Let f(x) = x64 + x27 + 1
The remainder when f(x) is divided by x + 1
is f (-1) = 1 - 1 + 1 = 1
6. If α, β, γ are the roots of x3 + bx + c = 0, then
Ans: 0
Solution:
Ans: 16
Solution:
Ans:
Solution:
The matrix is singular cos2θ − sin2θ = 0 cos2θ = 0 2θ = 
θ = 
or cos2 θ = sin2 θ θ =
Ans: 
Solution:
10. The number of non-injective functions from a set A containing 4 elements to set B containing 5 elements is
Ans: 505
Solution:
(Number of functions from A to B) - (Number of injections from A to B)
= 54 - 5P4 = 625 - 5! = 625 - 120 = 505
11. The number of ways in which 16 rupee coins can be distributed among three boys such that each boy does not receive less than three coins is..
Ans: 9C2
Solution:
Here x + y + z = 16
x, y, z ≥ 3 x - 3, y - 3, z - 3 ≥ 0
Put x - 3 = t, y - 3 = s, z - 3 = u
... t + 3 + s + 3 + u + 3 = 16 t + s + u = 7, t, s, u ≥ 0
required no. of ways 7 + 3 - 1C3-1 = 9C2
12. Assertion (A): 28C4 + 28C3 + 29C3 = 30C4
Reason (R): nCr−1 + nCr = n + 1Cr Which of the following is true
Ans: A and R are true and R is correct explanation of A.
Solution:
nCr−1 + nCr = n+1Cr
28C4 + 28C3 = 29C4
29C4 + 29C3 = 29C4
28C4 + 28C3 + 29C3 = 29
13. The highest exponent of 7 in 100! is
Ans: 16
Solution:
The highest exponent of 7 in 100!
= 14 + 2 + 0... = 16
14. The number of terms in the expansion of (x2 + 18x + 81)15 is
Ans: 31
Solution:
(x2 + 18x + 9)15 = [(x + 9)2]15 = (x + 9)30
No. of terms in (x + a)n is n + 1
... (n + 9)30 is 31
Ans: 
Solution:
Ans: n2
Solution:
Ans: 
Solution:
18. The value of k so that the system of linear equations x - y + 2z = 0, kx - y + z = 0, 3x + y - 3z = 0 does not possess a unique solution is
Ans: 5
Solution:
1 [3 - 1] + 1 [-3k - 3] + 2 [k + 3] = 0
2 - 3 (k + 1) + 2k + 6 = 0 -k + 5 = 0
k = 5
19. ABCDEF is a regular hexagon with centre 'O' as the origin of the coordinate axes. If the position vectors of A and B are 
and 
, then 
equals
Ans: 
