2. If ω is a complex cube root of unity, then 225 + (3ω + 8ω2)2 + (3ω2 + 8ω)2 =
1) 72 2) 192 3) 200 4) 248
Ans: 248
Sol: Given 225 + (3ω + 8ω2)2 + (3ω2 + 8ω)2
= 225 + 9ω2 + 64ω4+ 48ω3 + 9ω4 + 64ω2 + 48ω3
= 225 + (9 + 64)ω2 + (9 + 64)ω4 +96ω3
= 225 + 73 (ω2 + ω) + 96 (ω3 = 1)
= 321 - 73 = 248 (ω4 = ω)
ω2 + ω + 1 = 0
ω2 +ω = −1
1) x + y + 1 = 0 2) x + y - 1 = 0 3) x - y + 1 = 0 4) x - y - 1 = 0
Ans: x - y + 1 = 0
1) X - axis 2) Y - axis 3) X = 1 4) Y = 1
Ans: X - axis
1) x + y + 1 = 0 2) x + y - 1 = 0 3) x - y + 1 = 0 4) x - y - 1 = 0
Ans: x - y + 1 = 0
6. If α, β are complementary angles and tan (α - β) = 1, then sec 2β =
1) cosec 2β 2) tan 2β 3) sin 2β 4) cos 2β
Ans: cosec 2β
1) a rectangle 2) a square 3) a parallelogram 4) a rhombus
Ans: a rhombus
8. If f: R → R is defined by f(x) = x2 + 1, then the values of f-1(37) and f-1(-24) are respectively
1) {-6, 6}, φ 2) {-5, 5}, φ 3) {-√37, √37}, φ 4) {-√25, √25}, φ
Ans: {-6, 6}, φ
Sol: Let f-1 (37) = y 
f(y) = 37 y2 + 1
= 37 y2 = 36 y = ±6
f-1(-24) = y f(y) = -24 y2 + 1
= -26 y2 = -27
Which is not possible f(37) = {-6, 6}, f-1 (-24) = φ
10. The nth term in the series (1) + (3 + 5 + 7) + (9 + 11 + 13 + 15 + 17) + ... is
1) (2n − 1) [n2 − (n − 1)2] 2) (2n + 1) [n2 − (n − 1)2]
3) (2n − 1) [n2 + (n − 1)2] 4) (2n + 1) [n2 + (n − 1)2]
Ans: (2n − 1) [n2 + (n − 1)2]
11. The first term in the nth bracket of series (1) + (2 + 3 + 4) + (5 + 6 + 7 + 8 + 9) + ...
1) n2 + 2n + 2 2) n2 − 2n − 2 3) n2 − 2n −1 4) n2 − 2n + 2
Ans: n2 − 2n + 2
Sol: Number of terms in the nth bracket is (2n − 1)
First term in the nth bracket is
last term − No. of terms + 1
... n2 − (2n − 1) + 1
= n2 − 2n + 1 + 1 = n2 − 2n + 2
14. If α, β are the roots of the equation 6x2 − 5x + 1 = 0. Then tan−1α + tan−1β =
15. If the ratio of the roots of the equation ax2 + bx + c = 0 is m : n then which of the following is true?
16. The roots of the equation a(b − c) x2 + b(c − a) x + c (a − b) = 0 are
