Page 41 - ISC-12

2Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal

9. Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x-axis, y-axis

1 1 1
and z-axis, respectively, where O(0, 0, 0) is the origin. Let S , , be the centre of the cube and
2 2 2
−→
T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If ~p = SP,
−→ −→ −→
~
~q = SQ, ~r = SR and t = ST, then the value of (~p × ~q) × (~r × t) is [JEE 2018]
~

π

0
10. Let f : R → R be a differentiable function such that f(0) = 0, f = 3 and f (0) = 1. If
2
π/2
Z
0
g(x) = [f (t) cosec t − cot t cosec t f(t)] dt
x
π
i
for x ∈ 0, , then lim g(x) = [JEE 2017]
2 x→0
11. For a real number α, if the system
     
1 α α 2 x 1
 α 1 α  y = −1


  

α 2 α 1 z 1
2
of linear equations, has infinitely many solutions, then 1 + α + α = [JEE 2017]
12. Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number
of such words where no letter is repeated; and let y be the number of such words where exactly one
y
letter is repeated twice and no other letter is repeated. Then, = [JEE 2017]
9x

x x 2 1 + x 3

3
13. The total number of distinct x ∈ R for which 2x 4x 2 1 + 8x = 10 is [JEE 2016]

2
3x 9x 1 + 27x
3
x t 2
Z
14. The total number of distinct x ∈ [0, 1] for which 4 dt = 2x − 1 is [JEE 2016]
0 1 + t
2
x sin (βx)
15. Let α, β ∈ R be such that lim = 1. Then 6(α + β) equals [JEE 2016]
x → 0 αx − sin x
√ " #
−1 + 3 i √ (−z) r z 2s
16. Let z = , where i = −1, and r, s ∈ {1, 2, 3}. Let P = and I be the
2 z 2s z r
2
identity matrix of order 2. Then the total number of ordered pairs (r, s) for which P = −I is
[JEE 2016]

17. The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least
two heads is at least 0.96, is [JEE 2015]

2
18. If the normals of the parabola y = 4x drawn at the end points of its latus rectum are tangents to the
2
2
2
2
circle (x − 3) + (y + 2) = r , then the value of r is [JEE 2015]