Page 27 - ISC-12

4Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal
98 Z k+1 k + 1
P
14. If I = dx, then
k=1 k x(x + 1)
49 49
(A) I > log 99. (B) I < log 99. (C) I < . (D) I > .
e
e
50 50
[JEE 2017]

15. Consider a pyramid OPQRS located in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) with O as origin, and
OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square
with OP = 3. The point S is directly above the mid-point T of diagonal OQ such that TS = 3. Then
π
(A) the acute angle between OQ and OS is .
3
(B) the equation of the plane containing the triangle OQS is x − y = 0.
3
(C) the length of the perpendicular from P to the plane containing the triangle OQS is √ .
2
r
15
(D) the perpendicular distance from O to the straight line containing RS is .
2
[JEE 2016]

 
3 −1 −2
16. Let P = 2 0 α , where α ∈ R. Suppose Q = [q ij ] is a matrix such that PQ = kI, where


3 −5 0
k k 2
k ∈ R, k 6= 0 and I is the identity matrix of order 3. If q 23 = − and det (Q) = , then
8 2
(A) α = 0, k = 8. (B) 4α − k + 8 = 0.
13
9
(C) det (P adj (Q)) = 2 . (D) det (Q adj (P)) = 2 .
[JEE 2016]

17. A solution curve of the differential equation
dy
2
2
(x + xy + 4x + 2y + 4) − y = 0, x > 0,
dx
passes through the point (1, 3). Then the solution curve
(A) intersects y = x + 2 exactly at one point. (B) intersects y = x + 2 exactly at two points.

2
2
(C) intersects y = (x + 2) . (D) does NOT intersect y = (x + 3) .
[JEE 2016]

3
18. Let f : R → R, g : R → R and h : R → R be differentiable functions such that f(x) = x + 3x + 2,
g(f(x)) = x and h(g(g(x))) = x for all x ∈ R. Then

1
0
0
(A) g (2) = . (B) h (1) = 666. (C) h(0) = 16. (D) h(g(3)) = 36.
15
[JEE 2016]