Page 40 - ISC-12

Numerical Valued Answer Type

Each question in this section, when worked out will result in a numerical value.

1. The number of real solutions of the equation
∞ ∞
" # " #
∞
∞
X X x i π X x i X
sin −1 x i+1 − x = − cos −1 − − (−x) i
2 2 2
i=1 i=1 i=1 i=1

1 1
lying in the interval − , is [JEE 2018]
2 2

~
~
~
~
2. Let ~a and b be two unit vectors such that ~a · b = 0. For some x, y ∈ R, let ~c = x ~a + y b + ~a × b . If
~
2
|~c| = 2, and the vector ~c is inclined at the same angle α to both ~a and b, then the value of 8 cos α is
[JEE 2018]
3. The value of the integral
√
1/2
Z
1 + 3
dx
6
2
0 [(x + 1) (1 − x) ] 1/4
is [JEE 2018]
4. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {−1, 0, 1}. Then, the
maximum possible value of the determinant of P is [JEE 2018]
5. Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of
one-one functions from X to Y and β is the number of onto functions from Y to X, then the value of
1
(β − α) is [JEE 2018]
5!

6. Let f : R → R be a differentiable function with f(0) = 0. If y = f(x) satisfies the differential
equation
dy
= (2 + 5y)(5y − 2),
dx
then the value of lim f(x) is [JEE 2018]
x→∞
7. Let f : R → R be a differentiable function with f(0) = 1. and satisfying the equation
0
0
f(x + y) = f(x)f (y) + f (x)f(y) for all x, y ∈ R.
Then, the value of log (f(4)) is [JEE 2018]
e
8. Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment
PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies
on the z-axis. Let the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then

the length of PR is [JEE 2018]