Page 25 - ISC-12

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

(A) The curve y = f(x) passes through the point (1, 2).
(B) The curve y = f(x) passes through the point (2, −1).
√ π − 2
(C) The area of the region {(x, y) ∈ [0, 1] × R : f(x) ≤ y ≤ 1 − x } is .
2
4
√ π − 1
(D) The area of the region {(x, y) ∈ [0, 1] × R : f(x) ≤ y ≤ 1 − x } is .
2
4
[JEE 2018]

 
b 1
5. Let S be the set of all column matrices b 2 such that b 1 , b 2 , b 3 ∈ R and the system of equations (in
 
b 3
real variables)
−x + 2y + 5z = b 1 , 2x − 4y + 3z = b 2 , x − 2y + 2z = b 3

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least
 
b 1
one solution for each b 2 ∈ S?
 
b 3
(A) x + 2y + 3z = b 1 , 4y + 5z = b 2 and x + 2y + 6z = b 3 .
(B) x + y + 3z = b 1 , 5x + 2y + 6z = b 2 and −2x − y − 3z = b 3 .
(C) −x + 2y − 5z = b 1 , 2x − 4y + 10z = b 2 and x − 2y + 5z = b 3 .

(D) x + 2y + 5z = b 1 , 2x + 3z = b 2 and x + 4y − 5z = b 3 .
[JEE 2018]

6. Let f : (0, π) → R be twice differentiable function such that
f(x) sin t − f(t) sin x
2
lim = sin x for all x ∈ (0, π).
t→x t − x
π π

If f = − , then which of the following statement(s) is(are) TRUE?
6 12
π π

(A) f = √ .
4 4 2
x 4
2
(B) f(x) < − x for all x ∈ (0, π).
6
0
(C) There exists α ∈ (0, π) such that f (α) = 0.
π
π

(D) f 00 + f = 0.
2 2
[JEE 2018]
7. Let f : R → (0, 1) be a continuous function. Then, which of the following function(s) has(have) the
value zero at some point in the interval (0, 1)?
Z (π/2)−x
9
(A) x − f(x) (B) x − f(t) cos t dt
0
x π/2
Z Z
x
(C) e − f(t) sin t dt (D) f(x) + f(t) sin t dt
0 0
[JEE 2017]