Calculus
Satyajeet
Questions
sin x+sin 3x
1. What is the fundamental period of f (x) = cos
x+cos 3x :
(
2x + α2 , if x ≥ 2
2. Let f : ℜ → ℜ, f (x) = αx
2 + 10, if x < 2
If f (x) is onto, then α belongs to:
A. [1, 4]
B. [−2, 3]
C. (0, 3]
D. [2, 5]
3. If f (x) is an invertible function and g(x) = 2f (x) + 5, the value of g −1 (x) is:
A. 2f −1 (x) − 5 B. 2f −11(x)+5 C. 12 f −1 (x) + 5 D. f −1 x−5
2
(x−2)
4. limx→2+ {x} sin
(x−2)2 ={where {.} denotes fractional part of the function}:
A. 0
B. 2
C. 1
5. Evaluate limx→ π2
(1−sin x)(8x3 −π 3 ) cos x
.
(π−2x)4
6. limx→∞ x2 sin loge
2
D. does not exist
p
cos πx =
2
A. − π4
2
B. − π2
C. 0 D. − π8

π

sin 2 (x − [x]) , if x < 5
if x = 5
7. Let f (x) = 5(b − 1),

 2 |x2 −11x+24|
ab
,
if x > 5
x−3
If f (x) is continuous at x = 5, a, b ∈ ℜ. then {where [.] denotes the greatest integer function}:
A. a =
B. a =
C. a =
D. a =
8. If y =
A. y
x2
2
6
25
108 and b = 5
6
17
13 and b = 29
1
25
2 and b = 36
23
6
100 and b = 5
p
√
√
+ 12 x x2 + 1 + ln x + x2 + 1, then the value of xy ′ + ln y ′ is:
B. 2y
C. 0
9. If x + y = 3e2 . then
D. −2y
d
y
dx (x )
= 0 for x =:
10. From the point (1, 1) tangents are drawn to the curve represented parametrically as x = 2t − t2 and
y = t + t2 . the distance between the point of contacts is:
A.
√
2 43
9
B. 2
C.
√
2 53
9
D. 3
11. The distance of point P on the curve y = x3/2 which is nearest to the point M (4, 0) from the origin is:
q
q
q
q
112
100
101
112
A.
B.
C.
D.
27
27
9
9
1
12. Suppose the water is emptied from a spherical tank of radius 10cm. If the depth of the water in the tank
is 4cm and is decreasing at the rate of 2cm/sec, then the radius of the top surface of water is decreasing
at the rate of:
A. 1
B. 2/3
C. 3/2
D. 2
13. Given f ′ (1) = 1 and f (2x) = x, ∀x > 0. If f ′ (x) is differentiable then there exists a number c ∈ (2, 4)
such that f ′′ (c) equals
A. 1/4
B. −1/2
C. −1/4
14. The minimum value of
15.
16.
R
R
x+x2/3 +x1/6
dx
x(1+x1/3 )
D. −1/8
tan x+ π
6
tan x
, x ∈ 0, π3 is:
equals:
A.
3x2/3
4
+ 6tan−1 (x1/6 ) + C
B.
3x2/3
2
+ 6tan−1 (x1/6 ) + C
C.
3x2/3
10
+ 6tan−1 (x1/6 ) + C
D.
3x2/3
5
+ 6tan−1 (x1/6 ) + C
x2 +1
√
√
dx
x x2 +2x−1 1−x2 −x
A.
B.
C.
D.
equals:
q
2sin−1 x − x1 + 2 + C
q
2cos−1 x − x1 + 2 + C
q
sin−1 x − x1 + 2 + C
q
cos−1 x − x1 + 2 + C
17. If
R
x2 −4
x4 +9x2 +16 dx
= tan−1 ax +
18. If
R
1−7 cos2 x
dx
sin7 x cos2 x
=
f (x)
(sin x)7
b
x
+C, find a + b:
+ C, then f (x) is equal to:
A. sin x
B. cos x C. tan x D. cot x
R1
19. The value of I = −1 (1 + x)1/2 (1 − x)3/2 dx is:
20. The value of I =
A. −π log 2
R π/2
0
log sin xdx is:
B. π log 2
C.
π
2
log 2
D. − π2 log 2
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