MACMILLAN EDUCATION
M105(E)
Class 12 - Mathematics
Time Allowed: 40 minutes Maximum Marks: 85
20
General Instructions:
All Questions are Compulsory
Section A
⎧ 3,
⎪ if 0 ≤ x ≤ 1 [1]
1. All the points of discontinuity of the function f defined by f(x) = ⎨ 4, if 1 < x < 3 are
⎩
⎪
5, if 3 ≤ x ≤ 10
a) 0, 1, 3 b) 1, 3, 10
c) 3, 10 d) 1, 3
2. The value of k so that f defined by [1]
2
x if x≠
1
⎧
sin( ) 0
f(x) = ⎨ x
⎩
k if x= 0
is continuous at x = 0 is
a) 2 b)
1
c) 0 d) 1
3. The function f(x) = |cos x| is [1]
a) everywhere continuous but not b) either continuous or differentiable at (2n +
differentiable at (2n + 1) ,n∈Z 1) ,n∈Z
π π
2 2
c) neither continuous nor differentiable at (2n d) everywhere continuous and differentiable
+ 1) π
2
,n∈Z
4. If f(x) = |x| + |x - 2|, then [1]
a) f(x) is continuous at x = 0 and at x = 2 b) f(x) is continuous at x = 0 but not at x = 2
c) f(x) is continuous at x = -2 but not at x = 0 d) f(x) is continuous at x = 2 but not at x = 0
if x ≠
sin 5x
, 0 [1]
5. The value of k for which f(x) = {
3x
is continuous at x = 0 is
k, if x = 0
a) 0 b) 3
c) d)
1 5
3 3
6. The function f(x) = x |x|, x ∈ R is differentiable [1]
a) only at x = 1 b) only at x = 0
c) in R - {0} d) in R
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Mr.Balaji V & Mr.Yuvaraj S
7. If y = tan-1 (
1+x
2
) then
dy
= ? [1]
2
1−x dx
−2x
a) b)
2x
4 4
(1+x ) (1−x )
c) x
4
d) 2x
4
(1+x ) (1+x )
8. Let f(x) = x - |x| then f(x) is [1]
a) differentiable ∀ x ∈ R b) continuous ∀ x ∈ R and not differentiable at
x=0
c) discontinuous at x = 0 d) neither continuous nor differentiable at x =
0
2
9. Find
d y
, if x = at2, y = 2at. [1]
2
dx
a) 1
3
b) −1
2
2at 2at
c) −1
3
d) 0
2at
2
x −x−6
if x ≠ −2 [1]
10. The function f(x) = {
x+2
at x = −2 is
−5, if x = −2
a) f(2) = -5 b) f(-2) = 5
c) continuous d) not continuous
11. The number of points, where f (x) = [x], 0 < x < 3([⋅] denotes greatest integer function) is not differentiable [1]
is:
a) 1 b) 2
c) 3 d) 4
12. The function f(x) = |x| - x is: [1]
a) continuous but not differentiable at x = 0. b) differentiable but not continuous at x = 0.
c) continuous and differentiable at x = 0. d) neither continuous nor differentiable at x =
0.
13. If y = log (sin ex), then
dy
is: [1]
dx
a) cosec ex b) ex cot ex
c) cot ex d) ex cosec ex
14. d
( cos
−1
x) = −
1
where [1]
dx √1−x
2
a) −1 < x ⩽ 1 b) −1 ⩽ x < 1
c) -1 < x < 1 d) −1 ⩽ x ⩽ 1
15. The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at [1]
a) 1.5 b) 4
c) 1 d) -2
16. If y = 2x then
dy
= ? [1]
dx
a) x(2x-1) b) 2x (log 2)
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Mr.Balaji V & Mr.Yuvaraj S
x
c) 2
(log 2)
d) 3x (log 3)
17. Let f(x) = |sin x| Then [1]
a) f is everywhere differentiable b) f is everywhere continuous but not
differentiable at x = (2n + 1) , x ∈ Z
π
c) f is everywhere continuous but not d) f is everywhere differentiable x = (2n - 1) ,π
differentiable at x = nπ, n ∈ Z x≠Z
3 x + 5, x ≥ 2 [1]
18. The value of k for which f(x) = { 2
is a continuous function, is:
kx , x < 2
a) − 11
4
b) 11
c) 11
d) 4
11
4
1 + x, when x ≤ 2 [1]
19. The function f(x) = { is continuous and differentiable at x=2 ,yes or no
5 − x, when x > 2
a) Continuous but not differentiable at x = 2 b) Differentiable but not continuous at x = 2
c) Differentiable but continuous at x = 2 d) Continuous as well as differentiable at x = 2
20. If y = sin-1 (3x - 4x3) then
dy
= ? [1]
dx
a) − b)
3 3
√1+x2 √1−x2
c) 3
d) −4
√1+x2 √1−x2
−−−−− −
21. What is the rate of change of √x 2
+ 16 w.r.t. x2 at x = 3? [1]
a) 1
15
b) 1
20
c) 1
5
d) 1
10
22. If f(x) =
1−x
1
, then the set of points of discontinuity of the function f(f(f(x))) is [1]
a) {1} b) {1, 1}
c) {0, 1} d) {-1, 1}
3
−−−−−−
23. Let f(x) = c 2
3
− √x + x
2
then [1]
a) RHD at x ≠ 0 exists but LHD at x ≠ 0 b) LHD at x = 0 exists but RHD at x = 0 does
does not exists not exists
c) RHD at x = 0 exists but LHD at x = 0 does d) f(x) is differentiable at x = 0
not exists
24. sin |x| is a continuous function at x belongs to [1]
a) R b) R+
c) R - {0} d) R-
[1]
|x+2|
, x ≠ −2
25. If f(x) then f(x) is
−1
tan (x+2)
= {
2 , x = −2
a) not continuous at x = –2 b) continuous but derivable at x = –2
c) differentiable at x = –2 d) continuous at x = –2
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Mr.Balaji V & Mr.Yuvaraj S
