VELALAR VIDYALAYAA SENIOR SECONDARY SCHOOL
GRADE 12
MATHEMATICS (041)
TIMINGS: 1.30 HOURS MONTHLY TEST-3(i)
MAX.MARKS: 40
25.07.2025 SECTION-A
12*1=12
1. If θ=¿ sin−1 ¿ ¿
A. B. C. D.
π −7 π 2π −π
3 5 3 10
2. The principal value of the expression cos–1 [cos (– 680°)] is
A. B. C. D.
2π −2 π 34 π π
9 9 9 9
3. The value of cot ( cos−1 x ) is
A. B. C. D.
√1+ x 2 x 1 √1−x 2
x √1−x 2 x x
4. Assertion (A): The range of the function f ( x )=2sin x + , where x ∈ [ −1 ,1 ] is ,
[ ]
−1 3π π 5π
2 2 2
Reason (R): The range of the principal value branch of sin x is [ ]
−1 −π π
,
2 2
A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct
explanation of the Assertion (A).
B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion (A).
C. Assertion (A) is true, but Reason (R) is false
D. Assertion (A) is false, but Reason (R) is true.
5. The domain of sin−1 x is
A. [ 0 , 1 ] B.[ −1 ,1 ] C.
[ ] D.[ −2 ,2 ]
−1 1
,
2 2
6. The domain of y=cos−1 ( x 2−4 ) is
7. The domain of the function defined by f (x) =sin x +cos x is
A. [ 3 ,5 ] B.[ 0 , π ] C.[ −√ 5 ,−√ 3 ] ∩ [ −√ 5 , √ 3 ] D.[ −√ 5 ,−√ 3 ] ∪ [ √ 3 , √ 5 ]
−1
A. [ −1 ,1 ] B.[ −1 , π +1 ] C.(−∞ , ∞ ) D.∅
8. The function f (x) = [x], where [x] denotes the greatest integer function, is
continuous at
A. 2.3 B.-2 C.5 D.0
9. The function f (x) = |x| + |x+ 1| is
A. Continuous at x = 0 as well as at x =- 1. B. continuous at x = 1 but not
at x = 0.
C. continuous at x = 0 as well as at x = 1. D. continuous at x = 0 but
not at x = 1.
10. The function f ( x )= is
2
4−x
4−x
A. discontinuous at only one point B. discontinuous at exactly two
points
C. discontinuous at exactly three points D. none of these
11. The function f (x)=e|x| is
A. continuous everywhere but not differentiable at x = 0
B. continuous and differentiable everywhere
C. not continuous at x = 0
D. none of these.
12. The function f ( x )=
{ is continuous at x = 0, then the value of k is
sinx
+ cosx ,if x ≠ 0
x
k , if x=0
A. 3 B.2 C.1 D.1.5
SECTION -B 3*2=6
13. Find the value of tan (−1 )+ cos ( 12 )+ sin ( 12 ).
−1 −1 −1
14. Find the relationship between a and b so that the function f defined by
f ( x )= ax +1 , if x ≤−3 is continuous at x =−¿ 3.
{ bx+ 3 ,if x >−3
15. Differentiate with respect to x:
x
e
tanx
SECTION -C
4*3=12
16. Find for the following y=tan ( )
dy −1 2x
2
dx 1−x
17. Prove that the function f given by f(x) = [ x ], x ∈ R is not differentiable at x =
1.
18. Write the following function in the simplest form: tan−1 (√ 1−cosx
1+ cosx )
,
19. Prove that : tan √ x= cos
−1 1 −1 1−x
, x ∈[ 0 , 1]
2 1+ x
SECTION -D
5*2=10
{
1−cos 4 x
if x <0
20. Let f ( x )=
x2
a ,if x =0 For what value of a, f is continuous at x = 0?
√x , if x >0
√16 + √ x−4
21. Find : (A) y=tan ( )
3
dy −1 3 x−x 1 1
,− < x
dx 1−3 x
2
√ 3 √3
3M
(B)
{ 2M
2 x+3 , if −3 ≤ x ←2
Examine the differentiability of the function f defined by f ( x ) = x+1 , if −2 ≤ x <0
x +2 ,if 0≤ x ≤ 1
A.VASANTHAKUMAR/MT-3/GR-12/MATH/
2025-26
