Karnataka 2nd PUC Mathematics Continuity and Differentiability MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: CONTINUITY AND DIFFERENTIABILITY

SYNOPSIS
1) Continuity of a function: A function 𝑓(𝑥) is said to be continuous at 𝑥 = 𝑎 if
lim𝑥→𝑎− 𝑓(𝑥) = lim𝑥→𝑎+ 𝑓(𝑥) = 𝑓(𝑎) . Otherwise the function is discontinuous.
2) If 𝑓(𝑥) and 𝑔(𝑥) are two functions continuous at 𝑥 = 𝑎 then (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓𝑔)(𝑥)
𝑓
and (𝑔) (𝑥), 𝑔(𝑎) ≠ 0 are also continuous at 𝑥 = 𝑎.

3) If 𝑓(𝑥) and 𝑔(𝑥) are two functions such that (𝑓 ∘ 𝑔)(𝑥) is defined at 𝑥 = 𝑎 and 𝑔 is
continuous at 𝑥 = 𝑎 & 𝑓 is continuous at 𝑔(𝑎) then (𝑓 ∘ 𝑔)(𝑥) is continuous at 𝑥 = 𝑎.
0 ∞
4) Indeterminate forms: The forms 0 , ∞ , 0 × ∞, ∞ − ∞, 00 , 1∞ and ∞0 are called indeterminate

forms.
5) i) The greatest integer function:- 𝑥 ∈ 𝑅. Let [𝑥] denote the greatest integer in 𝑥 not
greater than 𝑥.
Example: [0] = 0, [1.43] = 1, [−2.314] = −3 etc...
ii) [𝑥] is continuous for all 𝑥 ∈ 𝑅 − 𝑍
iii) [𝑥] is discontinuous for all 𝑥 ∈ 𝑍
iv) If 𝑥 is any integer then [𝑥] = 𝑥, for very small r > 0 [𝑥 − r] = 𝑥 − 1 and [𝑥 + 𝑟] = 𝑥
6) L'Hospital's rule:
𝑢
- Step 1: first reduce the function to 𝑣 form.

- Step 2: apply the limits
- Step 3: after applying limits if we get indeterminate form then differentiate the
numerator and denominator separately w.r.t independent variable and apply the
limit
- Step 4: If we get finite unique quantity as the answer, that itself is its limit
otherwise repeat the step 3

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7) Differentiability of a function: A function f(x) is differentiable at x = a, if left hand
derivative at x = a is equal to right hand derivative x = a.
𝑓(𝑎 + ℎ) − 𝑓(𝑎) 𝑓(𝑎 + ℎ) − 𝑓(𝑎)
lim− = lim+ = 𝑓 ′ (𝑎)
ℎ→0 ℎ ℎ→0 ℎ
𝑓(𝑎 + ℎ) − 𝑓(𝑎)
i.e, lim = 𝑓 ′ (𝑎)
ℎ→0 ℎ
8) Derivatives of some standard functions:
d d −1
1. (x) = 1 13. dx (cos−1 x) = √1−x2
dx
d d 1
2. dx ( constant ) = 0 14. dx (tan−1 x) = 1+x2
d d −1
3. dx (log x) = 1/x 15. dx (cot −1 x) = 1+x2
d d 1
4. ax = ax . log a , (a > 0) 16. (sec −1 x) =
dx dx |x|√x2 −1
d
5. dx ex = ex d
17. dx (cosec −1 x) =
−1
|x|√x2 −1
d
6. dx (sin x) = cos x d d
18. dx (u ± v) = dx (u) ± dx (v)
d

d
7. dx (cos x) = −sin x d d
19. dx (Kf(x) = K dx f(x)(K is a scalar )
d
8. dx (tan x) = sec 2 x d d
20. dx (uv) = u dx (v) + v dx (u) where u
d

d
9. dx (sec x) = sec xtan x and v and functions of x
d d d
10. dx (cosecx) = −cosecxcot x d u
21. dx (v) = dx
v (u)−u
dx
(v)
where u and v
v2
d
11. dx (cot x) = −cosec 2 x are functions of x
d 1
12. dx (sin−1 x) = √1−x2

9) Derivative of parametric functions :
If x = g(t), y = f(t) are parametric equations of a relation
dy
dx dy dy f′ (t)
then dt = g ′ (t) and dt = f ′ (t) ⇒ dx = dx
dt
= g′ (t)
dt

10) Differentiation of a function with respect to another function
To find derivative of a function f(x) with respect to another function g(x),
du dy
Put u = f(x) and v = g(x), Then dx = f ′ (x) and dx = g ′ (x)
du du/dx f′ (x)
Therefore dv = dv/dx = g′ (x)

11) Implicit functions
Implicit functions are those in which y can't be expressed directly in terms of 'x' .
Ex: x 3 y + y 3 x = 5

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𝜕𝑓
𝑑𝑦 𝜕𝑓
Note: If 𝑓(𝑥, 𝑦, 𝑐) = 0 and 𝑓(𝑥, 𝑦) = 𝑐 then 𝑑𝑥 = − 𝜕𝑥
𝜕𝑓 , where 𝜕𝑥 is the derivative of 𝑓 with
𝜕𝑦

𝜕𝑓
respect to 𝑥 treating 𝑦 as constant and is the derivative of 𝑓 with respect to 𝑦 treating
𝜕𝑦

𝑥 as constant
12) Differentiation of inverse trigonometric functions by substitution
Expression Substitution

√𝑎2 − 𝑥 2 x = asin 𝜃 or x = acos 𝜃
√𝑎2 + 𝑥 2 x = atan 𝜃 or x = acot 𝜃
√x 2 − a2 x = asec 𝜃 or x = acosec𝜃
a−x a+x
+ x = atan 𝜃
a+x a−x
a−x a+x
√a+x or √a−x x = acos 2𝜃

Use the suitable substitution in case of problems involving inverse trigonometric
functions and then differentiate. If no substitution is found, differentiate the function
directly.
13) Logarithmic differentiation: It is used in the following cases:
1. 𝑦 = 𝑓(𝑥) 𝑔(𝑥)
f(x)g(x)h(x)…..
2. 𝑦 = p(x)q(x)r(x)…..

3. 𝑦 = f(x)g(x)h(x) … ….
14) Successive differentiation:
dy
Let y = f(x) be a function. If we find dx, then it is the first order derivative of y w.r.t. x. If
𝑑𝑦 𝑑 𝑑𝑦 𝑑2 𝑦
we find the derivative of 𝑑𝑥 , w.r.t.x we get 𝑑𝑥 (𝑑𝑥 ) which is 𝑑𝑥 2 called as second order

derivative of 𝑦 w.r.t 𝑥. Successive differentiation is the process of getting the higher
order derivatives.
d
f ′ (x) = dx f(x) (First derivative of f(x) w.r.t.' x ′ )
𝑑2 𝑑 𝑑𝑓(𝑥)
′′
⇒ 𝑓 (𝑥) = 2 𝑓(𝑥) = [ ] (Second derivative of 𝑓(𝑥) w. r. t.' 𝑥 ′
𝑑𝑥 𝑑𝑥 𝑑𝑥
15) Note:
 f(x) = |x − a0 | is continuous everywhere and not differentiable at 𝑥 = 𝑎0
dy x |𝑥|
 If y = |x|, x ≠ 0 then dx = |x| = 𝑥

 If y = [x], [x] is the greatest integer function then
𝑑𝑦 dy
i) = 0 if 𝑥 ∉ Z ii) does not exist if x ∈ Z
𝑑𝑥 dx
d
 f(x)f(x) = f(x)f(x) [1 + log f(x)]f ′ (x)
dx

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d
5. dx (odd function) = even function
d
6. dx (even function) = odd function
dy y
7. If x m y n = (x + y)m+n then dx = x
dy
8. If ax + ay = ax+y then dx = −ay−x

dy f′ (x)
9. If y = √f(x) + √f(x) + √f(x) + ⋯ … … ∞ then dx = 2y−1

dy
10. If y = √f(x)√f(x)√f(x) … … … ∞ then dx = f ′ (x)

x2 x3
11. ex = 1 + x + + + ⋯ … ….
2! 3!
x3 x5
12. sin x = x − 3! + 5! + ⋯ … …..
𝑥2 𝑥4
13. cos x = 1 − 2! + 4! − ⋯ … …..
1
14. = 1 + 𝑥 + 𝑥2 + 𝑥3 + ⋯ …
1−𝑥
1
15. = 1 − x + x2 − x3 + x4 − ⋯ …
1+x
x2 x3
16. log(1 + x) = x − + − ⋯ . . . . , |x| < 1
2 3

CLASSROOM CHALLENGE
CONTINUITY
 sin x
  cos x, x  0
1. The function f  x    x is continuous at 𝒙 = 𝟎, then the value of k
 k ,x 0

is (E)
(1) 3 (2) 2 (3) 1 (4) 1.5
 Kx 2 , x  2
2. If f  x   is continuous at x =2, then the value of K is (KCET2017)(E)
 3 , x2
(1) 4 (2) 4/3 (3) 3 /4 (4) 3
 3sin  x
 ,x  0
3. If f  x    5x is continuous at x=0, then k= (A)
 2k , x  0

3  3 3
(1) (2) (3) (4)
2 10 5 10
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 sin 5 x
 x 2  2 x , x  0
4. If f  x    is continuous at x=0, then k= (D)
 k1 , x0
 2
(1) -2 (2) 1 (3) 1/2 (4) 2
5. The function 𝒇(𝒙) = |𝒙| + |𝒙 − 𝟏|𝒊𝒔 (E)
(1) continuous at 𝑥 = 0 as well as 𝑥 = 1 (2) continuous at 𝑥 = 1 but not at 𝑥 = 0
(3) discontinuous at 𝑥 = 0 as well as 𝑥 = 1 (4) continuous 𝑎𝑡 𝑥 = 0 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑎𝑡 𝑥 = 1
x
6. f  x  x  is (E)
x
x
(1) discontinuous at the origin , because is discontinuous there
x
x
(2) discontinuous at the origin , because x and are discontinuous there
x
(3) continuous at the origin
(4) discontinuous at the origin , because x is discontinuous there

7. The function f  x    x  , where  x  denotes greatest integer function, is

(KCET2016) (E)
(1) continuous for all non-integral values of x
(2) continuous only at positive integral values of x
(3) continuous for all real values of x
(4) continuous only at rational values of x
8. The function f  x    x  , where  x  denotes greatest integer function, is continuous

at… (E)
(1) 1.5 (2) 1 (3) -2 (4) 4
 1
 x sin   , x  0
9. If f  x    x is continuous at x=0, then k= (E)
 k , x0

(1) 1 (2) -1 (3) 0 (4) 2
𝟏
10. The number of points at which the function 𝒇(𝒙) = 𝒙−[𝒙] is not continuous

is (where x denotes greatest integer function) (A)
1) 1 2) 2 3) 3 4) infinite
𝟒−𝒙𝟑
11. The function 𝒇(𝒙) = 𝟒𝒙−𝒙𝟑 𝒊𝒔 (A)

1) discontinuous at only one point 2) discontinuous at exactly two points
3) discontinuous at exactly three points 4) not discontinuous
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𝟏
12. If 𝒇(𝒙) = 𝒙−𝟏 ,then the points of discontinuity of 𝒇[𝒇(𝒙)] are (A)

1) 1 and 3 2) 1 and 2 3) 1 and -1 4) 1 and -2
𝒂𝒙 + 𝟕 if 𝒙 < 𝟏
13. If 𝒇(𝒙) = {𝟑𝒙 − 𝟏 if 𝒙 = 𝟏 is continuous at 𝒙 = 𝟏, then (KCET 2026) (A)
𝒙+𝟑
if 𝒙 > 𝟏
𝒃
(1) a = 5, b = 2 (2) a = −5, b = −2
(3) a = 5, b = −2 (4) 𝑎 = −5, 𝑏 = 2
14. The function 𝐟(𝐱) = 𝐜𝐨𝐭 𝐱 is discontinuous on every point of the set
(KCET 2023) (E)
𝜋 n𝜋
1) {x = (2n + 1) ; 𝜋 ∈ Z} 2) {x = n𝜋: n ∈ Z} 3) {x = ; n ∈ Z} 4) {x = 2n𝜋; n ∈ Z}
2 2
𝟏−𝐜𝐨𝐬 𝑲𝒙
, if 𝒙 ≠ 𝟎
15. If 𝒇(𝒙) = { 𝒙𝐬𝐢𝐧 𝒙 𝟏 is continuous at 𝒙 = 𝟎, then the value of K is
, if 𝒙 = 𝟎
𝟐
(KCET 2020) (D)
1
1) ±1 2) ± 2 3) 0 4) ±2

DIFFERENTIABILITY
16. The function represented by the graph is (E)
(1) continuous and differentiable at x=1
Y
(2) continuous but not differentiable at x=1
(3) neither continuous nor differentiable at x=1 1
(4) differentiable but not continuous at x=1 0 X
1 2
 x2 , x 1
17. f  x    is (A)
 2  x, x 1
(1) neither continuous nor differentiable at x = 1
(2) continuous and differentiable at x =1
(3) not continuous at x =1
(4) continuous but not differentiable at x=1
18. Let 𝒇(𝒙) = |𝐜𝐨𝐬 𝒙|. 𝑻𝒉𝒆𝒏 (KCET 2024) (A)
(1) 𝑓 is everywhere differentiable
(2) 𝑓 is everywhere continuous but not differentiable at 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍
𝜋
(3) 𝑓 is everywhere continuous but not differentiable at 𝑥 = (2𝑛 + 1) 2 , 𝑛 ∈ 𝑍

(4) 𝑓 is nowhere differentiable
19. The set of points where the function 𝒇(𝒙) = |𝟐𝒙 − 𝟏|𝒔𝒊𝒏𝒙 is differentiable is (E)
1
1) ℝ 2) ℝ − {2} 3) (0, ∞) 4) ℝ − {0}

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20. The function 𝒇(𝒙) = 𝒆|𝒙| is (A)
(1) continuous everywhere but not differentiable at 𝑥 = 0
(2) continuous and differentiable at 𝑥 = 0
(3) not continuous at 𝑥 = 0
(4) nowhere differentiable
e x  ax ,x 0
21. The function f  x    is differentiable at 𝒙 = 𝟎 .Then (KCET 2025) (D)
   
2
 b x 1 , x 0

(1) 𝑎 = 3, 𝑏 = 1 (2) 𝑎 = −3, 𝑏 = 1 (3) 𝑎 = 3, 𝑏 = −1 (4) 𝑎 = 1, 𝑏 = 1
 1x
 e  1 , x  0
22. The function 𝒇 defined by f  x    1x is (KCET 2025) (D)
 e  1
 0 , x  0

(1) not continuous at 𝑥 = 0
(2) differentiable at 𝑥 = 0
(3) differentiable at 𝑥 = 0, but not continuous at 𝑥 = 0
(4) continuous at 𝑥 = 0
23. Match the following: (KCET 2025) (D)
In the following, [𝒙] denotes the greatest integer less than or equal to 𝒙.

Column 1 Column 2

(a) x|x| (i) continuous in (−1,1)

(b) √|𝑥| (ii) differentiable in (-1,1)

(c) x + [x] (iii) strictly increasing in (−1,1)
(d)|𝑥 − 1| + |𝑥 + 1| (iv) not differentiable at, at least one point in (-1,1)
(1) a-iv, 𝑏 − 𝑖𝑖𝑖, 𝑐 − 𝑖, 𝑑 − 𝑖𝑖 (2) a - ii, b - iv, c - iii, d – i
(3) a - iii, 𝑏 − 𝑖𝑖, 𝑐 − i𝑣, 𝑑 − 𝑖 (4) 𝑎 − 𝑖, 𝑏 − 𝑖𝑖, 𝑐 − 𝑖𝑣, 𝑑 − 𝑖𝑖𝑖
DIFFERENTIATION

24. If the function g x  is defined by g  x  
x 200 x199 x198 x2
   ......  x  5 , then 𝒈𝟏 (𝟎) =
200 199 198 2
(KCET2015) (A)
(1) 5 (2) 100 (3) 200 (4) 1
25. If f (x) is an even function and 𝒇𝟏 (𝒙) exists, then 𝒇𝟏 (𝒆) + 𝒇𝟏 (−𝒆) is (A)
(1) > 0 (2) 0 (3) < 0 (4)  0

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𝒂 𝒃
26. If 𝒂 and 𝒃 are fixed non-zero constants, then the derivative of 𝒙𝟒 − 𝒙𝟐 + 𝐜𝐨𝐬 𝒙

is 𝒎𝒂 + 𝒏𝒃 − 𝒑, where (KCET 2021) (A)
−2 −4 2
1). 𝑚 = 4𝑥 3 ; 𝑛 = 𝑥 3 ; 𝑝 = sin 𝑥 2). 𝑚 = 𝑥 5 ; 𝑛 = 𝑥 3 ; 𝑝 = sin 𝑥
−4 −2 2
3). 𝑚 = 𝑥 5 ; 𝑛 = 𝑥 3 ; 𝑝 = − sin 𝑥 4). 𝑚 = 4𝑥 3 ; 𝑛 = 𝑥 3 ; 𝑝 = −sin 𝑥
𝝅
27. If 𝒇(𝒙) = |𝒄𝒐𝒔 𝒙 − 𝒔𝒊𝒏 𝒙|, then 𝒇𝟏 (𝟔 ) = (KCET2018) (A)
−1 1 −1 1
(1) 2 (1 + √3) (2) 2 (1 + √3) (3) 2 (1 − √3) (4) 2 (1 − √3)

 
28. If [x] represents the greatest integer function and f(x) = x - [x] – cos x then f 1   
2
(KCET2019) (A)
(1) 0 (2) 1 (3) 2 (4) does not exist
𝒅𝒚
29. If 𝒚 = (𝐜𝐨𝐬 𝒙𝟐 )𝟐 , then 𝒅𝒙 is equal to (KCET 2021) (A)

1) −4𝑥 sin 2𝑥 2 2) −𝑥sin 𝑥 2 3) −2𝑥 sin 2𝑥 2 4) −𝑥sin 2𝑥 2
30. If 𝐲 = 𝐚𝐬𝐢𝐧 𝐱 + 𝐛𝐜𝐨𝐬 𝐱, then 𝒚𝟐 + (𝒚′ )𝟐 is (KCET 2023) (D)
1) Function of x and y 2) Function of x
3) Constant 4) Function of 𝑦
𝐜𝐨𝐬 𝒙
31. If 𝒚 = 𝟏+𝐬𝐢𝐧 𝒙 then (KCET 2025) (D)
𝑑𝑦 −1 𝑑𝑦 1
(a) 𝑑𝑥 = 1+sin 𝑥 (b) 𝑑𝑥 = 1+sin 𝑥
𝑑𝑦 1 𝜋 𝑥 𝑑𝑦 1 𝜋 𝑥
(c) 𝑑𝑥 = − 2 sec 2 (4 − 2) (d) 𝑑𝑥 = 2 sec 2 ( 4 − 2)

(1) Only a is correct (2) Both a and c are correct
(3) Both b and d are correct (4) Only 𝑏 is correct
𝒅𝒚
32. If 𝒚 = (𝒙 − 𝟏)𝟐 (𝒙 − 𝟐)𝟑 (𝒙 − 𝟑)𝟓 then 𝒅𝒙 at 𝒙 = 𝟒 is equal to (KCET 2021) (D)

1) 108 2) 54 3) 36 4) 516
𝟏 𝟏
33. If 𝐟(𝐱) and 𝐠(𝐱) are two functions with 𝒈(𝒙) = 𝒙 − 𝒙 and 𝒇 ∘ 𝒈(𝒙) = 𝒙𝟑 − 𝒙𝟑, then 𝒇′ (𝒙) =

(KCET2023) (D)
1 1 3
1) 𝑥 2 − 𝑥 2 2)3𝑥 2 + 3 3)1 − 𝑥 2 4) 3𝑥 2 + 𝑥 4

34. Let the function satisfy the equation 𝒇(𝒙 + 𝒚) = 𝒇(𝒙)𝒇(𝒚) for all 𝒙, 𝒚 ∈ 𝑹, where
𝒇(𝟎) ≠ 𝟎. If 𝒇(𝟓) = 𝟑 and 𝒇′ (𝟎) = 𝟐, then 𝒇′ (𝟓) is (KCET 2024) (D)
1) 6 2) 0 3) 5 4) -6

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35. If 𝒇(𝟏) = 𝟏, 𝒇′ (𝟏) = 𝟑 then the derivative of 𝒇(𝒇(𝒇(𝒙))) + (𝒇(𝒙))𝟐 at 𝒙 = 𝟏 is
(KCET 2022) (D)
1) 10 2) 33 3) 35 4) 12

36. If 𝒚 = 𝒔𝒊𝒏[𝒕𝒂𝒏−𝟏 𝒙 + 𝒄𝒐𝒕−𝟏 𝒙] then dy  (E)
dx

(1) π/2 (2)-1 (3) 1 (4) 0
37. Differential coefficient of 𝐬𝐞𝐜(𝐭𝐚𝐧−𝟏 𝒙) 𝒊𝒔 (A)
𝑥 𝑥 1
1) √1+𝑥 2 2) 1+𝑥 2 3) 𝑥√1 + 𝑥 2 4) √1+𝑥 2
𝒅 𝒂−𝒙
38. 𝒅𝒙 (𝐭𝐚𝐧−𝟏 (𝟏+𝒂𝒙)) 𝒊𝒔 (A)
1 1 1 1 1
(1) − 1+𝑥 2 2) 𝑎−𝑥 2
3) 2
4) 1+𝑎2 − 1+𝑥 2
1+ (
1+𝑎𝑥
) √1− ( 𝑎−𝑥 )
1+𝑎𝑥

𝒔𝒊𝒏 𝒙+𝒄𝒐𝒔 𝒙 dy
39. If 𝒚 = 𝒕𝒂𝒏−𝟏 (𝒄𝒐𝒔 𝒙−𝒔𝒊𝒏 𝒙) then  (KCET2017) (A)
dx

(1) 0 2) 3) 1 4) 1/2
4
𝟐𝒙
40. If 𝒇(𝒙) = 𝐬𝐢𝐧−𝟏 (𝟏+𝒙𝟐 ), then 𝒇𝟏 (√𝟑) is (KCET2020) (A)
1 1 1 1
(1) − (2) − 2 (3) 2 (4)
√3 √3
𝒅𝒚
41. If 𝒚 = (𝟏 + 𝒙𝟐 )𝐭𝐚𝐧−𝟏 𝒙 − 𝒙 then 𝒅𝒙 is (KCET 2022) (A)
tan−1 𝑥
1) 2𝑥tan−1 𝑥 2) 3) 𝑥 2 tan−1 4) 𝑥tan−1 𝑥
𝑥

𝐝 𝟐+𝐱
42. 𝐝𝐱 [𝐜𝐨𝐬𝟐 (𝐜𝐨𝐭 −𝟏 √𝟐−𝐱)] is (KCET 2024) (D)

3 1 1 1
1) − 4 2) − 2 3) 2 4) 4
𝟐 𝒄𝒐𝒔 𝒙+𝟑 𝒔𝒊𝒏 𝒙 dy
43. If 𝒚 = 𝒄𝒐𝒔−𝟏 [ ], then is (D)
√𝟏𝟑 dx

1
(1) – 1 (2) 1 (3) (4) 2
2
𝟑𝒂𝟐 𝒙−𝒙𝟑 𝒅𝒚
44. If y = tan1 ( ) , 𝒕𝒉𝒆𝒏 = (D)
𝒂(𝒂𝟐 −𝟑𝒙𝟐 ) 𝒅𝒙

3a 2 3a a2 3
(1) (2) (3) (4)
a2  x2 a2  x2 a2  x2 a2  x2
𝟔𝒙
45. If 𝒇(𝒙) = 𝒔𝒊𝒏−𝟏 (𝟗𝒙𝟐 +𝟏), then f | (1)  (D)

2 3 3 2
(1) (2) (3) (4)
3 5 5 3

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𝟐𝒙+𝟏
46. If 𝒇(𝒙) = 𝒔𝒊𝒏−𝟏 [𝟏+𝟒𝒙 ] , then 𝒇𝟏 (𝟎) = (KCET2019) (D)

2 log 2 4 log 2
(1) 2 log 2 (2) log 2 (3) (4)
5 5
𝟏−𝒕 𝒅𝒚
47. If 𝒚 = 𝒔𝒊𝒏−𝟏 (𝟏+𝒕) and 𝒙 = √𝒕 , then 𝒅𝒙 = (D)

2 1 1 2
(1) (2) (3) (4)
1 x 1 x 2 1 x 1  x2
𝒅𝒚
48. If 𝒚 = 𝒔𝒊𝒏−𝟏 (𝟑−𝒙 ) then 𝒅𝒙= (D)

 log 3 3x log 3 3x log 3 log 3
(1) (2) (3) (4)
32 x  1 32 x  1 32 x  1 3x 32 x  1
𝟐𝒙 𝟏
49. If 𝒇(𝒙) = 𝐬𝐢𝐧−𝟏 (𝟏+𝒙𝟐 ), then 𝒇′ (𝟐) = (KCET 2026) (D)
8 5 4
(1) 5 (2) 8 (3) 5 (4) 0
𝒅𝒚
50. If 𝒚 = 𝟑√𝐭𝐚𝐧 𝒙 + 𝒚, then 𝒅𝒙 = (KCET 2026) (A)
tan 𝑥 sec2 𝑥 tan 𝑥 sec2 𝑥
(1) 3𝑦 2 −1 (2) 3𝑦−1 (3) 3𝑦−1 (4) 3𝑦 2 −1
𝒅𝒚
51. If 𝟐𝒙 + 𝟐𝒚 = 𝟐𝒙+𝒚 , then 𝒅𝒙 is (KCET2020) (A)
2𝑦 −1
(1) 2𝑥 −1 (2) 2𝑦−𝑥 (3) −2𝑦−𝑥 (4) 2𝑥−𝑦
dy
52. If 𝒄𝒐𝒔 𝒚 = 𝒙 𝒄𝒐𝒔(𝒂 + 𝒚) with 𝒄𝒐𝒔 𝒂 ≠ ±𝟏 then is equal to (KCET2018) (A)
dx
𝑠𝑖𝑛 𝑎 𝑐𝑜𝑠2 (𝑎+𝑦) 𝑐𝑜𝑠 𝑎 𝑐𝑜𝑠2 (𝑎+𝑦)
(1) 𝑐𝑜𝑠2 (𝑎+𝑦) (2) (3) 𝑠𝑖𝑛2 (𝑎+𝑦) (4)
𝑠𝑖𝑛 𝑎 𝑐𝑜𝑠 𝑎

𝒅𝒚
53. If 𝒚 = √𝒙 + √𝒙 + √𝒙+. . . . . . . . ∞, then 𝒅𝒙 is (KCET2018) (A)
1 𝑥 1 1
(1) 𝑥 𝑦 (2) 2𝑦−1 (3) 2𝑦+𝑥 (4) 2𝑦−1
√

dy
54. If 𝒚 = √𝒔𝒊𝒏 𝒙 + √𝒔𝒊𝒏 𝒙 + √𝒔𝒊𝒏 𝒙 +. . . . . . . . . . . 𝒕𝒐 𝒊𝒏𝒇 𝒊 𝒏𝒊𝒕𝒚, then  (A)
dx

2 y 1 y2 sin x cos x
(1) (2) (3) (4)
cos x cos x 2 y 1 2 y 1
dy
55. If 𝒚 = 𝒆𝒍𝒐𝒈 √𝟏+𝒕𝒂𝒏 𝒙
𝟐
then  (E)
dx

(1) sec2x (2) sec2x tan2x (3)-sin2x (4) secx tanx

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56. Which of the following is NOT correct for the features of exponential function
given by f  x   b x , where b  1? (E)

(1) The point (1,0) is always on the graph of the function.
(2) For very large negative values of x , the function is very close to 0.
(3) The range of the function is the set of all positive real numbers.
(4) The domain of the function is R (set of real numbers).
𝒅
57. For constant 𝒂, 𝒅𝒙 (𝒙𝒙 + 𝒙𝒂 + 𝒂𝒙 + 𝒂𝒂 ) is (KCET 2021) (E)

(1) 𝑥 𝑥 (1 + log 𝑥) + 𝑎𝑥 𝑎−1 (2) 𝑥 𝑥 (1 + log 𝑥) + 𝑎𝑥 𝑎−1 + 𝑎 𝑥 log 𝑎
(3) 𝑥 𝑥 (1 + log 𝑥) + 𝑎𝑎 (1 + log 𝑥) (4) 𝑥 𝑥 (1 + log 𝑥) + 𝑎𝑎 (1 + log 𝑎) + 𝑎𝑥 𝑎−1
d2y
58. If 𝒚 = 𝒍𝒐𝒈(𝒍𝒐𝒈 𝒙) then  (KCET2017) (A)
dx 2

(1)
1  log x  (2) 
1  log x  (3) 
1  log x  (4)
1  log x 
 x log x   x log x 
2 2 2 2
x log x x log x
𝒅𝒚
59. If 𝒚 = 𝒍𝒐𝒈𝟑 [ 𝒍𝒐𝒈𝒆 𝒙], then 𝒅𝒙 = (A)

log 3 log 3 e log 3 e log 3 e
(1) (2) (3) (4)
x log x 1  log x x log 3 x x log e x

60. Consider the following statements: (KCET 2021) (A)
𝑑𝑦 log10 𝑒 1
Statement 1: If 𝑦 = log10 𝑥 + log 𝑒 𝑥 then 𝑑𝑥 = +𝑥
𝑥
𝑑 log 𝑥 𝑑 log 𝑥
Statement 2: 𝑑𝑥 (log10 𝑥) = log 10 and 𝑑𝑥 (log 𝑒 𝑥) = log 𝑒

(1) Statement 1 is true; statement 2 is false
(2) Statement 1 is false; statement 2 is true
(3) Both statements 1 and 2 are true
(4)Both statements 1 and 2 are false
61. If f  x   log x2  log x  , then f 1  e   (D)

1 1
(1) (2) (3) 0 (4) 1
e 2e
dy
62. If 𝒙𝒎 𝒚𝒏 = (𝒙 + 𝒚)𝒎+𝒏 then  (KCET2016) (A)
dx
𝑦 𝑥+𝑦
(1) 𝑥𝑦 (2) 𝑥 (3) 𝑥𝑦 (4) 0

63. If 3 y x  6 x  y 5 , then
dy
 (KCET2019) (A)
dx
𝑥 𝑦
(1) 𝑦 (2) x+y (3) x-y (4) 𝑥

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x 3 y =  x + y  and 𝒙
n 𝒅𝒚
64. If − 𝒚 = 𝟎, then 𝒏 = (KCET 2026) (A)
𝒅𝒙
6 5 4
(1) 1 (2) 5 (3) 6 (4) 9
𝒅𝒚
65. If 𝒙𝒙 = 𝒚𝒚 , then 𝒅𝒙 is (A)
1+𝑙𝑜𝑔 𝑥 𝑥 −𝑥 −𝑦
(1) 1+𝑙𝑜𝑔 𝑦 (2) 1 + 𝑙𝑜𝑔 (𝑦) (3) (4)
𝑦 𝑥
𝒅𝒚
66. If (𝒙𝒆)𝒚 = 𝒆𝒙 , then 𝒅𝒙 is (KCET2020) (A)
𝑒𝑥 log 𝑥 1 log 𝑥
(1) 𝑥(𝑦−1) (2) (1+log 𝑥)2 (3)(1+log 𝑥)2 (4) (1+log 𝑥)

√𝒙√𝒙√𝒙…….. 𝒅𝟐 𝒚
67. If 𝒚 = 𝒆 , 𝒙 > 𝟏 then 𝒅𝒙𝟐 at 𝒙 = 𝐥𝐨𝐠 𝟑𝒆 is (KCET 2022) (A)

1) 3 2) 5 3) 0 4) 1
𝒅𝒚
68. If 𝒚 = 𝟐(𝒙𝟑𝒙 ), then 𝒅𝒙 at 𝒙 = 𝟏 is (KCET 2024) (A)

1) 2 2) 6 3) 3 4) 1
dy 3
69. If x = a cos4 , y = a sin4 , then at  = is (A)
dx 4
(1) 1 (2) 1 (3)  a2 (4) a2
𝒅𝒚 𝟐
70. If 𝒙 = 𝒂 𝒄𝒐𝒔𝟑 𝜽 , 𝒚 = 𝒂 𝒔𝒊𝒏𝟑 𝜽 then 1+(𝒅𝒙) = (A)

(1) 1 (2) sec2θ (3) tan2θ (4) tanθ
𝟐𝒕 𝟐𝒕 dy
71. If 𝒔𝒊𝒏 𝒙 = 𝟏+𝒕𝟐 , 𝒕𝒂𝒏 𝒚 = 𝟏−𝒕𝟐 then  (KCET2017) (A)
dx

(1) 0 (2) 2 (3) 1 (4) -1
72. The derivative of 𝐬𝐢𝐧 𝐱 with respect to 𝐥𝐨𝐠 𝒙 is (KCET 2025) (E)
cos 𝑥 cos 𝑥
(1) 𝑥cos 𝑥 (2) log 𝑥 (3) 𝑥
(4) cos 𝑥
𝟐𝐱 𝟐𝐱 𝐝𝐮
73. If 𝐮 = 𝐬𝐢𝐧−𝟏 (𝟏+𝐱𝟐 ) and 𝐯 = 𝐭𝐚𝐧−𝟏 (𝟏−𝐱𝟐 ), then 𝐝𝐯 = (KCET 2023) (A)
1−𝑥 2 1
1) 1+𝑥 2 2) 2 3) 1 4) 2

74. The differential coefficient of log10 x with respect to log x 10 is (KCET2016) (A)

(1)  log10 x  log 10
2 x2 2
(2) (3) 1 (4) x
100
𝒅𝒖
75. If 𝒖 = 𝒇(𝒙𝟑 ), 𝒗 = 𝒈(𝒙𝟐 ), 𝒇′ (𝒙) = 𝐜𝐨𝐬 𝒙 𝒂𝒏𝒅 𝒈′ (𝒙) = 𝐬𝐢𝐧 𝒙 𝒕𝒉𝒆𝒏 𝒅𝒗 = (D)
2
1) 𝑐𝑜𝑠 𝑥 3 𝑐𝑜𝑠𝑒𝑐 𝑥 3 2) 3 𝑠𝑖𝑛 𝑥 3 𝑠𝑒𝑐 𝑥 3
3
3) 𝑡𝑎𝑛 𝑥 4) 2 𝑥 𝑐𝑜𝑠 𝑥 3 𝑐𝑜𝑠𝑒𝑐 𝑥 2

2026-27 MATHEMATICS CET MATERIAL Page 12 of 19

𝟑𝒙 𝟑𝒙 𝒅𝟐 𝒚
76. If 𝒚 = 𝒄𝒐𝒔𝟐 𝟐 − 𝒔𝒊𝒏𝟐 𝟐 , then 𝒅𝒙𝟐 is (A)

(1) 3 1  y 2 (2) – 9y (3)9 y (4)  3 1  y 2
𝒙𝟐 𝒙𝟑 𝒅𝟐 𝒚
77. If 𝒚 = 𝟏 − 𝒙 + 𝟐! − 𝟑! +. . . .. then 𝒅𝒙𝟐 = (A)

(1) -x (2) x (3) y (4) -y
𝒏(𝒏−𝟏) 𝒏(𝒏−𝟏)(𝒏−𝟐)
78. If 𝒇(𝒙) = 𝟏 + 𝒏𝒙 + 𝒙𝟐 + 𝒙𝟑 + ⋯ . . +𝒙𝒏
𝟐 𝟔

then 𝒇′′ (𝟏) = (KCET 2023) (A)
1) 𝑛(𝑛 − 1)2𝑛 2) (𝑛 − 1)2𝑛−1 3) 2n−1 4) 𝑛(𝑛 − 1)2𝑛−2
𝒅𝟐 𝒚
79. If 𝒙 = 𝒂 𝒔𝒆𝒄𝟐 𝜽 , 𝒚 = 𝒂 𝒕𝒂𝒏𝟐 𝜽 𝒕𝒉𝒆𝒏 𝒅𝒙𝟐 = (KCET2019) (A)

(1) 2a (2) 1 (3) 0 (4) 4
𝟑 𝒅𝟐 𝒚
80. If 𝒚 = 𝟐𝒙𝒏+𝟏 + 𝒙𝒏, then 𝒙𝟐 𝒅𝒙𝟐 is (KCET2020) (D)
𝑑𝑦
(1) 𝑦 (2) 6𝑛(𝑛 + 1)𝑦 (3) 𝑛(𝑛 + 1)𝑦 (4) 𝑥 𝑑𝑥 + 𝑦

CLASSROOM CHALLENGE
1 2 3 4 5 6 7 8 9 10
2 3 4 4 1 1 1 1 3 4
11 12 13 14 15 16 17 18 19 20
3 2 4 2 1 2 4 3 2 1
21 22 23 24 25 26 27 28 29 30
2 1 2 4 2 2 1 3 3 3
31 32 33 34 35 36 37 38 39 40
2 4 2 1 2 4 1 1 3 3
41 42 43 44 45 46 47 48 49 50
1 4 2 2 3 2 4 1 1 4
51 52 53 54 55 56 57 58 59 60
3 2 4 4 4 1 2 2 4 1
61 62 63 64 65 66 67 68 69 70
2 2 4 3 1 2 1 2 1 2
71 72 73 74 75 76 77 78 79 80
3 1 3 1 4 2 3 4 3 3

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ADDITIONAL PROBLEMS
𝒔𝒊𝒏 𝟑𝒙
; 𝒙≠𝟎
1. If 𝒇(𝒙) = {𝒆𝟐𝒙−𝟏 𝒊𝒔 continuous at x=0, then k= (KCET2019) (A)
𝒌 − 𝟐; 𝒙 = 𝟎
1 3 7 9
(1) 2 (2) 2 (3) 2 (4) 5

1  cos x
 ,x  0
2. If f  x    x is continuous at x=0, then k= (E)
 k ,x 0

(1) 1/4 (2) -1/2 (3) 0 (4) 1/2
 log x
 ,x 1
3. If f  x    x  1 is continuous at x=1, then k= (KCET2018) (E)
 k , x  1

(1) e (2) 1 (3) -1 (4) 0
 x, if x is irrational
4. Let f  x    then f is (A)
 0, if x is rational
(1) continuous everywhere (2) discontinuous everywhere
(3) continuous only at x =0 (4) continuous at all rational numbers
 1
sin , x  0
5. The value of k which makes the function f  x    x continuous at 𝒙 = 𝟎, 𝒊𝒔
 k , x  0

(E)
1) 8 2) 1 3) −1 4) does not exist
√𝟏+𝒌𝒙−√𝟏−𝒌𝒙
𝒊𝒇 − 𝟏 ≤ 𝒙 < 𝟎
6. If 𝒇(𝒙) = { 𝒙
𝟐𝒙+𝟏
is continuous at x  0 then (KCET2018) (A)
𝒊𝒇 𝟎 ≤ 𝒙 ≤ 𝟏
𝒙−𝟏

(1) k=1 (2) k=-1 (3)k=0 (4) k=2

7. Let 𝒇(𝒙) = 𝟒𝒙−𝝅 , 𝒙 ≠ 𝟒 , 𝒙 ∈ 0,  . If 𝒇(𝒙) is continuous in [𝟎, 𝟐 )
𝟏−𝐭𝐚𝐧 𝒙 𝝅 𝝅
 2
𝝅
then 𝒇 (𝟒 )= (A)
1 1
1)− 2 2)2 3)1 4)−1

8. The function given by 𝒇(𝒙) = 𝐭𝐚𝐧 𝒙 𝒊𝒔 discontinuous on the set (E)
𝜋 𝑛𝜋
1) {𝑛𝜋: 𝑛 ∈ 𝑍} 2) {2𝑛𝜋 ∶ 𝑛 ∈ 𝑍} 3) {(2𝑛 + 1) 2 : 𝑛 ∈ 𝑍} 4) { 2 : 𝑛 ∈ 𝑍}
𝒙𝟐
9. If 𝒇(𝒙) = 𝟐𝒙 𝒂𝒏𝒅 𝒈(𝒙) = 𝟐 + 𝟏, then which of the following can be discontinuous

function? (E)
𝑔(𝑥)
1) 𝑓(𝑥) + 𝑔(𝑥) 2) 𝑓(𝑥) − 𝑔(𝑥) 3) 𝑓(𝑥). 𝑔(𝑥) 4) 𝑓(𝑥)

2026-27 MATHEMATICS CET MATERIAL Page 14 of 19

𝟏
10. If the function is 𝒇(𝒙) = 𝐱+𝟐, then one of the point of discontinuity of the

composite function 𝒚 = 𝒇(𝒇(𝒙)) is (KCET 2023) (A)
2 −5 1 5
1) 5 2) 2 3) 2 4) 2

 2 a  x,  a  x  a
11. If f  x   then which of the following is true? (A)
 3 x  2a, x  a
(1) f x  is not differentiable at x = a (2) f x  is discontinuous at x = a
(3) f x  is continuous at all x < a (4) f x  is differentiable at all x  a
12. The function 𝒇(𝒙) = 𝒙|𝒙|, ∀𝒙 ∈ 𝑹 𝒂𝒕 𝒙 = 𝟎 is (A)
1) continuous but not differentiable 2) not continuous
3) continuous and differentiable 4) differentiable but not continuous
𝐬𝐢𝐧{𝝅 [𝒙−𝝅]}
13. If f(𝒙) = , where [ .] denotes the greatest integer function. Then which of
𝟏+|𝒙|𝟐

the following is true (E)
1) 𝑓(𝑥) is not differentiable at 𝑥 = 0 2) 𝑓(𝑥) is not differentiable at 𝑥 = 𝜋
3) 𝑓(𝑥) is differentiable, ∀ 𝑥 ∈ 𝑅 3) 𝑓(𝑥) is not differentiable,∀ 𝑥 ∈ 𝐼
𝟏
𝒙 𝒔𝒊𝒏 (𝒙) , 𝒊𝒇 𝒙 ≠ 𝟎
14. If 𝒇(𝒙) = { , then at x = 0 the function f is (A)
𝟎, 𝒊𝒇 𝒙 = 𝟎
(1) not continuous (2) continuous and differentiable
(3) differentiable but not continuous (4) continuous but not differentiable
15. The function y = |x-1| + |x+1| is differentiable at all points except at 𝒙 = (A)
1) 0 2) ±1 3) 𝑅 4) ±2
16. The set of points where the function f given by 𝒇(𝒙) = |𝒙 − 𝟑| cos x is differentiable
is (E)
1) 𝑅 2) 𝑅 − {3} 3) (0, ∞) 4) 𝑅 − {1}
𝒙𝟑 − 𝟏, 𝟏 < 𝒙 < ∞
17. At 𝒙 = 𝟏, the function 𝒇(𝒙) = { is (KCET 2021) (A)
𝒙−𝟏, −∞<𝒙≤𝟏
(1) continuous and differentiable (2) continuous and non-differentiable
(3) discontinuous and differentiable (4) discontinuous and non-differentiable
𝒅𝒚
18. If 𝒚 = |𝒄𝒐𝒔 𝒙| + |𝒔𝒊𝒏 𝒙|then (𝒅𝒙) 𝟐𝝅 is (A)
𝒙=
𝟑

1 3
(1)
2
(2) 0 (3)
1
2
 3  1 (4) -1

 dy 
19. If y  f ( x 2  2) and 𝒇𝟏 (𝟑) = 5 then   = (A)
 dx  x 1
(1)10 (2) 15 (3) 25 (4) 5

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1
20. If g is the inverse function of f and f (x)  then g(x) is equal to (D)
1  xn
(1) 1 + (g(x))n (2) 1  g(x) (3) 1 + g(x) (4) 1 (g(x))n
𝟏+𝒙 𝒅𝒚
21. If 𝒔𝒆𝒄−𝟏 (𝟏−𝒚) = 𝒂, then 𝒅𝒙 = (A)

x 1 x 1 y 1 y 1
(1) (2) (3) (4)
y 1 y 1 x 1 x 1

22. If tan 1  x 2  y 2    then dy  (KCET2016) (A)
dx
x x
(1) xy (2)  xy (3) (4)
y y

1  (log x) 2 
1
23. If f ( x)  cos  2
, then f | (e)  (D)
1  (log x) 
1 2
(1) 1 (2) does not exist (3) (4)
e e
 e2 x  1 
24. If f  x   cos  2 x 1
 , then f (0) 
|
(D)
 e 1 
(1) – 2 (2) – 1 (3) 2 (4) 1
𝒅𝒚
25. If 𝒔𝒊𝒏 𝒚 = 𝒙 𝒔𝒊𝒏(𝒂 + 𝒚) , 𝒕𝒉𝒆𝒏 𝒅𝒙 is (A)
𝑠𝑖𝑛 𝑎 𝑠𝑖𝑛2 (𝑎+𝑦) 𝑠𝑖𝑛2 (𝑎−𝑦)
(1) (2) (3) 𝑠𝑖𝑛 𝑎 𝑠𝑖𝑛2 ( 𝑎 + 𝑦) (4)
𝑠𝑖𝑛2 (𝑎+𝑦) 𝑠𝑖𝑛 𝑎 𝑠𝑖𝑛 𝑎
𝒅𝒚
26. If 𝒚 = 𝟐𝐥𝐨𝐠 𝒙 , 𝒕𝒉𝒆𝒏 𝒅𝒙 𝒊𝒔 (A)
2log 𝑥 2log 𝑥 .log 2 2log 𝑥
(1) 2log 𝑥 . log 2 2) log 2 3) 4)
𝑥 𝑥

dy
27. If y = log | x | ( x  0) then  (E)
dx
1 1 −1
(1) doesn’t exist (2) |𝑥| (3) 𝑥 (4) 𝑥

dy
28. If x = log [ log y ] , then  (E)
dx

1 𝑥
(1) (2) 𝑒 𝑒 (3) y log y (4) 𝑒 𝑦 𝑙𝑜𝑔𝑦
y log y
𝒅𝒚
29. If yx = xy , then 𝒅𝒙 is (A)

y y(x log y  y) x x log y
(1) (2) (3) (4)
x x(y log x  x) y y log x

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30. Derivative of the function f(x) = log5 (log7 x), where x > 7 is: (D)
1 1
(1) (2)
x(log 5)(log 7)(log 7 x) x(log 5)(log 7)

1
(3) (4) none of these
x(log x)
𝒅𝒚 𝝅
31. If 𝒚 = 𝒙𝐬𝐢𝐧 𝒙 + (𝐬𝐢𝐧 𝒙)𝒙 then 𝒅𝒙 at 𝒙 = 𝟐 is (KCET 2022) (D)
4 𝜋 𝜋2
1) 𝜋 2) 𝜋log 2 3) 1 4) 2

dy
32. If 𝒙𝒚 = 𝒆𝒙−𝒚 then  (KCET2016) (D)
dx
ex 1 1 log x log x
(1) (2)  (3) (4)
x x y y x y log  x  y  1  log x 
2

 1  x2  dy
33. If y  log  2 
then  (D)
 1 x  dx

4 x 4 x3 1 4 x 3
(1) (2) (3) (4)
1  x4 1  x4 4  x4 1  x4

1  sin 4 x dy
34. If y  log , then at x = 0 is (D)
1  sin 4 x dx
1
(1) 1 (2) 4 (3) – 4 (4)
4
𝒅𝒚 𝒅𝟐 𝒚
35. If 𝒆𝒚 + 𝒙𝒚 = 𝒆 the ordered pair (𝒅𝒙 , 𝒅𝒙𝟐 ) at 𝒙 = 𝟎 is equal to (KCET 2022) (D)
1 1 −1 −1 1 −1 −1 1
1) (𝑒 , 𝑒 2 ) 2) ( 𝑒 , 𝑒 2 ) 3) (𝑒 , 𝑒 2 ) 4) ( 𝑒 , 𝑒 2 )
𝒅𝒚 𝝅
36. If 𝒙 = 𝒆𝜽 𝐬𝐢𝐧 𝜽, 𝒚 = 𝒆𝜽 𝐜𝐨𝐬 𝜽 where 𝜽 is a parameter, then 𝒅𝒙 at 𝜽 = 𝟒 is equal to

(KCET 2022) (A)
1 1 1
1) 0 2) 2 3) − 2 4) − 4
𝒅𝒚 𝟑𝝅
37. If 𝒚 = 𝒂𝐬𝐢𝐧𝟑 𝒕, 𝒙 = 𝒂𝐜𝐨𝐬𝟑 𝒕, then 𝒅𝒙 at 𝒕 = 𝟒 is (KCET 2025) (A)
1
(1) (2) −√3 (3) 1 (4) -1
√3
  𝐝𝐲
38. 𝐈𝐟 𝐱 = 𝐭 𝟐 + 𝐭 + 𝟏 𝐚𝐧𝐝 𝐲 = 𝐬𝐢𝐧 𝟐 𝐭 + 𝐜𝐨𝐬 𝟐 𝐭, 𝐭𝐡𝐞𝐧 𝐚𝐭 𝐭 = 𝟏, 𝐝𝐱 = (A)
   
(1) 2) 3) 4) −
3 4 2 6
𝒄 𝒅𝒚
39. If 𝒙 = 𝒄𝒕 and 𝒚 = 𝒕 then find 𝒅𝒙 at t = 2 (A)

(1) 0 (2)-1/4 (3) 4 (4) ¼

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𝟏− 𝒕𝟐 𝟐𝒕 𝒅𝒚
40. If 𝒙 = 𝟏+ 𝒕𝟐 𝒂𝒏𝒅 𝒚 = 𝟏+ 𝒕𝟐 , 𝒕𝒉𝒆𝒏 𝒅𝒙 = (A)
𝑥 𝑥 𝑦 𝑦
1) − 𝑦 2) 𝑦 3) − 𝑥 4) 𝑥

𝟏−𝒙𝟐 𝟐𝒙
41. Differential co-efficient of 𝒄𝒐𝒔−𝟏 (𝟏+𝒙𝟐 ) with respect to 𝒔𝒊𝒏−𝟏 (𝟏+𝒙𝟐 ) is (A)

1
(1) 1 (2) 1 (3)  (4) x
2

 1  x2 1 
42. Derivative of tan1   with respect to tan1 x is (A)
 x 
 
1
(1) 1 (2) 2 (3) (4) -1
2
𝟏−𝒙𝟐 𝟏−𝟑𝒙𝟐
43. The derivative of 𝒄𝒐𝒔−𝟏 (𝟏+𝒙𝟐 ) with respect to 𝒄𝒐𝒕−𝟏 (𝟑𝒙−𝒙𝟑 ) (A)

(1) 3/2 (2) 1 (3) 1/2 (4) 2/3
44. The derivative of 𝒂𝐬𝐞𝐜 𝒙 𝐰𝐢𝐭𝐡 𝐫𝐞𝐬𝐩𝐞𝐜𝐭 𝐭𝐨 𝒂𝐭𝐚𝐧 𝒙 , (𝒂 > 𝟎)𝒊𝒔 (A)
1)𝑠𝑖𝑛𝑥 𝑎 𝑠𝑒𝑐𝑥−𝑡𝑎𝑛𝑥 2) sec 𝑥 𝑎sec 𝑥−tan 𝑥
3) 𝑎 𝑠𝑒𝑐𝑥−tan 𝑥 4) sin 𝑥 𝑎𝑡𝑎𝑛 𝑥−sec 𝑥
𝟑
45. The differential coefficient of 𝒆𝒙 𝐰𝐢𝐭𝐡 𝐫𝐞𝐬𝐩𝐞𝐜𝐭 𝐭𝐨 𝐥𝐨𝐠 𝒙 𝒊𝒔 (A)
2 3 3 3 3
(1)3𝑥 𝑒 𝑥 + 3𝑥 2 2) 𝑒 𝑥 3) 3𝑥 3 𝑒 𝑥 4) 3𝑥 2 𝑒 𝑥
du
46. If u= f (x2) , v= g(x3) , f |(x)= sin x and g |(x)= cos x then  (D)
dv

3x cos( x3 ) 2sin 2 x 2sin( x 2 ) 3x sin( x 2 )
(1) (2) (3) (4)
2sin( x 2 ) 3x cos( x3 ) 3x cos( x3 ) 2 x cos( x3 )
47. If f (x) is a function such that f” (x) + f (x) = 0 and 𝒈(𝒙) = [𝒇(𝒙)]𝟐 + [𝒇′(𝒙)]𝟐 and
g (3) = 8 then g (8) = (A)
(1) 8 (2) 3 (3) 0 (4) 5
48. If 𝒇(𝒙) = 𝒃𝒆𝒂𝒙 + 𝒂𝒆𝒃𝒙 , 𝒕𝒉𝒆𝒏 𝒇” (𝟎) = (A)
(1) ab 2) 𝑎𝑏 (𝑎 + 𝑏) 3) 2ab 4) 0
49. Let f be a polynomial. Then the second derivative of f(ex) is (A)
(1) f (ex) e2x + f (ex) ex (2) f (ex) ex + f(ex)
(3) f (ex) e2x + f (ex) ex (4) f (ex)
50. If y=(tan-1x)2,then (x2+1)2y2+2x(x2+1)y1= (A)
(1) 2 (2) 4 (3) 1 (4) 0

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𝜽
51. If the parametric equation of a curve is given by 𝒙 = 𝐜𝐨𝐬 𝜽 + 𝐥𝐨𝐠 𝐭𝐚𝐧 𝟐 and 𝒚 = 𝐬𝐢𝐧 𝜽,
𝒅𝒚
then the points for which 𝒅𝒙 = 𝟎 are given by (KCET 2021) (D)
𝑛𝜋 𝜋
1)𝜃 = 2 , 𝑛 ∈ 𝑧 2)𝜃 = (2𝑛 + 1) 2 , 𝑛 ∈ 𝑧

3)𝜃 = (2𝑛 + 1)𝜋, 𝑛 ∈ 𝑧 4)𝜃 = 𝑛𝜋, 𝑛 ∈ 𝑧
𝒅𝟐 𝒚
52. If 𝒙 = 𝒂(𝒄𝒐𝒔 𝜽 + 𝜽 𝒔𝒊𝒏 𝜽), 𝒚 = 𝒂(𝒔𝒊𝒏 𝜽 − 𝜽 𝒄𝒐𝒔 𝜽), 𝒅𝒙𝟐 = (D)
𝑠𝑒𝑐 3 𝜃 𝑠𝑒𝑐 2 𝜃 𝑠𝑒𝑐 2 𝜃
(1) 𝑎𝜃 𝑐𝑜𝑠 3 𝜃 (2) (3) (4)
𝑎𝜃 𝜃 𝑎
𝒅𝟐 𝒚 𝝅
53. If 𝒙 = 𝟑 𝐜𝐨𝐬 𝒕 − 𝟐 𝒄𝒐𝒔𝟑 𝒕, 𝒚 = 𝟑 𝐬𝐢𝐧 𝒕 − 𝟐𝒔𝒊𝒏𝟑 𝒕 then 𝒅𝒙𝟐 at 𝒕 = 𝟔 𝒊𝒔 (D)
16 16 16 16
1) 3√3 2) − 3 3) 3 4) − 3√3

54. The second order derivative of 𝐜𝐨𝐬 −𝟏 (𝟒𝒙𝟑 − 𝟑𝒙) with respect to 𝐜𝐨𝐬 −𝟏 (𝟐𝒙𝟐 − 𝟏),
𝟏
where 𝟐 < 𝒙 < 𝟏 is (KCET 2026) (A)
−1 3 −3
(1) 0 (2) √1−x2 (3) 2 (4) 2

ADDITIONAL PROBLEMS
1 2 3 4 5 6 7 8 9 10
3 3 2 3 4 2 1 3 4 2
11 12 13 14 15 16 17 18 19 20
1 3 3 4 2 2 2 3 1 1
21 22 23 24 25 26 27 28 29 30
4 3 3 2 2 3 3 3 2 1
31 32 33 34 35 36 37 38 39 40
3 4 4 4 2 1 3 4 2 1
41 42 43 44 45 46 47 48 49 50
1 3 4 1 3 3 4 2 1 1
51 52 53 54
4 2 2 1

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