Continuity and Differentiability

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#599
If f(x)=∣x∣+∣x−1∣, then which of the following is correct?
(A) f(x) is both continuous and differentiable, at x=0 and x=1.
(B) f(x) is differentiable but not continuous, at x=0 and x=1.
(C) f(x) is continuous but not differentiable, at x=0 and x=1.
(D) f(x) is neither continuous nor differentiable, at x=0 and x=1.
MEDIUM
2 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#598
If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?

(A) A
(B) B
(C) C
(D) D
MEDIUM
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#597
If \(f(x)=\begin{cases}3x-2,&0 \lt x\le 1\\ 2x^{2}+ax,&1\lt x\lt 2\end{cases}\) is continuous for \(x\in(0,2)\), then a is equal to:
(A) -4
(B) \(-\frac{7}{2}\)
(C) -2
(D) -1
MEDIUM
4 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#596
The function f defined by \(f(x)=\begin{cases}x,&if~x\le1\\ 5,&if~x>1\end{cases}\) is not continuous at:
(A) \(x=0\)
(B) \(x=1\)
(C) \(x=2\)
(D) \(x=5\)
MEDIUM
5 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#595
If \(f(x)=\begin{cases}\frac{\sin^{2}ax}{x^{2}},&x\ne0\\ 1,&x=0\end{cases}\) is continuous at \(x=0\), then the value of a is:
(A) 1
(B) -1
(C) \(\pm1\)
(D) 0
MEDIUM
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#594
If \(f(x)=\{[x],x\in R\}\) is the greatest integer function, then the correct statement is:
(A) f is continuous but not differentiable at \(x=2\).
(B) f is neither continuous nor differentiable at \(x=2\).
(C) f is continuous as well as differentiable at \(x=2\).
(D) f is not continuous but differentiable at \(x=2\).
MEDIUM
7 Competency UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#593
If \(f(x)=\begin{cases}\frac{\log(1+ax)+\log(1-bx)}{x},&for~x\ne0\\ k&,for~x=0\end{cases}\) is continuous at \(x=0\), then the value of k is:
(A) a
(B) \(a+b\)
(C) \(a-b\)
(D) b
MEDIUM
8 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#592
If \( f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } x \geq 5 \end{cases} \) is continuous for all real numbers, then find the values of \(a\) and \(b\):
(A) \(a=3\), \(b=-8\)
(B) \(a=3\), \(b=8\)
(C) \(a=-3\), \(b=-8\)
(D) \(a=-3\), \(b=8\)
MEDIUM
9 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#591
A function \(f(x)=|1-x+|x||\) is:
(A) discontinuous at \(x=1\) only
(B) discontinuous at \(x=0\) only
(C) discontinuous at \(x=0,1\)
(D) continuous everywhere
MEDIUM
10 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#590
For what value of k, the function given below is continuous at \(x=0\) ? \(f(x)=\begin{cases}\frac{\sqrt{4+x}-2}{x},&x\ne0\\ k,&x=0\end{cases}\)
(A) 0
(B) \(\frac{1}{4}\)
(C) 1
(D) 4
MEDIUM
11 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#798
If $y=\frac{\cos x-\sin x}{\cos x+\sin x}$ then $\frac{dy}{dx}$ is:
(A) $-\sec^{2}(\frac{\pi}{4}-x)$
(B) $\sec^{2}(\frac{\pi}{4}-\pi)$
(C) $\log|\sec(\frac{\pi}{4}-x)|$
(D) $-\log|\sec(\frac{\pi}{4}-x)|$
MEDIUM
12 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#797
The value of k for which function $f(x)=\begin{cases}x^{2},&x\ge0\\ kx,&x<0\end{cases}$ is differentiable at $x=0$ is:
(A) 1
(B) 2
(C) any real number
(D) 0
MEDIUM
13 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#796
7. If $y=\sin^{2}(x^{3})$, then $\frac{dy}{dx}$ is equal to :
(A) $2\sin x^{3}\cos x^{3}$
(B) $3x^{3}\sin x^{3}\cos x^{3}$
(C) $6x^{2}\sin x^{3}\cos x^{3}$
(D) $2x^{2}\sin^{2}(x^{3})$
MEDIUM
14 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#795
6. The function $f(x)=|x|$ is
(A) continuous and differentiable everywhere.
(B) continuous and differentiable nowhere.
(C) continuous everywhere, but differentiable everywhere except at $x=0$.
(D) continuous everywhere, but differentiable nowhere.
MEDIUM
15 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023 AISSCE(Board Exam)
#794
The value of k for which$ f(x)=\begin{cases}3x+5,&x\ge2\\ kx^{2},&x<2\end{cases}$ is a continuous function, is :
(A) $-\frac{11}{4}$
(B) $\frac{4}{11}$
(C) 11
(D) $\frac{11}{4}$
MEDIUM
16 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#793
If x=A cos 4t + B sin 4t, then d²x/dt² is equal to
(A) x
(B) -x
(C) 16x
(D) -16x
MEDIUM
17 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#792
The function f(x)=[x], where [x] denotes the greatest integer less than or equal to x, is continuous at
(A) x=1
(B) x=1.5
(C) x=-2
(D) x=4
MEDIUM
18 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#791
The derivative of x²ⁿ w.r.t. x is
(A) x²ⁿ⁻¹
(B) 2x²ⁿ log x
(C) 2x²ⁿ(1+log x)
(D) 2x²ⁿ(1-log x)
EASY

Assertion & Reason

1 Knowledge REMEMBER ASSERTION_REASON 1 Mark(s) 2025 AISSCE(Board Exam)
#974
Assertion: Assertion (A): $f(x) = \begin{cases} x\sin\frac{1}{x}, & x\neq 0 \\ 0, & x=0 \end{cases}$ is continuous at $x=0$.
Reason: Reason (R): When $x \to 0$, $\sin\frac{1}{x}$ is a finite value between $-1$ and $1$.
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is NOT the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.
EASY

Very Short Answer (2 Mark)

1 Knowledge VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1383
Check the differentiability of f(x) at $x=-2$ if $f(x)=\begin{cases}2x-3,-3\le x\le-2\\ x+1,-2<x\le0\end{cases}$.
2 Knowledge APPLY VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1335
Check for differentiability of the function f defined by $f(x)=|x-5|$, at the point $x=5$.
MEDIUM
3 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1334
Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.
MEDIUM
4 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1288
Check whether the function $f(x)=x^{2}|x|$ is differentiable at $x=0$ or not.
MEDIUM
5 Knowledge REMEMBER VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1246
Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}$ at $x=1.$
MEDIUM
6 Knowledge APPLY VSA 2 Mark(s) 2023
#871
If $x=a\sin 2t, y=a(\cos 2t+\log\tan t)$ then find $\frac{dy}{dx}$
MEDIUM
7 Knowledge APPLY VSA 2 Mark(s) 2023
#870
If $y=x^{\frac{1}{x}}$ then find $\frac{dy}{dx}$ at $x=1$.
MEDIUM
8 Knowledge APPLY VSA 2 Mark(s) 2023
#869
22. If $(x^{2}+y^{2})^{2}=xy$, then find $\frac{dy}{dx}$
HARD
9 Competency APPLY VSA 2 Mark(s) 2023 AISSCE(Board Exam)
#868
If $y=(x+\sqrt{x^{2}-1})^{2}$;, then show that $(x^{2}-1)(\frac{dy}{dx})^{2}=4y^{2}.$
MEDIUM
10 Knowledge APPLY VSA 2 Mark(s) 2023 AISSCE(Board Exam)
#866
(a) If $f(x) = \begin{cases} x^2, & \text{if } x \geq 1 \\ x, & \text{if } x < 1 \end{cases}$, then show that $f$ is not differentiable at $x=1$.
OR
(b) Find the value(s) of '$\lambda$', if the function $f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2} & \text{if } x \neq 0 \\ 1 & \text{if } x=0 \end{cases}$ is continuous at $x=0$.
MEDIUM

Short Answer (3 Marks)

1 Knowledge UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1409
Check the differentiability of function $f(x)=x|x|$ at $x=0$.
MEDIUM
2 Knowledge REMEMBER SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1408
Find k so that $f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}$ is continuous at $x=-1$.
EASY
3 Knowledge REMEMBER SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1252
Find the value of a and b so that function f defined as : $$ f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2|} + b, & \text{if } x > 2 \end{cases} $$ is a continuous function.
MEDIUM
4 Competency APPLY SA 3 Mark(s) 2023 AISSCE(Board Exam)
#867
(a) Differentiate $\text{sec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)$ w.r.t. $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$.
OR
(b) If $y = \tan x + \sec x$, then prove that $\frac{d^2y}{dx^2} = \frac{\cos x}{(1-\sin x)^2}$.
MEDIUM