Section A — MCQ (Single Correct)
Question 1
Let $f(x) = \min(\{x\}, \{-x\})$, where $\{\cdot\}$ denotes the fractional part function. The number of points of non-differentiability for $f(x)$ inside the open interval $(0, 2)$ is:
A
$1$
B
$2$
C
$3$
D
$4$
Question 2
If $y = (x^2)^{x}$, then $\frac{dy}{dx}$ at $x = -1$ is equal to:
A
$2$
B
$-2$
C
$0$
D
$-1$
Question 3
Let $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$. The value of $f'''(x)$ is:
A
$6$
B
$12$
C
$0$
D
Constant and equal to 6
Question 4
If $x = \sec \psi - \cos \psi$ and $y = \sec^n \psi - \cos^n \psi$, then $(x^2 + 4)\left(\frac{dy}{dx}\right)^2$ equals:
A
$n^2 (y^2 + 4)$
B
$n^2 (y^2 - 4)$
C
$n^2 y^2$
D
$y^2 + 4$
Question 5
Let $f(x)$ be twice differentiable such that $f''(x) > 0$ for all $x \in [a, b]$. If $c$ is the point satisfying LMVT, then:
A
$c < \frac{a+b}{2}$
B
$c = \frac{a+b}{2}$
C
$c > \frac{a+b}{2}$
D
The position depends on the function chosen
Question 6
The second derivative $\frac{d^2x}{dy^2}$ expressed in terms of the derivatives of $y$ with respect to $x$ is:
A
$\left(\frac{d^2y}{dx^2}\right)^{-1}$
B
$-\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-3}$
C
$-\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-2}$
D
$\left(\frac{d^2y}{dx^2}\right)\left(\frac{dy}{dx}\right)^{-3}$
Question 7
Using Maclaurin series expansions, the coefficient of $x^5$ in the infinite series polynomial for $f(x) = \sin x$ is:
A
$\frac{1}{5}$
B
$\frac{1}{120}$
C
$-\frac{1}{120}$
D
$\frac{1}{24}$
Question 8
If $f(x + y) = f(x) \cdot f(y)$ for all real coordinates, and $f(0) = 1$ with $f'(0) = 2$, then $f(x)$ is equal to:
A
$e^{2x}$
B
$2e^x$
C
$e^x$
D
$x^2 + 1$
Question 9
Let $f(x) = \cos x \cdot \cos 2x \cdot \cos 4x \cdot \cos 8x$. The value of $f'\left(\frac{\pi}{4}\right)$ evaluates to:
A
$1$
B
$0$
C
$-1$
D
$\frac{1}{2}$
Question 10
The number of real roots of $f'(x) = 0$ for $f(x) = x(x+1)(x+2)(x+3)$ is:
A
$1$
B
$2$
C
$3$
D
$0$
Section B — Integer Type
Question 11 — Integer answer
If the $n$-th derivative of $y = x^{n}$ is given by $y^{(n)} = k!$, find the value of $k - n$.
Question 12 — Integer answer
Let $f(x) = \max(|x-1|, |x-2|, |x-3|)$. Find the total number of points across the real line where $f(x)$ fails to be differentiable.
Question 13 — Integer answer
Let $f(x)$ be differentiable such that $f(1) = 2$ and $f(4) = 8$. Find the minimum value of $f'(c)$ guaranteed by LMVT inside the interval $(1, 4)$.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): If a function is non-differentiable at $x = c$, then its graph must have a sharp corner at that point.Reason (R): Non-differentiability can also occur due to vertical tangents, cusps, or discontinuity gaps.
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
Solution: A is false, but R is true.
Question 15 — Assertion / Reason
Assertion (A): Cauchy's Mean Value Theorem simplifies to Lagrange's Mean Value Theorem when we choose $g(x) = x$.Reason (R): The derivative of $g(x) = x$ is a constant equal to 1, which reduces the denominator ratio to $(b-a)$.
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
Solution: Both A and R are true, and R is the correct explanation.