Section A — MCQ (Single Correct)
Question 1
If the subnormal to the curve $x y^n = a^{n+1}$ is of constant length, then the exponent parameter $n$ must equal:
A
$-2$
B
$-3$
C
$2$
D
$3$
Question 2
Let $f(x) = \int_0^x \sqrt{t} \cdot e^{-t} \, dt$ for $x > 0$. The maximum value of $f(x)$ is achieved at:
A
$x = 1/2$
B
$x = 1$
C
$x = 2$
D
$x = e$
Question 3
The total number of real roots for the transcendental equation $e^x = x^2$ across the real line is exactly:
A
$1$
B
$2$
C
$3$
D
$0$
Question 4
The shortest distance between the line $y - x = 1$ and the parabola $x = y^2$ is given by:
A
$\frac{3\sqrt{2}}{8}$
B
$\frac{2\sqrt{3}}{5}$
C
$\frac{1}{2}$
D
$\frac{3}{4}$
Question 5
If $f(x) = \frac{x^2 - 1}{x^2 + 1}$ for all real numbers, then the absolute minimum value of the function is:
A
$1$
B
$-1$
C
$0$
D
$-1/2$
Question 6
Evaluate the limit $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}$ using log transformations and L'Hôpital's Rule:
A
$e^{-1/6}$
B
$e^{-1/3}$
C
$e^{-1/2}$
D
$1$
Question 7
Let $f(x) = x^3 + b x^2 + c x + d$ with $b^2 < 3c$. The total number of critical points for this cubic polynomial is:
A
$0$
B
$1$
C
$2$
D
Infinite
Question 8
A right circular cylinder is inscribed inside a sphere of radius $R$. The maximum possible volume of the cylinder is:
A
$\frac{4\pi R^3}{3\sqrt{3}}$
B
$\frac{4\pi R^3}{3}$
C
$\frac{2\pi R^3}{3\sqrt{3}}$
D
$\frac{\pi R^3}{\sqrt{3}}$
Question 9
The function $f(x) = \frac{\ln x}{x}$ is concave downwards inside which interval?
A
$(0, e^{3/2})$
B
$(e^{3/2}, \infty)$
C
$(0, e)$
D
$(e, \infty)$
Question 10
If Rolle's Theorem holds for a differentiable function $f(x)$ on $[0, 1]$ with $f(0) = f(1) = 0$, then the equation $f'(x) + 2 f(x) = 0$ must have:
A
At least one real root in $(0, 1)$
B
No roots in $(0, 1)$
C
Exactly two roots in $(0, 1)$
D
No definitive conclusion can be drawn
Section B — Integer Type
Question 11 — Integer answer
If the curve $y = a x^3 + b x^2 + c x + 5$ has an inflection point at $P(1, 2)$ where the tangent line is horizontal, find the value of the coefficient constant $b$.
Question 12 — Integer answer
Find the total number of distinct real solutions to the equation $3x^5 + 15x^3 + 10x - 20 = 0$.
Question 13 — Integer answer
Find the maximum vertical distance measured between the curves $y = \sin x$ and $y = \cos x$ within the first quadrant interval $x \in [0, \pi/2]$.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): The curve $y = x^4$ has a critical point at $x = 0$, but this point does not qualify as an inflection point.Reason (R): The second derivative is $f''(x) = 12x^2 \ge 0$, which means it does not change sign across the origin.
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.
Question 15 — Assertion / Reason
Assertion (A): The shortest distance between two non-intersecting smooth curves always lies along their common normal line.Reason (R): The normal line represents the path of steepest ascent and descent perpendicular to the local boundaries.
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 (geometric property of shortest distance).