The equation of tangent to the curve x = a ^3 t, y = a ^3 t at t is : Option 1: x t + y t = a Option 2: x t - y t = a Option 3: x t + y t = a Option 4: x t - y t = a

Answer: x t + y t = a. Solution: dy/dx=(3a^2 t t)/(-3a^2 t t)=- t. Tangent at (a^3 t, a^3 t): y-a^3 t=- t(x-a^3 t). Simplifying: x t+y t=a t t. Dividing by t t gives x t+y t=a.

Calculus

Application of Derivatives

Medium

core

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

$x \sec t + y \csc t = a$

$x \sec t - y \csc t = a$

$x \csc t + y \sec t = a$

$x \csc t - y \sec t = a$

✅ Correct Option: 1

$\frac{dy}{dx}=\frac{3a\sin^2 t\cos t}{-3a\cos^2 t\sin t}=-\tan t$ .

Tangent at $(a\cos^3 t, a\sin^3 t)$ : $y-a\sin^3 t=-\tan t(x-a\cos^3 t)$ .

Simplifying: $x\sin t+y\cos t=a\sin t\cos t$ . Dividing by $\sin t\cos t$ gives $x\sec t+y\csc t=a$ .

2026: 21 May Shift 2

Consider the function $f(x) = -2x^3 + 3x^2 + 36x - 15$ , then which of the following are TRUE?

A. $f(x)$ increases on $(-2, 3)$

B. $f(x)$ decreases on $(-\infty, -2) \cup (3, \infty)$

C. $f(x)$ increases on $(-\infty, -2) \cup (3, \infty)$

D. $f(x)$ neither increases nor decreases on $(-2, 3)$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$ . The marginal cost when 10 items are produced is:

2022: 30 Aug Shift 1

Consider the function $f(x) = x^{\frac{1}{x}}$ . Its

2026: 30 May Shift 2

For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

2025: 26 May Shift 2

The function $f(x) = x^3 + 3x^2 + 4x + 4$ , $x \in \mathbb{R}$ (set of real numbers) :

2026: 31 May Shift 2

The rate of change in area of a triangle having sides 10cm and 12cm when the variable angle between them at an angle $60°$ , is:

The equation of tangent to the curve x=acos⁡3t,y=asin⁡3tx = a \cos^3 t, y = a \sin^3 tx=acos3t,y=asin3t at t is :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :