CUET · MATHS · PYQ PAPER 2025

The interval on which the function \(f(x)=x^4-\frac{x^3}{3}\) is strictly decreasing, is :

A \((4, \infty)\)
B \(\left(\frac{1}{4}, \infty\right)\)
C \(\left(-\infty, \frac{1}{4}\right)\)
D \(\left(0, \frac{1}{4}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\left(-\infty, \frac{1}{4}\right)\)

Step-by-step Solution

Detailed explanation

\(f'(x) = 4x^3 - x^2\) \(f'(x) = x^2(4x-1)\) For strictly decreasing, \(f'(x) \le 0\): \(x^2(4x-1) \le 0\) \(4x-1 \le 0 \quad (\text{since } x^2 \ge 0)\) \(x \le \frac{1}{4}\) Interval: \(\left(-\infty, \frac{1}{4}\right)\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

The tangent to the curve \(y=x^3-11 x+5\) at \(x=2\) is parallel to :
(A) \( y=x+1\)
(B) \(y=-x+2\)
(C) \(2 y=x+3\)
(D) \(2 y=2 x+4\)
Choose the correct answer from the options given below :

CUET 2023 Easy

Match List - I with List - II.

LIST - I	LIST - II
(A) \(x^x\) has a stationary point at \(x\) equal to	(I) \(e\)
(B) For \(x>0\), minimum value of \(x^x\)	(II) \(1 / e\)
(C) The greatest value of \(\left(\frac{1}{x}\right)^x\)	(III) \(e^{1 / e}\)
(D) The stationary point of \(\frac{\log x}{x}\) for \(x\), where \(x>0\), is	(IV) \(e^{-1 / e}\)

Choose the correct answer from the otions given below:

CUET 2023 Medium

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