CUET · MATHS · PYQ PAPER 2025

If \(y=3^x+e^x+x^x+x^3\), then the value of \(\frac{d y}{d x}\) at \(x=3\) is:

A \(e^3+27 \log _e 3+27\)
B \(e^3+54 \log _e 3+54\)
C \(e^3+54 \log _e 3+27\)
D \(e^3+27 \log _e 3+54\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(e^3+54 \log _e 3+54\)

Step-by-step Solution

Detailed explanation

\( \frac{dy}{dx} = 3^x \log 3 + e^x + x^x(\log x + 1) + 3x^2 \) \( \left.\frac{dy}{dx}\right|_{x=3} = 3^3 \log 3 + e^3 + 3^3(\log 3 + 1) + 3(3^2) \) \( = 27 \log 3 + e^3 + 27(\log 3 + 1) + 27 \) \( = 27 \log 3 + e^3 + 27 \log 3 + 27 + 27 \) \( = e^3 + 54 \log 3 + 54 \)

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

Match List-I with List-II

List - I	List - II
(A) \(\sin ^{-1}(-1)\)	(I) \(\frac{5 \pi}{6}\)
(B) \(\cot ^{-1}(-1)\)	(II) \(-\frac{\pi}{2}\)
(C) \(\sec ^{-1}\left(-\frac{2}{\sqrt{3}}\right)\)	(III) \(\frac{\pi}{4}\)
(D) \(\tan ^{-1}(1)\)	(IV) \(\frac{3 \pi}{4}\)

Choose the correct answer from the options given below :

CUET 2025 Hard

Explore more questions on app

From CUET

Explore more questions on app