CUET · MATHS · PYQ PAPER 2023

Match List I with List II
(List I) (List II)
A. \(\frac{d}{d x}\left(a^x\right)\) \(I.\) \(0\)
B. \(\frac{d}{d x}\left(x^a\right)\) \(II.\) \(x^x\left(1+\log _e x\right)\)
C. \(\frac{d}{d x}\left(a^a\right)\) \(III.\) \(a x^{a-1}\)
D. \(\frac{d}{d x}\left(x^x\right)\) \(IV.\) \(a^x \log _e a\)

Choose the correct answer from the options given below:

A \(A-III, B-I, C-II, D-IV\)
B \(A-IV, B-I, C-III, D-II\)
C \(A-III, B-I, C-IV, D-II\)
D \(A-IV, B-III, C-I, D-II\)

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Answer & Solution

Correct Answer

(D) \(A-IV, B-III, C-I, D-II\)

Step-by-step Solution

Detailed explanation

\( \frac{d}{d x}\left(a^x\right) = a^x \log_e a \). (A - IV) \( \frac{d}{d x}\left(x^a\right) = a x^{a-1} \). (B - III) \( \frac{d}{d x}\left(a^a\right) = 0 \). (C - I) Let \( y = x^x \). \( \log y = x \log x \) \( \frac{1}{y} \frac{dy}{dx} = \log x + 1 \)…

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.

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Match List-I with List-II:

List-I (AC Circuit)	List-II (Phase Angle)
(A) Phase Difference between current and voltage in a pure inductive ac circuit	(I) zero
(B) Phase Difference between current and voltage in a pure resistive ac circuit	(II) \(\pi/2\), current lags voltage
(C) Phase Difference between current and voltage in a pure capacitive ac circuit	(III) \(\tan^{-1}(\frac{X_L - X_C}{R})\)
(D) Phase Difference between current and voltage in a series LCR circuit	(IV) \(\pi/2\), current leads voltage

Choose the correct answer from the options given below

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(List I)	(List II)
A. \(\frac{d}{d x}\left(a^x\right)\)	\(I.\) \(0\)
B. \(\frac{d}{d x}\left(x^a\right)\)	\(II.\) \(x^x\left(1+\log _e x\right)\)
C. \(\frac{d}{d x}\left(a^a\right)\)	\(III.\) \(a x^{a-1}\)
D. \(\frac{d}{d x}\left(x^x\right)\)	\(IV.\) \(a^x \log _e a\)