CUET · MATHS · PYQ PAPER 2025

Match List-I with List-II (where \(c\) is an arbitrary constant)
List-I (Definite integral) List-II (Value)
(A) \(\int_1^e \frac{\log x}{x} d x\) (I) 4
(B) \(\int_{-2}^2 x^3\left(1-x^2\right) d x\) (II) \(\frac{1}{2}\)
(C) \(\int_1^2 x d x\) (III) \(0\)
(D) \(\int_{-2}^2|x| d x\) (IV) \(\frac{3}{2}\)
Choose the correct answer from the options given below :

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

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

(A) \(\int_1^e \frac{\log x}{x} d x\) \(= \left[ \frac{(\log x)^2}{2} \right]_1^e\) \(= \frac{(\log e)^2}{2} - \frac{(\log 1)^2}{2} = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}\) (B) \(\int_{-2}^2 x^3\left(1-x^2\right) d x\) \(x^3(1-x^2) = x^3 - x^5\) is an odd function.…

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 (Laws/properties)	List - II (Statements)
(A) Colligative properties	(I) for a solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in the solution.
(B) Henry's law	(II) the total pressure over the solution phase in the container will be the sum of the partial pressures of all the components of the solution.
(C) Raoult's law	(III) depends on the number of solute particles, irrespective of their nature.
(D) Dalton's law of partial pressures	(IV) at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas present above the surface of a liquid or solution.

Choose the correct answer from the options given below:

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List-I (Definite integral)	List-II (Value)
(A) \(\int_1^e \frac{\log x}{x} d x\)	(I) 4
(B) \(\int_{-2}^2 x^3\left(1-x^2\right) d x\)	(II) \(\frac{1}{2}\)
(C) \(\int_1^2 x d x\)	(III) \(0\)
(D) \(\int_{-2}^2\|x\| d x\)	(IV) \(\frac{3}{2}\)