CUET · MATHS · PYQ PAPER 2023

Match List - I with List - II. Evaluate the integrals.
\(List - I\) \(List - II\)
(A) \(\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\) \((I)\) \(2 \pi\)
(B) \(\int_0^{1 / 2} \frac{d x}{\sqrt{x-x^2}}\) \((II)\) \(\frac{\pi}{4}\)
(C) \(\int_{-\pi}^\pi x \sin x d x\) \((III)\) \(0\)
(D) \(\int_0^1 \frac{1}{1+x^2} d x\) \((IV)\) \(\frac{\pi}{2}\)

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

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

(A) \(I = \int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\) \(I = \int_0^{\pi / 2} \frac{\sin (\pi/2 - x)-\cos (\pi/2 - x)}{1+\sin (\pi/2 - x) \cos (\pi/2 - x)} d x = \int_0^{\pi / 2} \frac{\cos x-\sin x}{1+\cos x \sin x} d x = -I\) \(2I = 0 \Rightarrow I = 0\) (B)…

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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If \(A\) and \(B\) are independent events and \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) then
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List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{1}{2}\)
(B) \(P(\bar{A}) P(B)+P(A) P(\bar{B})\)	(II) \(\frac{1}{3}\)
(C) \(P(A \mid B)+P(B \mid A)\)	(III) \(\frac{1}{6}\)
(D) \(P(\overline{A \cap B})\)	(IV) \(\frac{5}{6}\)

Choose the correct answer from the options given below :

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Match List - I with List - II

List - I	List - II
(A) Simple microscope	(I) A concave mirror is used as an objective
(B) Compound microscope	(II) Two convex lenses both of small focal length
(C) Refracting Telescope	(III) Two convex lenses, one of small focal length other of large focal length
(D) Reflecting Telescope	(IV) Single convex lens of small focal length

Choose the correct answer from the options given below:

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\(List - I\)	\(List - II\)
(A) \(\int_0^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\)	\((I)\) \(2 \pi\)
(B) \(\int_0^{1 / 2} \frac{d x}{\sqrt{x-x^2}}\)	\((II)\) \(\frac{\pi}{4}\)
(C) \(\int_{-\pi}^\pi x \sin x d x\)	\((III)\) \(0\)
(D) \(\int_0^1 \frac{1}{1+x^2} d x\)	\((IV)\) \(\frac{\pi}{2}\)