CUET · MATHS · PYQ PAPER 2023

The interval in which where the function \(f(x)=x^3-3 x^2+4 x+1, x \in R\) is increasing, is:

A \([2,3]\)
B \([-2,4]\)
C \([0,3]\)
D R

Verified Solution

Answer & Solution

Correct Answer

(D) R

Step-by-step Solution

Detailed explanation

\(f'(x) = \frac{d}{dx}(x^3-3 x^2+4 x+1) = 3x^2-6x+4\) \(\Delta = (-6)^2 - 4(3)(4) = 36 - 48 = -12\) Since \(\Delta 0\), \(f'(x) > 0\) for all \(x \in R\). The function is increasing on \(R\).

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

\(y=a \cos x+b \sin x\) where \(a, b\) are arbitrary constants is a solution of the differential equation:

CUET 2023 Hard

Match List I with List II

LIST I	LIST II
A. \(\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3-x} + \sqrt{x}} dx\)	I. -\(\frac{1}{2}\log 3\)
B. \(\int_{1}^{2} x^2 dx\)	II. \(\frac{7}{3}\)
C. \(\int_{-4}^{-1} \frac{1}{x} dx\)	III. \(\frac{1}{2}\)
D. \(\int_{0}^{1} \frac{1}{2x-3} dx\)	IV. -\(\log 4\)

Choose the correct answer from the options given below:

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From CUET

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:

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