CUET · MATHS · PYQ PAPER 2023

For which values of \(a, f(x)=a(x+\sin x)\) is an increasing function?

A \(a \leq 0\)
B \(a \in(0, \infty)\)
C \(a \in[0, \infty)\)
D \(a \in R\) (set of real numbers)

Verified Solution

Answer & Solution

Correct Answer

(B) \(a \in(0, \infty)\)

Step-by-step Solution

Detailed explanation

\(f(x)=a(x+\sin x)\) Differentiating with respect to \(x:\) \(f^{\prime}(x)=a(1+\cos x)\) \((1+\cos x)\) The range of \(\cos x\) is \([-1,1]\). Therefore: The minimum value of \((1+\cos x)\) is \(1+(-1)=0\). The maximum value of \((1+\cos x)\) is \(1+1=2\). So,…

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 (Where R is the set of real numbers)

List-I	List-II
(A) \(sin\) \(x\) is continuous on :	(I) \(R -\{0\}\)
(B) \(tan\) \(x\) is continuous on :	(II) \(R\)
(C) \(\cot x\) is continuous on :	(III) \(R -\{n \pi: n \in Z \}\)
(D) \(x^{-n}, n \in N\) is continuous on:	(IV) \(R -\left\{(2 n+1) \frac{\pi}{2}: n \in Z \right\}\)

Choose the correct answer from the options given below :

CUET 2025 Hard

The area of the region (in square units) bounded by \(x=1, x=2\) and the curve \(y^2=4 x\) in the first quadrant is:

CUET 2025 Hard

Match the Matrix in List - I with its Adjoint's Determinant in List - II.
List - I (Matrix A) :

LIST 1	LIST 2
(A) \(\left[\begin{array}{ll}3 & 1 \\ 4 & 2\end{array}\right]\)	(I) 9
(B) \(\left[\begin{array}{cc}5 & -1 \\ 4 & 2\end{array}\right]\)	(II) 8
(C) \(\left[\begin{array}{cc}6 & -1 \\ 2 & 1\end{array}\right]\)	(III) 14
(D) \(\left[\begin{array}{ll}4 & 1 \\ 3 & 3\end{array}\right]\)	(IV) 2

Choose the correct answer from the options given below:

CUET 2025 Medium

The value of the constant \(\lambda\), so that the function given below is continuous at \(x=-1\) is :
\(f(x)=\left\{\begin{array}{ll}\frac{x^2-2 x-3}{x+1}, & x \neq-1 \\\lambda, & x=-1\end{array}\right.\)

CUET 2023 Hard

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