CUET · MATHS · PYQ PAPER 2023

Let \(f(x)\) satisfies the requirements of Lagrange's mean value theorem in \([0,2]\).
If \(f(0)=0\) and \(f^{\prime}(x) \leq \frac{1}{2}\) for all \(x \in[0,2]\), then :

A \(|f(x)|=2\)
B \(f(x)=2 x\)
C \(f(x) \leq 1\)
D \(f(x)=3\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(f(x) \leq 1\)

Step-by-step Solution

Detailed explanation

\(\text{By LMVT on } [0,x], x \in (0,2]: \exists c \in (0,x) \text{ s.t. } f'(c) = \frac{f(x) - f(0)}{x - 0}\) \(f'(c) = \frac{f(x) - 0}{x} = \frac{f(x)}{x}\) \(\text{Given } f'(c) \leq \frac{1}{2}\) \(\frac{f(x)}{x} \leq \frac{1}{2}\) \(f(x) \leq \frac{1}{2}x\)…

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

List - I (Matrix/Equations)	List - II (Values)
(A) \(\left[\begin{array}{cc}2 x+1 & 3 y \\ 0 & y^2-5 y\end{array}\right]=\left[\begin{array}{cc}x+3 & y^2+2 \\ 0 & -6\end{array}\right]\)	(I) \(x = 2, y = - 1\)
(B) \(\left[\begin{array}{ccc}1 & 2 & -1 \\ x & 0 & 3 \\ y & 3 & 4\end{array}\right]\) is symmetric	(II) \(x = 2, y = 2\)
(C) \(\left[\begin{array}{ll}x & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -2 & -3\end{array}\right]\left[\begin{array}{ll}5 & 2 \\ 0 & y\end{array}\right]=0\)	(III) \(x = - 2, y = 2\)
(D) \(\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}4 & 0 \\ -1 & y / 2\end{array}\right]\)	(IV) \(x = 2, y = 0\)

Choose the correct answer from the options given below:

CUET 2025 Easy

Let \(f:[-2,2] \rightarrow[-2,2]\) be a function defined by \(f(x)=x|x|\), then \(f\) is:

CUET 2023 Easy

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Let \(f(x)\) satisfies the requirements of Lagrange's mean value theorem in \([0,2]\). If \(f(0)=0\) and \(f^{\prime}(x) \leq \frac{1}{2}\) for all \(x \in[0,2]\), then :

Step-by-step Solution

Let \(f(x)\) satisfies the requirements of Lagrange's mean value theorem in \([0,2]\).
If \(f(0)=0\) and \(f^{\prime}(x) \leq \frac{1}{2}\) for all \(x \in[0,2]\), then :