CUET · MATHS · PYQ PAPER 2025

The values of \(\lambda\) for which the system of equation \(x+2 y+z=14,-x+y+z=10, x+\lambda y+z=2\) has unique solution is :

A \(R -\{2\}\), where R is set of real numbers.
B \(-2 \leq \lambda \leq 2\)
C \(-4<\lambda<4\)
D \(2\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(R -\{2\}\), where R is set of real numbers.

Step-by-step Solution

Detailed explanation

\(A = \begin{vmatrix} 1 & 2 & 1 \\ -1 & 1 & 1 \\ 1 & \lambda & 1 \end{vmatrix}\) \(\det(A) = 1(1 - \lambda) - 2(-1 - 1) + 1(-\lambda - 1)\) \(\det(A) = 1 - \lambda + 4 - \lambda - 1 = 4 - 2\lambda\) For a unique solution, \(\det(A) \neq 0\) \(4 - 2\lambda \neq 0\)…

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

The random variable X can take only the values 0, 1, 2. Given that \(P(X = 0) = P(X = 1) = p\) and \(E(X^2) = E(X) + 1\), then the value of p is:

CUET 2023 Medium

Match List-l with List-II

List - l	List - lI
(A) The chance of observing a specific outcome in an experiment	(I) Parameter
(B) A method used to estimate the population parameters	(II) Estimation
(C) A value that describes an entire population	(III) Statistic
(D) A measurable characteristic of a sample	(IV) Probability distribution

Choose the correct option:

CUET 2025 Hard

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