CUET · MATHS · PYQ PAPER 2023

If \(A = [a_{ij}]\) be a square matrix of order n, then the correct statements are:
A. A is a symmetric matrix if and only if \(A^T = A\).
B. A is a skew-symmetric matrix if and only if \(A^T = -A\).
C. If order of A is n x n, then order of \(A^T\) is m \(\times\) n (m\(\neq\)n).
D. If \(\begin{bmatrix} 1 & 2 \\ 3 & 2x \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 2x \end{bmatrix}\) then x = 4.
Choose the correct answer from the options given below.

A A, B, D only
B B, C, D only
C A, B only
D B, D only

Verified Solution

Answer & Solution

Correct Answer

(C) A, B only

Step-by-step Solution

Detailed explanation

A. Correct. B. Correct. C. Incorrect. If A is \(n \times n\), then \(A^T\) is \(n \times n\). D. Incorrect. \(2x = 2x\) is true for all \(x\). A, B only

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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List-I (Conditions)	List-II (Type of matrix)
(A) Square matrix \(A\)	(I) \(A=\left[a_{i j}\right]_{m \times m}\), where \(\left\{\begin{array}{ll}a_{i j}=0 & , i \neq j \\ a_{i j}=k & , i=j\end{array}\right.\) where \(k \neq 0\) is constant
(B) Scalar Matrix \(A\)	(II) \(A=\left[a_{i j}\right]_{m \times m}\)
(C) Diagonal matrix \(A\)	(III) \(A=\left[a_{i j}\right]_{m \times m}\), where \(\left\{\begin{array}{ll}a_{i j}=0 & , i \neq j \\ a_{i j}=1 & , i=j\end{array}\right.\)
(D) Identity matrix \(A\)	(IV) \(A=\left[a_{i j}\right]_{m \times m}\), where \(a_{i j}=0, i \neq j\)

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LIST I	LIST II
A. Green Revolution	I. Corrective therapy of hereditary disease
B. Gene therapy	II. Increased yield and better management of improved crop varieties
C. Genetic Engineering Approval Committee	III. Mass production of safe and more effective therapeutics
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LIST-I (Equation of curves)	LIST-II (Slope of tangent at \(x=2\))
(A) \(y=x^3-x\)	(I) 8
(B) \(y=(x-2)^2\)	(II) \(frac{2}{3}\)
(C) \(y=2 x^2+3\)	(III) \(11\)
(D) \(y=\sqrt{4 x+1}-7\)	(IV) \(0\)

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(A) Carbylamine reaction	(I)
(B) Gatterman Rection	(II)
(C) Sandmeyer reaction	(III)
(D) Hoffmann Bromamide reaction	(IV)