CUET · MATHS · PYQ PAPER 2023

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) be a \(3 \times 3\) matrix,
where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\).
If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is ___________ .

A \(2^{10}\)
B \(2^{11}\)
C \(2^{12}\)
D \(2^{13}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(2^{13}\)

Step-by-step Solution

Detailed explanation

\(\det(Q) = \det\begin{pmatrix} 2^{1+1}a_{11} & 2^{1+2}a_{12} & 2^{1+3}a_{13} \\ 2^{2+1}a_{21} & 2^{2+2}a_{22} & 2^{2+3}a_{23} \\ 2^{3+1}a_{31} & 2^{3+2}a_{32} & 2^{3+3}a_{33} \end{pmatrix}\)…

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 (Differential equation)	List-II (Order and Degree)
(A) \(\frac{d^3 y}{d x^3}+y^2+e^{d y / d x}=0\)	(I) order \(=3\), degree \(=1\)
(B) \(\left(\frac{d^2 y}{d x^2}\right)^3+\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}+1=0\)	(II) order \(=3\), degree not defined
(C) \(2 x^2 \frac{d^2 y}{d x^2}-3\left(\frac{d y}{d x}\right)^2+y=0\)	(III) order \(=2\), degree \(=3\)
(D) \(\frac{d^3 y}{d x^3}+2\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)	(IV) order \(=2\), degree \(=1\)

Choose the correct answer from the options given below :

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Match List - I with List - II.

List - I	List - II
(A) Australian marsupials	(I) Homologous organs
(B) Wings of butterfly and wings of a bird	(II) Natural selection
(C) Industrial melanism	(III) Adaptive radiation
(D) Forelimbs of Horse and Bat	(IV) Analogous organs

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X	-1	0	1	2	3
P(X)	K	\(\frac{1}{5}\)	2K	\(\frac{3}{10}\)	K

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) be a \(3 \times 3\) matrix,where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\).If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is ___________ .

Step-by-step Solution

Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) be a \(3 \times 3\) matrix,
where \(b_{i j}=2^{i+j} a_{i j}\) for \(1 \leq i, j \leq 3\).
If the determinant of \(P\) is 2 , then the determinant of the matrix \(Q\) is ___________ .