CUET UG 2025 — Mathematics Algebra

2025mcqmedium

If, in a pair of consecutive positive integers, both numbers are greater than 5 and their sum is less than 23, then the number of such pairs are:

Official previous-year question

Held on 16 May 2025 · Verified 13 Jul 2026.

Options

A
Two
B
Three
C
Four
D
Five

Did you get this right?

Solution

$(A - A^T)^T = A^T - (A^T)^T$

$= A^T - A$

$= -(A - A^T)$

Since the transpose equals the negative of itself, $A - A^T$ is skew-symmetric.

$(A) \rightarrow \text{(II)}$

$(AA^T)^T = (A^T)^T \cdot A^T$

$= A \cdot A^T$

$= AA^T$

Since the transpose equals itself, $AA^T$ is symmetric.

$(B) \rightarrow \text{(III)}$

Since $A \cdot A^{-1} = I$ , taking determinant on both sides:

$\det(A) \cdot \det(A^{-1}) = \det(I) = 1$

$\det(A^{-1}) = \dfrac{1}{\det(A)}$

$= [\det(A)]^{-1}$

$(C) \rightarrow \text{(IV)}$

From the standard identity:

$A^{-1} = \dfrac{\text{adj}(A)}{\det(A)}$

$\text{adj}(A) = \det(A) \cdot A^{-1}$

$= |A|\,A^{-1}$

$(D) \rightarrow \text{(I)}$

List-I	List-II
(A) $A - A^T$	(II) Skew-symmetric
(B) $AA^T$	(III) Symmetric
(C) $\det(A^{-1})$	(IV) $[\det(A)]^{-1}$
(D) $\text{adj}(A)$	(I) $\\|A\\|A^{-1}$

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Solution

$(A - A^T)^T = A^T - (A^T)^T$

$= A^T - A$

$= -(A - A^T)$

Since the transpose equals the negative of itself, $A - A^T$ is skew-symmetric.

$(A) \rightarrow \text{(II)}$

$(AA^T)^T = (A^T)^T \cdot A^T$

$= A \cdot A^T$

$= AA^T$

Since the transpose equals itself, $AA^T$ is symmetric.

$(B) \rightarrow \text{(III)}$

Since $A \cdot A^{-1} = I$ , taking determinant on both sides:

$\det(A) \cdot \det(A^{-1}) = \det(I) = 1$

$\det(A^{-1}) = \dfrac{1}{\det(A)}$

$= [\det(A)]^{-1}$

$(C) \rightarrow \text{(IV)}$

From the standard identity:

$A^{-1} = \dfrac{\text{adj}(A)}{\det(A)}$

$\text{adj}(A) = \det(A) \cdot A^{-1}$

$= |A|\,A^{-1}$

$(D) \rightarrow \text{(I)}$

List-I	List-II
(A) $A - A^T$	(II) Skew-symmetric
(B) $AA^T$	(III) Symmetric
(C) $\det(A^{-1})$	(IV) $[\det(A)]^{-1}$
(D) $\text{adj}(A)$	(I) $\\|A\\|A^{-1}$

CUET UG 2025 — Mathematics Algebra

Options

Did you get this right?

Solution

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CUET UG Algebra in other years

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Other Mathematics topics for CUET UG

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CUET UG 2025 — Mathematics Algebra

Options

Did you get this right?

Solution

Your note

CUET UG Algebra in other years

More CUET UG Algebra questions

Other Mathematics topics for CUET UG

Calculus

Applied-Mathematics

Statistics & Applications

Geometry

Arithmetic

Coordinate Geometry

Keep practicing Algebra