CUET · MATHS · PYQ PAPER 2025

If \(A\) is a square matrix such that \(A^2=A\) and \(I\) is the identity matrix of the same order as \(A\), then \((I+2 A)^2-5 A\) is equal to :

A \(I+2 A\)
B \(I+3 A\)
C \(I+A\)
D \(I\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(I+3 A\)

Step-by-step Solution

Detailed explanation

\((I+2A)^2 = I^2 + 2AI + 2AI + (2A)^2\) \(= I + 4A + 4A^2\) Given \(A^2=A\): \(= I + 4A + 4A\) \(= I + 8A\) \((I+2A)^2 - 5A = (I+8A) - 5A\) \(= I+3A\)

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	List - II
(A) The present value of an immediate perpetuity of ₹R payable at end of any year forever at rate of i per period is:	(I) \(\frac{R}{i}(1+i)^m\)
(B) Present value of deferred perpetuity is	(II) \(\frac{A i}{(1+i)^n-1}\)
(C) Periodic payment R in a sinking fund is given by formula (for Amount A)	(III) \(A(1+i)^{-n}\)
(D) The present value of redemption price A of a bond is given by	(IV) \(\frac{R}{i}\)

CUET 2023 Hard

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