CUET · MATHS · PYQ PAPER 2025

Evaluate the integral \(\int\left(x^4+x^2+1\right) d\left(x^2\right)\), where C is the constant of integration

A \(x^6+x^4+x^2+C\)
B \(\frac{x^6}{6}+\frac{x^4}{4}+\frac{x^2}{2}+C\)
C \(\frac{x^6}{3}+\frac{x^4}{2}+x^2+C\)
D \(x^6+x^5+x^4+x^3+x^2+x+C\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{x^6}{3}+\frac{x^4}{2}+x^2+C\)

Step-by-step Solution

Detailed explanation

\(\int\left(x^4+x^2+1\right) d\left(x^2\right) = \int\left((x^2)^2+x^2+1\right) d\left(x^2\right)\) \(= \frac{(x^2)^3}{3} + \frac{(x^2)^2}{2} + x^2 + C\) \(= \frac{x^6}{3} + \frac{x^4}{2} + x^2 + C\)

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

If a matrix \(A=\left[\begin{array}{cc}5 & -8 \\ -3 & 5\end{array}\right]\) then which of the following is / are TRUE?
(A) |A| = 1
(B) A is a singular matrix.
(C) \(-2 A=\left[\begin{array}{cc}10 & -16 \\ -6 & 10\end{array}\right]\)
(D) \(A I=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right] I\) is an identity matrix of order 2.
Choose the correct answer from the options given below:

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List-l (Statement)	List-ll (Its value)
(A) Degree of the differential equation \(\frac{d^2 y}{d x^2}=e^{d y / d x}\) is	(I) 2
(B) Order of the differential equation \(\left(\frac{d y}{d x}\right)^2+\frac{d^3 y}{d x^3}=0\) is	(II) not defined
(C) Degree of the differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2-5 x^2=0\)	(III) 3
(D) If \(p\) is the order and \(q\) is the degree of the differential equation \(\frac{d y}{d x}+3 y=e^x\), then \(p+q\) is	(IV) 1

Choose the correct answer from the options given below :

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LIST I	LIST II
A. If \(4 \sin ^{-1} x+\cos ^{-1} x=\pi\), then \(x\) equal to	I. \(\frac{\pi}{2}\)
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