CUET PG MCA Definite Integration Previous Year Question Paper and Solution with PDF

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CUET PG MCA Previous Year Questions (PYQs)

CUET PG MCA Definite Integration PYQ

Qus : 1 CUET PG MCA PYQ

List I		List II
A.	Dog : Rabies :: Mosquito :	I.	Bacteria
B.	Amnesia : Memory :: Paralysis :	II.	Liver
C.	Meningitis : Brain :: Cirrhosis :	III.	Movement
D.	Influenza : Virus :: Typhoid :	IV.	Malaria

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CUET PG MCA Previous Year PYQCUET PG MCA CUET 2023 PYQ

Solution

Dog causes Rabies, Mosquito causes Malaria → A-IV

Amnesia affects Memory, Paralysis affects Movement → B-III

Meningitis affects Brain, Cirrhosis affects Liver → C-II

Influenza is caused by Virus, Typhoid is caused by Bacteria → D-I

Correct matching: A-IV, B-III, C-II, D-I

Qus : 2 CUET PG MCA PYQ

Given below are two statements:

Statement I:

$\displaystyle \int_{-a}^{a} f(x),dx = \int_{0}^{a} [f(x)+f(-x)],dx$

Statement II:

$\displaystyle \int_{0}^{1} \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$

In the light of the above statements, choose the most appropriate answer from the options given below:

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CUET PG MCA Previous Year PYQ CUET PG MCA CUET 2023 PYQ

Solution

Statement I:

This is a standard property of definite integrals.

So Statement I is true.

Statement II:

Using AM ≥ GM:

$(1+x)(1+x^3) \le \left(\dfrac{(1+x)+(1+x^3)}{2}\right)^2$

So,

$\sqrt{(1+x)(1+x^3)} \le \dfrac{2 + x + x^3}{2}$

Integrating from $0$ to $1$:

$\displaystyle \int_0^1 \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$

Statement II is true.

Qus : 3 CUET PG MCA PYQ

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R. Assertion A: $\displaystyle \int_{-3}^{3} (x^3+5),dx = 30$ Reason R: $f(x)=x^3+5$ is an odd function. In the light of the above statements, choose the correct answer from the options given below:

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CUET PG MCA Previous Year PYQ CUET PG MCA CUET 2023 PYQ

Solution

$\displaystyle \int_{-3}^{3} x^3,dx = 0$ (odd function over symmetric limits) $\displaystyle \int_{-3}^{3} 5,dx = 5 \times 6 = 30$ So, $\displaystyle \int_{-3}^{3} (x^3+5),dx = 30$ ⇒ Assertion A is true. But $x^3+5$ is not an odd function (sum of odd and even function). So Reason R is false.

Qus : 4 CUET PG MCA PYQ

If $f(a+b-x)=f(x)$ then $\int ^b_axf(x)dx$ is equal to

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CUET PG MCA Previous Year PYQ CUET PG MCA CUET 2024 PYQ

Solution

We are asked to evaluate

$\displaystyle I = \int_a^b x f(x)\, dx \quad \text{given } f(a+b-x)=f(x).$

Step 1: Put substitution $t=a+b-x$. Then $dx=-dt$.

When $x=a \Rightarrow t=b$, when $x=b \Rightarrow t=a$.

So, \[ I = \int_a^b x f(x)\, dx = \int_b^a (a+b-t) f(t)(-dt) = \int_a^b (a+b-t) f(t)\, dt. \]

Step 2: Add both forms of $I$:

\[ 2I = \int_a^b [x f(x) + (a+b-x) f(x)] dx = \int_a^b (a+b) f(x)\, dx. \]

Step 3: Simplify:

\[ I = \frac{a+b}{2} \int_a^b f(x)\, dx. \]

Final Answer: $\displaystyle \frac{a+b}{2}\int_a^b f(x)\, dx$ → matches Option 1.

Qus : 5 CUET PG MCA PYQ

Match List – I with List – II

List - I	List - II
(A) $$\int ^{\pi/2}_0\frac{{\sin }^4x}{{\sin }^4x+{\cos }^4x}dx$$	(I) 0
(B) $$\int ^{\pi/3}_{\pi/6}\frac{1}{1+\sqrt[]{\tan x}}dx$$	(II) 0
(C) $$\int ^1_0x{e}^xdx$$	(III) $\frac{\pi}{12}$
(D) $$\int ^1_{-1}{x}^{109}{\cos }^{88}xdx$$	(IV) $\frac{\pi}{4}$

Choose the correct answer from the options given below:

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CUET PG MCA Previous Year PYQ CUET PG MCA CUET 2024 PYQ

Solution

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