CUET UG 2025 — Mathematics Statistics & Applications

2025mcqmedium

Which of the following are correct?

(A) A statistical hypothesis is a statement about the population parameter which may be true or false.

(B) In a statistical hypothesis testing procedure, a Type I error is made when the null hypothesis is accepted when it is false.

(C) In a statistical hypothesis testing procedure, a Type II error is made when the null hypothesis is rejected when it is true.

(D) The number of degrees of freedom of a statistic is the number of independent variates used to compute that statistic.

Choose the correct answer from the options given below:

Official previous-year question

Held on 16 May 2025 · Verified 13 Jul 2026.

Options

A
(A), (B) and (D) only
B
(A) and (D) only
C
(A), (B), (C) and (D)
D
(B) and (C) only

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Solution

The given differential equation is $\frac{dy}{dx} + y\tan x = \sec x$

This is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x) \cdot y = Q(x)$ where $P(x) = \tan x$ and $Q(x) = \sec x$ .

The integrating factor is $\displaystyle IF = e^{\int P(x)dx} = e^{\int \tan x dx}$

$\int \tan x \, dx = \int \frac{\sin x}{\cos x} dx$

Let $u = \cos x$ , then $du = -\sin x \, dx$

$= -\int \frac{1}{u} du$

$= -\ln|u|$

$= -\ln|\cos x|$

$= \ln|\sec x|$

Therefore $IF = e^{\ln|\sec x|} = \sec x$

Multiplying the entire equation by $\sec x$ :

$\sec x \cdot \frac{dy}{dx} + y \sec x \tan x = \sec^2 x$

The left side is the derivative of $(y \sec x)$ by the product rule.

$\frac{d}{dx}(y\sec x) = \sec^2 x$

Integrating both sides:

$\int \frac{d}{dx}(y\sec x) \, dx = \int \sec^2 x \, dx$

$y\sec x = \tan x + c$

$y\sec x - \tan x = c$

where $c$ is an arbitrary constant.

Your note

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Mr. X wishes to purchase a flat for Rs. 44,65,000 with a down payment of Rs. 10,00,000 and balance in equated monthly installments (EMI) for 25 years. If the bank charges 6 % per annum compounded monthly, the EMI is: [Given: $(1.005)^{300} =4.4650$]

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Solution

The given differential equation is $\frac{dy}{dx} + y\tan x = \sec x$

This is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x) \cdot y = Q(x)$ where $P(x) = \tan x$ and $Q(x) = \sec x$ .

The integrating factor is $\displaystyle IF = e^{\int P(x)dx} = e^{\int \tan x dx}$

$\int \tan x \, dx = \int \frac{\sin x}{\cos x} dx$

Let $u = \cos x$ , then $du = -\sin x \, dx$

$= -\int \frac{1}{u} du$

$= -\ln|u|$

$= -\ln|\cos x|$

$= \ln|\sec x|$

Therefore $IF = e^{\ln|\sec x|} = \sec x$

Multiplying the entire equation by $\sec x$ :

$\sec x \cdot \frac{dy}{dx} + y \sec x \tan x = \sec^2 x$

The left side is the derivative of $(y \sec x)$ by the product rule.

$\frac{d}{dx}(y\sec x) = \sec^2 x$

Integrating both sides:

$\int \frac{d}{dx}(y\sec x) \, dx = \int \sec^2 x \, dx$

$y\sec x = \tan x + c$

$y\sec x - \tan x = c$

where $c$ is an arbitrary constant.

CUET UG 2025 — Mathematics Statistics & Applications

Options

Did you get this right?

Solution

Your note

CUET UG Statistics & Applications in other years

More CUET UG Statistics & Applications questions

Other Mathematics topics for CUET UG

Algebra

Calculus

Applied-Mathematics

Geometry

Arithmetic

Coordinate Geometry

Keep practicing Statistics & Applications

CUET UG 2025 — Mathematics Statistics & Applications

Options

Did you get this right?

Solution

Your note

CUET UG Statistics & Applications in other years

More CUET UG Statistics & Applications questions

Other Mathematics topics for CUET UG

Algebra

Calculus

Applied-Mathematics

Geometry

Arithmetic

Coordinate Geometry

Keep practicing Statistics & Applications