Algebra

Relations & Functions

Medium

core

Consider a function $f: \left[0, \dfrac{\pi}{2}\right] \to R$ given by $f(x) = \sin x$ and $g: \left[0, \dfrac{\pi}{2}\right] \to R$ given by $g(x) = \cos x$ , where $R$ is set of real numbers then which of the following are correct?

A. $f$ is one-one

B. $g$ is one-one

C. $f + g$ is one-one

D. $f + g$ is not one-one

Choose the correct answer from the options given below:

A, B and C only

A, B and D only

A and B only

C only

✅ Correct Option: 2

On $\left[0,\frac{\pi}{2}\right]$ , $\sin x$ is strictly increasing and $\cos x$ strictly decreasing, so both are one-one, making A and B true. However $f+g=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)$ takes the value $1$ at both $x=0$ and $x=\frac{\pi}{2}$ , so it is not one-one: D is true and C false.

2025: 15 May Shift 2

A relation R in the set A = {1, 2,3, 4} is given by R = {(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)} is

2026: 19 May Shift 2

Match the LIST-I with LIST-II

[Where R and Z denotes set of real numbers and set of integers respectively]

	LIST-I Function		LIST-II Nature
A.	$f: R \to R, f(x) = \sin^2 x + \cos^2 x$	I.	Bijective
B.	$f: Z \to Z, f(x) = x^3$	II.	Surjective but not injective
C.	$f: R \to R, f(x) = x^3 - x$	III.	Injective but not surjective
D.	$f: R \to R, f(x) = 3 - 4x$	IV.	Neither injective nor surjective

Choose the correct answer from the options given below:

2022: 6 Aug Shift 2

If a relation R is defined on the set $X = \{1, 2, 3, 4\}$ as $R = \{(1,1), (2,2), (3,4), (4,3)\}$ , then R is

2022: 30 Aug Shift 1

Which of the following is true on the basis of above diagram?

2022: 4 Aug Shift 1

A function f: R $\to$ R is given by $f(x) = x^3 + 3$ . If $f(x) = -24$ , then the value of x is:

2022: 25 May Shift 1

The range of the function $f(x) = \frac{1}{3 - \sin 4x}$ is:

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)