Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is...
CBSE CLASS 12 • MATHEMATICS • CHAPTER 01
Relations and Functions
298 distinct questions across 404 board paper set appearances (2022–2026).
The function $f(x) = \frac{x}{2} + \frac{2}{x}$ has a local minima at $x$ equal to : (A) $2$ (B) $1$ (C) $0$ (D) $-2$
(a) Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let L be the set of all rail lines on the railway track and R be the relation on L defined by R = {(l_1, l...
(a) Verify whether the function f defined by
$f(x) = \begin{cases} x \sin\left$ \frac{1}{x} \right$, & x \neq 0 \\ 0, & x = 0 \end{cases}$
is continuous at $x = 0$ or not.
OR
(b) Check for differentiability of the function f defined by $f(x) = |x - 5|$, at the point $x = 5$.
A function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = x^{2} - 4x + 5$ is : (A) injective but not surjective. (B) surjective but not injective. (C) both injective and surjective. (D) neither injective nor surjective.
(a) Find the absolute maximum value of $f(x) = \cos x + \sin^2 x$, $x \in [0, \pi]$
OR
(b) If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.
The value of k for which the function $f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & x \neq 0 \\ k(x + 1), & x = 0 \end{cases}$ is a continuous function, is :
(A) $\frac{1}{4}$
(B) $2$
(C) $\frac{1}{2}$
(D) $0$
If f(x) = $\begin{cases} \frac{\sin x}{x} + \cos x, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at x = 0, then the value of k is : (A) 0 (B) - 2 (C) -1 (D) 2
The greatest integer function, f(x) = [x], 0 < x < 3 is not differentiable at how many points ? (A) At only one point (B) At only two points (C) At no point (D) At three points
Assertion (A) : A function f : N $\rightarrow$ N given by f(x) = $x^3$ + 2, $\forall$ x $\in$ N is one-one but not onto.
Reason (R) : Since $\forall$ y $\in$ N (Codomain), there does not exist x = (y - 2)$^{1/3}$ in N (Domain) such that f(x) = $x^3$ + 2 = y.
If $f(x) = \begin{cases} \frac{x^{2} - 4x - 5}{x + 1}, & x \neq -1 \\ k, & x = -1 \end{cases}$ is continuous at $x = -1$, then the value of $k$ is :
(A) Any real value
(B) $6$
(C) $-1$
(D) $-6$
Absolute minimum value of f(x) = $(x - 2)^2 + 5$ in the interval $[-3, 2]$ is : (A) $-3$ (B) 2 (C) 5 (D) 30
Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation R is defined on the set A of thumb pins as
$$R = \{(P, Q) : P, Q \in A \text{ and distance between } P \text{ and } Q \text{ is less than } 3\text{ cm}\}$$
Determine if $R$ is an equivalence relation.
(a) If $R$ is a relation on set of natural numbers $\mathbb{N}$ such that
$R = \{(x, y) : x, y \in \mathbb{N}, x + 2y = 25\}$,
then find the Domain and Range of the relation $R$. Also, determine if it is an equivalence relation.
OR
(b) Determine whether the function $f : \mathbb{R} \to \mathbb{R}$ d...
During the festival season, there was a mela organized by the Resident Welfare Association at a park, near the society. The main attraction of the mela was a huge swing installed at one corner of the park. The swing is traced to follow the path of a parabola given by $x^2 = y$.
Based on the above in...
The absolute maximum value of function $f(x) = x^3 - 3x + 2$ in $[0, 2]$ is :
(A) 0
(B) 2
(C) 4
(D) 5
(a) If $f : R^+ \to R$ is defined as $f(x) = \log_a x$ ($a > 0$ and $a \neq 1$), prove that $f$ is a bijection. ($R^+$ is a set of all positive real numbers.) OR (b) Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. A relation $R$ from $A$ to $B$ is defined as $R = \{(x, y) : x + y = 6, x \in A, y \in B\...
If f(x) = $\begin{cases} \frac{\sin^2 ax}{x^2}, & x \neq 0 \\ 1, & x = 0 \end{cases}$
is continuous at x = 0, then the value of a is :
(A) 1
(B) -1
(C) $\pm 1$
(D) 0
If f(x) = \begin{cases} \frac{\log (1 + ax) + \log (1 - bx)}{x}, & \text{for } x \neq 0 \\ k & , \text{for } x = 0 \end{cases} is continuous at x = 0, then the value of k is : (A) a (B) a + b (C) a - b (D) b
(a) A student wants to pair up natural numbers in such a way that they satisfy the equation $2x + y = 41$, $x, y \in \text{N}$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or n...
Domain of $f(x) = \cos^{-1} x + \sin x$ is : (A) $R$ (B) $(-1, 1)$ (C) $[-1, 1]$ (D) $\phi$
The function $f(x) = kx - \sin x$ is strictly increasing for
(A) $k > 1$
(B) $k < 1$
(C) $k > -1$
(D) $k < -1$
The value of k for which function $f(x) = \begin{cases} x^2, & x \geq 0 \\ kx, & x < 0 \end{cases}$ is differentiable at $x = 0$ is :
(a) $1$
(b) $2$
(c) any real number
(d) $0$
A and B are skew-symmetric matrices of same order. AB is symmetric, if :
(a) AB = O
(b) AB = - BA
(c) AB = BA
(d) BA = O
An organization conducted bike race under two different categories - Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college proje...
(a) Show that the relation $S$ in set $\mathbb{R}$ of real numbers defined by $S = \{(a, b) : a \leq b^3, a \in \mathbb{R}, b \in \mathbb{R}\}$ is neither reflexive, nor symmetric, nor transitive.
OR
(b) Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b) : \tex...
(a) Check whether $f: \mathbb{R} - \{3\} \to \mathbb{R}$ defined as $f(x) = \frac{x - 2}{x - 3}$ is onto or not.
OR
(b) Check whether $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ (where $\mathbb{Z}$ is the set of integers) defined as $f(x, y) = (2y, 3x)$ is injective or not.
Find the sub-interval of $(0, \infty)$ in which $f(x) = x^2 e^{-x}$ is increasing.
(a) Show that the function $f(x) = \begin{cases} \frac{\cos x}{-x+\frac{\pi}{2}}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$. OR (b) Find whether the function $f(x) = \begin{cases} x-1, & x < 2 \\ 2x-3, & x \geq 2 \end{cases}$ at $x = 2$ is dif...
(a) Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. A function $f : A \rightarrow B$ is defined by $f(x) = \left$ \frac{x - 2}{x - 3} \right$$. Find whether $f$ is one-one and onto.
OR
(b) Let $n$ be a fixed positive integer. A relation $R$ is defined in set $Z$ such that $R = \{(x, y) ...
Assertion $(A) :$ A relation $R$ on the set $\{1, 2, 3\}$ defined as $R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\}$ is an equivalence relation.
Reason $(R) :$ A relation that is reflexive, symmetric and transitive is an equivalence relation.
A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition.
7 students out of 80 are selected for the project and are categorized into two sets such that :
Girl ...
For $f(x) = -|x + 1| + 5$, which of the following statements is true ?
(A) Max $f(x) = 0$ at $x = 4$.
(B) Max $f(x) = 5$ at $x = -1$.
(C) Max $f(x) = 4$ at $x = 0$.
(D) $f(x)$ has no maximum value.
The domain of $f(x) = \cos^{-1} (2x)$ is :
(A) $[-1, 1]$
(B) $\left[0, \frac{1}{2}\right]$
(C) $[-2, 2]$
(D) $\left[-\frac{1}{2}, \frac{1}{2}\right]$
Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$, defined as $f(x) = x^3$.
Assertion (A) : $f(x)$ is a one-one function.
Reason (R) : $f(x)$ is a one-one function, if co-domain = range.
**Assertion (A):** $f(x) = \begin{cases} 3x - 8, & x \le 5 \\ 2k, & x > 5 \end{cases}$ is continuous at $x = 5$ for $k = \frac{5}{2}$.
**Reason (R):** For a function f to be continuous at $x = a$, $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$.
A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set A = {1, 2, 3} :
R_1 = {(2, 3), (3, 2)}
R_2 = {(1, 2), (1, 3), (3, 2)}
R_3 = {(1, 2), (2, 1), (1, 1)}
R_4 = {(1, 1), (1, 2), (...
The function $f(x) = x^2 - 4x + 6$ is increasing in the interval
(A) $(0, 2)$
(B) $(-\infty, 2]$
(C) $[1, 2]$
(D) $[2, \infty)$
A school is organizing a debate competition with participants as speakers $S = \{S_1, S_2, S_3, S_4\}$ and these are judged by judges $J = \{J_1, J_2, J_3\}$. Each speaker can be assigned one judge. Let $R$ be a relation from set $S$ to $J$ defined as $R = \{(x, y) : \text{speaker } x \text{ is judg...
If $f(x) = \begin{cases} 3x - 2, & 0 < x \leq 1 \\ 2x^2 + ax, & 1 < x < 2 \end{cases}$ is continuous for $x \in (0, 2)$, then $a$ is equal to: (A) $-4$ (B) $-\frac{7}{2}$ (C) $-2$ (D) $-1$
(a) Find $k$ so that $f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1}, & x \neq -1 \\ k, & x = -1 \end{cases}$ is continuous at $x = -1$. OR (b) Check the differentiability of function $f(x) = x|x|$ at $x = 0$.
The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :
(A) $1 < \lambda < \sqrt{2}$
(B) $\lambda \ge 1$
(C) $\lambda \ge \sqrt{2}$
(D) $\lambda < 1$
The function f defined by f(x) = $\begin{cases} x, & \text{if x } \le 1 \\ 5, & \text{if x } > 1 \end{cases}$ is not continuous at : (A) x = 0 (B) x = 1 (C) x = 2 (D) x = 5
(a) Find the least value of 'a' so that $f(x) = 2x^2 - ax + 3$ is an increasing function on $[2, 4]$. OR (b) If $f(x) = x + \frac{1}{x}, x \ge 1$, show that f is an increasing function.
Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where R = {(x, y) : x, y are Roll Numbers of students such that y = 3x}. List the elements of R. Is the relation R reflexive, symmetric and transitive ? Justify your answer.
Assertion (A) : Let $f(x) = e^x$ and $g(x) = \log x$. Then $(f + g) x = e^x + \log x$ where domain of $(f + g)$ is $R$.
Reason (R) : $ ext{Dom}(f + g) = ext{Dom}(f) \cap ext{Dom}(g)$.
Find the domain of $f(x) = \sin^{-1} (-x^2)$.
Let $f : A \to B$ be defined by $f(x) = \frac{x - 2}{x - 3}$, where $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Discuss the bijectivity of the function.
If $f(x) = -2x^8$, then the correct statement is :
(A) $f'\left(\frac{1}{2}\right) = f'\left$-\frac{1}{2}\right$$
(B) $f'\left(\frac{1}{2}\right) = -f'\left$-\frac{1}{2}\right$$
(C) $-f'\left(\frac{1}{2}\right) = f\left$-\frac{1}{2}\right$$
(D) $f\left(\frac{1}{2}\right) = -f\left$-\frac{1}{2}\right...
(b) Find the domain of the function $f(x) = \sin^{-1}(x^2 - 4)$. Also, find its range.
Show that $f(x) = e^x - e^{-x} + x - \tan^{-1} x$ is strictly increasing in its domain.
(a) Show that a function $f: R \to R$ defined by $f(x) = \frac{2x}{1+x^2}$ is neither one-one nor onto. Further, find set A so that the given function $f: R \to A$ becomes an onto function. OR (b) A relation R is defined on $N \times N$ (where $N$ is the set of natural numbers) as : $(a, b) R (c, d)...
The number of points of discontinuity of $f(x) = \begin{cases} |x|+3, & \text{if } x \leq -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x+2, & \text{if } x \geq 3 \end{cases}$ is :
(A) 0
(B) 1
(C) 2
(D) infinite
(a) Find the intervals in which the function $f(x) = \frac{\log x}{x}$ is strictly increasing or strictly decreasing.
OR
(b) Find the absolute maximum and absolute minimum values of the function $f$ given by $f(x) = \frac{x}{2} + \frac{2}{x}$, on the interval $[1, 2]$.
Assertion (A) : The relation $R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in \mathbb{N}\}$ is not a reflexive relation.
Reason (R) : The number '2n' is composite for all natural numbers n.
Check whether the relation $S$ in the set of real numbers $R$ defined by $S = \{(a, b) : \text{where } a - b + \sqrt{2} \text{ is an irrational number}\}$ is reflexive, symmetric or transitive.
Find the values of a and b so that the following function is differentiable for all values of $x$ : $f(x) = \begin{cases} ax + b , & x > -1 \\ bx^2 - 3 , & x \leq -1 \end{cases}$
OR (b) Let S be the relation defined by S = {(l_1, l_2) : l_1 is perpendicular to l_2} check whether the relation S is symmetric and transitive.
The greatest integer function defined by $f(x) = [x]$, $1 < x < 3$ is not differentiable at $x =$ (A) 0 (B) 1 (C) 2 (D) $\frac{3}{2}$
Given that $f(x) = \frac{\log x}{x}$, find the point of local maximum of f(x).
If $f(x) = a(x - \cos x)$ is strictly decreasing in $\mathbb{R}$, then 'a' belongs to
(a) $\{0\}$
(b) $(0, \infty)$
(c) $(-\infty, 0)$
(d) $(-\infty, \infty)$
(a) A function $f : A \to B$ defined as $f(x) = 2x$ is both one-one and onto. If $A = \{1, 2, 3, 4\}$, then find the set B. OR (b) Evaluate : $\sin^{-1} \left$\sin \frac{3\pi}{4}\right$ + \cos^{-1} \left$\cos \frac{3\pi}{4}\right$ + \tan^{-1} (1)$
A function f: [-4, 4] $\rightarrow$ [0, 4] is given by f(x) = $\sqrt{16 - x^2}$. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = $\sqrt{7}$.
The function $f(x) = [x]$, where $[x]$ denotes the greatest integer less than or equal to $x$, is continuous at
(a) $x = 1$
(b) $x = 1.5$
(c) $x = -2$
(d) $x = 4$
The interval in which the function $f(x) = 2x^3 + 9x^2 + 12x - 1$ is decreasing, is
(a) $(-1, \infty)$
(b) $(-2, -1)$
(c) $(-\infty, -2)$
(d) $[-1, 1]$
(a) Simplify :
$$\tan^{-1}\left(\frac{\cos x}{1 - \sin x}\right)$$
OR
(b) Prove that the greatest integer function $f: \mathbb{R} \to \mathbb{R}$, given by $f(x) = [x]$, is neither one-one nor onto.
In an agricultural institute, scientists do experiments with varieties of seeds to grow them in different environments to produce healthy plants and get more yield. A scientist observed that a particular seed grew very fast after germination. He had recorded growth of plant since germination and he ...
If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?
(A) $AB$
(B) $AB + BA$
(C) $(A + B)^2$
(D) $A - B$
The least value of $f(x) = x^3 - 12x$, $x \in [0, 3]$ is
(A) $-16$
(B) $-9$
(C) $0$
(D) $16$
(a) A relation R is defined on $\mathbb{Z}$, the set of integers, as
$R = \{(x, y) : |x - y| \text{ is divisible by a prime number } p', x, y \in \mathbb{Z}\}$
check whether R is an equivalence relation or not.
OR
(b) A function $f : \text{R} - \left\{\\frac{3}{5}\right\} \longrightarrow \text{R} - ...
The absolute maximum value of $f(x) = x^2 + 1$ in $[-5, 2]$ is
(A) $26$
(B) $1$
(C) $5$
(D) $2$
(a) A relation $\mathrm{R}$ is defined on $\mathbb{Z}$, the set of integers, as
$\mathrm{R} = \{(x, y) : |x-y| \text{ is divisible by a prime number 'p', } x, y \in \mathbb{Z}\}$
check whether $\mathrm{R}$ is an equivalence relation or not.
OR
(b) A function $f : \mathbb{R} - \left\{\frac{3}{5}\ri...
The least value of $f(x) = e^{-x}$ in $[0, 3]$ is
(A) $e^{-3}$
(B) $-1$
(C) $1$
(D) $-e^3$
(a) Check whether function $f(x)$ defined as
$f(x) = \begin{cases} \frac{|x - 3|}{2(x - 3)}, & x < 3 \\ \frac{x - 6}{6}, & x \ge 3 \end{cases}$ is continuous at $x = 3$ or not ?
OR
(b) If $\sqrt{3}(x^2 + y^2) = 4xy$, then find $\frac{\text{d}y}{\text{d}x}$ at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2...
(a) A relation R is defined on $\mathbb{Z}$, the set of integers, as<br>$\text{R} = \{(x, y) : |x - y| \text{ is divisible by a prime number 'p', } x, y \in \mathbb{Z}\}$<br>check whether R is an equivalence relation or not.<br>OR<br>(b) A function $f : \text{R} - \left\{\frac{3}{5}\right\} \longrig...
A relation $R$ on set $A = \{1, 2, 3\}$ defined as $R = \{(1, 1), (2, 2), (1, 2)\}$ is
(A) Reflexive only
(B) Reflexive and Transitive
(C) Symmetric and Transitive
(D) Transitive only
(a) Find the sub intervals in which $f(x) = \cot^{-1}(\sin x + \cos x)$, $x \in (0, \pi)$ is increasing and decreasing.
OR
(b) A rectangle of perimeter $36\text{ cm}$ is revolved around one of its sides to sweep out a cylinder of maximum volume.
Find the dimensions of the rectangle.
Find the domain of $g(x) = \cos^{-1}(x^2 - 1)$. Hence, find the value of $x$ for which $g(x) = \frac{\pi}{3}$.
Also, write the range of $\cos^{-1}x$ other than its principal branch.
Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.
The equation of one such track is given as follows :
$f(x) = \begin{cases} x^4 - 4x^2 + 4, & 0 \le x < 3 \\ x^2 + 40, & x \ge 3 \end{cases}$
Based on gi...
A relation R on set A = \{1, 2, 3\} defined as R = \{(1, 2), (2, 1), (2, 2)\} is
(A) Reflexive only
(B) Reflexive and Transitive
(C) Symmetric and Transitive
(D) Symmetric only
(a) Check whether $\text{f} : \text{R} - \{3\} \to \text{R}$ defined as $\text{f}(x) = \frac{x - 2}{x - 3}$ is onto or not.
(b) Check whether $\text{f} : \text{Z} \times \text{Z} \to \text{Z} \times \text{Z}$ (where $\text{Z}$ is the set of integers) defined as $\text{f}(x, y) = (2y, 3x)$ is injective or not.
(a) Find the sub-interval of $(0, \pi)$ in which $f(x) = \tan^{-1}(\sin x - \cos x)$ is increasing and decreasing.
A relation R on set $ ext{A} = \{1, 2, 3\}$ is defined as $\text{R} = \{(1, 3), (3, 3), (1, 1), (2, 2), (3, 1)\}$ is
(A) only reflexive and symmetric
(B) reflexive only
(C) only reflexive and transitive
(D) reflexive, symmetric and transitive
Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = x^3$, then $f$ is :
(A) one-one onto function
(B) many-one function
(C) one-one but not onto function
(D) neither one-one nor onto function
The value of k for which $f(x) = \begin{cases} \frac{\sin 7x}{5x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$ is :
(A) $\frac{5}{7}$
(B) $\frac{7}{5}$
(C) $0$
(D) $\frac{1}{5}$
Assertion (A) : f(x) = e^{-x} is a decreasing function, where x \in \mathbb{R}. \n\n Reason (R) : If f'(x) < 0, then f(x) is a decreasing function.
Two friends Ramesh and Divya have started playing Ludo at their house. While rolling the die, they observed each time, that the possible outcomes are $\{1, 2, 3, 4, 5, 6\}$. Let A be the set of players and B be the set of all possible outcomes, so $A = \{ ext{Ramesh, Divya}\}$ and $B = \{1, 2, 3, 4,...
The domain of $f(x) = \cos^{-1}(2x - 5)$ is :
(A) $[-1, 1]$
(B) $[4, 6]$
(C) $[-7, -3]$
(D) $[2, 3]$
Find the sub-interval(s) of $\left(0, \frac{\pi}{2}\right)$ in which $f(x) = \tan x - 4x$ is increasing.
A relation $R$ on $A = \{1, 2, 3\}$ is defined as $R = \{(1, 1), (3, 3), (1, 2)\}$. Is $R$ a symmetric relation ? Justify. Write the smallest relation set $R_1$ such that $R \cup R_1$ becomes an equivalence relation on the set $\{1, 2, 3\}$.
Show that the function $f : \text{R} \rightarrow \text{R}$ given by $f(x) = \frac{x}{\sqrt{1 + x^2}}$ is one-one, but not onto.
Show that $f : R \to R$ defined as $f(x) = \frac{x}{\sqrt{1 + x^2}}$ is one-one but not onto.
(a) Find the value of $\sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))]$.\nOR\n(b) A relation $\text{R}$ on $\text{A} = \{1, 2, 3\}$ is defined as $\text{R} = \{(1, 1), (3, 3), (1, 2)\}$. Is $\text{R}$ a symmetric relation ? Justify. Write the smallest relation set $\text{R}_1$ such that $\text{R} \cu...
Show that $f : R_+ \to [-5, \infty)$ given by $f(x) = 4x^2 + 4x - 5$ is both one-one and onto where $R_+ = [0, \infty)$. Also, find $p \in R_+$ such that $f(p) = 3$.
Find the sub-interval of $(0, \infty)$ in which $f(x) = x \log x$ is increasing.
(a) Find the value of $\sin[\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))]$. OR (b) A relation R on A = {1, 2, 3} is defined as R = {(1, 1) (3, 3), (1, 2)}. Is R a symmetric relation ? Justify. Write the smallest relation set $R_1$ such that $R \cup R_1$ becomes an equivalence relation on the set {1, 2, 3...
Show that a function $f : R_+ \rightarrow A \subset R$, defined as $f(x) = 4x^2 + 12x + 15$ is one-one. Find set $A$ so that $f$ is onto where $R_+ = [0, \infty)$. Also, find if there exists $a \in R_+$ such that $f(a) = 7$. Justify.
Find the values of x for which $f(x) = x^x, x > 0$ is increasing.
Find the interval(s) in which the function $f(x) = \frac{x}{\log x}$, where $x \in (0, 1) \cup (1, \infty)$, is increasing.
(a) Show that the function $f(x) = \begin{cases} \frac{\cos x}{-x+\frac{\pi}{2}}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.
OR (b) Find whether the function $f(x) = \begin{cases} x - 1, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$ at $x = 2$ is differentiable or not.
Find the interval(s) for which the function $f(x) = \frac{x}{3} + \frac{3}{x}, x \neq 0$ is increasing.
(a) Let $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. A function $f : A \to B$ is defined by $f(x) = \left$ \frac{x - 2}{x - 3} \right$$. Find whether $f$ is one-one and onto. OR (b) Let $n$ be a fixed positive integer. A relation $R$ is defined in set $\mathbb{Z}$ such that $R = \{(x, y) :...
Determine the values of $x$ for which $f(x) = \frac{x-3}{x+1}, x \neq -1$ is an increasing function.
Let R be a relation from A $\rightarrow$ B such that R = {(G$_1$, B$_1$), (G$_2$, B$_2$), (G$_3$, B$_2$), (G$_4$, B$_3$), (G$_1$, B$_2$)}. Is R an injective function ? Justify your answer.
Let the relation R from A $\rightarrow$ A be such that R = {(x, y), x, y $\in$ A, x and y are students from the same colony in the city} Verify if R is an equivalence relation.
Verify if any function f : B $\rightarrow$ A is bijective. Give reason to support your answer.
Determine the interval(s) in which $f(x) = 5x^{3/2} - 3x^{5/2}$, $x > 0$ is increasing.
Determine the interval(s) in which $f(x) = \frac{x}{3} + \frac{3}{x}, x \neq 0$ is increasing.
(a) Find the intervals in which the function $f(x) = 1 - 12x - 9x^2 - 2x^3$ is strictly increasing or decreasing.
OR
(b) Find the maximum and minimum values of the function $f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the interval $[0, 3]$.
Consider the set $A = \mathbb{N} \times \mathbb{N}$ as the set of all ordered pairs of natural numbers. Let R be a relation in A defined by $(a, b) \text{ R } (c, d)$, iff $ad = bc$, $\forall \text{ a, b, c, d} \in \mathbb{N}$. Prove that R is an equivalence relation.
Let $A = \mathbb{R} - \{2\}$ and $B = \mathbb{R} - \{1\}$. If $f : A \to B$ is a function defined by $f(x) = \frac{x - 1}{x - 2}$, then show that $f$ is one-one and onto.
For $f(x) = -|x + 1| + 5$, which of the following statements is true ?
(A) $\text{Max } f(x) = 0 \text{ at } x = 4$.
(B) $\text{Max } f(x) = 5 \text{ at } x = -1$.
(C) $\text{Max } f(x) = 4 \text{ at } x = 0$.
(D) $f(x)$ has no maximum value.
(a) Determine $a$ and $b$ if $f(x)$ is a continuous function, where
$$f(x) = \begin{cases} 2, & x \leq 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \geq 5 \end{cases}$$
OR
(b) If $2y^3 + y = 2x$, then show $\frac{\text{d}^2y}{\text{dx}^2} = \frac{-48y}{(6y^2 + 1)^3}$.
Assertion (A) : Let R be a relation on set A of all books in a school library given by $\text{R} = \{(x, y) : x, y \in \text{A}, x \text{ and } y \text{ have the same number of pages}\}$ and S is a relation on set A from the same library such that $\text{S} = \{(p, q) : p \text{ and } q \text{ are M...
(a) Determine a and b if $f(x)$ is a continuous function, where
$$f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases}$$
OR
(b) If $2y^3 + y = 2x$, then show $\frac{\text{d}^2y}{\text{dx}^2} = \frac{-48y}{(6y^2 + 1)^3}$.
Find the sum of maximum and minimum values of $f(x) = \sin x (1 + \cos x)$ in the interval $[0, \pi]$.
(a) Determine $a$ and $b$ if $f(x)$ is a continuous function, where \\[0.5em] $f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases}$ \\[0.5em] \textbf{OR} \\[0.5em] (b) If $2y^3 + y = 2x$, then show $\frac{\text{d}^2y}{\text{d}x^2} = \frac{-48y}{(6y^2 + 1)^3}$.
Find the product of the maximum and minimum values of the function $f(x) = \cos x - \frac{1}{2} \cos 2x$, $x \in [0, \frac{\pi}{2}]$.
If a function defined by $f(x) = \begin{cases} kx + 1, & x \le \pi \\ \cos x &, x > \pi \end{cases}$ is continuous at $x = \pi$, then the value of $k$ is\n(A) $\pi$\n(B) $\frac{-1}{\pi}$\n(C) $0$\n(D) $\frac{-2}{\pi}$
If $f(x) = x \tan^{-1} x$, then $f^{\prime}(1)$ is equal to\n(A) $\frac{\pi}{4} - \frac{1}{2}$\n(B) $\frac{\pi}{4} + \frac{1}{2}$\n(C) $-\frac{\pi}{4} - \frac{1}{2}$\n(D) $-\frac{\pi}{4} + \frac{1}{2}$
A function $f(x) = 10 - x - 2x^2$ is increasing on the interval\n(A) $\left(-\infty, -\frac{1}{4}\right)$\n(B) $\left(-\infty, \frac{1}{4}\right)$\n(C) $\left[-\frac{1}{4}, \infty\right)$\n(D) $\left[-\frac{1}{4}, \frac{1}{4}\right]$
Case Study -1
A city's traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled $A, B, C, D,$ and $E$.
The department has collected the following data regarding one-way traffic flow betw...
The function $f(x) = a^x$ is increasing on $\text{R}$, if :
(A) $a > 0$
(B) $a > 1$
(C) $a < 0$
(D) $0 < a < 1$
A function $f : \text{R} \rightarrow \text{R}$ is defined as $f(x) = x^3 + 1$. The function $f$ has :
(A) no maximum value
(B) no minimum value
(C) both maximum and minimum values
(D) neither maximum nor minimum value
Fill in the blanks :
(a) t-distribution curve is symmetrical about the line __________.
(b) The variable t of t-distribution lies between __________.
(c) The mean of the t-distribution is __________.
(d) The variance of the t-distribution is __________.
Let $A = \{a, b\}$, then the number of reflexive relations defined on $A$ is :
(A) $16$
(B) $8$
(C) $4$
(D) $2$
If $f(x) = x^2 + ax + 3$ is strictly increasing in the interval $(3, 4)$, then the minimum value of $a$ is :
(A) $-6$
(B) $-8$
(C) $6$
(D) $8$
(a) Find the value of k, so that $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 3, & \text{if } x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.
Let $A$ be the set of all positive integers and a relation $R$ on $A \times A$ is defined by $(a, b) \, R \, (c, d) \iff ad = bc$, for all $(a, b), (c, d) \in A \times A$. Show that $R$ is an equivalence relation on $A \times A$.
If $\frac{d}{dx} f(x) = 3x^2 - \frac{3}{x^4}$ such that f(1) = 0, then f(x) is :
(A) $6x + \frac{12}{x^5}$
(B) $x^4 - \frac{1}{x^3} + 2$
(C) $x^3 + \frac{1}{x^3} - 2$
(D) $x^3 + \frac{1}{x^3} + 2$
Consider the function $\text{f} : \mathbb{R} \longrightarrow \mathbb{R}$, defined as $\text{f(x)} = \text{x}^3$.
Assertion (A) : $\text{f(x)}$ is a one-one function.
Reason (R) : $\text{f(x)}$ is a one-one function, if co-domain = range.
A function $f : R_+ \to R$ (where $R_+$ is the set of all non-negative real numbers) defined by $f(x) = 4x + 3$ is :
(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto
(a) A relation $R$ on set $A = \{1, 2, 3, 4, 5\}$ is defined as $R = \{(x, y) : |x^2 - y^2| < 8\}$. Check whether the relation $R$ is reflexive, symmetric and transitive.\nOR\n(b) A function $f$ is defined from $R \to R$ as $f(x) = ax + b$, such that $f(1) = 1$ and $f(2) = 3$. Find function $f(x)$. ...
Find the intervals in which function $f(x) = 5x^{\frac{3}{2}} - 3x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.
Assertion (A) : Let Z be the set of integers. A function f : Z $\rightarrow$ Z defined as f(x) = 3x - 5, $\forall$x $\in$ Z is a bijective.
Reason (R) : A function is a bijective if it is both surjective and injective.
Assertion (A) : f(x) = $\begin{cases} 3x - 8, & x \le 5 \\ 2k, & x > 5 \end{cases}$ is continuous at x = 5 for k = $\frac{5}{2}$.
Reason (R) : For a function f to be continuous at x = a, $\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)$.
Find the values of 'a' for which f(x) = $x^2 - 2ax + b$ is an increasing function for x > 0.
Find the absolute maximum and absolute minimum of function $f(x) = 2x^3 - 15x^2 + 36x + 1$ on $[1, 5]$.
Assertion (A) : Let $\mathbb{Z}$ be the set of integers. A function $f : \mathbb{Z} \to \mathbb{Z}$ defined as $f(x) = 3x - 5, \forall x \in \mathbb{Z}$ is a bijective.
Reason (R) : A function is a bijective if it is both surjective and injective.
Find the values of '$a$' for which $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$ is decreasing on $\mathbb{R}$.
Find the values of ‘a’ for which $f(x) = \sin x - ax + b$ is increasing on R.
OR
(b) If $f(x) = \begin{cases} 2x - 3, & -3 \leq x \leq -2 \\ x + 1, & -2 < x \leq 0 \end{cases}$
Check the differentiability of $f(x)$ at $x = -2$.
Let $R$ be a relation defined over $N$, where $N$ is set of natural numbers, defined as "$mRn$ if and only if $m$ is a multiple of $n$, $m, n \in N$." Find whether $R$ is reflexive, symmetric and transitive or not.
Find the values of 'a' for which $f(x) = \sin x - ax + b$ is increasing on $\mathbb{R}$.
Prove that $f : \text{N} \to \text{N}$ defined as $f(x) = ax + b$ ($a, b \in \text{N}$) is one-one but not onto.
Which of the following functions from $Z$ to $Z$ is both one-one and onto ?
(A) $f(x) = 2x - 1$
(B) $f(x) = 3x^2 + 5$
(C) $f(x) = x + 5$
(D) $f(x) = 5x^3$
Show that the function $f(x) = (x - 1)^{\frac{1}{3}}$ is not differentiable at $x = 1$.
Rajesh, a student of Class-XII, visited an exhibition with his family. There he saw a huge swing and found that it traced the path of a parabola $y = x^2$. The following questions came to his mind. Answer the questions : (i) Let $f : R \to R$ be a function defined as $f(x) = x^2$. Find whether $f$ i...
For real $\text{x}$, let $\text{f}(\text{x}) = \text{x}^3 + 5\text{x} + 1$. Then :
(A) $\text{f}$ is one-one but not onto on $\text{R}$
(B) $\text{f}$ is onto on $\text{R}$ but not one-one
(C) $\text{f}$ is one-one and onto on $\text{R}$
(D) $\text{f}$ is neither one-one nor onto on $\text{R}$
The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are: (A) $1 < \lambda < \sqrt{2}$ (B) $\lambda \geq 1$ (C) $\lambda \geq \sqrt{2}$ (D) $\lambda < 1$
If f: N -> W is defined as f(n) = { n/2, if n is even ; 0, if n is odd } , then f is : (A) injective only (B) surjective only (C) a bijection (D) neither surjective nor injective
The function f defined by f(x) = { x, if x ≤ 1 ; 5, if x > 1 } is not continuous at : (A) x = 0 (B) x = 1 (C) x = 2 (D) x = 5
(a) Find the least value of 'a' so that $\text{f(x) = 2x}^2 - \text{ax} + 3$ is an increasing function on $[2, 4]$. OR (b) If $\text{f(x) = x} + \frac{1}{\text{x}}, \text{x} \geq 1$, show that f is an increasing function.
If f(x) = $\begin{cases} \frac{1-\sin^3 x}{3 \cos^2 x}, & \text{for } x \neq \frac{\pi}{2} \\ k, & \text{for } x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then the value of k is : (A) $\frac{3}{2}$ (B) $\frac{1}{6}$ (C) $\frac{1}{2}$ (D) 1
For real x, let f(x) = $x^3 + 5x + 1$. Then : (A) f is one-one but not onto on R (B) f is onto on R but not one-one (C) f is one-one and onto on R (D) f is neither one-one nor onto on R
If f(x) = $\begin{cases} 3x - 2, & 0 < x \le 1 \\ 2x^2 + ax, & 1 < x < 2 \end{cases}$ is continuous for x $\in$ (0, 2), then a is equal to : (A) $-$4 (B) $-\frac{7}{2}$ (C) $-$2 (D) $-$1
If f : N $\rightarrow$ W is defined as f(n) = $\begin{cases} \frac{n}{2}, & \text{if n is even} \\ 0, & \text{if n is odd} \end{cases}$, then f is : (A) injective only (B) surjective only (C) a bijection (D) neither surjective nor injective
If $f: \mathbb{N} \to \mathbb{W}$ is defined as
$f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ 0, & \text{if } n \text{ is odd} \end{cases}$,
then $f$ is :
(A) injective only
(B) surjective only
(C) a bijection
(D) neither surjective nor injective
For real $x$, let $f(x) = x^3 + 5x + 1$. Then :
(A) $f$ is one-one but not onto on $\mathbb{R}$
(B) $f$ is onto on $\mathbb{R}$ but not one-one
(C) $f$ is one-one and onto on $\mathbb{R}$
(D) $f$ is neither one-one nor onto on $\mathbb{R}$
Assertion $(A)$ : Let $A = \{x \in \mathbb{R} : -1 \le x \le 1\}$. If $f : A \to A$ be defined as $f(x) = x^2$, then $f$ is not an onto function.
Reason $(R)$ : If $y = -1 \in A$, then $x = \pm\sqrt{-1} \notin A$.
Find the domain of the function $f(x) = \cos^{-1}(x^2 - 4)$.
Find the value of 'a' for which $f(x) = \sqrt{3} \sin x - \cos x - 2ax + 6$ is decreasing in $\mathbb{R}$.
Let A be the set of 30 students of class XII in a school. Let f : A $\rightarrow$ N, N is a set of natural numbers such that function f(x) = Roll Number of student x. On the basis of the given information, answer the following : (i) Is f a bijective function? (ii) Give reasons to support your answer...
Assertion (A) : Let $A = \{x \in \mathbb{R} : -1 \le x \le 1\}$. If $f : A \to A$ be defined as $f(x) = x^2$, then $f$ is not an onto function.\nReason (R) : If $y = -1 \in A$, then $x = \pm\sqrt{-1} \notin A$.
Show that $f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in $\left[0, \frac{\pi}{4}\right]$.
Let A be the set of 30 students of class XII in a school. Let $\text{f} : \text{A} \to \text{N}$, N is a set of natural numbers such that function $\text{f(x)} = \text{Roll Number of student x}$. On the basis of the given information, answer the following : (i) Is f a bijective function ?
(iii) (a) Let R be a relation defined by the teacher to plan the seating arrangement of students in pairs, where $\text{R} = \{(\text{x, y}) : \text{x, y are Roll Numbers of students such that y} = 3\text{x}\}$. List the elements of R. Is the relation R reflexive, symmetric and transitive ? Justify ...
Assertion (A) : Let $A = \{x \in \mathbb{R} : -1 \leq x \leq 1\}$. If $f : A \to A$ be defined as $f(x) = x^2$, then $f$ is not an onto function.
Reason (R) : If $y = -1 \in A$, then $x = \pm\sqrt{-1} \notin A$.
Find the interval/intervals in which the function $f(x) = \sin 3x - \cos 3x$, $0 < x < \frac{\pi}{2}$ is strictly increasing.
Is $f$ a bijective function ?
Determine the values of $x$ for which $f(x) = \frac{x-4}{x+1}$, $x \neq -1$ is an increasing or a decreasing function.
Determine those values of x for which $f(x) = \frac{2}{x} - 5$, $x \neq 0$ is increasing or decreasing.
$f(x) = x^x$ has a critical point at :
(A) $x = e$
(B) $x = e^{-1}$
(C) $x = 0$
(D) $x = 1$
Assertion (A) : Let $f(x) = e^x$ and $g(x) = \log x$. Then $(f + g)x = e^x + \log x$ where domain of $(f + g)$ is $\mathbb{R}$.
Reason (R) : $ ext{Dom}(f + g) = ext{Dom}(f) \cap ext{Dom}(g)$.
If $f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } 5 \leq x \end{cases}$ is continuous in $\mathbb{R}$, then the values of $a$ and $b$ are :
If $f(x) = -2x^8$, then the correct statement is :
If $f : \mathbb{R} \rightarrow \mathbb{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is :
(A) a decreasing function
(B) an increasing function
(C) maximum at $x = \frac{\pi}{2}$
(D) maximum at $x = 0$
(a) Show that the function $f : R \to R$ defined by $f(x) = 4x^3 - 5$, $\forall \ x \in R$ is one-one and onto.
OR
(b) Let $R$ be a relation defined on a set $N$ of natural numbers such that $R = \{(x, y) : xy \text{ is a square of a natural number}, x, y \in N\}$. Determine if the relation $R$ is...
If $f(x) = \begin{cases} \frac{\sin^2 ax}{x^2} & , \text{if } x \neq 0 \\ 1 & , \text{if } x = 0 \end{cases}$ is continuous at $x = 0$, then the value of 'a' is :
(A) $\pm 1$
(B) -1
(C) 0
(D) 1
If $f : \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is :
(A) a decreasing function
(B) an increasing function
(C) maximum at $x = \frac{\pi}{2}$
(D) maximum at $x = 0$
If $R$ be a relation defined as $aRb$ iff $|a - b| > 0$ $a, b \in \mathbb{R}$ then $R$ is :
(A) reflexive
(B) symmetric
(C) transitive
(D) symmetric and transitive
27. (a) Show that the function $f : R \rightarrow R$ defined by $f(x) = 4x^3 - 5$, $\forall \ x \in R$ is one-one and onto.\nOR\n(b) Let R be a relation defined on a set N of natural numbers such that $R = \{(x, y) : xy \text{ is a square of a natural number, } x, y \in N\}$. Determine if the relati...
If $f(x) = \begin{cases} 3ax - b &, x > 1 \\ 11 &, x = 1 \\ -5ax - 2b &, x < 1 \end{cases}$ is continuous at $x = 1$, then the values of a and b are :
(A) a = 3, b = 5
(B) a = 8, b = -1
(C) a = 1, b = -8
(D) a = -3, b = 5
If $f : \text{R} \to \text{R}$ is defined as $f(x) = 2x - \sin x$, then $f$ is :
(A) a decreasing function
(B) an increasing function
(C) maximum at $x = \frac{\pi}{2}$
(D) maximum at $x = 0$
Let $f : A \rightarrow B$ be defined by $f(x) = \frac{x - 2}{x - 3}$, where $A = R - \{3\}$ and $B = R - \{1\}$. Discuss the bijectivity of the function.
(a) Show that the function $f : R \rightarrow R$ defined by $f(x) = 4x^3 - 5$, $\forall\ x \in R$ is one-one and onto.
OR
(b) Let R be a relation defined on a set N of natural numbers such that $R = \{(x, y) : xy \text{ is a square of a natural number}, x, y \in N\}$. Determine if the relation R is ...
A relation $R$ defined on a set of human beings as
$R = \{(x, y) : x \text{ is } 5\text{ cm shorter than } y\}$
is :
(A) reflexive only
(B) reflexive and transitive
(C) symmetric and transitive
(D) neither transitive, nor symmetric, nor reflexive
(a) Check the differentiability of function $f(x) = [x]$ at $x = -3$, where $[\cdot]$ denotes greatest integer function.
OR
(b) If $x^{1/3} + y^{1/3} = 1$, find $\frac{\text{d}y}{\text{d}x}$ at the point $\left(\frac{1}{8}, \frac{1}{8}\right)$.
(a) A relation $R$ on set $A = \{1, 2, 3, 4, 5\}$ is defined as $R = \{(x, y) : |x^2 - y^2| < 8\}$. Check whether the relation $R$ is reflexive, symmetric and transitive.
OR
(b) A function $f$ is defined from $R \rightarrow R$ as $f(x) = ax + b$, such that $f(1) = 1$ and $f(2) = 3$. Find function ...
Find the domain and range of $f(x) = 2 \sin^{-1} (1 - x)$.
Assertion (A): The function $f: \mathbb{R} - \left\{(2n + 1)\frac{\pi}{2} : n \in \mathbb{Z}\right\} \to (-\infty, -1] \cup [1, \infty)$ defined by $f(x) = \sec x$ is not one-one function in its domain.
Reason (R): The line $y = 2$ meets the graph of the function at more than one point.
If $f: R \rightarrow R$ is defined by $f(x) = |x|^3$, show that $f''(x)$ exists for all real $x$ and find it.
Write the smallest equivalence relation on G.
Ravi defines a relation from B to B as $R_1 = \{(b_1, b_2), (b_2, b_1)\}$. Write the minimum ordered pairs to be added in $R_1$ so that it becomes (A) reflexive but not symmetric, (B) reflexive and symmetric but not transitive.
If the track of the final race (for the biker $b_1$) follows the curve $x^2 = 4y$; (where $0 \le x \le 20\sqrt{2} \& 0 \le y \le 200$), then state whether the track represents a one-one and onto function or not. (Justify).
Let $f : R_+ \to [- 5, \infty)$ be defined as $f(x) = 9x^2 + 6x - 5$, where $R_+$ is the set of all non-negative real numbers. Then, $f$ is :
(A) one-one
(B) onto
(C) bijective
(D) neither one-one nor onto
The number of points of discontinuity of $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$ is :
(A) $0$
(B) $1$
(C) $2$
(D) \text{infinite}
(a) If $f(x) = |\tan 2x|$, then find the value of $f'(x)$ at $x = \frac{\pi}{3}$.
Which of the following statements is not true about equivalence classes $A_i$ (i = 1, 2, .... n) formed by an equivalence relation R defined on a set A ?
(A) $\bigcup_{i=1}^{n} A_i = A$
(B) $A_i \cap A_j \neq \phi, i \neq j$
(C) $x \in A_i \text{ and } x \in A_j \Rightarrow A_i = A_j$
(D) All elemen...
(a) Check the differentiability of $f(x) = |\cos x|$ at $x = \frac{\pi}{2}$.
A relation $R$ is defined on $N \times N$ (where $N$ is the set of natural numbers) as: $(a, b) R (c, d) \iff a - c = b - d$. Show that $R$ is an equivalence relation.
Let $f : \text{R}_+ \to [- 5, \infty)$ be defined as $f(x) = 9x^2 + 6x - 5$, where $\text{R}_+$ is the set of all non-negative real numbers. Then, f is :
(A) one-one
(B) onto
(C) bijective
(D) neither one-one nor onto
Determine whether the function $f(x) = x^{2} - 6x + 3$ is increasing or decreasing in $[4, 6]$.
(a) If $f(x) = |\tan 2x|$, then find the value of $f'(x)$ at $x = \frac{\pi}{3}$. OR (b) If $y = \text{cosec}(\cot^{-1}x)$, then prove that $\sqrt{1 + x^{2}} \frac{\text{dy}}{\text{dx}} - x = 0$.
(a) Show that a function $f : R \to R$ defined as $f(x) = x^2 + x + 1$ is neither one-one nor onto. Also, find all the values of $x$ for which $f(x) = 3$.
OR
(b) A relation $R$ is defined on $N \times N$ (where $N$ is the set of natural numbers) as $(a, b) R (c, d) \iff \frac{a}{c} = \frac{b}{d}$. S...
The value of k, for which f(x) = $\begin{cases} \frac{\sqrt{3} \cos x + \sin x}{3x + \frac{\pi}{2}}, & x \neq -\frac{\pi}{3} \\ k, & x = -\frac{\pi}{3} \end{cases}$ is continuous at x = $-\frac{\pi}{3}$, is : (A) $\frac{2}{3}$ (B) $-\frac{2}{3}$ (C) $\frac{3}{2}$ (D) 6
(a) Let $A = \{x \mid x \in \mathbb{Z}, 0 \leq x \leq 12\}$. Show that the relation $R = \{(a, b) : a, b \in A, (a - b) \text{ is divisible by } 4\}$ is an equivalence relation. Find the set of elements related to $2$.
(b) Let $A = \mathbb{R} - \{4\}$ and $B = \mathbb{R} - \{1\}$ and let function $f : A \to B$ be defined as $f(x) = \frac{x - 3}{x - 4}$ for $\forall x \in A$. Show that $f$ is one-one and onto.
Let R_+ denote the set of all non-negative real numbers. Then the function f : R_+ \to R_+ defined as f(x) = x^2 + 1 is :
(A) one-one but not onto
(B) onto but not one-one
(C) both one-one and onto
(D) neither one-one nor onto
A relation $R$ on set $A = \{- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4\}$ be defined as $R = \{(x, y) : x + y \text{ is an integer divisible by } 2\}$. Show that $R$ is an equivalence relation. Also, write the equivalence class $[2]$.
A function $f: R \rightarrow A$ defined as $f(x) = x^2 + 1$ is onto, if $A$ is :
(A) $(-\infty, \infty)$
(B) $(1, \infty)$
(C) $[1, \infty)$
(D) $[-1, \infty)$
A relation $R$ on set $A = \{x : -10 \le x \le 10, x \in \mathbb{Z}\}$ is defined as $R = \{(x, y) : (x - y) \text{ is divisible by } 5\}$. Show that $R$ is an equivalence relation. Also, write the equivalence class $[5]$.
Let $Z$ denote the set of integers, then function $f : Z \to Z$ defined as $f(x) = x^3 - 1$ is :
(A) both one-one and onto
(B) one-one but not onto
(C) onto but not one-one
(D) neither one-one nor onto
Find the value of $a$ and $b$ so that function $f$ defined as : $f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & \text{if } x < 2 \\ a + b, & \text{if } x = 2 \\ \frac{x - 2}{|x - 2|} + b, & \text{if } x > 2 \end{cases}$ is a continuous function.
Find the absolute maximum and absolute minimum values of the function $f$ given by $f(x) = \frac{x}{2} + \frac{2}{x}$, on the interval $[1, 2]$.
Let A = R - {3} and B = R - {a}. Find the value of ‘a’ such that the function f : A $\rightarrow$ B defined by f(x) = $\frac{x - 2}{x - 3}$ is onto. Also, check whether the given function is one-one or not.
For what value of k, the function given below is continuous at $x = 0$ ?
$f(x) = \begin{cases} \frac{\sqrt{4+x}-2}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$
(A) 0
(B) $\frac{1}{4}$
(C) 1
(D) 4
Assertion (A) : The relation $R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in N\}$ is not a reflexive relation.
Reason (R) : The number '$2n$' is composite for all natural numbers $n$.
Find the interval in which the function $f(x) = x^{4} - 4x^{3} + 10$ is strictly decreasing.
(a) Let $A = R - \{5\}$ and $B = R - \{1\}$. Consider the function $f : A \to B$, defined by $f(x) = \frac{x - 3}{x - 5}$. Show that f is one-one and onto. OR (b) Check whether the relation S in the set of real numbers R defined by $S = \{(a, b) : \text{where } a - b + \sqrt{2} \text{ is an irration...
The value of constant c that makes the function f defined by f(x) = \begin{cases} x^2 - c^2, & \text{if } x < 4 \\ cx + 20, & \text{if } x \geq 4 \end{cases} continuous for all real numbers is : (A) -2 (B) -1 (C) 0 (D) 2
Show that $f(x) = \frac{4 \sin x}{2 + \cos x} - x$ is an increasing function of $x$ in $\left[0, \frac{\pi}{2}\right]$.
Let $A = R - \{5\}$ and $B = R - \{1\}$. Consider the function $f : A \rightarrow B$, defined by $f(x) = \frac{x - 3}{x - 5}$. Show that $f$ is one-one and onto.
Show that the function f given by $f(x) = \sin x + \cos x$, is strictly decreasing in the interval $\left$ \frac{\pi}{4}, \frac{5\pi}{4} \right$$.
Let $A = R - \{5\}$ and $B = R - \{1\}$. Consider the function $f : A \to B$, defined by $f(x) = \frac{x - 3}{x - 5}$. Show that $f$ is one-one and onto.
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x) = x^2 - 4x + 5$ is:
(A) injective but not surjective.
(B) surjective but not injective.
(C) both injective and surjective.
(D) neither injective nor surjective.
OR Let S be the relation defined by $S = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2\}$ check whether the relation S is symmetric and transitive.
Verify whether the function $f$ defined by $f(x) = \begin{cases} x \sin\left(\frac{1}{x}\right) , x \neq 0 \\ 0 \quad \quad \quad , x = 0 \end{cases}$ is continuous at $x = 0$ or not.
Check for differentiability of the function $f$ defined by $f(x) = |x - 5|$, at the point $x = 5$.
(b) Let S be the relation defined by S = {(l_1, l_2) : l_1 is perpendicular to l_2} check whether the relation S is symmetric and transitive.
The number of discontinuities of the function f given by $f(x) = \begin{cases} x+2, & \text{if} & x < 0 \\ e^x, & \text{if} & 0 \leq x \leq 1 \\ 2-x, & \text{if} & x > 1 \end{cases}$ is : (A) 0 (B) 1 (C) 2 (D) 3
Check whether the relation S in the set of all real numbers (\mathbb{R}) defined by S = \{(a, b) : a \leq b^3\} is reflexive, symmetric or transitive.
If $f(x) = x^x$, then $f'(e)$ is equal to :
(A) $0$
(B) $2$
(C) $e^e$
(D) $2e^e$
The function f given by $f(x) = x^3 - 3x^2 + 3x$, $x \in \mathbb{R}$ is increasing on : (A) $[1, \infty)$ (B) $(1, \infty)$ (C) $(-\infty, \infty)$ (D) $(-\infty, 1)$
The number of discontinuities of the function f given by
$f(x) = \begin{cases} x+2, & \text{if } x < 0 \\ e^x, & \text{if } 0 \le x \le 1 \\ 2-x, & \text{if } x > 1 \end{cases}$
is :
(A) 0
(B) 1
(C) 2
(D) 3
Prove that the relation R in the set of integers Z defined as R = {(a, b) : 2 divides (a + b)} is an equivalence relation. Also, determine [3].
The value of $k$ for which $f(x) = \begin{cases} 3x + 5, & x \geq 2 \\ kx^2, & x < 2 \end{cases}$ is a continuous function, is :
Show that the function $f(x) = \frac{16 \sin x}{4 + \cos x} - x$, is strictly decreasing in $\left$\frac{\pi}{2}, \pi
ight$$.
A relation R is defined on a set of real numbers $\mathbb{R}$ as R = {(x, y) : x . y is an irrational number}. Check whether R is reflexive, symmetric and transitive or not.
For what value of k may the function $f(x) = \begin{cases} k(3x^2 - 5x), & x \le 0 \\ \cos x, & x > 0 \end{cases}$ become continuous ?
(a) 0
(b) 1
(c) $-\frac{1}{2}$
(d) No value
Assertion (A) : The number of onto functions from a set P containing 5 elements to a set Q containing 2 elements is 30.
Reason (R) : Number of onto functions from a set containing m elements to a set containing n elements is $n^m$.
Find the sub-intervals in which $f(x) = \log(2 + x) - \frac{x}{2 + x}$, $x > -2$ is increasing or decreasing.
A function $f: A \to B$ defined as $f(x) = 2x$ is both one-one and onto. If $A = \{1, 2, 3, 4\}$, then find the set B.
A relation R is defined on a set of real numbers $\mathbb{R}$ as R = $\{(x, y) : x \cdot y$ is an irrational number$\}$. Check whether R is reflexive, symmetric and transitive or not.
If f(x) = a(x - cos x) is strictly decreasing in \mathbb{R}, then 'a' belongs to
(a) {0}
(b) (0, \infty)
(c) (-\infty, 0)
(d) (-\infty, \infty)
A relation $R$ is defined on a set of real numbers $\mathbb{R}$ as $R = \{(x, y) : x \cdot y \text{ is an irrational number}\}$. Check whether $R$ is reflexive, symmetric and transitive or not.
The set of all points where the function $f(x) = x + |x|$ is differentiable, is
(a) $(0, \infty)$
(b) $(-\infty, 0)$
(c) $(-\infty, 0) \cup (0, \infty)$
(d) $(-\infty, \infty)$
ASSERTION (A): The relation $f : \{1,2,3,4\} \rightarrow \{x,y,z,p\}$ defined by $f = \{(1,x),(2,y),(3,z)\}$ is a bijective function. REASON (R): The function $f : \{1,2,3\} \rightarrow \{x,y,z,p\}$ such that $f = \{(1,x),(2,y),(3,z)\}$ is one-one.
Find the interval/s in which the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = xe^x$, is increasing.
If $f(x) = \frac{1}{4x^2+2x+1}; x \in \mathbb{R}$, then find the maximum value of $f(x)$. OR Find the maximum profit that a company can make, if the profit function is given by $P(x) = 72 + 42x - x^2$, where $x$ is the number of units and $P$ is the profit in rupees.
Check whether the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3+x$, has any critical point/s or not ? If yes, then find the point/s.
Let $\mathbb{N}$ be the set of all natural numbers and $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b) R (c,d) \iff ad = bc$ for all $(a,b), (c,d) \in \mathbb{N} \times \mathbb{N}$. Show that $R$ is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. Also, find the equiv...
If $f(x) = \begin{cases} ax + b ; & 0 < x \le 1 \\ 2x^2 - x ; & 1 < x < 2 \end{cases}$ is a differentiable function in $(0, 2)$, then find the values of $a$ and $b$.
If $\frac{d}{dx}f(x) = 2x + \frac{3}{x}$ and $f(1) = 1$, then $f(x)$ is
(A) $x^{2} + 3 \log |x| + 1$
(B) $x^{2} + 3 \log |x|$
(C) $2 - \frac{3}{x^{2}}$
(D) $x^{2} + 3 \log |x| - 4$
A function $f: [-4, 4] \to [0, 4]$ is given by $f(x) = \sqrt{16 - x^2}$. Show that $f$ is an onto function but not a one-one function. Further, find all possible values of 'a' for which $f(a) = \sqrt{7}$.
Number of symmetric matrices of order $3 \times 3$ with each entry 1 or $-1$ is
(A) 512
(B) 64
(C) 8
(D) 4
(b) If $f(x) = \begin{cases} ax + b & ; \; 0 < x \le 1 \\ 2x^2 - x & ; \; 1 < x < 2 \end{cases}$ is a differentiable function in $(0, 2)$, then find the values of $a$ and $b$.
The domain of the function $\cos^{-1} x$ is :
(a) $[0, \pi]$
(b) $(- 1, 1)$
(c) $[- 1, 1]$
(d) $[-\frac{\pi}{2}, \frac{\pi}{2}]$
The value of k for which the function f given by f(x) = $\begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 5, & \text{if } x = \frac{\pi}{2} \end{cases}$ is continuous at x = $\frac{\pi}{2}$, is: (a) 6 (b) 5 (c) $\frac{5}{2}$ (d) 10
(a) A relation R in the set A = {5, 6, 7, 8, 9} is given by R = {(x, y) : |x - y| is divisible by 2}. Write R in roster form and prove that R is an equivalence relation. Also, find the elements related to element 7. OR (b) Let A = ℝ - {3} and B = ℝ - {1} be two sets. Prove that the function f : A → ...
Show that a function f : ℝ → ℝ defined as f(x) = \frac{5x - 3}{4} is both one-one and onto.
If f(x) = a(tan x - cot x), where a > 0, then find whether f(x) is increasing or decreasing function in its domain.
Consider the statement "There exists at least one value of $b \in \mathbb{R}$ for which $f(x) = \frac{b}{x}$, $b \neq 0$ is strictly increasing in $\mathbb{R} - \{0\}$."
State True or False. Justify.
Prove that a function $f : [0, \infty) \rightarrow [- 5, \infty)$ defined as $f(x) = 4x^{2} + 4x - 5$ is both one-one and onto.
Find the maximum and minimum values of the function given by $f(x) = 5 + \sin 2x$.
(a) If $\mathbb{N}$ denotes the set of all natural numbers and $R$ is the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) R (c, d)$, if $ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation.
OR
(b) Let $f : \mathbb{R} - \left\{-\frac{4}{3}\right\} \to \mathbb{R}$ be a funct...
Find the interval in which the function $f(x) = 2x^3 - 3x$ is strictly increasing.
(a) If $\text{N}$ denotes the set of all natural numbers and $\text{R}$ is the relation on $\text{N} \times \text{N}$ defined by $(a, b)\text{ R }(c, d)$, if $ad(b + c) = bc(a + d)$. Show that $\text{R}$ is an equivalence relation.
OR
(b) Let $f: \mathbb{R} - \left\{-\frac{4}{3}\right\} \to \mathbb{...
If $f(x) = 2|x| + 3|\sin x| + 6$, then the right hand derivative of $f(x)$ at $x = 0$ is :
(a) $6$
(b) $5$
(c) $3$
(d) $2$
(a) If $N$ denotes the set of all natural numbers and $R$ is the relation on $N imes N$ defined by $(a, b) R (c, d)$, if $ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation.
OR
(b) Let $f : \mathbb{R} - \left\{ -\frac{4}{3}
ight\}
ightarrow \mathbb{R}$ be a function defined as $f(...
Case Study - 2
37. Let $f(x)$ be a real valued function. Then its
• Left Hand Derivative (L.H.D.) : $L f'(a) = \lim_{h o 0} \frac{f(a - h) - f(a)}{-h}$
• Right Hand Derivative (R.H.D.) : $R f'(a) = \lim_{h o 0} \frac{f(a + h) - f(a)}{h}$
Also, a function $f(x)$ is said to be differentiable at $x...
Select the correct option out of the four given options :
Let $A = \{3, 5\}$. Then number of reflexive relations on A is
(a) 2
(b) 4
(c) 0
(d) 8
Write the domain and range (principle value branch) of the following functions : f(x) = $\tan^{-1} x$
(a) If f(x) = $\begin{cases} x^{2} , \text{ if } x \geq 1 \\ x , \text{ if } x < 1 \end{cases}$ then show that f is not differentiable at x = 1. OR (b) Find the value(s) of '$\lambda$', if the function f(x) = $\begin{cases} \frac{\sin^{2} \lambda x}{x^{2}} , \text{ if } x \neq 0 \\ 1 , \text{ if } x...
Let $A = \{3, 5\}$. Then number of reflexive relations on A is
(a) $2$
(b) $4$
(c) $0$
(d) $8$
The function $f(x) = x | x |$, $x \in \mathbf{R}$ is differentiable
(a) only at $x = 0$
(b) only at $x = 1$
(c) in $\mathbf{R}$
(d) in $\mathbf{R} - \{0\}$
Draw the graph of the principal branch of the function $f(x) = \cos^{-1} x$.
(a) If $f(x) = \begin{cases} x^2 , & \text{if } x \ge 1 \\ x , & \text{if } x < 1 \end{cases}$, then show that $f$ is not differentiable at $x = 1$.
OR
(b) Find the value(s) of '$\lambda$', if the function
$f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2} , & \text{if } x \neq 0 \\ 1 , & \text{if }...
The function f(x) = x^x, x > 0 is decreasing in the interval :
(a) (-∞, e)
(b) (0, e)
(c) (0, 1/e)
(d) [1/e, ∞)
Function f is defined as
f(x) = \begin{cases} 2x + 2, & \text{if } x < 2 \\ k, & \text{if } x = 2 \\ 3x, & \text{if } x > 2 \end{cases}
Find the value of k for which the function f is continuous at x = 2.
Find the intervals in which the function f(x) = x^4 - 4x^3 + 4x^2 + 15, is strictly increasing.
Find the points at which the function $f(x) = \frac{4 + x^2}{4x - x^3}$ is discontinuous.
Case Study - 3
In an agricultural institute, scientists do experiments with varieties of seeds to grow them in different environments to produce healthy plants and get more yield.
A scientist observed that a particular seed grew very fast after germination. He had recorded growth of plant since germ...
The function f(x) = $\begin{cases} 3ax + b, & \text{if} & x > 1 \\ 11, & \text{if} & x = 1 \\ 5ax - 2b, & \text{if} & x < 1 \end{cases}$ is continuous at x = 1. Find the values of a and b.
Find the interval in which the function f(x) = $x^3 + \frac{1}{x^3}$, x $\neq$ 0 is decreasing.
(a) Simplify : $\tan^{-1}\left(\frac{\cos x}{1 - \sin x}\right)$ OR (b) Prove that the greatest integer function f : $\mathbb{R} \rightarrow \mathbb{R}$, given by f(x) = [x], is neither one-one nor onto.
Let $R$ be the relation defined in the set $A = \{1, 2, 3, 4, 5, 6, 7\}$ by $R = \{(a, b) : \text{both } a \text{ and } b \text{ are either odd or even}\}$. Show that $R$ is an equivalence relation. Hence, find the elements of equivalence class $[1]$.
The domain of the function $\sin^{-1}(2x)$ is:
(a) $[-1, 1]$
(b) $[0, 1]$
(c) $[-\frac{1}{2}, \frac{1}{2}]$
(d) $$-\frac{1}{2}, \frac{1}{2}$$
The value of k for which the function f, given by
$f(x) = \begin{cases} kx + 1, & \text{if } x \le \pi \\ \cos x, & \text{if } x > \pi \end{cases}$
is continuous at $x = \pi$, is :
(a) $\frac{2}{\pi}$
(b) $-\frac{\pi}{2}$
(c) $-\frac{2}{\pi}$
(d) $\frac{\pi}{2}$
(a) Let the relation $R$ be given as
$R = \{(x, y) : x, y \in \mathbb{N} ext{ and } x + 3y = 12\}$. Find the domain and range of $R$.
OR
(b) Show that the function $f : \mathbb{R} o \mathbb{R}$, $f(x) = x^4$ is neither one-one nor onto.
Find the intervals in which the function given by $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$ is (a) increasing (b) decreasing.
(a) Let R be a relation in $\mathbb{R}$, the set of all real numbers, defined by R = {(a, b) : a $\le$ b$^3$}. Show that R is neither reflexive, nor symmetric and nor transitive. OR (b) Let set A = {1, 2, 3, ..., 10} and R be a relation in A $\times$ A, defined by (a, b) R (c, d) $\iff$ a + d = b + ...
Find the value of $\sin^{-1}[\sin$\frac{13\pi}{7}$]$
OR
Prove that the function f is surjective, where $f: N \to N$ such that $f(n) = \begin{cases} \frac{n + 1}{2}, \text{ if } n \text{ is odd} \\ \frac{n}{2}, \text{ if } n \text{ is even} \end{cases}$
Is the function injective? Justify your answer.
Define the relation R in the set $N \times N$ as follows:
For $(a, b), (c, d) \in N \times N$, $(a, b)\text{ R }(c, d)$ iff $ad = bc$. Prove that R is an equivalence relation in $N \times N$.
Given a non-empty set $X$, define the relation $R$ in $P(X)$ as follows: For $A, B \in P(X)$, $(A, B) \in R$ iff $A \subset B$. Prove that $R$ is reflexive, transitive and not symmetric.
Case-Study 1: Read the following passage and answer the questions given below.
The temperature of a person during an intestinal illness is given by
$f(x) = -0.1x^2 + mx + 98.6, 0 \le x \le 12$, m being a constant, where $f(x)$ is the temperature in $^{\circ}\text{F}$ at $x$ days.
(i) Is the functio...