Relations and Functions Advanced Problems for JEE Main and WBJEE

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25 Multiple Choice Questions (MCQs) on Relations and Functions for students of class XI and XII preparing for board examinations or JEE Mains, IIT Advanced WBJEE or any other competitive entrance examination.

👨‍🏫 Author: Singh

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Prime Maths

Advanced Algebra - Relations and Functions

Test your understanding of core concepts.

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Question 1

The number of reflexive relations on a set \(A\) of \(n\) elements is equal to

A\( 2^{n^2}\) B\( n^2\) C\( 2^{n(n-1)} \) D\( n^2-n\)

For the relation to be reflexive, \((a_i,a_i) \in A \times A \quad \forall a_i \in A\) So the from the remaining \(n^2 - n\) elements, any number of elements can be choosen to form a subset of \(A \times A\)

Question 2

Let \(X\) be a non-void set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on \(X\), then [ \( \circ\) denotes composition ]

A \(\rho_1 \circ \rho_2\) is transitive relation B \(\rho_1 \circ \rho_2\) is not transitive relation C \(\rho_1 \circ \rho_2\) is equivalence relation D \(\rho_1 \circ \rho_2\) is not any relation on \(X\)

Let's define a set \(X = \{1, 2, 3, 4, 5\}\). Now, let's create two relations on \(X\), \(\rho_1\) and \(\rho_2\), both of which are transitive:Let \(\rho_1 = \{(1, 2), (3, 4)\}\) Let \(\rho_2 = \{(2, 3), (4, 5)\}\) (Note: These are trivially transitive). Now, let's find the composition \(\rho_1 \circ \rho_2\). By definition of composition, \((x, z) \in \rho_1 \circ \rho_2\) if there is some intermediate element \(y\) such that \((x, y) \in \rho_1\) and \((y, z) \in \rho_2\). So, the resulting composition is:\(\rho_1 \circ \rho_2 = \{(1, 3), (3, 5)\}\) which is clearly not transitive.

Question 3

If one root of \(x^2 + px - q^2 = 0\), \(p\) and \(q\) are real, be less than \(2\) and other be greater than \(2\), then

A \( 4+2p+q^2 > 0 \) B \( 4+2p+q^2 \\< 0 \) C \( 4+2p-q^2 > 0 \) D \( 4+2p-q^2 \\< 0 \)

Let \(y=x^2+px-q^2\) as \(x \rightarrow \pm \infty, y \rightarrow \infty\) thus the resulting parabola is open in the positive direction of \(y\)-axis. As \(2\), lies between the roots, \(y\) is less than \(0\) at \(x=2\).

Question 4

If \(R\) and \(Q\) are equivalence relations on set \(A\), then which of the following is not an equivalence relation

A\(R^{-1} \cap Q^{-1}\) B\(Q^{-1}\) C\(R \cup Q\) D\(R \cap Q\)

Let \( A=\{1,2,3\}, P =\{(1,1),(2,2),(3,3),(2,3),(3,2)\}\) and \(Q =\{(1,1),(2,2),(3,3),(1,2),(2,1)\}\). It is easy to see that both \(P \text{ and } Q\) are equivalence relation on \(A\) but \(P \cup Q\) is not an equivalence relation.

Question 5

Let \( \rho \) be a relation defined on set of natural numbers \( \mathbb{N} \), as \( \rho = \{(x, y) \in \mathbb{N} \times \mathbb{N} : 2x + y = 41\} \). Then domain A and range B are

A\( A \subset \{x \in \mathbb{N} : 1 \le x \le 20\} \) and \( B \subset \{y \in \mathbb{N} : 1 \le y \le 39\} \) B\( A = \{x \in \mathbb{N} : 1 \le x \le 15\} \) and \( B = \{y \in \mathbb{N} : 2 \le y \le 30\} \) C\( A \equiv \mathbb{N}, B \equiv \mathbb{Q} \) D\( A \equiv \mathbb{Q}, B \equiv \mathbb{Q} \)

See that \(y= 41 - 2x\) and for \(x \in \{1,2,...,20 \}\), \( y \in \{1,3,5,...,39\}\)

Question 6

In \(\mathbb{R}\), a relation \(\rho\) is defined as follows: \(\forall a,b \in \mathbb{R}\), \(a\rho b \quad \text{holds iff} \quad a^2 -4ab+3b^2=0 \)

A\(\rho\) is equivalence relation B\(\rho\) is only symmetric C\(\rho\) is only reflexive D\(\rho\) is only transitive

See that \(a^2-4ab+3b^2 =0 \implies a=b\) or \(a=3b\)

Question 7

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \frac{e^{|x|} - e^{-x}}{e^x + e^{-x}} \), then

A \( f \) is both one-one and onto B \( f \) is one-one but not onto C \( f \) is onto but not one-one D \( f \) is neither one-one nor onto

\Injectivity (One-One): For \( x < 0 \), \( |x| = -x \). $$ f(x) = \frac{e^{-x} - e^{-x}}{e^x + e^{-x}} = 0 $$ Since \( f(-1) = f(-2) = 0 \), multiple inputs yield the same output. Therefore, \( f \) is not one-one. 2. Surjectivity (Onto): For \( x \ge 0 \), \( |x| = x \). $$ f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$ The range for \( x \ge 0 \) is \( [0, 1) \) ( use limits), the total range of \( f \) is \( [0, 1) \). Since the range \( [0, 1) \neq \) codomain \( \mathbb{R} \), \( f \) is not onto.

Question 8

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = 2026x^3+2025x^2+2024x+2023 \), then

A \( f \) is both one-one and onto B \( f \) is one-one but not onto C \( f \) is onto but not one-one D \( f \) is neither one-one nor onto

It is easy to see that \(f\) is injective. For surjectivity, think about the nature of roots of a cubic equation.

Question 9

Let \(u + v + w = 3\), \(u, v, w \in \mathbb{R}\) and \(f(x) = ux^2 + vx + w\) be such that \(f(x + y) = f(x) + f(y) + xy\), \(\forall x, y \in \mathbb{R}\). Then \(v\) is equal to

A\(\frac{5}{2}\) B\(\frac{1}{2}\) C\(\frac{1}{\sqrt{2}}\) D\(3\)

Start by substituting \(f(x) = ux^2 + vx + w\) directly into the given equation \(f(x + y) = f(x) + f(y) + xy\). Expand the left side, and then compare the coefficients of the \(xy\) terms and the constant terms on both sides. This will reveal the values of \(u\) and \(w\) first, which you can then use with the condition \(u + v + w = 3\) to find \(v\).

Question 10

Let \(X\) and \(Y\) be two non-empty sets. Let \(f:X \rightarrow Y\) be a function. For \(A \subset X\) and \(B \subset Y\) , define \(f(A)=\{f(x):x \in A\}\) and \(f^{-1}(B)=\{x \in X : f(x) \in B \}\), then

A\(f(f^{-1}(B))=B\) B\(f(f^{-1}(B)) \subset B\) C\(f^{-1}(f(A))=A\) D\(f^{-1}(f(A)) \subset A\)

Collection of all pre-images will be mapped into \(B\). Not every element in \(Y\) has a pre-image. Additional exercise, what will happen if the function is surfective?

Question 11

Range of the function \(f(x)= \frac{x^2+x+2}{x^2+x+1}\ \: x \in \mathbb{R}\)

A\((1,\infty)\) B\((1, 1.5]\) C\((1,\frac{7}{3}]\) D\([1,\frac{7}{3}]\)

Rewrite the function as \(f(x)=1+\frac{1}{x^2+x+1}\)

Question 12

Let \(X=\{v,i,n,o,d\}\) and \(Y=\{p,m\}\). The number of onto ( surjective) functions from \(X\) to \(Y\) is

A\(32\) B\(28 \) C\(30\) D\(36\)

To find the number of onto functions, first calculate the total number of unrestricted mappings from \(\{v, i, n, o, d\}\) to \(\{p, m\}\), which is \(2^5\). Then, simply subtract the cases where the mapping is not onto—specifically, the exact number of scenarios where all \(5\) elements map exclusively to just a single output in the codomain.

Question 13

Find the natural number \(a\) for which \(\sum_{k=1}^{n} f(a+k) = 16(2^n-1)\) where the function \(f\) satisfies the relation \(f(x+y)=f(x)f(y)\) for all natural numbers \(x,y\) and further , \(f(1)=2\).

A\(11\) B\(5\) C\(7\) D\(3\)

Hint not yet added.

Question 14

The domain of the definition of the function \(f(x)=\frac{1}{4-x^2} + log_{10} (x^3-x)\) is

A \( (-1,0) \cup (1,2) \cup (3,\infty) \) B \( (-2,-1) \cup (-1,0) \cup (2,\infty) \) C \( (-1,0) \cup (1,2) \cup (2,\infty) \) D \( (1,2) \cup (2,\infty)\)

Critical points of \(x^3-x\) are \(-1,0\) and \(1\)

Question 15

Let \(f(x)= a^x \quad (a > 0) \) be written as \(f(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) is an even function and \(f_2(x)\) is an odd function. Then \(f_1(x+y)+f_1(x-y)\) equals

A\(2f_1(x+y)f_2(x-y)\) B\(2f_1(x+y)f_1(x-y)\) C\(2f_1(x)f_2(x)\) D\(2f_1(x)f_1(x)\)

see that \(f(x)=a^x=f_1(x)+f_2(x)\) and \(\f(-x)=a^{-x}=f_1(x)-f_2(x))

Question 16

Let \(g(x) = 1 + x - [x]\) and \(f(x) = \begin{cases} -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0 \end{cases}\), then for all \(x\), \(f[g(x)]\) is equal to

A\(x\) B\(1\) C\(f(x)\) D\(g(x)\)

Note that \(x-[x] = \{x\}\) ( fractional part of \(x\) )

Question 17

Let \(\mathbb{N}\) be the set of natural numbers and two functions \(f\) and \(g\) defined as \(f,g:\mathbb{N} \rightarrow \mathbb{N}\) such that \(f(x) = \begin{cases} \frac{n+1}{2}, & \text{ if n is odd} \\ \frac{n}{2}, & \text{ if n is even} \end{cases}\) and \(g(n)=n-(-1)^n\). Then \(f\circ g\) is

Aone-one but not onto Bonto but not one-one Cboth one-one and onto Dneither one-one nor onto

Start by evaluating the composite function for the first couple of natural numbers. What exactly are the outputs for \(n = 1\) and \(n = 2\) when you calculate \((f \circ g)(n)\)? If you choose an even natural number for your initial input, say \(n = 2k\), what does the composite function \((f \circ g)(2k)\) ultimately simplify to?

Question 18

If \(f\) is an even function defined on the interval \((-5, 5)\), the four real values of \(x\) satisfying the equation: \(f(x) = f\left(\frac{x+1}{x+2}\right)\) are

A\(x = \frac{-1 \pm \sqrt{7}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\) B\(x = \frac{-1 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 \pm \sqrt{5}}{2}\) C\(x = \frac{-2 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\) D\(x = \frac{-1 \pm \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 \pm \sqrt{5}}{2}\)

Since the function \(f\) is an even function, the condition \(f(a) = f(b)\) implies that either \(a = b\) or \(a = -b\).

Question 19

Let \(f:X \rightarrow X\) be such that \(f(f(x))=x\), for all \(x \in X\) and \( X \subset R \), then

A\(f\) is one-to-one B\(f\) is onto C\(f\) is one-to-one not onto D\(f\) is onto but not one-to-one

Hint not yet added.

Question 20

Let \( f : R \rightarrow R \) be such that \( f \) is injective and \( f(x)f(y) = f(x+y) \) for \( \forall x, y \in R \). If \( f(x) \), \( f(y) \), \( f(z) \) are in G.P., then \( x, y, z \) are in

AAP always and \(f(0)=1\) BGP always and \(f(0)=0\) CAP depending on the value of \( x, y, z \) DGP depending on the value of \( x, y, z \)

Hint not yet added.

Question 21

The domain of definition of \( f(x) = \sqrt{\frac{1 - |x|}{2 - |x|}} \) is

A\( (-\infty, -1) \cup (2, \infty) \) B\( [-1, 1] \cup (2, \infty) \cup (-\infty, -2) \) C\( (-\infty, 1) \cup (2, \infty) \) D\( [-1, 1] \cup (2, \infty) \)

Hint not yet added.

Question 22

Let \( f(x) = ax^2 + bx + c \), \( g(x) = px^2 + qx + r \) such that \( f(1) = g(1) \), \( f(2) = g(2) \) and \( f(3) - g(3) = 2 \). Then \( f(4) - g(4) \) is

A4 B5 C6 D7

Hint not yet added.

Question 23

Let \( R \) be the set of real numbers and the functions \( f : R \rightarrow R \) and \( g : R \rightarrow R \) be defined by \( f(x) = x^2 + 2x - 3 \) and \( g(x) = x + 1 \). Then, the value of \( x \) for which \( f(g(x)) = g(f(x)) \) is

A\( -1 \) B\( 0 \) C\( 1 \) D\( 2 \)

Hint not yet added.

Question 24

The minimum value of the function \(f(x)=2|x-1|+|x-2|\) is

A\( 0 \) B\( 1 \) C\( 2 \) D\( 3 \)

Sketch the graph.

Question 25

Let \(\mathrm{A}=\{-3,-2,-1,0,1,2,3\}\) and \(R\) be a relation on \(A\) defined by \(x R y\) if and only if \(2 x-y \in\{0,1\}\). Let \(l\) be the number of elements in \(R\). Let \(m\) and \(n\) be the minimum number of elements required to be added to \(R\) to make it a reflexive and symmetric relation, respectively. Then \(l+\mathrm{m}+ \mathrm{n}\) is equal to :-

A\(18\) B\(15\) C\(16\) D\(17\)

Hint not yet added.

Prime Maths - Excellence in Mathematical Education

© 2026 Prime Maths | Created by Vinod Singh

Circles for ICSE and CBSE | 25 Conceptual Problems

Prime Maths

Chapter Test Circles for CBSE, ICSE | Class X

👨‍🏫 Author: Singh

📞 WA: +91-9038126497

Prime Maths

Geometry - Circles

This quiz provides a thorough evaluation of key concepts and advanced skills in Circle Geometry, tailored for both CBSE and ICSE curriculums. It covers all core theorems and properties, including: Angle properties (Angle at the centre, Angles in the same segment). Cyclic Quadrilateral properties. Intersecting chord Theorem. Tangent-Secant theorems (Alternate Segment Theorem, Tangent-Radius properties). Perfect for exam preparation, this practice test is designed to challenge your understanding and refine your problem-solving techniques. CIRCLES | CBSE | ICSE . 25 multiple-choice questions designed to test both theoretical understanding and practical application.

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Question 1

From a point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle with centre at \(O\) and radius \(r\). If \(OP=2r\), then \(\triangle APB\) is:

Aan isosceles triangle Ba right angle triangle Can equilateral triangle Da scalene triangle

Question 2

In the given figure, \(AB\) is a chord of length \(16\) cm of a circle of radius \(10\) cm. The tangents at \(A\) and \(B\) interesct at point \(P\). The length of \(PA\) is:

A\(\frac{37}{7} \quad cm\) B\(\frac{37}{3} \quad cm\) C\(\frac{40}{7} \quad cm\) D\(\frac{40}{3} \quad cm\)

Question 3

The angle subtended by a tangent \(MN\) at the centre \(O\) of a circle, intercepted between two parallel tangets \(AM\) and \(BN\) of the same circle is:

A\(75^{\circ}\) B\(60^{\circ}\) C\(45^{\circ}\) D\(90^{\circ}\)

Question 4

In the figure \(AB\) is a common tangent of two circles intersecting at \(C\) and \(D\). The value of \(\angle ACB+ \angle ADB\) is:

A\(180^{\circ}\) B\(220^{\circ}\) C\(160^{\circ}\) D\(280^{\circ}\)

Question 5

In the given figure \(O\) is the centre of the circle. \(AB\) and \(AC\) are tangents drawn to the circle from point \(A\). If \(\angle BAC = 65^{\circ}\), then the measure of \(\angle BOC \) is:

A\(100^{\circ}\) B\(135^{\circ}\) C\(135^{\circ}\) D\(115^{\circ}\)

Question 6

\(AB\) is the diameter of the circle with centre \(O\). From a point \(P\) on the circle a perpendicular \(PN\) is drawn on \(AB\). Which of the following is true?

A\(PB^2 = AN.BN\) B\(PB^2 = AB.BN\) C\(PN^2 = AB.BN\) D\(PN^2 = AB.AP\)

Question 7

From an external point \(P\) of a circle with centre \(O\), two tangents \(PS\) and \(PT\) are drawn. \(QS\) is a chord of the circle parallel to \(PT\). If \( \angle SPT = 80^{\circ}\), then the value of \(\angle QST \) is

A\(55^{\circ}\) B\(45^{\circ}\) C\(50^{\circ}\) D\(60^{\circ}\)

Question 8

In the figure given below, \(PA\) is a tangent to the circle with centre \(O\) and \(PCB\) is a straight line. The measure on \(\angle OBC \) is

A\(20^{\circ}\) B\(30^{\circ}\) C\(40^{\circ}\) D\(35^{\circ}\)

Question 9

In the figure given below, a circle with centre \(O\) inscribed inside triangle \(LMN\). \(A\) and \(B\) are the points of tangency. The measure on \(\angle ANB \) is

A\(70^{\circ}\) B\(80^{\circ}\) C\(60^{\circ}\) D\(50^{\circ}\)

Question 10

In the diagram given below, \(\angle EDC = 90^{\circ}\). The tangent drawn to the circle at \(C\) makes an angle of \(50^{\circ}\) with \(AB\) produced.The measure on \(\angle ACB \) is

A\(60^{\circ}\) B\(50^{\circ}\) C\(30^{\circ}\) D\(40^{\circ}\)

Question 11

In the figure given below, \(CD\) is the diameter of the circle which meets the chord \(AB\) at \(P\) such that \(AP = BP = 12 \quad cm\). If \(DP = 8\) cm, find the radius of the circle is

A\(17 \) cm B\(11\) cm C\(7\) cm D\(13\) cm

Question 12

\(ABCD\) is a cyclic quadrilateral. If \(AD=AB\), \(\angle DAC = 60^{\circ}\), \(\angle BDC=50^{\circ}\) then the measure of \(\angle ACD\) is

A\(25^{\circ}\) B\(20^{\circ}\) C\(35^{\circ}\) D\(45^{\circ}\)

Question 13

In the figure given below, two circles touch each other at point \(A\). \(PQ\) is a direct common tangent, the point of contacts being \(P\) and \(Q\) respectively. The measure of \(\angle PAQ\) is

A\(90^{\circ}\) B\(45^{\circ}\) C\(120^{\circ}\) DCannot be determined

Question 14

In the diagram given below, \(O\) is centre of circle. The tangent \(PT\) meets the diameter \(RQ\) produced at \(P\). If \(PT = 6\) cm, \(QR = 9 \) cm, the length of \(PQ\) is ( Hint \(\triangle PQT \sim \triangle PTR\))

A\(1.5\) cm B\(2\) cm C\(3\) cm D\(3.5 \) cm

Question 15

In the given figure AC is the diameter of the circle with centre \(O\). \(CD\) is parallel to \(BE\). If \(\angle AOB = 80^{\circ}\) and \(\angle ACE = 20^{\circ}\), the sum of the angles \(\angle BEC\), \(\angle BCD\) and \(\angle CED\) is

A\(280^{\circ}\) B\(160^{\circ}\) C\(120^{\circ}\) D\(180^{\circ}\)

Question 16

In the given figure, \(O\) is the centre of the circle and \(AB\) is a tangent to the circle at \(B\).If \(\angle PQB =55^{\circ}\), sum of \(x,y\) and \(z\) is

A\(160^{\circ}\) B\(140^{\circ}\) C\(180^{\circ}\) D\(120^{\circ}\)

Question 17

In the figure given below, \(O\) is the centre of the circle and \(SP\) is a tangent.If \(\angle SRT = 65^{\circ}\), find the sum of \(x, y\) and \(z\).

A\(120^{\circ}\) B\(115^{\circ}\) C\(135^{\circ}\) D\(150^{\circ}\)

Question 18

The radii of two circles with center at \(A\) and \(B\) are \(11\) cm and \(6\) cm respectively. If \(PQ\) is the common tangent of the circles and \(AB = 13\) cm, length of PQ is

A13 cm B12 cm C17 cm D8.5 cm

Question 19

Suppose \(Q\) is a point on the circle with centre \(P\) and radius \(1\), as shown in the figure; \(R\) is a point outside thr circle such that \(QR = 1\) and \(\angle QRP = 2^{\circ}\). Let \(S\) be the point where the segment \(RP\) intersects the given circle. Then measure of \(\angle RQS\) equals.

A\(86^{\circ}\) B\(87^{\circ}\) C\(88^{\circ}\) D\(89^{\circ}\)

Question 20

The chords \(PQ\) and \(RS\) of a circle are extended to meet at the point \(Q\). If \(PQ = 6\) cm, \(OQ = 8\) cm, \(OS = 7\) cm, then length of \(RS\)

A\(12\) cm B\(9\) cm C\(10\) cm D\(16\) cm

Question 21

\(ABC\) is a triangle in which \(AB = 4\) cm, \(BC = 5\) cm and \(AC = 6\) cm. A circle is drawn to touch side \(BC\) at \(P\), side \(AB\) extended at \(Q\) and side \(AC\) extended at \(R\). Then, \(AQ\) equals:

A\(7\) cm B\(7.5\) cm C\(6.5\) cm D\(15\) cm

Question 22

A line from one vertex \(A\) of an equilateral \(\triangle ABC\) meets the opposite side \(BC\) in \(P\) and the circumcircle of \(\triangle ABC\) in \(Q\).If \(BQ = 4\) cm and \(CQ = 3\) cm, then \(PQ\) is equal to

A\(7 \) cm B\(\frac{4}{3}\) cm C\(\frac{12}{7}\) cm D\(2 \) cm

Question 23

In the figure, \(AB, AC\) and \(BC\) are three tangents touching the circle at \(D\), \(E\) and \(F\) respectively. If \(AC = 24\) cm, \(BC = 18\) cm and \( \angle ACB = 90^{\circ}\),the radius of the circle is

A\(3\) cm B\(4\) cm C\(5\) cm D\(6\) cm

Question 24

In the figure, \(ST\) is a tangents to the smaller circle, \(ABC\) is a straight line. If \(\angle TAD = 2x\) and \(\angle DPC = 3x\), find \(x\)

A\(30^{\circ}\) B\(36^{\circ}\) C\(40^{\circ}\) D\(42^{\circ}\)

Question 25

In the adjoining figure, \(ABC\) is a triangle in which, \(\angle B = 90^{\circ}\) and its incircle \(C_1\) has radius \(3\) units. A circle \(C_2\) of radius \(1\) unit touches sides \(AC\), \(BC\) and the circle \(C_1\). Then length \(AB\) is equal to

A\(3+6\sqrt{3}\) B\(10+3\sqrt{2}\) C\(10+2\sqrt{3}\) D\(9+3\sqrt{3}\)

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