Quiz Tweek

Relation and Function Set-2

Test your knowledge on Relations And Functions from Mathematics, Class 12.

60

Minutes

40

Questions

1 / -0

Marking Scheme

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Questions in this Quiz

Q1: Let $f : R \rightarrow R$ be defined by $f(x) = 3x^2 – 5$ and $g : R \rightarrow R$ by $g(x) = \frac{2}{x^2 + 1}$ then $gof$ is
  • $\frac{4x^2 - 30x + 26}{9x^2 - 30x + 26}$
  • $\frac{4x^2 - 3}{9x^2 + 6x - 26}$
  • $\frac{4x^2 + 2x - 3}{9x^2 - 30x + 2}$
  • $\frac{4}{9x^4 - 30x^2 + 26}$
Q2: If the set $A$ contains 5 elements and set $B$ contains 6 elements, then the number of one-one and onto mapping from $A$ to $B$ is
  • 720
  • 120
  • 0
  • None of these
Q3: Let $f : \rightarrow$ be defined by $f(x) = \{x, \text{ if } x \text{ is rational} ; 1-x, \text{ if } x \text{ is irrational}\}$. Then $(fof)x$ is
  • constant
  • $1 + x$
  • $x$
  • None of these
Q4: Let $f : [2, \infty) \rightarrow R$ be the function defined by $f(x) = x^2 – 4x + 5$, then the range of $f$ is
  • $R$
  • $[1, \infty)$
  • $[4, \infty)$
  • $[5, \infty)$
Q5: Let $f : R \rightarrow R$ be defined by $f(x) = \{2x^2 : x > 3; x^2 : 1 \leq x \leq 3; 3x : x < 1\}$. Then $f(– 1) + f(2) + f(4)$ is
  • 9
  • 14
  • 5
  • None of these
Q6: The relation $R$ is defined on the set of natural numbers as $\{(a, b) : a = 2b\}$. Then, $R^{-1}$ is given by
  • $\{(2, 1), (4, 2), (6, 3),\ldots.\}$
  • $\{(1, 2), (2, 4), (3, 6), \ldots.\}$
  • $R^{-1}$ is not defiend
  • None of these
Q7: Let $f : R \rightarrow R$ be a function defined by $f(x) = \frac{e^{|x|}-e^{-x}}{e^{x}+e^{-x}}$ then $f(x)$ is
  • one-one onto
  • one-one but not onto
  • onto but not one-one
  • None of these
Q8: Let $g(x) = x^2 – 4x – 5$, then
  • $g$ is one-one on $R$
  • $g$ is not one-one on $R$
  • $g$ is bijective on $R$
  • None of these
Q9: The function $f : R \rightarrow R$ given by $f(x) = x^3 – 1$ is
  • a one-one function
  • an onto function
  • a bijection
  • neither one-one nor onto
Q10: Let $f : [0, \infty) \rightarrow$ be defined by $f(x)=\frac{2 x}{1+x}$, then $f$ is
  • one-one but not onto
  • onto but not one-one
  • both one-one and onto
  • neither one-one nor onto
Q11: If $N$ be the set of all-natural numbers, consider $f : N \rightarrow N$ such that $f(x) = 2x, \forall x \in N$, then $f$ is
  • one-one onto
  • one-one into
  • many-one onto
  • None of these
Q12: Let $A = \{x : -1 \leq x \leq 1\}$ and $f : A \rightarrow A$ is a function defined by $f(x) = x |x|$ then $f$ is
  • a bijection
  • injection but not surjection
  • surjection but not injection
  • neither injection nor surjection
Q13: Let $f : R \rightarrow R$ be a function defined by $f(x) = x^3 + 4$, then $f$ is
  • injective
  • surjective
  • bijective
  • none of these
Q14: If $f(x) = (ax^2 – b)^3$, then the function $g$ such that $f\{g(x)\} = g\{f(x)\}$ is given by
  • $g(x)=\left(\frac{b-x^{1 / 3}}{a}\right)^{1 / 2}$
  • $g(x)=\frac{1}{\left(a x^{2}+b\right)^{3}}$
  • $g(x)=\left(a x^{2}+b\right)^{1 / 3}$
  • $g(x)=\left(\frac{x^{1 / 3}+b}{a}\right)^{1 / 2}$
Q15: If $f : [1, \infty) \rightarrow [2, \infty)$ is given by $f(x) = x + \frac{1}{x}$, then $f^{-1}$ equals to
  • $\frac{x+\sqrt{x^{2}-4}}{2}$
  • $\frac{x}{1+x^{2}}$
  • $\frac{x-\sqrt{x^{2}-4}}{2}$
  • $1+\sqrt{x^{2}-4}$
Q16: Let $f(x) = x^2 – x + 1, x \geq \frac{1}{2}$, then the solution of the equation $f(x) = f^{-1}(x)$ is
  • $x = 1$
  • $x = 2$
  • $x = \frac{1}{2}$
  • None of these
Q17: Which one of the following function is not invertible?
  • $f : R \rightarrow R, f(x) = 3x + 1$
  • $f : R \rightarrow [0, \infty), f(x) = x^2$
  • $f : R^+ \rightarrow R^+, f(x) = \frac{1}{x^{3}}$
  • None of these
Q18: The inverse of the function $y=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}$ is
  • $\log _{10}(2-x)$
  • $\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right)$
  • $\frac{1}{2} \log _{10}(2 x-1)$
  • $\frac{1}{4} \log \left(\frac{2 x}{2-x}\right)$
Q19: If $f : R \rightarrow R$ defind by $f(x) = \frac{2 x-7}{4}$ is an invertible function, then find $f^{-1}$.
  • $\frac{4 x+5}{2}$
  • $\frac{4 x+7}{2}$
  • $\frac{3 x+2}{2}$
  • $\frac{9 x+3}{5}$
Q20: Consider the function $f$ in $A = R – \{\frac{2}{3}\}$ defiend as $f(x)=\frac{4 x+3}{6 x-4}$. Find $f^{-1}$.
  • $\frac{3+4 x}{6 x-4}$
  • $\frac{6 x-4}{3+4 x}$
  • $\frac{3-4 x}{6 x-4}$
  • $\frac{9+2 x}{6 x-4}$
Q21: If $f$ is an invertible function defined as $f(x) = \frac{3 x-4}{5}$, then $f^{-1}(x)$ is
  • $5x + 3$
  • $5x + 4$
  • $\frac{5 x+4}{3}$
  • $\frac{3 x+2}{3}$
Q22: If $f : R \rightarrow R$ defined by $f(x) = \frac{3 x+5}{2}$ is an invertible function, then find $f^{-1}$.
  • $\frac{2 x-5}{3}$
  • $\frac{x-5}{3}$
  • $\frac{5 x-2}{3}$
  • $\frac{x-2}{3}$
Q23: Let $f : R \rightarrow R, g : R \rightarrow R$ be two functions such that $f(x) = 2x – 3, g(x) = x^3 + 5$. The function $(fog)^{-1} (x)$ is equal to
  • $\left(\frac{x+7}{2}\right)^{1 / 3}$
  • $\left(x-\frac{7}{2}\right)^{1 / 3}$
  • $\left(\frac{x-2}{7}\right)^{1 / 3}$
  • $\left(\frac{x-7}{2}\right)^{1 / 3}$
Q24: Let $*$ be a binary operation on set of integers $I$, defined by $a * b = a + b – 3$, then find the value of $3 * 4$.
  • 2
  • 4
  • 7
  • 6
Q25: If $*$ is a binary operation on set of integers $I$ defined by $a * b = 3a + 4b – 2$, then find the value of $4 * 5$.
  • 35
  • 30
  • 25
  • 29
Q26: Let $*$ be the binary operation on $N$ given by $a * b = HCF (a, b)$ where, $a, b \in N$. Find the value of $22 * 4$.
  • 1
  • 2
  • 3
  • 4
Q27: Consider the binary operation $*$ on $Q$ defind by $a * b = a + 12b + ab$ for $a, b \in Q$. Find $2 * \frac{1}{3}$.
  • $\frac{20}{3}$
  • 4
  • 18
  • $\frac{16}{3}$
Q28: The domain of the function $f(x)=\frac{1}{\sqrt{\{\sin x\}+\{\sin (\pi+x)\}}}$ where $\{.\}$ denotes fractional part, is
  • $[0, \pi]$
  • $(2n + 1) \pi/2, n \in Z$
  • $(0, \pi)$
  • None of these
Q29: Range of $f(x)=\sqrt{(1-\cos x) \sqrt{(1-\cos x) \sqrt{(1-\cos x) \ldots \ldots \infty}}}$
  • $[0,1]$
  • $(0, 1)$
  • $[0,2]$
  • $(0, 2)$
Q30: Let $A = \{1, 2, 3, \ldots n\}$ and $B = \{a, b\}$. Then the number of surjections from $A$ into $B$ is
  • $^n P_2$
  • $2^n – 2$
  • $2^n – 1$
  • none of these
Q31: Let $T$ be the set of all triangles in the Euclidean plane and let a relation $R$ on $T$ be defined as $a R b$, if $a$ is congruent to $b$, $\forall a, b \in T$. Then $R$ is
  • Reflexive but not transitive
  • Transitive but not symmetric
  • Equivalence
  • None of these
Q32: Consider the non-empty set consisting of children in a family and a relation $R$ defined as $a R b$, if $a$ is brother of $b$. Then $R$ is
  • symmetric but not transitive
  • transitive but not symmetric
  • neither symmetric nor transitive
  • both symmetric and transitive
Q33: The maximum number of equivalence relations on the set $A = \{1, 2, 3\}$ are
  • 1
  • 2
  • 3
  • 5
Q34: If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$, then $R$ is
  • reflexive
  • transitive
  • symmetric
  • None of these
Q35: Let us define a relation $R$ in $R$ as $a R b$ if $a \geq b$. Then $R$ is
  • an equivalence relation
  • reflexive, transitive but not symmetric
  • symmetric, transitive but not reflexive
  • neither transitive nor reflexive but symmetric.
Q36: Let $A = \{1, 2, 3\}$ and consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$, then $R$ is
  • reflexive but not symmetric
  • reflexive but not transitive
  • symmetric and transitive
  • neither symmetric nor transitive.
Q37: The identity element for the binary operation $*$ defined on $Q \sim \{0\}$ as $a * b = \frac{a b}{2}$ $\forall a, b \in Q \sim \{0\}$ is
  • 1
  • 0
  • 2
  • None of these
Q38: The function $f : A \rightarrow B$ defined by $f(x) = 4x + 7, x \in R$ is
  • one-one
  • Many-one
  • Odd
  • Even
Q39: The smallest integer function $f(x) = [x]$ is
  • One-one
  • Many-one
  • Both (a) & (b)
  • None of these
Q40: The function $f : R \rightarrow R$ defined by $f(x) = 3 – 4x$ is
  • Onto
  • Not onto
  • None one-one
  • None of these