If $A$, $B$ and $C$ are the elements of Boolean algebra, simplify the expression $(A' + B')(A + C') + B'(B + C)$. Draw the simplified circuit.
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Relations and Functions - ISC Class 12 Mathematics Questions with Answers, Page 4
62 past-paper questions on Relations and Functions from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 61-62 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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Prove that locus of $z$ is circle and find its centre and radius if $\frac{z-i}{z-1}$ is purely imaginary.
Given: $\frac{z-i}{z-1}$ is purely imaginary.
To show: Locus of $z$ is circle with centre $(1/2, 1/2)$ and radius $1/\sqrt{2}$.
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Questions on other pages on Relations and Functions
- Let $g(x) = x - 6\sqrt{x} + 1, 0 \le x \le 9$. Find $g^{-1}(x)$ and verify $(g \circ g^{-1})(4) = 4$. 2027 · 1 mark · Short answer
- A relation $R$ is defined on $\mathbb{Z}$ as $a R b$ if and only if $a^2 - 7ab + 6b^2 = 0$. Then $R$ is: 2027 · 1 mark · MCQ
- Does $f(x)$ have an inverse function? Explain your answer. 2027 · 1 mark · Short answer
- Find the value of $x$ such that $g^{-1}(x) = h^{-1}(x)$. 2027 · 1 mark · Short answer
- State the natural domain of $f(x)$. 2027 · 1 mark · Short answer
- Let $h(x) = x - 6\sqrt{x} + 1, x \ge 9$. Find $h^{-1}(x)$ and verify $(h \circ h^{-1})(16) = 16$. 2027 · 1 mark · Short answer
- The curves given above can be considered as $f(x)$ and $f^{-1}(x)$. Observe the given graph: 2027 · 1 mark · Assertion-reason
- Find the range of the function shown in the graph below: 2026 · 1 mark · Numerical
- $R$ is not an equivalence relation. If $A = \{1, 2, 3\}$ and the relation $R = \{(1, 1), (2, 2), (3, 3), (2, 3)\}$, then 2026 · 1 mark · Assertion-reason
- Find the range of the function $f(x) = \frac{1}{3 - 2\sin x}$. 2026 · 2 marks · Numerical
- For what values of $x$ is the function $f(x) = \frac{4x+3}{6x-4}$ invertible? Also, find the inverse function. 2026 · 2 marks · Numerical
- Prove that the function $f: \mathbb{Z} \to \mathbb{Z}$, defined by $f(x) = x - 5$ is a bijective function. 2026 · 2 marks · Derivation
- Let $f(x) = 4 - (x - 7)^3$ be an invertible function, then find $f^{-1}(x)$. 2026 · 2 marks · Numerical
- If set $A$ contains four elements and set $B$ contains five elements, then the number of one-one and onto mapping from $A \to B$… 2026 · 1 mark · MCQ
- A relation $R$ on the set $A = \{a, b, c\}$ is defined by $R = \{(a, b), (b, a)\}$. Is the relation $R$ symmetric? Justify. 2026 · 1 mark · Short answer
- The curve in the graph below is not a one-one function: Assertion (A): The curve in the graph below is not a one-one function… 2025 · 1 mark · Assertion-reason
- “A function $f$ is called a self-inverse function if $f^{-1}(x) = f(x)$ for all values of $x$ in the domain.” Let… 2025 · 2 marks · Short answer
- Answer the following questions: If a real-valued function is given by: $f(x) = \sqrt{25 - x^2}$ is an onto function, then find… 2025 · 4 marks · Long answer
- If $h(x) = 4^x$ and $h^{-1}(x) = 2$, then value of $x$ is: 2025 · 1 mark · MCQ
- $f(x) = x^2 + 2x - 3$ for $x \ge -1$. Given that the minimum value of $x^2 + 2x - 3$ occurs when $x = -1$, explain why $f(x)$ has… 2025 · 1 mark · Short answer
- Explain why $x + 4y = 12$, $x, y \in \mathbb{N}$ is NOT a symmetric relation. 2025 · 1 mark · Short answer
- Statement I: For any two real numbers $a$ and $b$, we define $a\text{ R }b$ if $\sec^2 a - \tan^2 b = 1$. Then, R is transitive… 2025 · 1 mark · MCQ
- The values of two functions $f$ and $g$ for certain values of $x$ are given in the following table: $x$ -2 0 3 $f(x)$ -12 -4 8… 2025 · 2 marks · Short answer
- If $f: [1, \infty) \to [2, \infty)$ is given by $f(x) = x + \frac{1}{x}$ then, find $\frac{d}{dx}f^{-1}(x)$. 2025 · 4 marks · Short answer
- Consider the following graph of $f(x)$, $f: \mathbb{R} \to \mathbb{R}$: Statement 1: The function is one-one function. Statement… 2025 · 1 mark · MCQ
- If Set $A$ has $m$ elements, Set $B$ has $n$ elements and $n < m$, then the number of one-one function(s) from $A \to B$ is zero. 2025 · 1 mark · Assertion-reason
- A part of the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 8, x \in \mathbb{R}$ is shown below: Answer the following… 2025 · 4 marks · Long answer
- Statement I: $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \frac{1}{x} - 2$, is neither injective nor surjective. Statement II… 2025 · 1 mark · MCQ
- If $R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x - 3$, find the range of $R^{-1}$. 2025 · 1 mark · One word
- The relation $f: \{m, n, p, q\} \to \{11, 12, 13, 14\}$ defined by $f = \{(m, 11), (n, 12), (p, 13)\}$ is a bijective function… 2025 · 1 mark · Assertion-reason
- Write the smallest equivalence relation from the set $A$ to $A$, where $A = \{1, 2, 3\}$. 2025 · 1 mark · One word
- Let $L$ be a set of all straight lines in a plane. The relation $R$ on $L$ defined as 'perpendicular to' is: 2024 · 1 mark · MCQ
- Which one of the following graphs is a function of $x$? Graph A Graph B 2024 · 1 mark · Short answer
- Let $f: \mathbb{R} - \left\{-\frac{1}{3}\right\} \to \mathbb{R} - \{0\}$ be defined as $f(x) = \frac{5}{3x + 1}$ is invertible… 2024 · 2 marks · Numerical
- Let $A$ be a non-empty set. Statement 1: Identity relation on $A$ is Reflexive. Statement 2: Every Reflexive relation on $A$ is… 2024 · 1 mark · MCQ
- If $f: \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = \frac{2x - 7}{4}$, show that $f(x)$ is one - one and onto. 2024 · 2 marks · Derivation
- Write the smallest equivalence relation on the set $A = \\{a, b, c\\}$ 2023 · 1 mark · Short answer
- Let $A$ be the set of all $50$ cards numbered from $1$ to $50$. Let $f: A \to \mathbb{N}$ be a function defined by $f(x) =$ card… 2023 · 1 mark · MCQ
- If $f(x) = [4 - (x - 7)^3]^{\frac{1}{5}}$ is a real invertible function, then find $f^{-1}(x)$. 2023 · 2 marks · Short answer
- A relation $R$ on $\{1, 2, 3\}$ is given by $R = \{(1, 1), (2, 2), (1, 2), (3, 3), (2, 3)\}$. Then the relation $R$ is: 2023 · 1 mark · MCQ
- Which of the following is NOT an equivalence relation on $\mathbb{Z}$? 2023 · 1 mark · MCQ
- Let $f(x) = x^3$ be a function with domain $\{0, 1, 2, 3\}$. Then domain of $f^{-1}$ is: 2023 · 1 mark · MCQ
- Let $A = \mathbb{R} - \{2\}$ and $B = \mathbb{R} - \{1\}$. If $f : A \to B$ is a function defined by $f(x) = \frac{x - 1}{x - 2}$… 2023 · 2 marks · Derivation
- Let $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = 2x - 3$. Find $f^{-1}(x)$ domain and range of $f^{-1}(x)$ 2023 · 2 marks · Short answer
- If set A contains 5 elements and set B contains 6 elements, then the number of one-one onto mappings from A to B is: 2022 · 2 marks · MCQ
- The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \sin(3x + 2)$, $\forall x \in \mathbb{R}$ is: 2022 · 2 marks · MCQ
- Consider the mapping $f: A \to B$ is defined by $f(x) = \frac{x-1}{x-2}$ such that $f(x)$ is one-one onto. Based on the above… 2022 · 8 marks · Case based
- Let A be the set of all students of a boy’s school. Then the relation R in A is defined by… 2022 · 2 marks · MCQ
- Let $R$ be the relation in the set $N$, given by $R = \{(x, y) : x = y + 3 \wedge y > 5\}$. Choose the correct answer from the… 2021 · 1 mark · MCQ
- Write total number of functions from set A to set B, where $A = \{1, 2, 3\}$, $B = \{p, q, r, s\}$. 2021 · 1 mark · Short answer
- If $A = \{1, 2, 3\}$, $B = \{x, y\}$, then the number of functions from $A$ to $B$ is: 2021 · 1 mark · MCQ
- Show that the function $f : N \to N$, defined by $f(x) = 5x - 3, \forall x \in N$, is one-one function but not onto function. 2021 · 2 marks · Derivation
- If the function $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \frac{3x + 4}{5x - 7}, x \ne \frac{7}{5}$ and… 2020 · 4 marks · Derivation
- Determine whether the binary operation $$ on $\mathbb{R}$ defined by $a b = |a - b|$ is commutative. Also, find the value of… 2020 · 2 marks · Short answer
- If $f : A \to A$ and $A = \mathbb{R} - \left\\{\frac{8}{5}\right\\}$, show that the function $f(x) = \frac{8x + 3}{5x - 8}$ is… 2019 · 4 marks · Short answer
- If $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ and $g : \mathbb{R} \to \mathbb{R}$, $g(x) = 2x^2 + 1$, and $\mathbb{R}$ is the… 2019 · 2 marks · Short answer
- The binary operation $* : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined as $a * b = 2a + b$. Find $(2 * 3) * 4$. 2018 · 2 marks · Short answer
- A binary operation $$ defined on $\mathbb{Q} - \{1\}$ is given by $a b = a + b - ab$. Find the identity element. 2018 · 2 marks · Short answer
- If the function $f(x) = \sqrt{2x-3}$ is invertible then find its inverse. Hence prove that $(f \circ f^{-1})(x) = x$. 2018 · 4 marks · Derivation
- Let $R^+$ be the set of all positive real numbers and $f: R^+ \to [4, \infty): f(x) = x^2 + 4$. Show that inverse of $f$ exists… 2018 · 4 marks · Derivation
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