MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 26(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section C, question 26(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(b) · 3 marks · Short answerRelations and Functions

A function f is defined from $R \to R$ as $f (x) = a x + b$ , such that $f (1) = 1$ and $f (2) = 3$ . Find function f(x). Hence, check whether function f(x) is one-one and onto or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Substitute x = 1 and x = 2 into ax + b to get two linear equations.

Hint 2

For one-one, start from $f (x_{1}) = f (x_{2})$ .

Hint 3

For onto, solve $y = 2 x - 1$ for $x$ and check it lies in R.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$f (1) = 1 \Rightarrow a + b = 1$ ; $f (2) = 3 \Rightarrow 2 a + b = 3$ . Subtracting, $a = 2$ , $b = - 1$ . So $f (x) = 2 x - 1$ .
1 mark
One-one: Let $f (x_{1}) = f (x_{2})$ for $x_{1}, x_{2} \in R$ . Then $2 x_{1} - 1 = 2 x_{2} - 1 \Rightarrow x_{1} = x_{2}$ . Hence f is one-one.
1 mark
Onto: Let $y \in R$ (codomain) and $y = 2 x - 1 \Rightarrow x = \frac{y + 1}{2} \in R$ (domain). Then $f (\frac{y + 1}{2}) = 2 \cdot \frac{y + 1}{2} - 1 = y$ . So every $y$ has a pre-image; f is onto.
1 mark

Answer: f(x) = 2x − 1; f is both one-one and onto.

Where the marks are

Solves for a = 2, b = −1 to get f(x) = 2x − 1 (1 mark)
Shows one-one (1 mark)
Shows onto by finding pre-image $(y + 1) /2$ (1 mark)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 26(b)

Hints

Step-by-step solution

Revise the chapter

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