MathsCBSE Class 12 · Sample paper, session 2026-27 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2026-27: question 32(B)

From the official CBSE sample question paper for 2026-27, section D, question 32(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(B) · 5 marks · Long answerRelations and Functions

A relation $R : (- \infty, \infty) \to [0, \infty)$ is defined as $R = {(x, y) : y = x^{2} + 5}$
Draw its graph and check whether it is a function or not.
If the given relation is a function, then examine whether the function is injective, and whether it is surjective or not.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Use the vertical line test on the parabola to decide if it is a function.

Hint 2

For one-one, compare $f (1)$ and $f (- 1)$ ; for onto, compare the range with the co-domain $[0, \infty)$ .

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.

Graph: $y = x^{2} + 5$ is an upward-opening parabola, symmetric about the y-axis, with vertex $(0, 5)$ . It passes through $(\pm 1, 6)$ , $(\pm 2, 9)$ and $(\pm 3, 14)$ and lies entirely above the line $y = 5$ .
1 mark
Every vertical line $x = a$ meets the graph in exactly one point, so every element of $(- \infty, \infty)$ has a unique image $x^{2} + 5 \in [0, \infty)$ . Hence $R$ is a function; call it $f (x) = x^{2} + 5$ .
1 mark
Injective: $f (- 1) = (- 1)^{2} + 5 = 6$ and $f (1) = 1^{2} + 5 = 6$ . Since $- 1 \neq = 1$ but $f (- 1) = f (1)$ , $f$ is many-one, so it is not injective.
1½ marks
Surjective: The range of $f$ is $[5, \infty)$ , which is not equal to the co-domain $[0, \infty)$ . For example, $y = 1$ would need $x^{2} = - 4$ , which has no real solution. So $f$ is not surjective.
1½ marks

Answer: $R$ is a function; it is neither injective nor surjective.

Where the marks are

Draws the correct graph of $y = x^{2} + 5$ (1 mark)
Justifies that $R$ is a function (1 mark)
Shows it is not injective, e.g. $f (- 1) = f (1)$ (1½ marks)
Shows it is not surjective, since the range $[5, \infty)$ is not the co-domain $[0, \infty)$ (1½ marks)

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 paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 32(B)

Hints

Step-by-step solution

Revise the chapter

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