According to a psychologist, the ability of a person to understand spatial concepts is given by A=1/3√(t), where t is the age in years, t[5,18]. Show that the rate of increase of the ability to understand spatial concepts decreases with age in between 5 and 18.

(d²A)/(dt²)=-(1)/(12t t)<0, so the rate of increase decreases with age.

MathsCBSE Class 12 · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 27

From the official CBSE sample question paper for 2024-25, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerCompetency-basedApplication of Derivatives

According to a psychologist, the ability of a person to understand spatial concepts is given by $A = \frac{1}{3} t$ , where $t$ is the age in years, $t \in [5, 18]$ . Show that the rate of increase of the ability to understand spatial concepts decreases with age in between 5 and 18.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

The 'rate of increase' is $\frac{d A}{d t}$ .

Hint 2

To show a quantity decreases, check the sign of its derivative, i.e. $\frac{d ^{2} A}{d t ^{2}}$ .

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.

$A = \frac{1}{3} t \Rightarrow \frac{d A}{d t} = \frac{1}{3} \cdot \frac{1}{2} t^{- 1/2} = \frac{1}{6 t}$ , the rate of increase of ability.
1 mark
$\frac{d ^{2} A}{d t ^{2}} = \frac{1}{6} (- \frac{1}{2}) t^{- 3/2} = - \frac{1}{12 t t}$ .
1 mark
For $t \in (5, 18)$ , $t t > 0$ , so $\frac{d ^{2} A}{d t ^{2}} < 0$ .
½ mark
Hence $\frac{d A}{d t}$ is decreasing: the rate of increase of the ability to understand spatial concepts slows down with age between 5 and 18.
½ mark

Answer: $\frac{d ^{2} A}{d t ^{2}} = - \frac{1}{12 t t} < 0$ , so the rate of increase decreases with age.

Where the marks are

$\frac{d A}{d t} = \frac{1}{6 t}$ (1 mark)
$\frac{d ^{2} A}{d t ^{2}} = - \frac{1}{12 t t}$ (1 mark)
Shows it is negative on $(5, 18)$ (½ mark)
Conclusion (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Rate of Change of Quantities

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics sample paper 2024-25: question 27

Hints

Step-by-step solution

Revise the chapter

More questions from Application of Derivatives