MathsCBSE Class 12 · Board exam 2023, set 65/5/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2023: question 32(b)

Set in the CBSE Class 12 board exam of 2023, paper 65/5/1, 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 answerApplication of Derivatives

Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/5/1.

Hints

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

Hint 1

Use $x + y = 5$ to write the sum of cubes as a function of $x$ alone.

Hint 2

Set the derivative to zero and confirm a minimum with the second derivative before finding $x^{2} + y^{2}$ .

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.

Let the numbers be $x$ and $y$ with $x + y = 5$ , so $y = 5 - x$ .
½ mark
Sum of cubes: $S = x^{3} + y^{3} = x^{3} + (5 - x)^{3}$
1 mark
$\frac{d S}{d x} = 3 x^{2} - 3 (5 - x)^{2} = 3 (x - 5 + x) (x + 5 - x) = 15 (2 x - 5)$
1 mark
Setting $\frac{d S}{d x} = 0$ gives $x = \frac{5}{2}$ .
½ mark
$\frac{d ^{2} S}{d x ^{2}} = 6 x + 6 (5 - x) = 30 > 0$ , so $S$ is minimum at $x = \frac{5}{2}$ .
1 mark
The numbers are $\frac{5}{2}$ and $\frac{5}{2}$ .
Sum of squares $= \frac{25}{4} + \frac{25}{4} = \frac{25}{2}$
1 mark

Answer: Sum of squares = $\frac{25}{2}$ (numbers are $\frac{5}{2}$ and $\frac{5}{2}$ )

Where the marks are

Expresses $y = 5 - x$ (½ mark)
Forms $S = x^{3} + (5 - x)^{3}$ (1 mark)
Gets $\frac{d S}{d x} = 15 (2 x - 5)$ (1 mark)
Critical point $x = \frac{5}{2}$ (½ mark)
Second derivative test shows a minimum (1 mark)
Sum of squares $= \frac{25}{2}$ (1 mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Maxima and Minima

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 paper2023 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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