MathsCBSE Class 12 · Sample paper, session 2024-25 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2024-25: question 33

From the official CBSE sample question paper for 2024-25, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerCompetency-basedMatrices

The equation of the path traversed by the ball headed by the footballer is $y = a x^{2} + b x + c$ ; (where $0 \leq x \leq 14$ and $a, b, c \in R$ and $a \neq = 0$ ) with respect to a XY-coordinate system in the vertical plane. The ball passes through the points $(2, 15)$ , $(4, 25)$ and $(14, 15)$ . Determine the values of a, b and c by solving the system of linear equations in a, b and c, using matrix method. Also find the equation of the path traversed by the ball.

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

Each given point gives one equation in $a, b, c$ .

Hint 2

Write the system as $A X = B$ and use $X = \frac{1}{∣ A ∣} (adj A) B$ .

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.

Substituting the points in $y = a x^{2} + b x + c$ : $4 a + 2 b + c = 15$ , $16 a + 4 b + c = 25$ , $196 a + 14 b + c = 15$ .
1 mark
Matrix form $A X = B$ with $A = 4161962414111$ , $X = a b c$ , $B = 152515$ .
½ mark
$∣ A ∣ = 4 (4 - 14) - 2 (16 - 196) + 1 (224 - 784) = - 40 + 360 - 560 = - 240 \neq = 0$ , so $A^{- 1}$ exists.
½ mark
Cofactors: $A_{11} = - 10, A_{12} = 180, A_{13} = - 560, A_{21} = 12, A_{22} = - 192, A_{23} = 336, A_{31} = - 2, A_{32} = 12, A_{33} = - 16$ . So $adj A = - 10 180 - 560 12 - 192 336 - 2 12 - 16$ .
1 mark
$X = A^{- 1} B = - \frac{1}{240} - 10 180 - 560 12 - 192 336 - 2 12 - 16 152515 = - \frac{1}{240} 120 - 1920 - 240$ .
1 mark
Hence $a = - \frac{1}{2}$ , $b = 8$ , $c = 1$ .
½ mark
Path of the ball: $y = - \frac{1}{2} x^{2} + 8 x + 1$ .
½ mark

Answer: $a = - \frac{1}{2}, b = 8, c = 1$ ; path $y = - \frac{1}{2} x^{2} + 8 x + 1$

Where the marks are

Forms the three linear equations (1 mark)
Writes $A X = B$ (½ mark)
$∣ A ∣ = - 240 \neq = 0$ (½ mark)
Correct adj $A$ (1 mark)
Computes $X = A^{- 1} B$ (1 mark)
$a = - \frac{1}{2}, b = 8, c = 1$ (½ mark)
Equation of path (½ mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Matrices, 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 33

Hints

Step-by-step solution

Revise the chapter

More questions from Matrices