CBSE Class 12 Maths Board Paper 2024

Try 3 Class 12 questions like these

Written and checked by us, not taken from the CBSE paper. Tap "Worked answer" to see every step.

1. Continuity and Differentiability · 4 marks

   Crop-spraying drone. A drone flies over a field so that, \(t\) seconds after take-off, its horizontal distance is \(x = 4t\) metres and its height is \(y = 8t - t^2\) metres \((0 \le t \le 8)\).

   1. Find \(\dfrac{dy}{dx}\) in terms of \(t\). [1 mark]

   2. Find the slope of the flight path at \(t = 2\). [1 mark]

   3. Find \(\dfrac{d^2y}{dx^2}\) and state what its sign tells you about the path. [2 marks]

   Worked answer

   1. \(\dfrac{dy}{dx} = \dfrac{8 - 2t}{4} = \dfrac{4 - t}{2}\) A1

   2. \(\dfrac{4 - 2}{2} = 1\) A1

   3. \(\dfrac{d^2y}{dx^2} = \dfrac{\frac{d}{dt}\left(\frac{4-t}{2}\right)}{dx/dt} = \dfrac{-1/2}{4} = -\dfrac18\) M1
      It is negative (constant), so the slope keeps decreasing: the path bends downward throughout A1

2. Integrals · 3 marks

   Find \(\displaystyle\int\frac{x^2 - 3}{x^4 + 5x^2 + 9}\,dx\).

   Worked answer

   Divide by \(x^2\): \(\displaystyle\int\frac{1 - \frac{3}{x^2}}{x^2 + 5 + \frac{9}{x^2}}dx\); put \(t = x + \frac3x\), \(dt = \left(1 - \frac{3}{x^2}\right)dx\), and \(x^2 + \frac{9}{x^2} = t^2 - 6\), so the denominator is \(t^2 - 1\) M1
   \(\displaystyle\int\frac{dt}{t^2 - 1} = \frac12\log\left|\frac{t - 1}{t + 1}\right|\) A1
   \(= \dfrac12\log\left|\dfrac{x^2 - x + 3}{x^2 + x + 3}\right| + C\) A1

3. Vector Algebra · 2 marks

   A vector \(\overrightarrow{PQ}\) of length \(14\) units has direction cosines \(\tfrac27, -\tfrac37, \tfrac67\). If \(P\) is the point \((1, 0, -2)\), find the coordinates of \(Q\) and the direction cosines of \(\overrightarrow{QP}\).

   Worked answer

   \(\overrightarrow{PQ} = 14\left(\tfrac27\hat{i} - \tfrac37\hat{j} + \tfrac67\hat{k}\right) = 4\hat{i} - 6\hat{j} + 12\hat{k}\), so \(Q = (5, -6, 10)\) A1
   \(\overrightarrow{QP} = -\overrightarrow{PQ}\): direction cosines \(-\tfrac27,\ \tfrac37,\ -\tfrac67\) A1

More on these topics: Continuity and Differentiability · Integrals · Vector Algebra

What the paper covers

Class 12 Mathematics (041) is a 3-hour, 80-mark theory paper plus 20 marks of internal assessment. Unit weightage in the CBSE curriculum:

Chapter links will appear once the CBSE course pages are live. We haven't mapped this paper question by question yet.

Formula booklet

CBSE Mathematics (Class 10 and Class 12) · CBSE · No official formula booklet

CBSE does not publish a formula booklet and none is given in the exam: formulas must be learnt. Use our one-page sheets to revise them.

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