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NDA 1 2025 Maths, question 39

Topic: Trigonometry

Passage · Q39 to Q40

The top (M)(M) of a tower is observed from three points P,Q,RP,Q,R lying in a horizontal straight line which passes directly along the foot (N)(N) of the tower. The angles of elevations of MM from P,Q,RP,Q,R are 30∘30^{\circ}, 45∘45^{\circ} and 60∘60^{\circ} respectively. Let PQ=aPQ = a and QR=bQR = b.

Q39

What is PNPN equal to?

  1. (a) (3−32)a\left( \frac{3 - \sqrt{3}}{2} \right)a
  2. (b) (3+32)a\left( \frac{3 + \sqrt{3}}{2} \right)aCorrect answer
  3. (c) (3−34)a\left( \frac{3 - \sqrt{3}}{4} \right)a
  4. (d) (3+34)a\left( \frac{3 + \sqrt{3}}{4} \right)a

Solution.

Answer: (b) (3+32)a\left( \frac{3 + \sqrt{3}}{2} \right)a

  1. Let h=MNh = MN. Then PN=hcot⁡30∘=3hPN = h\cot 30^{\circ} = \sqrt{3}h and QN=hcot⁡45∘=hQN = h\cot 45^{\circ} = h.
  2. With P, Q, R on the same side of the tower, a=PN−QN=(3−1)ha = PN - QN = (\sqrt{3} - 1)h, so h=a3−1h = \frac{a}{\sqrt{3} - 1}.
  3. PN=3h=3a3−1=3+32aPN = \sqrt{3}h = \frac{\sqrt{3}a}{\sqrt{3} - 1} = \frac{3 + \sqrt{3}}{2}a.

Trap: The passage does not say the points are on the same side; if P and Q were on opposite sides, a=PN+QNa = PN + QN and you would get (a). Rising angles 30°, 45°, 60° mean the observer is walking towards the tower on one side.