By signing in you confirm you are 18 or older, or have your parent’s or guardian’s consent.

NDA 1 2025 Maths, question 40

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.

Q40

What is MNMN equal to?

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

Solution.

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

  1. With h=MNh = MN: QN=hQN = h and RN=hcot⁡60∘=h/3RN = h\cot 60^{\circ} = h/\sqrt{3}.
  2. On the same side, b=QN−RN=h(1−13)=h3−13b = QN - RN = h\left( 1 - \frac{1}{\sqrt{3}} \right) = h\frac{\sqrt{3} - 1}{\sqrt{3}}.
  3. So MN=h=33−1b=3+32bMN = h = \frac{\sqrt{3}}{\sqrt{3} - 1}b = \frac{3 + \sqrt{3}}{2}b.

Trap: Placing Q and R on opposite sides of the tower gives (b); as in question 39, the points are on one side.