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NECO 2022 · Paper 2 · Q3

  1. (a)

    Calculate the value of pp if the distance between the points (p,5)(p, 5) and (5,9)(5, 9) is 5. (Enter both values.)

    Separate values with commas, e.g. 3, −2

  2. (b)

    Find the angle between two lines whose slopes are 12 and 25\frac25 (to 2 decimal places).

Worked solution (try it first)

(a)

  1. Use the distance formula: (5−p)2+(9−5)2=52(5 - p)^2 + (9 - 5)^2 = 5^2.
  2. Simplify the known squares: (5−p)2+16=25(5 - p)^2 + 16 = 25.
  3. Subtract 16: (5−p)2=9(5 - p)^2 = 9.
  4. Take square roots: 5−p=±35 - p = \pm 3.
  5. So p=2p = 2 or p=8p = 8.
  6. Both points are 5 units from (5,9)(5, 9).

(b)

  1. For slopes m1m_1 and m2m_2, the angle θ\theta between the lines satisfies tan⁡θ=∣m1−m21+m1m2∣\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|.
  2. Numerator: 12−25=58512 - \frac25 = \frac{58}{5}.
  3. Denominator: 1+12×25=1+2451 + 12 \times \frac25 = 1 + \frac{24}{5}
    =295= \frac{29}{5}.
  4. Divide 585\frac{58}{5} by 295\frac{29}{5}: the fives cancel and 58÷29=258 \div 29 = 2, so tan⁡θ=2\tan\theta = 2.
  5. So θ=tan⁡−12≈63.43∘\theta = \tan^{-1} 2 \approx 63.43^\circ.

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