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NECO 2022 · Paper 2 · Q9

  1. (a)

    Make MM the subject of the formula M=N−M2f+1d−NM = \sqrt{\frac{N - M^2 f + 1}{d - N}}. M=M =

  2. (a)(hence)

    Hence, evaluate MM if N=3N = 3, f=4f = 4 and d=8d = 8.

  3. (b)

    If RR varies inversely as the cube of SS and R=8R = 8 when S=4S = 4, find SS when R=512216R = \frac{512}{216}.

  4. (c)

    Find the derivative of y=x43y = \sqrt[3]{x^4} with respect to xx. dydx=\frac{dy}{dx} =

Worked solution (try it first)

(a)

  1. Square both sides: M2=N−M2f+1d−NM^2 = \frac{N - M^2 f + 1}{d - N}.
  2. Multiply both sides by (d−N)(d - N): M2(d−N)=N−M2f+1M^2(d - N) = N - M^2 f + 1.
  3. Collect the M2M^2 terms on the left: M2(d−N)+M2f=N+1M^2(d - N) + M^2 f = N + 1.
  4. Factorise: M2(d−N+f)=N+1M^2(d - N + f) = N + 1.
  5. Divide and take the square root: M=N+1d−N+fM = \sqrt{\frac{N + 1}{d - N + f}}.
  6. Substitute N=3N = 3, f=4f = 4, d=8d = 8: M=3+18−3+4M = \sqrt{\frac{3 + 1}{8 - 3 + 4}}
    =49= \sqrt{\frac49}
    =23= \frac23.

(b)

  1. Inverse variation with the cube: R=kS3R = \frac{k}{S^3}.
  2. Use R=8R = 8, S=4S = 4: 8=k648 = \frac{k}{64}, so k=512k = 512.
  3. Put in R=512216R = \frac{512}{216}: 512216=512S3\frac{512}{216} = \frac{512}{S^3}, so S3=216S^3 = 216.
  4. Take the cube root: S=6S = 6.

(c)

  1. Write the root as a power: y=x4/3y = x^{4/3}.
  2. Bring down the power and reduce it by 1: dydx=43x1/3\frac{dy}{dx} = \frac43 x^{1/3}
    =43x3= \frac43\sqrt[3]{x}.

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