Index Number: \( \qquad \) Question No.: \( \qquad \) Do not write in is margin. Sechon B elovits Answer four questions only from this section, with at least one question from each part. All questions carry equal mairks. Pakt I Pure Mathematics 9. (a) If the area of a circle with radius 9 cm is increasing at the rate of \( 33 \mathrm{~cm}^{2} \mathrm{~s}^{-1} \), find the rate at which the radius is increasing, leaving your answer in terms of \( \pi \). (b) (i) Find the binomial expansion of \( (2 x+1)^{6} \) in descending powers of \( x \). (ii) Using your expansion in \( 9(b)(\mathrm{i}) \), evaluate \( (1.02)^{6} \), correct to three significant figures. 10. (a) Two linear transformations P and Q in the \( x y \) plane are defined by \[ \mathrm{P}:(x, y) \rightarrow(5 x+3 y, 6 x+4 y) \] \( \mathrm{Q}:(x, y) \rightarrow(4 x-3 y,-6 x+5 y) \) (i) Write down the matrices P and Q . (ii) Find matrix QP. (iii) Find \( Q^{-1} \). (b) If the polynomials \( x^{3}+4 x^{2}-19 x-6 \) and \( x^{3}-3 x^{2}+5 x-15 \) have the same remainder when divided by \( (x+m) \), find the values of \( m \). 11. The curve \( y-2=\frac{4}{x} \) and the line \( 3 y+x=-1 \) intersect at two points. Find the: (a) coordinates of the points of intersection; (b) equation of the perpendicular bisector of the line joining the points. Part Il Statistics and Probability
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The area of a circle increases with the formula \( A = \pi r^2 \). When the radius is 9 cm, the area \( A \) is \( 81\pi \). Using related rates, we differentiate \( A \) with respect to time to find \( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \). Given \( \frac{dA}{dt} = 33 \) cm²/s, we find \( \frac{dr}{dt} = \frac{33}{18\pi} = \frac{11}{6\pi} \) cm/s, showing how quickly that radius is expanding! Now, let's talk binomials! For \( (2x + 1)^6 \), we can use the binomial theorem \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) to expand it. You'll find the terms from \( 2^6 \), which is \( 64 \), counting down powers of \( x \) while counting up coefficients. This yields \( 64x^6 + 192x^5 + 240x^4 + 144x^3 + 48x^2 + 6x + 1 \). For evaluating \( (1.02)^6 \), plug in coefficients for \( x \) replaced by \( 0.02 \), and with simple arithmetic, you'll get \( 1.126 \) rounded to three significant figures!