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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Let's solve the first question from Part I, which is question 9(a).

### Problem Statement
We need to find the rate at which the radius of a circle is increasing when the area of the circle is increasing at a rate of $$ 33 \, \text{cm}^2/\text{s} $$. The radius of the circle is given as $$ r = 9 \, \text{cm} $$.

### Known Conditions
1. The area $$ A $$ of a circle is given by the formula:
   $$
   A = \pi r^2
   $$
2. The rate of change of the area with respect to time is given as:
   $$
   \frac{dA}{dt} = 33 \, \text{cm}^2/\text{s}
   $$

### Step-by-Step Solution

1. **Differentiate the Area Formula**:
   We differentiate the area $$ A $$ with respect to time $$ t $$:
   $$
   \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt}
   $$
   Here, $$ \frac{dr}{dt} $$ is the rate at which the radius is increasing.

2. **Substitute Known Values**:
   We know $$ \frac{dA}{dt} = 33 \, \text{cm}^2/\text{s} $$ and $$ r = 9 \, \text{cm} $$. Substituting these values into the differentiated equation gives:
   $$
   33 = 2\pi (9) \frac{dr}{dt}
   $$

3. **Solve for $$ \frac{dr}{dt} $$**:
   Rearranging the equation to solve for $$ \frac{dr}{dt} $$:
   $$
   \frac{dr}{dt} = \frac{33}{2\pi (9)}
   $$

4. **Calculate $$ \frac{dr}{dt} $$**:
   Now we can simplify this expression:
   $$
   \frac{dr}{dt} = \frac{33}{18\pi} = \frac{11}{6\pi}
   $$

### Final Answer
The rate at which the radius is increasing is:
$$
\frac{dr}{dt} = \frac{11}{6\pi} \, \text{cm/s}
$$

Now, let's proceed to question 9(b)(i) regarding the binomial expansion of $$ (2x + 1)^6 $$.

### Problem Statement
Find the binomial expansion of $$ (2x + 1)^6 $$ in descending powers of $$ x $$.

### Step-by-Step Solution

1. **Use the Binomial Theorem**:
   The binomial theorem states that:
   $$
   (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
   $$
   For our case, $$ a = 2x $$, $$ b = 1 $$, and $$ n = 6 $$.

2. **Write the Expansion**:
   The expansion will be:
   $$
   (2x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} (2x)^{6-k} (1)^k
   $$

3. **Calculate Each Term**:
   We will calculate the terms for $$ k = 0 $$ to $$ k = 6 $$:
   - For $$ k = 0 $$: $$ \binom{6}{0} (2x)^6 = 1 \cdot 64x^6 = 64x^6 $$
   - For $$ k = 1 $$: $$ \binom{6}{1} (2x)^5 = 6 \cdot 32x^5 = 192x^5 $$
   - For $$ k = 2 $$: $$ \binom{6}{2} (2x)^4 = 15 \cdot 16x^4 = 240x^4 $$
   - For $$ k = 3 $$: $$ \binom{6}{3} (2x)^3 = 20 \cdot 8x^3 = 160x^3 $$
   - For $$ k = 4 $$: $$ \binom{6}{4} (2x)^2 = 15 \cdot 4x^2 = 60x^2 $$
   - For $$ k = 5 $$: $$ \binom{6}{5} (2x)^1 = 6 \cdot 2x = 12x $$
   - For $$ k = 6 $$: $$ \binom{6}{6} (2x)^0 = 1 $$

4. **Combine All Terms**:
   Combining all the terms gives:
   $$
   (2x + 1)^6 = 64x^6 + 192x^5 + 240x^4 + 160x^3 + 60x^2 + 12x + 1
   $$

### Final Answer
The binomial expansion of $$ (2x + 1)^6 $$ in descending powers of $$ x $$ is:
$$
64x^6 + 192x^5 + 240x^4 + 160x^3 + 60x^2 + 12x + 1
$$

Next, let's move on to question 9(b)(ii) to evaluate $$ (1.02)^6 $$ using the expansion we just found.

### Problem Statement
Using the expansion from 9(b)(i), evaluate $$ (1.02)^6 $$, correct to three significant figures.

### Step-by-Step Solution

1. **Substitute $$ x = 0.02 $$**:
   We substitute $$ x = 0.02 $$ into the expansion:
   $$
   (2(0.02) + 1)^6 = (0.04 + 1)^6 = (1.04)^6
   $$

2. **Evaluate the Expansion**:
   Using the expansion:
   $$
   64(0.02)^6 + 192(0.02)^5 + 240(0.02)^4 + 160(0.02)^3 + 60(0.02)^2 + 12(0.02) + 1
   $$

3. **Calculate Each Term**:
   - $$ 64(0.02)^6 = 64 \cdot 0.000064 = 0.004096 $$
   - $$ 192(0.02)^5 = 192 \cdot 0.00032 = 0.06144 $$
   - $$ 240(0.02)^4 = 240 \cdot 0.0016 = 0.384 $$
   - $$ 160(0.
The radius is increasing at a rate of $$ \frac{11}{6\pi} $$ cm/s. The binomial expansion of $$ (2x + 1)^6 $$ is $$ 64x^6 + 192x^5 + 240x^4 + 160x^3 + 60x^2 + 12x + 1 $$. Using this expansion, $$ (1.02)^6 $$ is approximately 1.126.

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