QCE Math Methods 2025 Paper 1

Question 1 (4 marks): Differentiate $f(x) = x^2\ln(3x)$ and evaluate $f'(1)$. Give an exact answer.

This is a product of two functions, so we apply the product rule. The second factor, $\ln(3x)$, also requires the chain rule to differentiate correctly.

Step 1: Identify the product and differentiate $\ln(3x)$

Set $u = x^2$ and $v = \ln(3x)$. The derivative of $u$ is $u' = 2x$. For $v$, we apply the chain rule: $\dfrac{d}{dx}\ln(3x) = \dfrac{1}{3x}\cdot 3 = \dfrac{1}{x}$. The factor of $3$ in the numerator and denominator cancel, leaving $v' = \dfrac{1}{x}$.

Step 2: Apply the product rule

The product rule gives $f'(x) = u'v + uv'$:

The second term simplifies because $x^2 \cdot \dfrac{1}{x} = x$.

Step 3: Substitute $x = 1$ for the exact value

This is already in exact form. The common technology-free slip is writing $\dfrac{d}{dx}\ln(3x) = \dfrac{1}{3x}$ (forgetting to multiply by the inner derivative), which gives the wrong answer.

Final answer: $f'(1) = 2\ln 3 + 1$.

Question 2 (5 marks): Evaluate $\displaystyle\int_{0}^{2}\left(3x^2 - 4x + 1\right)\,dx$ and state what the result represents if the integrand is non-negative on the interval.

We apply the fundamental theorem of calculus: antidifferentiate, then evaluate at the upper and lower limits and subtract. Showing each term separately earns the available method marks.

Step 1: Antidifferentiate term by term

Raise each power by one and divide by the new power:

The constant of integration drops out when we evaluate a definite integral, because it cancels in the subtraction.

Step 2: Substitute the limits and compute

Evaluate at $x = 2$ first, then at $x = 0$, and subtract:

Step 3: Interpret the result geometrically

The definite integral equals $2$. If the integrand is non-negative everywhere on $[0,2]$, this value equals the area enclosed between the curve $y = 3x^2 - 4x + 1$ and the $x$-axis. In fact, the curve dips below the axis between its roots, so the value here is the net signed area rather than the total enclosed area. A point worth checking before writing "area" is whether the integrand changes sign on the interval.

Final answer: $\displaystyle\int_{0}^{2}(3x^2 - 4x + 1)\,dx = 2$.

Question 3 (6 marks): A biased coin lands heads with probability $p = 0.4$. It is tossed 5 times. Let $X$ be the number of heads. (a) Write the distribution of $X$. (b) Find $P(X = 2)$ exactly. (c) State the mean $E(X)$.

This question tests three aspects of the binomial distribution in sequence: naming the distribution, computing a specific probability, and recalling the mean formula.

Step 1: Name the distribution

Because each toss is independent, there are a fixed five tosses, and the probability of heads is the same constant $p = 0.4$ on every toss, $X$ follows a binomial distribution: $X \sim \text{Bin}(5, 0.4)$.

Step 2: Calculate $P(X = 2)$ exactly

Apply the binomial formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ with $k=2$, $n=5$, $p=0.4$, and $1-p=0.6$:

So $P(X=2) = 0.3456$ (or $\tfrac{432}{1250}$ exactly). The recurring error is swapping $p$ and $1-p$ in the exponents; here heads is the "success", so $p=0.4$ carries the power $k=2$ and $0.6$ carries the remaining power $3$.

Step 3: State the mean

For any binomial distribution, the mean is $E(X) = np$:

Final answer: $X \sim \text{Bin}(5, 0.4)$; $P(X=2) = 0.3456$; $E(X) = 2$.