Edexcel A Level Maths exam readiness check

Question 1

Find the coefficient of $x^{3}$ in the binomial expansion of $(2 - 3x)^{6}$.

Accepted Answer

Term in $x^{3}$: $\binom{6}{3}(2)^{3}(-3x)^{3} = 20 	imes 8 	imes (-27)x^{3} = -4320x^{3}$.

The coefficient is $-4320$.

Question 2

The curve $C$ has equation $y = x^{2}\ln x$, $x > 0$.

Find an equation of the tangent to $C$ at the point where $x = e$, giving your answer in the form $y = mx + c$ with exact values.

Accepted Answer

$\dfrac{dy}{dx} = 2x\ln x + x^{2}\cdot\dfrac{1}{x} = 2x\ln x + x$.

At $x = e$: $\dfrac{dy}{dx} = 2e\ln e + e = 2e + e = 3e$ and $y = e^{2}\ln e = e^{2}$.

Tangent: $y - e^{2} = 3e(x - e) \Rightarrow y = 3ex - 3e^{2} + e^{2} = 3ex - 2e^{2}$.

Question 3

Solve $3\sin^{2}	heta - \cos	heta - 1 = 0$ for $0 \le 	heta < 360°$.

Give your answers correct to 1 decimal place where appropriate.

Accepted Answer

$3(1 - \cos^{2}	heta) - \cos	heta - 1 = 0 \Rightarrow 3\cos^{2}	heta + \cos	heta - 2 = 0$.

$(3\cos	heta - 2)(\cos	heta + 1) = 0$.

$\cos	heta = 	frac23 \Rightarrow 	heta = 48.19° 	ext{ or } 311.81°$.

$\cos	heta = -1 \Rightarrow 	heta = 180°$.

So $	heta = 48.2°,\ 180°,\ 311.8°$ (1 d.p.).

Question 4

Use the substitution $u = 1 + x^{2}$ to evaluate $\displaystyle\int_{0}^{2} \frac{x}{\sqrt{1 + x^{2}}}\,dx$.

Give your answer in exact form.

Accepted Answer

$u = 1 + x^{2} \Rightarrow du = 2x\,dx \Rightarrow x\,dx = 	frac12 du$.

Limits: $x = 0 \Rightarrow u = 1$; $x = 2 \Rightarrow u = 5$.

$\displaystyle\int_{0}^{2}\frac{x}{\sqrt{1+x^{2}}}dx = 	frac12\int_{1}^{5}u^{-1/2}du = \left[u^{1/2}ight]_{1}^{5} = \sqrt{5} - 1$.

Question 5

The times taken by students to complete a task are normally distributed with mean $\mu$ and standard deviation $4.5$ minutes.

12% of students take longer than 32 minutes.

Find $\mu$, correct to 1 decimal place.

Accepted Answer

Let $T \sim N(\mu, 4.5^{2})$. $P(T > 32) = 0.12$, so $P(Z < z) = 0.88$ and $z = 1.1750$.

$\dfrac{32 - \mu}{4.5} = 1.1750$

$32 - \mu = 5.2875 \Rightarrow \mu = 26.7125 = 	extbf{26.7 minutes}$ (1 d.p.).

Edexcel A Level Maths — free exam readiness check