Edexcel A Level Mathematics (AS & A2) - 20 Structured Questions

Edexcel A Level Mathematics (AS & A2): 20 Structured Questions

This document contains 20 structured questions designed for A Level Mathematics students (AS & A2) following the Edexcel syllabus. It covers key topics from Pure Mathematics, Statistics, and Mechanics, suitable for revision and exam practice.

Instructions:

• Attempt all questions.
• Show clear working for all calculations.
• Unless otherwise stated, give answers to 3 significant figures where appropriate.
• Assume $g = 9.8\text{ m/s}^2$ where necessary.

Questions:

1. Solve the inequality $2x^2 - 7x + 3 < 0$.

2. Find the gradient of the curve $y = 3x^2 \ln(x)$ at $x=e$.

3. Solve $2 \sin(x - 30^\circ) = \sqrt{3}$ for $0^\circ \le x < 360^\circ$.

4. The function $f(x) = (x+3)^2 - 5$ is defined for $x \in \mathbb{R}$. Find the range of $f(x)$. Determine $f^{-1}(x)$ for $x \ge -5$, given that $x \ge -3$ for $f(x)$ to have an inverse.

5. Given that $y = \mathrm{e}^{2x} \sin(3x)$, show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^{2x}(2 \sin(3x) + 3 \cos(3x))$.

6. Evaluate $\int_{1}^{2} (3x^2 + \frac{1}{x}) \, \mathrm{d}x$.

7. Given vectors $\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\mathbf{b} = \mathbf{i} + 5\mathbf{j} - 2\mathbf{k}$, find the magnitude of $\mathbf{a} + \mathbf{b}$.

8. Find the first three terms, in ascending powers of $x$, of the binomial expansion of $(2 - 3x)^5$.

9. Prove by contradiction that if $n^2$ is even, then $n$ must be even.

10. A curve has equation $x^2 + y^2 - 4x + 2y = 15$. Find the equation of the tangent to the curve at the point $(5, 2)$.

11. The rate of increase of the radius $r$ of a sphere is $0.2 \text{ cm/s}$. Find the rate of increase of its volume when its radius is $5 \text{ cm}$. (Volume $V = \frac{4}{3}\pi r^3$).

12. Use the trapezium rule with 4 strips to estimate $\int_0^2 \sqrt{1 + x^2} \, \mathrm{d}x$. Give your answer to 3 significant figures.

13. A bag contains 7 red and 3 blue counters. Two counters are drawn without replacement. Find the probability that both counters are the same colour.

14. The weights of packets of biscuits are normally distributed with a mean of 500g and a standard deviation of 10g. Find the probability that a randomly chosen packet weighs less than 495g.

15. A biased coin is tossed 10 times. The probability of getting a head is $p$. If $X$ is the number of heads, and $P(X=0) = 0.05$, find $p$ to 3 decimal places.

16. A survey claims that 30% of students use a particular social media platform. In a random sample of 50 students, 10 are found to use the platform. Test, at the 5% significance level, whether there is evidence that the proportion of students using the platform is less than 30%.

17. A particle moves in a straight line from rest with constant acceleration $a \text{ m/s}^2$. If it travels 40m in the first 5 seconds, find the value of $a$.

18. A car of mass 1200 kg is towing a trailer of mass 800 kg. The car engine produces a driving force of 4000 N. Resistances to motion are 500 N on the car and 300 N on the trailer. Find the acceleration of the system.

19. A uniform rod AB of length 3m and mass 10kg rests horizontally on two supports at C and D. AC = 0.5m and AD = 2.5m. A mass of 5kg is placed at B. Find the reactions at C and D.

20. A stone is projected vertically upwards from a point A with speed $20 \text{ m/s}$. Find the maximum height reached by the stone above A.

Answer Key

1. $\frac{1}{2} < x < 3$

2. $9e$

3. $x = 90^\circ, 150^\circ$

4. Range: $f(x) \ge -5$. Inverse: $f^{-1}(x) = \sqrt{x+5} - 3$.

5. Proof by product rule: $\frac{\mathrm{d}y}{\mathrm{d}x} = 2\mathrm{e}^{2x}\sin(3x) + 3\mathrm{e}^{2x}\cos(3x) = \mathrm{e}^{2x}(2 \sin(3x) + 3 \cos(3x))$

6. $7 + \ln 2$

7. $\sqrt{26}$

8. $32 - 240x + 720x^2 + \dots$

9. Assume $n$ is odd. Then $n=2k+1$ for some integer $k$. $n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$, which is odd. This contradicts the assumption that $n^2$ is even. Therefore, if $n^2$ is even, $n$ must be even.

10. $y = -x + 7$

11. $20\pi \text{ cm}^3\text{/s}$ (approx $62.8 \text{ cm}^3\text{/s}$)

12. $2.98$ (3 s.f.)

13. $\frac{8}{15}$

14. $0.3085$

15. $p = 0.259$ (3 d.p.)

16. Let $p$ be the proportion. $H_0: p = 0.30$, $H_1: p < 0.30$. $X \sim B(50, 0.30)$. $P(X \le 10) \approx 0.0160$. Since $0.0160 < 0.05$, reject $H_0$. There is evidence that the proportion is less than 30%.

17. $a = 3.2 \text{ m/s}^2$

18. $a = 1.6 \text{ m/s}^2$

19. $R_C = 36.8 \text{ N}$ (3 s.f.), $R_D = 110 \text{ N}$ (3 s.f.)

20. $20.4 \text{ m}$ (3 s.f.)