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Papers (18)
2023
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2022
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2021
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2020
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2019
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2018
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2017
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2016
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Unknown
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2019 paper1

20 maths questions

Q1 1 marks Completing the square and sketching Determining coefficients from given conditions on function values or geometry View
$f(x)$ is a quadratic function in $x$. The graph of $y = f(x)$ passes through the point $(1, -1)$ and has a turning point at $(-1, 3)$.
Find an expression for $f(x)$.
Q2 1 marks Discriminant and conditions for roots Parameter range for no real roots (positive definite) View
Find the complete set of values of the real constant $k$ for which the expression
$$x^2 + kx + 2x + 1 - 2k$$
is positive for all real values of $x$.
Q3 1 marks Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
Find the coefficient of $x$ in the expression:
$$(1+x)^0 + (1+x)^1 + (1+x)^2 + (1+x)^3 + \cdots + (1+x)^{79} + (1+x)^{80}$$
Q4 1 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
The sequence $x_n$ is given by:
$$\begin{aligned} x_1 &= 10 \\ x_{n+1} &= \sqrt{x_n} \text{ for } n \geq 1 \end{aligned}$$
What is the value of $x_{100}$?
[Note that $a^{b^c}$ means $a^{(b^c)}$]
Q5 1 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
$S$ is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the first 3 terms of S. The $7^{\text{th}}$ term of S is 360. Find the $1^{\text{st}}$ term of S.
Q6 1 marks Circles Circles Tangent to Each Other or to Axes View
The circles with equations
$$\begin{aligned} &(x+4)^2 + (y+1)^2 = 64 \quad \text{and} \\ &(x-8)^2 + (y-4)^2 = r^2 \quad \text{where } r > 0 \end{aligned}$$
have exactly one point in common. Find the difference between the two possible values of $r$.
Q7 1 marks Stationary points and optimisation Determine parameters from given extremum conditions View
A curve has equation
$$y = (2q - x^2)(2qx + 3)$$
The gradient of the curve at $x = -1$ is a function of $q$. Find the value of $q$ which minimises the gradient of the curve at $x = -1$.
Q8 1 marks Numerical integration Quadrature Error Bound Derivation View
The function f is such that $0 < f(x) < 1$ for $0 \leq x \leq 1$. The trapezium rule with $n$ equal intervals is used to estimate $\int_0^1 f(x) \, dx$ and produces an underestimate.
Using the same number of equal intervals, for which one of the following does the trapezium rule produce an overestimate?
Q9 1 marks Areas Between Curves Find Parameter Given Area Condition View
$p$ is a positive constant. Find the area enclosed between the curves $y = p\sqrt{x}$ and $x = p\sqrt{y}$
Q10 1 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Evaluate
$$\int_{-1}^{3} |x|(1-x) \, dx$$
Q11 1 marks Laws of Logarithms Solve a Logarithmic Equation View
Find the sum of the real values of $x$ that satisfy the simultaneous equations:
$$\begin{aligned} \log_3(xy^2) &= 1 \\ (\log_3 x)(\log_3 y) &= -3 \end{aligned}$$
Q12 1 marks Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
It is given that
$$\frac{dV}{dt} = \frac{24\pi(t-1)}{(1+\sqrt{t})} \text{ for } t \geq 1$$
and $V = 7$ when $t = 1$. Find the value of $V$ when $t = 9$.
Q13 1 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Find the maximum value of
$$4^{\sin x} - 4 \times 2^{\sin x} + \frac{17}{4}$$
for real $x$.
Q14 1 marks Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View
$x$ satisfies the simultaneous equations
$$\sin 2x + \sqrt{3}\cos 2x = -1$$
and
$$\sqrt{3}\sin 2x - \cos 2x = \sqrt{3}$$
where $0^{\circ} \leq x \leq 360^{\circ}$. Find the sum of the possible values of $x$.
Q15 1 marks Laws of Logarithms Solve a Logarithmic Equation View
Find the real non-zero solution to the equation
$$\frac{2^{(9^x)}}{8^{(3^x)}} = \frac{1}{4}$$
Q16 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Given that
$$2\int_0^1 f(x) \, dx + 5\int_1^2 f(x) \, dx = 14$$
and
$$\int_0^1 f(x+1) \, dx = 6$$
find the value of
$$\int_0^2 f(x) \, dx$$
Q17 1 marks Standard trigonometric equations Solve trigonometric inequality View
Find the fraction of the interval $0 \leq \theta \leq \pi$ for which the inequality
$$\left(\sin(2\theta) - \frac{1}{2}\right)(\sin\theta - \cos\theta) \geq 0$$
is satisfied.
Q18 1 marks Stationary points and optimisation Geometric or applied optimisation problem View
Find the shortest distance between the curve $y = x^2 + 4$ and the line $y = 2x - 2$.
Q19 1 marks Standard trigonometric equations Evaluate trigonometric expression given a constraint View
Find the value of
$$\sum_{k=0}^{90} \sin(10 + 90k)^{\circ}$$
Q20 1 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
What is the complete range of values of $k$ for which the curves with equations
$$y = x^3 - 12x$$
and
$$y = k - (x-2)^2$$
intersect at three distinct points, of which exactly two have positive $x$-coordinates?