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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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Unknown
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2017 paper1

20 maths questions

Q1 1 marks Standard Integrals and Reverse Chain Rule Antiderivative with Initial Condition View
Given that
$$\frac { d y } { d x } = 3 x ^ { 2 } - \frac { 2 - 3 x } { x ^ { 3 } } , \quad x \neq 0$$
and $y = 5$ when $x = 1$, find $y$ in terms of $x$.
A $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { - 2 } - 3 x ^ { - 1 } + 6 \frac { 2 } { 3 }$
B $y = x ^ { 3 } + \frac { 1 } { 2 } x ^ { - 2 } - 3 x ^ { - 1 } + 6 \frac { 1 } { 2 }$
C $y = x ^ { 3 } + x ^ { - 2 } - 3 x ^ { - 1 } + 6$
D $y = x ^ { 3 } + x ^ { - 2 } - x ^ { - 1 } + 4$
E $y = x ^ { 3 } + 2 x ^ { - 2 } - x ^ { - 1 } + 3$
F $y = 3 x ^ { 3 } + x ^ { - 2 } - x ^ { - 1 } + 2$
Q2 1 marks Product & Quotient Rules View
The function $f$ is given by
$$f ( x ) = \left( \frac { 2 } { x } - \frac { 1 } { 2 x ^ { 2 } } \right) ^ { 2 } \quad ( x \neq 0 )$$
What is the value of $f ^ { \prime \prime } ( 1 )$ ?
A - 3
B - 1
C 5
D 17
E 29
F 80
Q3 1 marks Areas by integration View
A line $l$ has equation $y = 6 - 2 x$
A second line is perpendicular to $l$ and passes through the point $( - 6,0 )$.
Find the area of the region enclosed by the two lines and the $x$-axis.
A $16 \frac { 1 } { 5 }$
B 18
C $21 \frac { 3 } { 5 }$
D 27
E $\quad 40 \frac { 1 } { 2 }$
Q4 1 marks Factor & Remainder Theorem Remainder by Linear Divisor View
When $\left( 3 x ^ { 2 } + 8 x - 3 \right)$ is multiplied by $( p x - 1 )$ and the resulting product is divided by $( x + 1 )$, the remainder is 24 .
What is the value of $p$ ?
A - 4
B 2
C 4
D $\frac { 8 } { 7 }$
E $\frac { 11 } { 4 }$
Q5 1 marks Inequalities Simultaneous/Compound Quadratic Inequalities View
$S$ is the complete set of values of $x$ which satisfy both the inequalities
$$x ^ { 2 } - 8 x + 12 < 0 \text { and } 2 x + 1 > 9$$
The set $S$ can also be represented as a single inequality.
Which one of the following single inequalities represents the set $S$ ?
A $\left( x ^ { 2 } - 8 x + 12 \right) ( 2 x + 1 ) < 0$
B $\left( x ^ { 2 } - 8 x + 12 \right) ( 2 x + 1 ) > 0$
C $x ^ { 2 } - 10 x + 24 < 0$
D $x ^ { 2 } - 10 x + 24 > 0$
E $\quad x ^ { 2 } - 6 x + 8 < 0$
F $\quad x ^ { 2 } - 6 x + 8 > 0$
G $x < 2$
H $x > 6$
Q6 1 marks Circles Tangent Lines and Tangent Lengths View
A tangent to the circle $x ^ { 2 } + y ^ { 2 } = 144$ passes through the point $( 20,0 )$ and crosses the positive $y$-axis.
What is the value of $y$ at the point where the tangent meets the $y$-axis?
A 12
B 15
C $\frac { 49 } { 3 }$
D 20
E $\frac { 64 } { 3 }$
F $\frac { 80 } { 3 }$
Q7 1 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
The first three terms of an arithmetic progression are $p , q$ and $p ^ { 2 }$ respectively, where $p < 0$
The first three terms of a geometric progression are $p , p ^ { 2 }$ and $q$ respectively.
Find the sum of the first 10 terms of the arithmetic progression.
A $\frac { 23 } { 8 }$
B $\frac { 95 } { 8 }$
C $\frac { 115 } { 8 }$
D $\frac { 185 } { 8 }$
Q8 1 marks Quadratic trigonometric equations View
Find the complete set of values of $x$, with $0 \leq x \leq \pi$, for which
$$( 1 - 2 \sin x ) \cos x \geq 0$$
A $0 \leq x \leq \frac { \pi } { 6 } , \frac { \pi } { 2 } \leq x \leq \frac { 5 \pi } { 6 }$
B $0 \leq x \leq \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \leq x \leq \pi$
C $\frac { \pi } { 6 } \leq x \leq \frac { \pi } { 2 } , \quad \frac { 5 \pi } { 6 } \leq x \leq \pi$
D $\frac { \pi } { 6 } \leq x \leq \frac { 5 \pi } { 6 }$
Q9 1 marks Circles Inscribed/Circumscribed Circle Computations View
A circle has equation $x ^ { 2 } + y ^ { 2 } - 18 x - 22 y + 178 = 0$
A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle.
What is the area of the hexagon?
A 6
B $6 \sqrt { 3 }$
C 18
D $18 \sqrt { 3 }$
E 36
F $36 \sqrt { 3 }$
G 48
H $48 \sqrt { 3 }$
Q10 1 marks Stationary points and optimisation Geometric or applied optimisation problem View
A curve $C$ has equation $y = f ( x )$ where
$$f ( x ) = p ^ { 3 } - 6 p ^ { 2 } x + 3 p x ^ { 2 } - x ^ { 3 }$$
and $p$ is real.
The gradient of the normal to the curve $C$ at the point where $x = - 1$ is $M$.
What is the greatest possible value of $M$ as $p$ varies?
A $- \frac { 3 } { 2 }$
B $- \frac { 2 } { 3 }$
C $- \frac { 1 } { 2 }$
D $\frac { 1 } { 4 }$
E $\frac { 2 } { 3 }$
F $\frac { 3 } { 2 }$
Q11 1 marks Fixed Point Iteration View
The sequence $x _ { n }$ is defined by the rules
$$\begin{aligned} x _ { 1 } & = 7 \\ x _ { n + 1 } & = \frac { 23 x _ { n } - 53 } { 5 x _ { n } + 1 } \end{aligned}$$
The first three terms in the sequence are $7,3,1$
What is the value of $x _ { 100 }$ ?
A - 5
B 0
C 1
D 3
E 7
Q12 1 marks Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The polynomial function $f ( x )$ is such that $f ( x ) > 0$ for all values of $x$.
Given $\int _ { 2 } ^ { 4 } f ( x ) d x = A$, which one of the following statements must be correct?
A $\int _ { 0 } ^ { 2 } [ f ( x + 2 ) + 1 ] d x = A + 1$
B $\quad \int _ { 0 } ^ { 2 } [ f ( x + 2 ) + 1 ] d x = A + 2$
C $\int _ { 2 } ^ { 4 } [ f ( x + 2 ) + 1 ] d x = A + 1$
D $\int _ { 2 } ^ { 4 } [ f ( x + 2 ) + 1 ] d x = A + 2$
E $\quad \int _ { 4 } ^ { 6 } [ f ( x + 2 ) + 1 ] d x = A + 1$
F $\quad \int _ { 4 } ^ { 6 } [ f ( x + 2 ) + 1 ] d x = A + 2$
Q13 1 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of $( a + b x ) ^ { 5 }$ the coefficient of $x ^ { 4 }$ is 8 times the coefficient of $x ^ { 2 }$.
Given that $a$ and $b$ are non-zero positive integers, what is the smallest possible value of $a + b$ ?
A 3
B 4
C 5
D 9
E 13
F 17
Q14 1 marks Exponential Functions Exponential Equation Solving View
The solution of the simultaneous equations
$$\begin{array} { r } 2 ^ { x } + 3 \times 2 ^ { y } = 3 \\ 2 ^ { 2 x } - 9 \times 2 ^ { 2 y } = 6 \end{array}$$
is $x = p , y = q$.
Find the value of $p - q$
A $\frac { 5 } { 12 }$
B $\frac { 7 } { 3 }$
C $\log _ { 2 } \frac { 5 } { 12 }$
D $\log _ { 2 } \frac { 7 } { 3 }$
E $\log _ { 2 } 9$
F $\quad \log _ { 2 } 15$
Q15 1 marks Numerical integration Quadrature Error Bound Derivation View
It is given that $f ( x ) = - 2 x ^ { 2 } + 10$
Consider the following three curves:
(1) $y = f ( x )$
(2) $y = f ( x + 1 )$
(3) the curve $y = f ( x + 1 )$ reflected in the line $y = 6$
The trapezium rule is used to estimate the area under each of these three curves between $x = 0$ and $x = 1$.
State whether the trapezium rule gives an overestimate or underestimate for each of these areas.
(1)(2)(3)
Aunderestimateunderestimateunderestimate
Bunderestimateunderestimateoverestimate
Cunderestimateoverestimateunderestimate
Dunderestimateoverestimateoverestimate
Eoverestimateunderestimateunderestimate
Foverestimateunderestimateoverestimate
Goverestimateoverestimateunderestimate
Hoverestimateoverestimateoverestimate
Q16 1 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
The functions $f$ and $g$ are given by $f ( x ) = 3 x ^ { 2 } + 12 x + 4$ and $g ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 9 x - 8$.
What is the complete set of values of $x$ for which one of the functions is increasing and the other decreasing?
A $x \geq - 1$
B $x \leq - 1$
C $\quad - 3 \leq x \leq - 2 , x \geq - 1$
D $x \leq - 2 , x \geq - 1$
E $\quad x \leq - 3 , - 2 \leq x \leq - 1$
F $x \leq - 3 , x \geq - 2$
G $\quad - 2 \leq x \leq - 1$
Q17 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The two functions $F ( n )$ and $G ( n )$ are defined as follows for positive integers $n$ :
$$\begin{aligned} & F ( n ) = \frac { 1 } { n } \int _ { 0 } ^ { n } ( n - x ) d x \\ & G ( n ) = \sum _ { r = 1 } ^ { n } F ( r ) \end{aligned}$$
What is the smallest positive integer $n$ such that $G ( n ) > 150$ ?
A 22
B 23
C 24
D 25
E 26
Q18 1 marks Function Transformations View
The graph of $y = \log _ { 10 } x$ is translated in the positive $y$-direction by 2 units. This translation is equivalent to a stretch of factor $k$ parallel to the $x$-axis. What is the value of $k$ ?
A 0.01
B $\log _ { 10 } 2$
C $\quad 0.5$
D 2
E $\quad \log _ { 2 } 10$
F 100
Q19 1 marks Inequalities Solve Polynomial/Rational Inequality for Solution Set View
The set of solutions to the inequality $x ^ { 2 } + b x + c < 0$ is the interval $p < x < q$ where $b , c , p$ and $q$ are real constants with $c < 0$.
In terms of $p , q$ and $c$, what is the set of solutions to the inequality $x ^ { 2 } + b c x + c ^ { 3 } < 0$ ?
A $\frac { p } { c } < x < \frac { q } { c }$
B $\frac { q } { c } < x < \frac { p } { c }$
C $p c < x < q c$
D $q c < x < p c$
E $p c ^ { 2 } < x < q c ^ { 2 }$
F $q c ^ { 2 } < x < p c ^ { 2 }$
Q20 1 marks Sine and Cosine Rules Prove an inequality or ordering relationship in a triangle View
The lengths of the sides $Q R , R P$ and $P Q$ in triangle $P Q R$ are $a , a + d$ and $a + 2 d$ respectively, where $a$ and $d$ are positive and such that $3 d > 2 a$.
What is the full range, in degrees, of possible values for angle $P R Q$ ?
A $0 <$ angle $P R Q < 60$
B 0 < angle $P R Q < 120$
C 60 < angle $P R Q < 120$
D 60 < angle $P R Q < 180$
E 120 < angle $P R Q < 180$