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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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2021 paper1

20 maths questions

Q1 1 marks Circles Intersection of Circles or Circle with Conic View
Two circles have the same radius. The centre of one circle is $( - 2,1 )$. The centre of the other circle is $( 3 , - 2 )$. The circles intersect at two distinct points. What is the equation of the straight line through the two points at which the circles intersect?
A $3 x - 5 y = 4$ B $3 x + 5 y = - 1$ C $5 x - 3 y = - 4$ D $5 x - 3 y = - 1$ E $\quad 5 x - 3 y = 1$ F $5 x - 3 y = 4$ G $5 x + 3 y = 1$
Q2 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The curve $y = x ^ { 3 } - 6 x + 3$ has turning points at $x = \alpha$ and $x = \beta$, where $\beta > \alpha$. Find
$$\int _ { \alpha } ^ { \beta } x ^ { 3 } - 6 x + 3 \mathrm {~d} x$$
A $- 8 \sqrt { 2 }$ B - 10 C $- 10 + 6 \sqrt { 2 }$ D 0 E $\quad 12 - 8 \sqrt { 2 }$ F $\quad 6 \sqrt { 2 }$ G 12
Q3 1 marks Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
An arithmetic progression and a convergent geometric progression each have first term $\frac { 1 } { 2 }$
The sum of the second terms of the two progressions is $\frac { 1 } { 2 }$ The sum of the third terms of the two progressions is $\frac { 1 } { 8 }$ What is the sum to infinity of the geometric progression?
A - 2 B - 1 C $- \frac { 1 } { 2 }$ D $- \frac { 1 } { 3 }$ E $\frac { 1 } { 3 }$ F $\quad \frac { 1 } { 2 }$ G 1 H 2
Q4 1 marks Exponential Functions MCQ on Function Properties View
Find the minimum value of the function
$$2 ^ { 2 x } - 2 ^ { x + 3 } + 4$$
A - 16 B - 12 C - 8 D 0 E 4 F $\quad 20$
Q5 1 marks Proof Computation of a Limit, Value, or Explicit Formula View
The function f is such that
$$\mathrm { f } ( m n ) = \begin{cases} \mathrm { f } ( m ) \mathrm { f } ( n ) & \text { if } m n \text { is a multiple of } 3 \\ m n & \text { if } m n \text { is not a multiple of } 3 \end{cases}$$
for all positive integers $m$ and $n$. Given that $\mathrm { f } ( 9 ) + \mathrm { f } ( 16 ) - \mathrm { f } ( 24 ) = 0$, what is the value of $\mathrm { f } ( 3 )$ ? A $\frac { 8 } { 3 }$ B $2 \sqrt { 2 }$ C 3 D $\frac { 16 } { 5 }$ E $3 \sqrt { 2 }$ F 4
Q6 1 marks Reciprocal Trig & Identities View
The function f is given by
$$\mathrm { f } ( x ) = \frac { \cos x + 3 } { 7 + 5 \cos x - \sin ^ { 2 } x }$$
Find the positive difference between the maximum and the minimum values of $\mathrm { f } ( x )$.
A 0 B $\frac { 1 } { 3 }$ C $\frac { 1 } { 2 }$ D $\frac { 2 } { 3 }$ E 1 F 2
Q7 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The function f is such that $\mathrm { f } ( 0 ) = 0$, and $x \mathrm { f } ( x ) > 0$ for all non-zero values of $x$. It is given that
$$\int _ { - 2 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 4$$
and
$$\int _ { - 2 } ^ { 2 } | \mathrm { f } ( x ) | \mathrm { d } x = 8$$
Evaluate
$$\int _ { - 2 } ^ { 0 } \mathrm { f } ( | x | ) \mathrm { d } x$$
A - 8 B - 6 C - 4 D - 2 E 2 F $\quad 4$ G 6 H 8
Q8 1 marks Discriminant and conditions for roots Condition for repeated (equal/double) roots View
The line $y = 2 x + 3$ meets the curve $y = x ^ { 2 } + b x + c$ at exactly one point. The line $y = 4 x - 2$ also meets the curve $y = x ^ { 2 } + b x + c$ at exactly one point. What is the value of $b - c$ ?
A - 9 B - 5.5 C - 1 D 5 E 6 F 14
Q9 1 marks Areas by integration View
Find the area enclosed by the graph of
$$| x | + | y | = 1$$
A $\frac { 1 } { 2 }$ B 1 C 2 D 4 E $\frac { 1 } { 2 } \sqrt { 2 }$ F $\quad \sqrt { 2 }$ G $2 \sqrt { 2 }$
Q10 1 marks Numerical integration Riemann Sum Computation from a Given Formula View
Use the trapezium rule with 3 strips to estimate
$$\int _ { \frac { 1 } { 2 } } ^ { 2 } 2 \log _ { 10 } x \mathrm {~d} x$$
A $\log _ { 10 } \frac { \sqrt { 6 } } { 2 }$
B $\log _ { 10 } \frac { 3 } { 2 }$
C $\log _ { 10 } \frac { 9 } { 4 }$
D $\log _ { 10 } 3$
E $\quad \log _ { 10 } \frac { 81 } { 16 }$
F $\log _ { 10 } \frac { \sqrt { 23 } } { 2 }$
Q11 1 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
The function f is given by
$$\mathrm { f } ( x ) = x ^ { \frac { 1 } { 7 } } \left( x ^ { 2 } - x + 1 \right)$$
Find the fraction of the interval $0 < x < 1$ for which $\mathrm { f } ( x )$ is decreasing.
A $\frac { 2 } { 15 }$
B $\frac { 1 } { 5 }$
C $\frac { 1 } { 3 }$
D $\frac { 1 } { 2 }$
E $\frac { 2 } { 3 }$
F $\frac { 4 } { 5 }$
G $\frac { 13 } { 15 }$
Q12 1 marks Stationary points and optimisation Determine parameters from given extremum conditions View
The minimum value of the function $x ^ { 4 } - p ^ { 2 } x ^ { 2 }$ is - 9 $p$ is a real number.
Find the minimum value of the function $x ^ { 2 } - p x + 6$
A - 3 B $6 - \frac { 3 \sqrt { 2 } } { 2 }$ C $\frac { 3 } { 2 }$ D 3 E $\frac { 9 } { 2 }$ F $6 + \frac { 3 \sqrt { 2 } } { 2 }$
Q13 1 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
The function f is such that, for every integer $n$
$$\int _ { n } ^ { n + 1 } \mathrm { f } ( x ) \mathrm { d } x = n + 1$$
Evaluate
$$\sum _ { r = 1 } ^ { 8 } \left( \int _ { 0 } ^ { r } \mathrm { f } ( x ) \mathrm { d } x \right)$$
A 36 B 84 C 120 D 165 E 204 F 288
Q14 1 marks Trigonometric equations in context View
This question uses radians. Find the number of distinct values of $x$ that satisfy the equation
$$( x + 1 ) ( 3 - x ) = 2 ( 1 - \cos ( \pi x ) )$$
A 2 B 3 C 4 D 5 E 6 F 7
Q15 1 marks Areas by integration View
The diagram shows the graph of $y = \mathrm { f } ( x )$.
The graph consists of alternating straight-line segments of gradient 1 and - 1 and continues in this way for all values of $x$.
The function g is defined as
$$\mathrm { g } ( x ) = \sum _ { r = 1 } ^ { 10 } f \left( 2 ^ { r - 1 } x \right)$$
Find the value of
$$\int _ { 0 } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x$$
A $\frac { 1023 } { 1024 }$ B $\frac { 1023 } { 512 }$ C 5 D 10 E $\frac { 55 } { 2 }$ F 55
Q16 1 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
Consider the expansion of
$$( a + b x ) ^ { n }$$
The third term, in ascending powers of $x$, is $105 x ^ { 2 }$ The fourth term, in ascending powers of $x$, is $210 x ^ { 3 }$ The fourth term, in descending powers of $x$, is $210 x ^ { 3 }$ Find the value of $\frac { a } { b } ^ { 2 }$
A $\frac { 1 } { 4 }$
B $\frac { 4 } { 9 }$
C $\frac { 25 } { 36 }$
D $\frac { 5 } { 6 }$
E 1
Q17 1 marks Circles Circle Identification and Classification View
Which of the following sketches shows the graph of
$$\sin \left( x ^ { 2 } + y ^ { 2 } \right) = \frac { 1 } { 2 }$$
where $x ^ { 2 } + y ^ { 2 } \leq 8 \pi$ ?
Q18 1 marks Straight Lines & Coordinate Geometry Reflection and Image in a Line View
The curve with equation
$$x = y ^ { 2 } - 6 y + 11$$
is rotated $90 ^ { \circ }$ clockwise about the point $P$ to give the curve $C$. $P$ has $x$-coordinate - 2 and $y$-coordinate 3 . What is the equation of $C$ ?
A $y = - x ^ { 2 } - 4 x - 3$ B $y = - x ^ { 2 } - 4 x - 5$ C $y = - x ^ { 2 } - 6 x - 7$ D $y = - x ^ { 2 } - 6 x - 11$ E $y = x ^ { 2 } - 4 x + 5$ F $y = x ^ { 2 } + 4 x + 3$ G $y = x ^ { 2 } - 6 x + 11$ H $y = x ^ { 2 } + 6 x + 7$
Q19 1 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The equation
$$\sin ^ { 2 } \left( 4 ^ { \cos \theta } \times 60 ^ { \circ } \right) = \frac { 3 } { 4 }$$
has exactly three solutions in the range $0 ^ { \circ } \leq \theta \leq x ^ { \circ }$ What is the range of all possible values of $x$ ?
A $90 \leq x < 120$ B $90 \leq x < 270$ C $120 \leq x < 240$ D $270 \leq x < 300$ E $\quad 300 \leq x < 360$ F $\quad 450 \leq x < 630$
Q20 1 marks Circles Circle Equation Derivation View
Find the length of the curve with equation
$$2 \log _ { 10 } ( x - y ) = \log _ { 10 } ( 2 - 2 x ) + \log _ { 10 } ( y + 5 )$$
A 5 B 10 C 15 D $3 \pi$ E $9 \pi$ F $12 \pi$