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2023
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2020
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2016 paper2

20 maths questions

Q1 1 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Find the value of
$$\int _ { 1 } ^ { 2 } \left( x ^ { 2 } - \frac { 4 } { x ^ { 2 } } \right) ^ { 2 } d x$$
Q2 1 marks Chain Rule Straightforward Polynomial or Basic Differentiation View
$$f ( x ) = \frac { \left( x ^ { 2 } + 5 \right) ( 2 x ) } { \sqrt [ 4 ] { x ^ { 3 } } } , \quad x > 0$$
Which one of the following is equal to $f ^ { \prime } ( x )$ ?
Q3 1 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
What is the value, in radians, of the largest angle $x$ in the range $0 \leq x \leq 2 \pi$ that satisfies the equation $8 \sin ^ { 2 } x + 4 \cos ^ { 2 } x = 7$ ?
Q4 1 marks Proof View
Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The following statements are attached to the urns.
Urn P This urn contains one or four balls.
Urn Q This urn contains two or four balls.
Urn R This urn contains more than two balls and fewer than five balls.
Urn S This urn contains one or two balls.
Urn T This urn contains fewer than three balls.
Exactly one of the urns has a true statement attached to it.
Which urn is it?
Q5 1 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
Consider the statement:
() A whole number $n$ is prime if it is 1 less or 5 less than a multiple of 6 .
How many counterexamples to () are there in the range $0 < n < 50$ ?
Q6 1 marks Chain Rule Iterated/Nested Exponential Differentiation View
The sequence of functions $f _ { 1 } ( x ) , f _ { 2 } ( x ) , f _ { 3 } ( x ) , \ldots$ is defined as follows:
$$\begin{aligned} f _ { 1 } ( x ) & = x ^ { 10 } \\ f _ { n + 1 } ( x ) & = x f _ { n } ^ { \prime } ( x ) \text { for } n \geq 1 \end{aligned}$$
where $f _ { n } ^ { \prime } ( x ) = \frac { d f _ { n } ( x ) } { d x }$
Find the value of
$$\sum _ { n = 1 } ^ { 20 } f _ { n } ( x )$$
Q7 1 marks Laws of Logarithms Prove a Logarithmic Identity View
The four real numbers $a , b , c$, and $d$ are all greater than 1 .
Suppose that they satisfy the equation $\log _ { c } d = \left( \log _ { a } b \right) ^ { 2 }$.
Use some of the lines given to construct a proof that, in this case, it follows that
$$( * ) \log _ { b } d = \left( \log _ { a } b \right) \left( \log _ { a } c \right)$$
(1) Let $x = \log _ { a } b$ and $y = \log _ { a } c$
(2) $d = \left( c ^ { x } \right) ^ { 2 }$
(3) $d = c ^ { \left( x ^ { 2 } \right) }$
(4) $d = b ^ { x y }$
(5) $d = \left( a ^ { y } \right) ^ { \left( x ^ { 2 } \right) }$
(6) $d = \left( \left( a ^ { y } \right) ^ { x } \right) ^ { 2 }$
(7) $d = \left( a ^ { x } \right) ^ { x y }$
(8) $d = a ^ { \left( y ^ { 2 x } \right) }$
(9) $d = a ^ { \left( x ^ { 2 } y \right) }$
Q8 1 marks Inequalities Ordering and Sign Analysis from Inequality Constraints View
A region is defined by the inequalities $x + y > 6$ and $x - y > - 4$
Consider the three statements:
$1 x > 1$
$2 y > 5$
$3 ( x + y ) ( x - y ) > - 24$
Which of the above statements is/are true for every point in the region?
Q9 1 marks Proof Proof of Equivalence or Logical Relationship Between Conditions View
Triangles $A B C$ and $X Y Z$ have the same area.
Which of these extra conditions, taken independently, would imply that they are congruent?
(1) $A B = X Y$ and $B C = Y Z$
(2) $A B = X Y$ and $\angle A B C = \angle X Y Z$
(3) $\angle A B C = \angle X Y Z$ and $\angle B C A = \angle Y Z X$
Q10 1 marks Inequalities Ordering and Sign Analysis from Inequality Constraints View
In this question $x$ and $y$ are non-zero real numbers.
Which one of the following is sufficient to conclude that $x < y$ ?
Q11 1 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View
$f ( x )$ is a polynomial with real coefficients.
The equation $f ( x ) = 0$ has exactly two real roots, $x = - p$ and $x = p$, where $p > 0$.
Consider the following three statements:
$1 \quad f ^ { \prime } ( x ) = 0$ for exactly one value of $x$ between $- p$ and $p$
2 The area between the curve $y = f ( x )$, the $x$-axis and the lines $x = - p$ and $x = p$ is given by $2 \int _ { 0 } ^ { p } f ( x ) \mathrm { d } x$
3 The graph of $y = - f ( - x )$ intersects the $x$-axis at the points $x = - p$ and $x = p$ only
Which of the above statements must be true?
Q12 1 marks Arithmetic Sequences and Series Find Common Difference from Given Conditions View
The first term of an arithmetic sequence is $a$ and the common difference is $d$.
The sum of the first $n$ terms is denoted by $S _ { n }$.
If $S _ { 8 } > 3 S _ { 6 }$, what can be deduced about the sign of $a$ and the sign of $d$ ?
Q13 1 marks Proof True/False Justification View
In this question, $a , b$, and $c$ are positive integers.
The following is an attempted proof of the false statement:
If $a$ divides $b c$, then $a$ divides $b$ or $a$ divides $c$.
['$a$ divides $b c$' means '$a$ is a factor of $b c$']
Which line contains the error in this proof?
1. The statement is equivalent to if $a$ does not divide $b$ and $a$ does not divide $c$ then $a$ does not divide $b c$'.
2. Suppose $a$ does not divide $b$ and $a$ does not divide $c$. Then the remainder when dividing $b$ by $a$ is $r$, where $0 < r < a$, and the remainder when dividing $c$ by $a$ is $s$, where $0 < s < a$.
3. So $b = a x + r$ and $c = a y + s$ for some integers $x$ and $y$.
4. Thus $b c = a ( a x y + x s + y r ) + r s$.
5. So the remainder when dividing $b c$ by $a$ is $r s$.
6. Since $r > 0$ and $s > 0$, it follows that $r s > 0$.
7. Hence $a$ does not divide $b c$.
Q14 1 marks Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
$f ( x ) = a x ^ { 4 } + b x ^ { 3 } + c x ^ { 2 } + d x + e$, where $a , b , c , d$, and $e$ are real numbers.
Suppose $f ( x ) = 1$ has $p$ distinct real solutions, $f ( x ) = 2$ has $q$ distinct real solutions, $f ( x ) = 3$ has $r$ distinct real solutions, and $f ( x ) = 4$ has $s$ distinct real solutions.
Which one of the following is not possible?
Q15 1 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
Consider the quadratic $f ( x ) = x ^ { 2 } - 2 p x + q$ and the statement:
$\left( ^ { * } \right) f ( x ) = 0$ has two real roots whose difference is greater than 2 and less than 4.
Which one of the following statements is true if and only if (*) is true?
Q16 1 marks Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
In the figure, $P Q R S$ is a trapezium with $P Q$ parallel to $S R$.
The diagonals of the trapezium meet at $X$.
$U$ lies on $S P$ and $T$ lies on $R Q$ such that $U T$ is a line segment through $X$ parallel to $P Q$.
The length of $P Q$ is 12 cm and the length of $S R$ is 3 cm .
What, in centimetres, is the length of UT?
Q17 1 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Consider these simultaneous equations, where $c$ is a constant:
$$\begin{aligned} & y = 3 \sin x + 2 \\ & y = x + c \end{aligned}$$
Which of the following statements is/are true?
1 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there is at least one solution with $- \pi < x < 0$.
2 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there are no solutions with $- \pi < x < 0$.
3 For some value of $c$ : there is exactly one solution with $0 \leq x \leq \pi$ and there are no solutions with $x > \pi$.
Q18 1 marks Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Consider this statement about a function $f ( x )$ :
$\left( ^ { * } \right)$ If $( f ( x ) ) ^ { 2 } \leq 1$ for all $- 1 \leq x \leq 1$ then $\int _ { - 1 } ^ { 1 } ( f ( x ) ) ^ { 2 } \mathrm {~d} x \leq \int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x$
Which one of the following functions provides a counterexample to (*)?
Q19 1 marks Proof View
Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.
Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively.
This is the side elevation of the object.
This is the front elevation of the object.
This is the plan view of the object.
How many cubes were used to construct the object?
Q20 1 marks Circles Circle Equation Derivation View
Each interior angle of a regular polygon with $n$ sides is $\frac { 3 } { 4 }$ of each interior angle of a second regular polygon with $m$ sides.
How many pairs of positive integers $n$ and $m$ are there for which this statement is true?