Questions - SPS SPS SM - A-Level Maths

SPS SPS SM 2020 June Q1

1. A curve has equation $$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Hence find the range of values of $x$ for which $y$ is increasing. Write your answer in set notation.

SPS SPS SM 2020 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-04_556_1052_1119_552} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Diagram not drawn to scale Figure 1 shows the plan view of a design for a stage at a concert.
The stage is modelled as a sector $B C D F$, of a circle centre $F$, joined to two congruent triangles $A B F$ and $E D F$. Given that
$A F E$ is a straight line $$\begin{aligned} & A F = F E = 10.7 \mathrm {~m}
& B F = F D = 9.2 \mathrm {~m} \end{aligned}$$ angle $B F D = 1.82$ radians
find

the perimeter of the stage, in metres, to one decimal place,
the area of the stage, in $\mathrm { m } ^ { 2 }$, to one decimal place.

SPS SPS SM 2020 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-05_702_700_278_712} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 is a sketch showing the line $l _ { 1 }$ with equation $y = 2 x - 1$ and the point $A$ with coordinates $( - 2,3 )$. The line $l _ { 2 }$ passes through $A$ and is perpendicular to $l _ { 1 }$

Find the equation of $l _ { 2 }$ writing your answer in the form $y = m x + c$, where $m$ and $c$ are constants to be found. The point $B$ and the point $C$ lie on $l _ { 1 }$ such that $A B C$ is an isosceles triangle with $A B = A C = 2 \sqrt { 13 }$
Show that the $x$ coordinates of points $B$ and $C$ satisfy the equation $$5 x ^ { 2 } - 12 x - 32 = 0$$ Given that $B$ lies in the 3rd quadrant
find, using algebra and showing your working, the coordinates of $B$.

SPS SPS SM 2020 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-06_803_816_269_676} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { g } ( x )$.
The curve has a single turning point, a minimum, at the point $M ( 4 , - 1.5 )$.
The curve crosses the $x$-axis at two points, $P ( 2,0 )$ and $Q ( 7,0 )$.
The curve crosses the $y$-axis at a single point $R ( 0,5 )$.

State the coordinates of the turning point on the curve with equation $y = 2 \mathrm {~g} ( x )$.
State the largest root of the equation $$\mathrm { g } ( x + 1 ) = 0$$
State the range of values of $x$ for which $\mathrm { g } ^ { \prime } ( x ) \leqslant 0$ Given that the equation $\mathrm { g } ( x ) + k = 0$, where $k$ is a constant, has no real roots,
state the range of possible values for $k$. Use the binomial expansion to find, in ascending powers of $x$, the first four terms in the expansion of $$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.

SPS SPS SM 2020 June Q6

6. A company which makes batteries for electric cars has a 10 -year plan for growth.

In year 1 the company will make 2600 batteries
In year 10 the company aims to make 12000 batteries

In order to calculate the number of batteries it will need to make each year, from year 2 to year 9 , the company considers the following model: \section*{the number of batteries made will increase by the same percentage each year} Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10.

SPS SPS SM 2020 June Q7

7.

Solve, for $- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }$, the equation, $$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(a) A student's attempt at the question
"Solve, for $- 90 ^ { \circ } < x < 90 ^ { \circ }$, the equation $7 \tan x = 8 \sin x$ " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x
7 \times \frac { \sin x } { \cos x } = 8 \sin x
7 \sin x = 8 \sin x \cos x
7 = 8 \cos x
\cos x = \frac { 7 } { 8 }
x = 29.0 ^ { \circ } \text { (to } 3 \mathrm { sf } \text { ) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$

SPS SPS SM 2020 June Q8

8. Prove by contradiction that there are no positive integers $a$ and $b$ with $a$ odd such that $$a + 2 b = \sqrt { 8 a b }$$

SPS SPS SM 2020 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-09_639_1007_808_561} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, $P$, on 1st June each year for $t$ years after 1996. Scientists counted the number of red squirrels in the wood, $P$, on 1st June each year for $t$ years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = a b ^ { t }$$ where $a$ and $b$ are constants.
The line $l$, shown in Figure 1, illustrates the linear relationship between $\log _ { 10 } P$ and $t$ over a period of 20 years. Using the information given on the graph and using the model,
find a complete equation for the model giving the value of b to 4 significant figures.

SPS SPS SM 2020 June Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-10_883_885_283_644} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with equation $y = 3 x - 2 \sqrt { x } , x \geqslant 0$ and the line $l$ with equation $y = 8 x - 16$ The line cuts the curve at point $A$ as shown in Figure 3.

Using algebra, find the $x$ coordinate of point $A$.
(5)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-10_656_814_1786_662} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The region $R$ is shown unshaded in Figure 4. Identify the inequalities that define $R$.

SPS SPS SM 2020 June Q11

11.

Sketch the curve with equation $$y = k - \frac { 1 } { 2 x } \quad \text { where } k \text { is a positive constant }$$ State, in terms of $k$, the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. The straight line $l$ has equation $y = 2 x + 3$
Given that $l$ cuts the curve in two distinct places,
find the range of values of $k$, writing your answer in set notation.

SPS SPS SM 2020 June Q12

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-11_572_675_1098_772} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} \section*{In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.} Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2 ^ { 2 x }$$ The point $P ( a , 96 \sqrt { 2 } )$ lies on the curve.

Find the exact value of $a$. The curve with equation $y = 3 \times 2 ^ { 2 x }$ meets the curve with equation $y = 6 ^ { 3 - x }$ at the point $Q$.
Show that the $x$ coordinate of $Q$ is $$\frac { 3 + 2 \log _ { 2 } 3 } { 3 + \log _ { 2 } 3 }$$

SPS SPS SM 2020 June Q13

13. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-12_803_981_301_651} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a sketch of the curve $C$ with equation $y = ( x - 2 ) ^ { 2 } ( x + 3 )$
The region $R$, shown shaded in Figure 5, is bounded by $C$, the vertical line passing through the maximum turning point of $C$ and the $x$-axis. Find the exact area of $R$.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

SPS SPS SM 2020 December Q1

The curve C is defined by the equations $y = x - 4 \sqrt { x }$, $x \geq 0$
a) Find $\frac { d y } { d x }$
b) Find the coordinates of the turning point of $C$
c) Find the coordinates of the two $x$-intercepts of $C$.
The cubic polynomial $f ( \mathrm { x } )$ is defined by $f ( x ) = x ^ { 3 } + k x ^ { 2 } + 9 x - 20$
a) Given that $( x - 5 )$ is a factor of $f ( x )$, find the value of $k$.
b) Show clearly that there is only one real solution to the equations $f ( x ) = 0$
c) Given also that the function $g ( x )$ is defined as $g ( x ) = \log _ { 2 } x$, $x > 0$

Solve $f g ( x ) = 0$
3)
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-05_537_990_137_338} The diagram above shows part of the curve with equation $y = k \sin \left( x + \frac { \pi } { 3 } \right)$
The curve meets the y -axis at $( 0 , \sqrt { 3 } )$ and the x -axis at $( p , 0 )$ and $( q , 0 )$
a) Find the value of the constant $k$
b) Find the value of $p$ and the value of $q$.
4) On the axes provided, sketch the curve $y = \tan \left( \frac { x } { 2 } \right) , - 2 \pi \leq x \leq 2 \pi$ Mark clearly the coordinates of any points the curves crosses the coordinate axes and the equations of any asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-05_661_979_2131_568}
5)
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-06_643_661_132_667} The diagram above shows the cross-section of a small shed.
The straight line $A B$ is vertical and has length 2.12 m . The straight line $A D$ is horizontal and has length 1.86 m . The curve $B C$ is an arc of a circle with centre $A$, and $C D$ is a straight line.
Given that the size of angle BAC is 0.65 radians:
a) Find the size of angle CAD giving your answer in radians to 2 dp .
b) Find the area of the cross-section $A B C D$
c) Find the perimeter of the cross-section ABCD
6) Prove, from first principles, that if $f ( x ) = 3 x ^ { 2 }$ then the derivative $f ^ { \prime } ( x )$ is given by $f ^ { \prime } ( x ) = 6 x$
7) Given $f ( x ) = x ^ { 2 } + 1 , \quad x < - 1$ Find $f ^ { - 1 } ( x )$ stating its domain and range
8) The curve $C$ has equation $y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , \quad x > 0$ The points P and Q lie on C and have x -coordinates 1 and 2 respectively.
a) Show that the length of PQ is V 170 .
b) Show that the tangents to C at P and Q are parallel.
c) Find an equation for the normal to C at P , giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
a) Solve, for $0 \leq x < 360 ^ { \circ }$, giving your answers to 1 decimal place. $$5 \sin 2 x = 2 \cos 2 x$$ b) Solve for $0 \leq x \leq 4 \pi$ giving your answers in radians to 3 significant figures. $$4 \sin ^ { 2 } x = 6 - 9 \cos x$$

SPS SPS SM 2020 December Q10

10.

\includegraphics[max width=\textwidth, alt={}]{a202ddae-5ecd-4803-9a05-33c37d1880cd-10_597_533_155_760}

The diagram above shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height h cm . The cross section is a sector of a circle. The sector has radius r cm and angle 1 radian. The volume of the box is $300 \mathrm {~cm} ^ { 3 }$
a) Show that the surface area of the box, $S \mathrm {~cm} ^ { 2 }$, is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$ b) Hence find the value of $r$ and the value of $h$ which minimises the surface area of the box.

SPS SPS SM 2021 January Q1

1. $$f ( x ) = 6 x + 9 \sqrt { x } - \frac { 4 } { x ^ { 2 } } , x > 0 .$$ Find a fully simplified expression for $$\int f ( x ) d x$$

SPS SPS SM 2021 January Q2

2.

\includegraphics[max width=\textwidth, alt={}]{fbe229f8-d390-487d-87fc-90edc50c3325-2_449_1130_842_440}

The figure above shows the graph of a curve with equation $y = f ( x )$. The curve meets the $x$ axis at $( - 3,0 )$ and the $y$ axis at $( 0,2 )$. The curve has a maximum at $( 3,4 )$ and a minimum at $( - 3,0 )$. The line with equation $y = 2$ is a horizontal asymptote to the curve. Sketch on separate diagrams the graph of ...
a) $\ldots \quad y = f ( x + 3 )$.
b) $. . \quad y = f ( x ) - 2$.
c) $\ldots \quad y = \frac { 1 } { 2 } f ( x )$. Each of the sketches must include

the coordinates of any points where the graph meets the coordinate axes.
the coordinates of any minimum or maximum points of the curve.
any asymptotes to the curve, clearly labelled.

SPS SPS SM 2021 January Q3

3. a) Solve the linear inequality $$6 - 2 ( x + 2 ) < 10 .$$ b) Solve the quadratic inequality $$( x + 1 ) ^ { 2 } \geq 4 x + 9$$ c) Hence determine the range of values of $x$ that satisfy both the inequalities of part (a) and part (b).

SPS SPS SM 2021 January Q4

4.
\includegraphics[max width=\textwidth, alt={}, center]{fbe229f8-d390-487d-87fc-90edc50c3325-3_554_988_1153_486} The figure above shows the graph of the curve with equation $y = f ( x )$. The curve crosses the $x$ axis at the points $( - 4,0 ) , ( 2,0 )$ and $( 4,0 )$, and the $y$ axis at the point $( 0,16 )$. Determine the equation of $f ( x )$ in the form $$f ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where $a , b , c$ and $d$ are constants.

SPS SPS SM 2021 January Q5

5. A curve $C$ and a straight line $L$ have respective equations $$y = x ^ { 2 } - 4 x - 5 \text { and } y = 2 x - 14$$ a) Find the coordinates of any points of intersection between $C$ and $L$.
b) Sketch in the same diagram the graph of $C$ and the graph of $L$. The sketch must include of any points of intersection between the graph of $C$ and the coordinate axes, and any points of intersection between the graph of $L$ and the coordinate axes.

SPS SPS SM 2021 January Q6

6. Solve the following trigonometric equation in the range given. $$\tan ( 5 y - 35 ) ^ { \circ } = - 2 - \sqrt { 3 } , \quad 0 \leq y < 90$$

SPS SPS SM 2021 January Q7

7. A circle whose centre is at $( 3 , - 5 )$ has equation $$x ^ { 2 } + y ^ { 2 } - 6 x + a y = 15$$ where $a$ is a constant.
a) Find the value of $a$.
b) Determine the radius of the circle.

SPS SPS SM 2021 January Q8

8. Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate.
a) $\quad 7 ^ { x } = 10$.
b) $\quad \log _ { 2 } y = \frac { 9 } { \log _ { 2 } y }$.

SPS SPS SM 2021 January Q9

9. The total cost $C$, in $\pounds$, for a certain car journey, is modelled by $$C = \frac { 200 } { V } + \frac { 2 V } { 25 } , V > 30 ,$$ where $V$ is the average speed in miles per hour.
a) Find the value of $V$ for which $C$ is stationary.

SPS SPS SM 2020 October Q1

Simplify fully the following expressions:
i. $\frac { 7 y ^ { 13 } } { 35 y ^ { 7 } }$
ii. $6 x ^ { - 2 } \div x ^ { - 5 }$
A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots$ is defined by $u _ { 1 } = 7$ and $u _ { n + 1 } = u _ { n } + 4$ for $n \geq 1$.
i. State what type of sequence this is.
ii. Find $u _ { 17 }$.
i. Write $3 x ^ { 2 } - 6 x + 1$ in the form $p ( x + q ) ^ { 2 } + r$, where $p , q$ and $r$ are integers.
ii. Solve $3 x ^ { 2 } - 6 x + 1 \leq 0$, giving your answer in set notation. In this question you must show detailed reasoning.
i. Express $\frac { \sqrt { 2 } } { 1 - \sqrt { 2 } }$ in the form $c + d \sqrt { } e$, where $c$ and $d$ are integers and $e$ is a prime number.
ii. Solve the equation $\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8$.
Let $a = \log _ { 2 } x , b = \log _ { 2 } y$ and $c = \log _ { 2 } z$.

Express $\log _ { 2 } ( x y ) - \log _ { 2 } \left( \frac { z } { x ^ { 2 } } \right)$ in terms of $a , b$ and $c$.

SPS SPS SM 2020 October Q6

6. i. A student was asked to solve the equation $2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0$. The student's attempt is written out below. $$\begin{gathered} 2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0
2 ^ { 2 x } + 2 ^ { 4 } - 9 \left( 2 ^ { x } \right) = 0
\text { Let } y = 2 ^ { x }
y ^ { 2 } - 9 y + 8 = 0
( y - 8 ) ( y - 1 ) = 0
y = 8 \text { or } y = 1
\text { So } x = 3 \text { or } x = 0 \end{gathered}$$ Identify the two mistakes that the student has made.
ii. Solve the equation $2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0$, giving your answer in exact form.

Questions — SPS SPS SM (145 questions)

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