Questions - SPS - A-Level Maths

SPS SPS SM 2022 January Q2

2. A curve has equation $y = 16 x + \frac { 1 } { x ^ { 2 } }$. Find
(A) $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
(B) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q3

3. Triangle ABC is shown in Fig. 1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-06_517_652_237_845} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Find the perimeter of triangle ABC .
(3)
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q4

4. Find $$\int \frac { 2 x ^ { 2 } + 6 x - 5 } { 3 \sqrt { x ^ { 3 } } } d x$$ simplifying your answer.
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q5

5. Prove, from first principles, that the derivative of $x ^ { 3 }$ is $3 x ^ { 2 }$
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-12_570_922_118_374} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the graph of $y = \mathrm { f } ( x )$.

Write down the number of solutions that exist for the equation
1. $\mathrm { f } ( x ) = 1$,
2. $\mathrm { f } ( x ) = - x$.
Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
1. $\quad y = \mathrm { f } ( x - 2 )$,
2. $y = \mathrm { f } ( 2 x )$.
  [0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q7

7. Prove by contradiction that $\sqrt [ 3 ] { 2 }$ is an irrational number.
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q8

8. (a) Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for $0 \leqslant x < 90 ^ { \circ }$ $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q9

9. $$\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$

Evaluate $\mathrm { f } ( 1 )$ and hence state a linear factor of $\mathrm { f } ( x )$.
Show that $\mathrm { f } ( x )$ can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where $p$ and $q$ are integers to be found.
Sketch the curve $y = \mathrm { f } ( x )$ in the space provided.
Using integration, find the area of the region enclosed by the curve $y = \mathrm { f } ( x )$ and the $x$-axis.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]

SPS SPS SM 2022 January Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4eb48b49-816b-4a08-9f7f-c20313c4d1c9-22_659_970_141_614} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of part of the curve $C$ with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point $P ( 4,6 )$ lies on $C$.
The line $l$ is the normal to $C$ at the point $P$.
The region $R$, shown shaded in Figure 4, is bounded by the line $l$, the curve $C$, the line with equation $x = 2$ and the $x$-axis. Show that the area of $R$ is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]

SPS SPS FM 2022 February Q1

1. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { l l } 4 & 1
0 & 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 1 & 1
0 & - 1 \end{array} \right)$.

Find $\mathbf { A } + 3 \mathbf { B }$.
Show that $\mathbf { A } - \mathbf { B } = k \mathbf { I }$, where $\mathbf { I }$ is the identity matrix and $k$ is a constant whose value should be stated.

SPS SPS FM 2022 February Q2

2. The complex numbers $3 - 2 \mathrm { i }$ and $2 + \mathrm { i }$ are denoted by $z$ and $w$ respectively. Find, giving your answers in the form $x + \mathrm { i } y$ and showing clearly how you obtain these answers,

$2 z - 3 w$,
$( \mathrm { i } z ) ^ { 2 }$,
$\frac { z } { w }$.

SPS SPS FM 2022 February Q3

3. The diagram shows the curve $y = 4 - x ^ { 2 }$ and the line $y = x + 2$.
\includegraphics[max width=\textwidth, alt={}, center]{bcba22d8-5f22-4576-b57c-7fdd05d128ad-1_344_349_993_1372}

Find the $x$-coordinates of the points of intersection of the curve and the line.
Use integration to find the area of the shaded region bounded by the line and the curve.

SPS SPS FM 2022 February Q4

4. The transformation S is a shear parallel to the $x$-axis in which the image of the point ( 1,1 ) is the point $( 0,1 )$.

Draw a diagram showing the image of the unit square under S .
Write down the matrix that represents S .

SPS SPS FM 2022 February Q5

5.

Sketch the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$, and state the coordinates of any point where the curve crosses an axis.
Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$, the axes, and the line $x = 2$.
The point $P$ on the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ has $y$-coordinate equal to $\frac { 1 } { 6 }$. Prove that the $x$-coordinate of $P$ may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$ \section*{6.} In an Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$ are given by $$| z | = 2 \quad \text { and } \quad \arg z = \frac { 1 } { 3 } \pi$$ respectively.
Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.
Hence find, in the form $x + \mathrm { i } y$, the complex number representing the point of intersection of $C _ { 1 }$ and $C _ { 2 }$.

SPS SPS FM 2022 February Q7

7. The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)$.

Find $\mathbf { A } ^ { 2 }$ and $\mathbf { A } ^ { 3 }$.
Hence suggest a suitable form for the matrix $\mathbf { A } ^ { n }$.
Use induction to prove that your answer to part (ii) is correct.

SPS SPS FM 2022 February Q8

8.

Expand $( 1 - 3 x ) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Find the coefficient of $x ^ { 2 }$ in the expansion of $\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }$ in ascending powers of $x$.

SPS SPS FM 2022 February Q9

9. The position vectors of three points $A , B$ and $C$ relative to an origin $O$ are given respectively by $$\text { and } \quad \begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } ,
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } . \end{aligned}$$

Find the angle between $A B$ and $A C$.
Find the area of triangle $A B C$.

SPS SPS SM 2022 February Q1

1.

Evaluate $27 ^ { - \frac { 2 } { 3 } }$.
Express $5 \sqrt { 5 }$ in the form $5 ^ { n }$.
Express $\frac { 1 - \sqrt { 5 } } { 3 + \sqrt { 5 } }$ in the form $a + b \sqrt { 5 }$.

SPS SPS SM 2022 February Q2

2.

Solve the equation $x ^ { 4 } - 10 x ^ { 2 } + 25 = 0$.
Given that $y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the number of stationary points on the curve $y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3$.

SPS SPS SM 2022 February Q3

3. Solve each of the following equations, for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.

$2 \sin ^ { 2 } x = 1 + \cos x$.
$\sin 2 x = - \cos 2 x$.

SPS SPS SM 2022 February Q4

4.

By expanding the brackets, show that $$( x - 4 ) ( x - 3 ) ( x + 1 ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
Sketch the curve $$y = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$ giving the coordinates of the points where the curve meets the axes. Label the curve $C _ { 1 }$.
On the same diagram as in part (ii), sketch the curve $$y = - x ^ { 3 } + 6 x ^ { 2 } - 5 x - 12$$ Label this curve $C _ { 2 }$.

SPS SPS SM 2022 February Q5

5. The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }$, and the curve passes through the point $( 4,5 )$. Find the equation of the curve. \section*{6.} The diagram shows the curve $y = 4 - x ^ { 2 }$ and the line $y = x + 2$.
\includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_408_435_27_1560}

Find the $x$-coordinates of the points of intersection of the curve and the line.
Use integration to find the area of the shaded region bounded by the line and the curve.

SPS SPS SM 2022 February Q7

7. The diagram shows a triangle $A B C$, and a sector $A C D$ of a circle with centre $A$. It is given that $A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}$, angle $A B C = 0.8$ radians and angle $D A C = 1.7$ radians. The shaded segment is bounded by the line $D C$ and the $\operatorname { arc } D C$.

Show that the length of $A C$ is 7.90 cm , correct to 3 significant figures.
Find the area of the shaded segment.
Find the perimeter of the shaded segment.
\includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_641_1360_1219_310}

SPS SPS SM 2022 February Q8

8. The diagram shows the graph of $y = \mathrm { f } ( x )$, where

Evaluate $\mathrm { ff } ( - 3 )$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$. $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
Sketch the graph of $y = \mathrm { f } ^ { - 1 } ( x )$. Indicate the coordinates of the points where the graph meets the axes.
\includegraphics[max width=\textwidth, alt={}, center]{6831b037-2ec7-499a-8bf6-62728a5681b7-2_476_490_2014_1352}

SPS SPS FM Pure 2022 June Q1

(a) For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
1. $\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x$;
  (3 marks)
2. $\quad \int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x$.
  (3 marks)
  (b) Explain briefly why the integrals in part (a) are improper integrals.
  (1 mark)
  [0pt] [BLANK PAGE]
\section*{2.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-06_592_1027_258_593}
\end{figure} Figure 1 shows part of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The graph consists of two line segments that meet at the point $( 1 , a ) , a < 0$. One line meets the $x$-axis at $( 3,0 )$. The other line meets the $x$-axis at $( - 1,0 )$ and the $y$-axis at $( 0 , b ) , b < 0$. In separate diagrams, sketch the graph with equation
(a) $y = \mathrm { f } ( x + 1 )$,
(b) $y = \mathrm { f } ( | x | )$. Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that $\mathrm { f } ( x ) = | x - 1 | - 2$, find
(c) the value of $a$ and the value of $b$,
(d) the value of $x$ for which $\mathrm { f } ( x ) = 5 x$.
[0pt] [BLANK PAGE]
3. (a) Show on an Argand diagram the locus of points given by $$| z - 10 - 12 i | = 8$$ Set $A$ is defined by $$A = \left\{ z : 0 \leqslant \arg ( z - 10 - 10 i ) \leqslant \frac { \pi } { 2 } \right\} \cap \{ z : | z - 10 - 12 i | \leqslant 8 \}$$ (b) Shade the region defined by $A$ on your Argand diagram.
(c) Determine the area of the region defined by $A$.
[0pt] [BLANK PAGE]
4. The curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at $x = \alpha$
(a) Show that $\alpha$ is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for $\alpha$.
Starting with $x _ { 1 } = 0.3$
(b) calculate, giving each answer to 4 decimal places,
the value of $x _ { 2 }$
the value of $x _ { 4 }$ Using a suitable interval and a suitable function that should be stated,
(c) show that $\alpha$ is 0.341 to 3 decimal places.
[0pt] [BLANK PAGE]
5. The triangle $T$ has vertices at the points $( 1 , k ) , ( 3,0 )$ and $( 11,0 )$, where $k$ is a positive constant. Triangle $T$ is transformed onto the triangle $T ^ { \prime }$ by the matrix $$\left( \begin{array} { c c } 6 & - 2
1 & 2 \end{array} \right)$$ Given that the area of triangle $T ^ { \prime }$ is 364 square units, find the value of $k$.
[0pt] [BLANK PAGE]
6. The complex number $w$ is given by $$w = 10 - 5 \mathrm { i }$$ (a) Find $| w |$.
(b) Find arg $w$, giving your answer in radians to 2 decimal places. The complex numbers $z$ and $w$ satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$ (c) Use algebra to find $z$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real numbers. Given that $$\arg ( \lambda + 9 i + w ) = \frac { \pi } { 4 }$$ where $\lambda$ is a real constant,
(d) find the value of $\lambda$.
[0pt] [BLANK PAGE]
7. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-16_634_1025_191_479}
Figure 1 shows the finite region $R$, which is bounded by the curve $y = x \mathrm { e } ^ { x }$, the line $x = 1$, the line $x = 3$ and the $x$-axis. The region $R$ is rotated through 360 degrees about the $x$-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
[0pt] [BLANK PAGE]
8. With respect to a fixed origin $O$, the line $l$ has equation $$\mathbf { r } = \left( \begin{array} { c } 13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r }

Questions — SPS (1106 questions)

Browse by module

Browse by board