Questions - SPS SPS FM Pure

SPS SPS FM Pure 2021 May Q4

4. You are given that the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0
0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0
0 & 0 & 1 \end{array} \right)$, where $a$ is a positive constant, represents the transformation $R$ which is a reflection in 3-D.

State the plane of reflection of R .
Determine the value of $a$.
With reference to R explain why $\mathbf { A } ^ { 2 } = \mathbf { I }$, the $3 \times 3$ identity matrix.
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SPS SPS FM Pure 2021 May Q5

5 marks

5. Express $\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }$ in partial fractions.
[0pt] [5]
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SPS SPS FM Pure 2021 May Q6

6. A circle $C$ in the complex plane has equation $| z - 2 - 5 i | = a$. The point $z _ { 1 }$ on $C$ has the least argument of any point on $C$, and $\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }$.
Prove that $a = \frac { 3 \sqrt { 2 } } { 2 }$.
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SPS SPS FM Pure 2021 May Q7

7. The region $R$ between the $x$-axis, the curve $y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }$ and the lines $x = \sqrt { p }$ and $x = \sqrt { 3 p }$, where $p$ is a positive parameter, is rotated by $2 \pi$ radians about the $x$-axis to form a solid of revolution $S$.

Find and simplify an algebraic expression, in terms of $p$, for the exact volume of $S$.
Given that $R$ must lie entirely between the lines $x = 1$ and $x = \sqrt { 48 }$ find in exact form
- the greatest possible value of the volume of $S$
- the least possible value of the volume of $S$.
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SPS SPS FM Pure 2021 May Q8

8 marks

Let $C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos ( r \theta )$. Show that $C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos ( 10 \theta )$.
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During an industrial process substance $X$ is converted into substance $Z$. Some of the substance $X$ goes through an intermediate phase, and is converted to substance $Y$, before being converted to substance $Z$. The situation is modelled by

$$\frac { d y } { d t } = 0.3 x - 0.2 y \quad \text { and } \quad \frac { d z } { d t } = 0.2 y + 0.1 x$$ where $x , y$ and $z$ are the amounts in kg of $X , Y$ and $Z$ at time $t$ hours after the process starts. Initially there is 10 kg of substance $X$ and nothing of substance $Y$ and $Z$. The amount of substance $X$ decreases exponentially. The initial rate of decrease is 4 kg per hour.

Show that $x = A e ^ { - 0.4 t }$, stating the value of $A$.
Show that $\frac { d x } { d t } + \frac { d y } { d t } + \frac { d z } { d t } = 0$. Comment on this result in the context of the industrial process.
Express $y$ in terms of $t$.
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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)
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\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 $\mathrm { f } ( x ) = 5 x$.
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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 $A$.
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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.
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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 $k$.
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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 $\lambda$.
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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.
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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 }

SPS SPS FM Pure 2022 June Q3

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 $A$.
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SPS SPS FM Pure 2022 June Q4

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 $x = \alpha$

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 $x _ { 1 } = 0.3$
calculate, giving each answer to 4 decimal places,
1. the value of $x _ { 2 }$
2. the value of $x _ { 4 }$ Using a suitable interval and a suitable function that should be stated,
show that $\alpha$ is 0.341 to 3 decimal places.
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SPS SPS FM Pure 2022 June Q5

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 $k$.
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SPS SPS FM Pure 2022 June Q6

6. The complex number $w$ is given by $$w = 10 - 5 \mathrm { i }$$

Find $| w |$.
Find arg $w$, giving your answer in radians to 2 decimal places. The complex numbers $z$ and $w$ satisfy the equation $$( 2 + \mathrm { i } ) ( z + 3 \mathrm { i } ) = w$$
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,
find the value of $\lambda$.
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SPS SPS FM Pure 2022 June Q7

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.
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SPS SPS FM Pure 2022 June Q8

8. With respect to a fixed origin $O$, the line $l$ has equation $$\mathbf { r } = \left( \begin{array} { c } 13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates $( 3 , - 2,6 )$.
The point $P$ has position vector ( $- \boldsymbol { i } + 2 \boldsymbol { k }$ ) relative to $O$.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$, find the coordinates of the two possible positions of $B$.
(5)
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SPS SPS FM Pure 2022 June Q9

9. Prove by induction that for $n \in \mathbb { Z } ^ { + }$ $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
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SPS SPS FM Pure 2022 June Q10

10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point $P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)$ is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

Show that the equation of the normal to the curve at $P$ can be written as $3 y - x = \frac { 7 \sqrt { 3 } } { 2 }$
Show that the Cartesian equation of the curve may be written as $a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0$ where $a$, $b$ and $c$ are integers to be found.
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SPS SPS FM Pure 2022 June Q11

11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which $y = 0$ at $x = \frac { \pi } { 3 }$, giving your answer in the form $\sec ^ { 2 } x = \mathrm { g } ( y )$.
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SPS SPS FM Pure 2022 June Q13

13
8
1 \end{array} \right) + \lambda \left( \begin{array} { r } 2
2
- 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point $A$ lies on $l$ and has coordinates $( 3 , - 2,6 )$.
The point $P$ has position vector ( $- \boldsymbol { i } + 2 \boldsymbol { k }$ ) relative to $O$.
Given that vector $\overrightarrow { P A }$ is perpendicular to $l$, and that point $B$ is a point on $l$ such that $\angle B P A = 45 ^ { \circ }$, find the coordinates of the two possible positions of $B$.
(5)
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9. Prove by induction that for $n \in \mathbb { Z } ^ { + }$ $$\mathrm { f } ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
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10. The curve defined by the parametric equations $$x = 2 \cos \theta , y = 3 \sin ( 2 \theta ) \text { and } \theta \in [ 0,2 \pi ]$$ is shown below.
The point $P \left( \sqrt { 3 } , \frac { 3 \sqrt { 3 } } { 2 } \right)$ is marked on the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-22_604_826_518_758}

Show that the equation of the normal to the curve at $P$ can be written as $3 y - x = \frac { 7 \sqrt { 3 } } { 2 }$
Show that the Cartesian equation of the curve may be written as $a y ^ { 2 } + b x ^ { 4 } + c x ^ { 2 } = 0$ where $a$, $b$ and $c$ are integers to be found.
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11. Solve the differential equation $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which $y = 0$ at $x = \frac { \pi } { 3 }$, giving your answer in the form $\sec ^ { 2 } x = \mathrm { g } ( y )$.
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12. A linear transformation T of the $x - y$ plane has an associated matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l c } \lambda & k
1 & \lambda - k \end{array} \right)$, and $\lambda$
and $k$ are real constants. and $k$ are real constants.
You are given that $\operatorname { det } \mathbf { M } > 0$ for all values of $\lambda$.
1. Find the range of possible values of $k$.
2. What is the significance of the condition $\operatorname { det } \mathbf { M } > 0$ for the transformation T ? For the remainder of this question, take $k = - 2$.
Determine whether there are any lines through the origin that are invariant lines for the transformation T .
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13. (i) Show that $\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta$.
(ii) Hence solve the equation $\sin 3 \theta = \cos 2 \theta$ for $0 \leqslant \theta \leqslant 2 \pi$.
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SPS SPS FM Pure 2022 June Q14

14. Using an appropriate substitution, or otherwise, show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$ [BLANK PAGE]
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SPS SPS FM Pure 2021 September Q1

(a) The equation $\mathrm { e } ^ { - x } - 2 + \sqrt { x } = 0$ has a single root, $\alpha$.

Show that $\alpha$ lies between 3 and 4 .
(b) Use the recurrence relation $x _ { n + 1 } = \left( 2 - \mathrm { e } ^ { - x _ { n } } \right) ^ { 2 }$, with $x _ { 1 } = 3.5$, to find $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
(c) The diagram below shows parts of the graphs of $y = \left( 2 - \mathrm { e } ^ { - x } \right) ^ { 2 }$ and $y = x$, and a position of $x _ { 1 }$. On the diagram, draw a staircase or cobweb diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{f5cae2a4-a0f4-4227-a773-fcdecd87cb46-04_1180_1502_808_374}

SPS SPS FM Pure 2021 September Q2

2. (a) Find the binomial expansion of $( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
(2 marks)
(b) (i) Find the binomial expansion of $( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
(ii) Given that $\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }$, use your binomial expansion from part (b)(i) to obtain an approximation to $\sqrt [ 3 ] { \frac { 2 } { 7 } }$, giving your answer to six decimal places.
(2 marks)

SPS SPS FM Pure 2021 September Q3

3. The diagram below shows the graphs of $y = | 2 x - 3 |$ and $y = | x |$.
\includegraphics[max width=\textwidth, alt={}, center]{f5cae2a4-a0f4-4227-a773-fcdecd87cb46-08_645_1256_246_497}

Find the $x$-coordinates of the points of intersection of the graphs of $y = | 2 x - 3 |$ and $y = | x |$.
Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$

SPS SPS FM Pure 2021 September Q4

4. By forming and solving a quadratic equation, solve the equation $$8 \sec x - 2 \sec ^ { 2 } x = \tan ^ { 2 } x - 2$$ in the interval $0 < x < 2 \pi$, giving the values of $x$ in radians to three significant figures.

SPS SPS FM Pure 2021 September Q5

5.

1. By writing $\ln x$ as $( \ln x ) \times 1$, use integration by parts to find $\int \ln x \mathrm {~d} x$.
2. Find $\int ( \ln x ) ^ { 2 } \mathrm {~d} x$.
Use the substitution $u = \sqrt { x }$ to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$

SPS SPS FM Pure 2021 September Q6

The functions $f$ and $g$ are defined with their respective domains by

$$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } - 6 x + 5 , & \text { for } x \geqslant 3
\mathrm {~g} ( x ) = | x - 6 | , & \text { for all real values of } x \end{array}$$

Find the range of f .
The inverse of f is $\mathrm { f } ^ { - 1 }$. Find $\mathrm { f } ^ { - 1 } ( x )$. Give your answer in its simplest form.
1. Find $\mathrm { gf } ( x )$.
2. Solve the equation $\operatorname { gf } ( x ) = 6$.

SPS SPS FM Pure 2021 September Q7

7. The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 1 , - 5,6 )$ and $( - 4,5 , - 1 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3
- 2
4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7
- 7
5 \end{array} \right]$.

Show that the point $C$ lies on the line $l$.
Find a vector equation of the line that passes through points $A$ and $B$.
The point $D$ lies on the line through $A$ and $B$ such that the angle $C D A$ is a right angle. Find the coordinates of $D$.
The point $E$ lies on the line through $A$ and $B$ such that the area of triangle $A C E$ is three times the area of triangle $A C D$. Find the coordinates of the two possible positions of $E$.
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SPS SPS FM Pure 2021 September Q8

8. (a) It is given that $z = x + y \mathrm { i }$, where $x$ and $y$ are real numbers.

Write down, in terms of $x$ and $y$, an expression for $( z - 2 \mathrm { i } ) ^ { * }$.
(1 mark)
Solve the equation $$( z - 2 i ) ^ { * } = 4 i z + 3$$ giving your answer in the form $a + b \mathrm { i }$.
(b) It is given that $p + q \mathrm { i }$, where $p$ and $q$ are real numbers, is a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$. Without finding the values of $p$ and $q$, state why $p - q \mathrm { i }$ is not a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$.
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Questions — SPS SPS FM Pure (237 questions)

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