Questions - SPS - A-Level Maths

SPS SPS FM 2020 December Q13

13. A series is given by $$\sum _ { r = 1 } ^ { k } 9 ^ { r - 1 }$$ i. Write down a formula for the sum of this series.
ii. Prove by induction that $P ( n ) = 9 ^ { n } - 8 n - 1$ is divisible by 64 if $n$ is a positive integer greater than 1 . Spare space for extra working. Spare space for extra working.

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 FM 2020 October Q1

i. Find the binomial expansion of $( 2 + x ) ^ { 5 }$, simplifying the terms.
ii. Hence find the coefficient of $y ^ { 3 }$ in the expansion of $\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }$.
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 FM 2020 October Q3

3. i. Give full details of a sequence of two transformations needed to transform the graph $y = | x |$ to the graph of $y = | 2 ( x + 3 ) |$.
ii. Solve $| x | > | 2 ( x + 3 ) |$, giving your answer in set notation.

SPS SPS FM 2020 October Q4

4. Prove by induction that, for $n \geq 1 , \sum _ { r = 1 } ^ { n } r ( 3 r + 1 ) = n ( n + 1 ) ^ { 2 }$.

SPS SPS FM 2020 October Q5

5. The diagram shows triangle $A B C$, with $A B = x \mathrm {~cm} , A C = ( x + 2 ) \mathrm { cm } , B C = 2 \sqrt { 7 } \mathrm {~cm}$ and angle $C A B = 60 ^ { \circ }$.
i. Find the value of $x$.
ii. Find the area of triangle $A B C$, giving your answer in an exact form as simply as possible.

SPS SPS FM 2020 October Q6

6. Prove by contradiction that $\sqrt { 7 }$ is irrational.

SPS SPS FM 2020 October Q7

7. A curve has equation $y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }$.
i. Find $\frac { d y } { d x }$.
ii. Hence sketch the gradient function for the curve.
iii. Find the equation of the tangent to the curve $y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }$ at $x = 4$.

SPS SPS FM 2020 October Q8

8. The equation of a circle is $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
i. Find the centre and radius of the circle.
ii. Find the coordinates of any points where the line $y = 2 x - 3$ meets the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
iii. State what can be deduced from the answer to part ii. about the line $y = 2 x - 3$ and the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$.
iv. The point $A ( - 1,5 )$ lies on the circumference of the circle $x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0$. Given that $A B$ is a diameter of the circle, find the coordinates of $B$.

SPS SPS FM 2020 October Q9

9. In this question you must show detailed reasoning. A sequence $t _ { 1 } , t _ { 2 } , t _ { 3 } \ldots$ is defined by $t _ { n } = 5 - 2 n$.
Use an algebraic method to find the smallest value of $N$ such that $$\sum _ { n = 1 } ^ { \infty } 2 ^ { t _ { n } } - \sum _ { n = 1 } ^ { N } 2 ^ { t _ { n } } < 10 ^ { - 8 }$$

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.

SPS SPS SM 2020 October Q7

7. i. Sketch the curves $y = \frac { 3 } { x ^ { 2 } }$ and $y = x ^ { 2 } - 2$ on the axes provided below.
\includegraphics[max width=\textwidth, alt={}, center]{e1b41613-a703-4eb3-9760-7b47b1dad099-06_849_921_1683_644}
ii. In this question you must show detailed reasoning. Find the exact coordinates of the points of interception of the curves $y = \frac { 3 } { x ^ { 2 } }$ and $y = x ^ { 2 } - 2$.
iii. Hence, solve the inequality $\frac { 3 } { x ^ { 2 } } \leq x ^ { 2 } - 2$, giving your answer in interval notation.

SPS SPS SM 2020 October Q9

9. In this question you must show detailed reasoning. Solve the following simultaneous equations: $$\begin{gathered} \left( \log _ { 3 } x \right) ^ { 2 } + \log _ { 3 } \left( y ^ { 2 } \right) = 5
\log _ { 3 } \left( \sqrt { 3 } x y ^ { - 1 } \right) = 2 \end{gathered}$$

SPS SPS SM 2020 October Q10

In this question you must show detailed reasoning.

A sequence $t _ { 1 } , t _ { 2 } , t _ { 3 } \ldots$ is defined by $t _ { n } = 25 \times 0.6 ^ { n }$.
Use an algebraic method to find the smallest value of $N$ such that $$\sum _ { n = 1 } ^ { \infty } t _ { n } - \sum _ { n = 1 } ^ { N } t _ { n } < 10 ^ { - 4 }$$

Questions — SPS (1106 questions)

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SPS SPS FM 2020 December Q13

SPS SPS SM 2020 December Q1

SPS SPS SM 2020 December Q10

SPS SPS SM 2021 January Q1

SPS SPS SM 2021 January Q2

SPS SPS SM 2021 January Q3

SPS SPS SM 2021 January Q4

SPS SPS SM 2021 January Q5

SPS SPS SM 2021 January Q6

SPS SPS SM 2021 January Q7

SPS SPS SM 2021 January Q8

SPS SPS SM 2021 January Q9

SPS SPS FM 2020 October Q1

SPS SPS FM 2020 October Q3

SPS SPS FM 2020 October Q4

SPS SPS FM 2020 October Q5

SPS SPS FM 2020 October Q6

SPS SPS FM 2020 October Q7

SPS SPS FM 2020 October Q8

SPS SPS FM 2020 October Q9

SPS SPS SM 2020 October Q1

SPS SPS SM 2020 October Q6

SPS SPS SM 2020 October Q7

SPS SPS SM 2020 October Q9

SPS SPS SM 2020 October Q10