Questions - A-Level Maths

Express each of the vectors $\overrightarrow { A M }$ and $\overrightarrow { G M }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
Use a scalar product to find angle $G M A$ correct to the nearest degree.

CAIE P1 2019 June Q8

8 marks Standard +0.3

The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme $A$ is to increase the amount of waste recycled each month by 0.16 tonnes.
Scheme $B$ is to increase the amount of waste recycled each month by $6 \%$ of the amount recycled in the previous month.
The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme $A$
Scheme $B$ $\_\_\_\_$

CAIE P1 2019 June Q9

7 marks Moderate -0.3

9 The function f is defined by $\mathrm { f } ( x ) = 2 - 3 \cos x$ for $0 \leqslant x \leqslant 2 \pi$.

State the range of f .
Sketch the graph of $y = \mathrm { f } ( x )$. The function g is defined by $\mathrm { g } ( x ) = 2 - 3 \cos x$ for $0 \leqslant x \leqslant p$, where $p$ is a constant.
State the largest value of $p$ for which g has an inverse.
For this value of $p$, find an expression for $\mathrm { g } ^ { - 1 } ( x )$.

CAIE P1 2019 June Q10

9 marks Moderate -0.8

10 A curve for which $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5$ has a stationary point at $( 3,6 )$.

Find the equation of the curve.
Find the $x$-coordinate of the other stationary point on the curve.
Determine the nature of each of the stationary points.
\includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve $y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }$ and a point $P ( 2,1 )$ lying on the curve. The normal to the curve at $P$ intersects the $x$-axis at $Q$.
Show that the $x$-coordinate of $Q$ is $\frac { 16 } { 9 }$.
Find, showing all necessary working, the area of the shaded region.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P1 2019 June Q1

3 marks Moderate -0.5

1 Find the coefficient of $x$ in the expansion of $\left( \frac { 2 } { x } - 3 x \right) ^ { 5 }$.

CAIE P1 2019 June Q2

5 marks Moderate -0.8

2 Two points $A$ and $B$ have coordinates $( 1,3 )$ and $( 9 , - 1 )$ respectively. The perpendicular bisector of $A B$ intersects the $y$-axis at the point $C$. Find the coordinates of $C$.

CAIE P1 2019 June Q3

5 marks Moderate -0.8

3 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }$. The point $P ( 2,9 )$ lies on the curve.

A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. [2]
Find the equation of the curve.

CAIE P1 2019 June Q4

5 marks Moderate -0.3

4 Angle $x$ is such that $\sin x = a + b$ and $\cos x = a - b$, where $a$ and $b$ are constants.

Show that $a ^ { 2 } + b ^ { 2 }$ has a constant value for all values of $x$.
In the case where $\tan x = 2$, express $a$ in terms of $b$.

CAIE P1 2019 June Q5

5 marks Standard +0.3

5
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter $A B$, centre $O$ and radius $r$. The point $C$ lies on the circumference and angle $A O C = \theta$ radians. The perimeter of sector $B O C$ is twice the perimeter of sector $A O C$. Find the value of $\theta$ correct to 2 significant figures.

CAIE P1 2019 June Q6

7 marks Moderate -0.8

6 The equation of a curve is $y = 3 \cos 2 x$ and the equation of a line is $2 y + \frac { 3 x } { \pi } = 5$.

State the smallest and largest values of $y$ for both the curve and the line for $0 \leqslant x \leqslant 2 \pi$.
Sketch, on the same diagram, the graphs of $y = 3 \cos 2 x$ and $2 y + \frac { 3 x } { \pi } = 5$ for $0 \leqslant x \leqslant 2 \pi$.
State the number of solutions of the equation $6 \cos 2 x = 5 - \frac { 3 x } { \pi }$ for $0 \leqslant x \leqslant 2 \pi$.

CAIE P1 2019 June Q7

7 marks Moderate -0.8

7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 2 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 2 x + 3 } { x - 1 } , \quad x \in \mathbb { R } , x \neq 1 \end{aligned}$$

Obtain expressions for $\mathrm { f } ^ { - 1 } ( x )$ and $\mathrm { g } ^ { - 1 } ( x )$, stating the value of $x$ for which $\mathrm { g } ^ { - 1 } ( x )$ is not defined. [4]
Solve the equation $\operatorname { fg } ( x ) = \frac { 7 } { 3 }$.

CAIE P1 2019 June Q8

8 marks Moderate -0.3

8 The position vectors of points $A$ and $B$, relative to an origin $O$, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 6 \\ - 2 \\ - 6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ k \\ - 3 \end{array} \right)$$ where $k$ is a constant.

Find the value of $k$ for which angle $A O B$ is $90 ^ { \circ }$.
Find the values of $k$ for which the lengths of $O A$ and $O B$ are equal.
The point $C$ is such that $\overrightarrow { A C } = 2 \overrightarrow { C B }$.
In the case where $k = 4$, find the unit vector in the direction of $\overrightarrow { O C }$.

CAIE P1 2019 June Q9

8 marks Challenging +1.2

9 The curve $C _ { 1 }$ has equation $y = x ^ { 2 } - 4 x + 7$. The curve $C _ { 2 }$ has equation $y ^ { 2 } = 4 x + k$, where $k$ is a constant. The tangent to $C _ { 1 }$ at the point where $x = 3$ is also the tangent to $C _ { 2 }$ at the point $P$. Find the value of $k$ and the coordinates of $P$.

CAIE P1 2019 June Q10

10 marks Standard +0.3

In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is $a$.
1. Show that the common difference of the progression is $\frac { 1 } { 3 } a$.
2. Given that the tenth term is 36 more than the fourth term, find the value of $a$.
The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12 , find the value of the fifth term.

CAIE P1 2019 June Q11

12 marks Standard +0.3

11
\includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve $y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }$ and the minimum point $M$.

Find expressions for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\int y \mathrm {~d} x$.
Find the coordinates of $M$.
The shaded region is bounded by the curve, the $y$-axis and the line through $M$ parallel to the $x$-axis.
Find, showing all necessary working, the area of the shaded region.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P1 2019 June Q1

5 marks Easy -1.2

1 The function f is defined by $\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 8$ for $x \in \mathbb { R }$.

Express $x ^ { 2 } - 4 x + 8$ in the form $( x - a ) ^ { 2 } + b$.
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-02_67_1569_397_328}
Hence find the set of values of $x$ for which $\mathrm { f } ( x ) < 9$, giving your answer in exact form.

CAIE P1 2019 June Q2

5 marks Moderate -0.3

In the binomial expansion of $\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 5 }$, the first three terms are $32 x ^ { 5 } - 40 x ^ { 3 } + 20 x$. Find the remaining three terms of the expansion.
Hence find the coefficient of $x$ in the expansion of $\left( 1 + 4 x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 5 }$.
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-04_385_655_262_744} The diagram shows triangle $A B C$ which is right-angled at $A$. Angle $A B C = \frac { 1 } { 5 } \pi$ radians and $A C = 8 \mathrm {~cm}$. The points $D$ and $E$ lie on $B C$ and $B A$ respectively. The sector $A D E$ is part of circle with centre $A$ and is such that $B D C$ is the tangent to the $\operatorname { arc } D E$ at $D$.
Find the length of $A D$.
Find the area of the shaded region.

CAIE P1 2019 June Q4

5 marks Standard +0.3

4 The function f is defined by $\mathrm { f } ( x ) = \frac { 48 } { x - 1 }$ for $3 \leqslant x \leqslant 7$. The function g is defined by $\mathrm { g } ( x ) = 2 x - 4$ for $a \leqslant x \leqslant b$, where $a$ and $b$ are constants.

Find the greatest value of $a$ and the least value of $b$ which will permit the formation of the composite function gf.
It is now given that the conditions for the formation of gf are satisfied.
Find an expression for $\operatorname { gf } ( x )$.
Find an expression for $( \mathrm { gf } ) ^ { - 1 } ( x )$.

CAIE P1 2019 June Q5

7 marks Moderate -0.3

5 Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg . At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg . It is given that his weekly weight loss follows an arithmetic progression.

Write down an expression for his total weight loss after $x$ weeks.
He reaches his 13 kg target during week $n$. Use your answer to part (i) to find the value of $n$.
Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.
Calculate his total weight loss after 20 weeks and show that he can never reach his target.

CAIE P1 2019 June Q6

7 marks Moderate -0.3

6
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-08_739_867_260_641} The diagram shows a solid figure $A B C D E F$ in which the horizontal base $A B C$ is a triangle right-angled at $A$. The lengths of $A B$ and $A C$ are 8 units and 4 units respectively and $M$ is the mid-point of $A B$. The point $D$ is 7 units vertically above $A$. Triangle $D E F$ lies in a horizontal plane with $D E , D F$ and $F E$ parallel to $A B , A C$ and $C B$ respectively and $N$ is the mid-point of $F E$. The lengths of $D E$ and $D F$ are 4 units and 2 units respectively. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ respectively.

Find $\overrightarrow { M F }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
Find $\overrightarrow { F N }$ in terms of $\mathbf { i }$ and $\mathbf { j }$.
Find $\overrightarrow { M N }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
Use a scalar product to find angle $F M N$.

CAIE P1 2019 June Q7

9 marks Moderate -0.8

7 The coordinates of two points $A$ and $B$ are $( 1,3 )$ and $( 9 , - 1 )$ respectively and $D$ is the mid-point of $A B$. A point $C$ has coordinates $( x , y )$, where $x$ and $y$ are variables.

State the coordinates of $D$.
It is given that $C D ^ { 2 } = 20$. Write down an equation relating $x$ and $y$.
It is given that $A C$ and $B C$ are equal in length. Find an equation relating $x$ and $y$ and show that it can be simplified to $y = 2 x - 9$.
Using the results from parts (ii) and (iii), and showing all necessary working, find the possible coordinates of $C$.

CAIE P1 2019 June Q8

8 marks Moderate -0.3

8 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a x + b$. The curve has stationary points at $( - 1,2 )$ and $( 3 , k )$. Find the values of the constants $a , b$ and $k$.

CAIE P1 2019 June Q9

10 marks Moderate -0.3

9
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-14_558_963_260_589} The function $\mathrm { f } : x \mapsto p \sin ^ { 2 } 2 x + q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y = \mathrm { f } ( x )$.

In terms of $p$ and $q$, state the range of f .
State the number of solutions of the following equations.
(a) $\mathrm { f } ( x ) = p + q$
(b) $\mathrm { f } ( x ) = q$
(c) $\mathrm { f } ( x ) = \frac { 1 } { 2 } p + q$
For the case where $p = 3$ and $q = 2$, solve the equation $\mathrm { f } ( x ) = 4$, showing all necessary working.

CAIE P1 2019 June Q10

13 marks Standard +0.3

10
\includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-16_600_593_262_774} The diagram shows part of the curve with equation $y = ( 3 x + 4 ) ^ { \frac { 1 } { 2 } }$ and the tangent to the curve at the point $A$. The $x$-coordinate of $A$ is 4 .

Find the equation of the tangent to the curve at $A$.
Find, showing all necessary working, the area of the shaded region.
L
A point is moving along the curve. At the point $P$ the $y$-coordinate is increasing at half the rate at which the $x$-coordinate is increasing. Find the $x$-coordinate of $P$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P1 2016 March Q1

4 marks Moderate -0.8

Find the coefficients of $x ^ { 4 }$ and $x ^ { 5 }$ in the expansion of $( 1 - 2 x ) ^ { 5 }$.
It is given that, when $( 1 + p x ) ( 1 - 2 x ) ^ { 5 }$ is expanded, there is no term in $x ^ { 5 }$. Find the value of the constant $p$.

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