1.05f - OCR Spec

CAIE P1 2021 June Q4

3 marks Moderate -0.8

4 \includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-05_677_1591_260_278} The diagram shows part of the graph of $y = a \tan ( x - b ) + c$.
Given that $0 < b < \pi$, state the values of the constants $a , b$ and $c$.

CAIE P1 2022 June Q2

3 marks Easy -1.2

2 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-04_657_1253_269_431} The diagram shows part of the curve with equation $y = p \sin ( q \theta ) + r$, where $p , q$ and $r$ are constants.

State the value of $p$.
State the value of $q$.
State the value of $r$.

CAIE P1 2023 June Q7

5 marks Standard +0.3

7 A curve has equation $y = 2 + 3 \sin \frac { 1 } { 2 } x$ for $0 \leqslant x \leqslant 4 \pi$.

State greatest and least values of $y$.
Sketch the curve. \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for $0 \leqslant x \leqslant 4 \pi$.

CAIE P1 2024 June Q2

5 marks Moderate -0.5

2

\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575} The diagram shows the curve $y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)$ where $k$ is a positive constant and $x$ is measured in radians. The curve crosses the $x$-axis at point $A$ and $B$ is a minimum point. Find the coordinates of $A$ and $B$.
Find the exact value of $t$ that satisfies the equation $$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}

CAIE P1 2020 November Q4

3 marks Moderate -0.8

4 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-05_615_1169_260_488} In the diagram, the lower curve has equation $y = \cos \theta$. The upper curve shows the result of applying a combination of transformations to $y = \cos \theta$. Find, in terms of a cosine function, the equation of the upper curve.

CAIE P1 2020 November Q11

11 marks Moderate -0.3

11 A curve has equation $y = 3 \cos 2 x + 2$ for $0 \leqslant x \leqslant \pi$.

State the greatest and least values of $y$.
Sketch the graph of $y = 3 \cos 2 x + 2$ for $0 \leqslant x \leqslant \pi$.
By considering the straight line $y = k x$, where $k$ is a constant, state the number of solutions of the equation $3 \cos 2 x + 2 = k x$ for $0 \leqslant x \leqslant \pi$ in each of the following cases.
1. $k = - 3$
2. $k = 1$
3. $k = 3$ Functions $\mathrm { f } , \mathrm { g }$ and h are defined for $x \in \mathbb { R }$ by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\ & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\ & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
Describe fully a sequence of transformations that maps the graph of $y = \mathrm { f } ( x )$ on to $y = \mathrm { g } ( x )$.
Describe fully a sequence of transformations that maps the graph of $y = \mathrm { f } ( x )$ on to $y = \mathrm { h } ( x )$. [2]
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 2021 November Q5

5 marks Moderate -0.5

5 \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of $y = a \cos ( b x ) + c$.

Find the values of the positive integers $a , b$ and $c$.
For these values of $a$, $b$ and $c$, use the given diagram to determine the number of solutions in the interval $0 \leqslant x \leqslant 2 \pi$ for each of the following equations.
1. $a \cos ( b x ) + c = \frac { 6 } { \pi } x$
2. $a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x$ The diagram shows a metal plate $A B C$ in which the sides are the straight line $A B$ and the arcs $A C$ and $B C$. The line $A B$ has length 6 cm . The arc $A C$ is part of a circle with centre $B$ and radius 6 cm , and the arc $B C$ is part of a circle with centre $A$ and radius 6 cm .

CAIE P1 2023 November Q8

5 marks Standard +0.3

8 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577} The diagram shows part of the graph of $y = \sin ( a ( x + b ) )$, where $a$ and $b$ are positive constants.

State the value of $a$ and one possible value of $b$.
Another curve, with equation $y = \mathrm { f } ( x )$, has a single stationary point at the point $( p , q )$, where $p$ and $q$ are constants. This curve is transformed to a curve with equation $$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of $p$ and $q$.

CAIE P1 2002 June Q6

7 marks Easy -1.2

6 The function f , where $\mathrm { f } ( x ) = a \sin x + b$, is defined for the domain $0 \leqslant x \leqslant 2 \pi$. Given that $\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2$ and that $\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8$,

find the values of $a$ and $b$,
find the values of $x$ for which $\mathrm { f } ( x ) = 0$, giving your answers in radians correct to 2 decimal places,
sketch the graph of $y = \mathrm { f } ( x )$.

CAIE P1 2003 June Q6

5 marks Moderate -0.3

6

Sketch the graph of the curve $y = 3 \sin x$, for $- \pi \leqslant x \leqslant \pi$. The straight line $y = k x$, where $k$ is a constant, passes through the maximum point of this curve for $- \pi \leqslant x \leqslant \pi$.
Find the value of $k$ in terms of $\pi$.
State the coordinates of the other point, apart from the origin, where the line and the curve intersect.

CAIE P1 2005 June Q7

7 marks Standard +0.3

7 A function f is defined by f : $x \mapsto 3 - 2 \sin x$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.

Find the range of f .
Sketch the graph of $y = \mathrm { f } ( x )$. A function g is defined by $\mathrm { g } : x \mapsto 3 - 2 \sin x$, for $0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }$, where $A$ is a constant.
State the largest value of $A$ for which g has an inverse.
When $A$ has this value, obtain an expression, in terms of $x$, for $\mathrm { g } ^ { - 1 } ( x )$.

CAIE P1 2009 June Q4

6 marks Moderate -0.8

4 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_561_1210_895_465} The diagram shows the graph of $y = a \sin ( b x ) + c$ for $0 \leqslant x \leqslant 2 \pi$.

Find the values of $a , b$ and $c$.
Find the smallest value of $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ for which $y = 0$.

CAIE P1 2013 June Q5

6 marks Moderate -0.3

5

Sketch, on the same diagram, the curves $y = \sin 2 x$ and $y = \cos x - 1$ for $0 \leqslant x \leqslant 2 \pi$.
Hence state the number of solutions, in the interval $0 \leqslant x \leqslant 2 \pi$, of the equations
1. $2 \sin 2 x + 1 = 0$,
2. $\sin 2 x - \cos x + 1 = 0$.

CAIE P1 2014 June Q1

2 marks Easy -1.2

1 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-2_750_1287_258_427} The diagram shows part of the graph of $y = a + b \sin x$. State the values of the constants $a$ and $b$. [2

CAIE P1 2017 June Q5

7 marks Moderate -0.8

5 The equation of a curve is $y = 2 \cos x$.

Sketch the graph of $y = 2 \cos x$ for $- \pi \leqslant x \leqslant \pi$, stating the coordinates of the point of intersection with the $y$-axis. Points $P$ and $Q$ lie on the curve and have $x$-coordinates of $\frac { 1 } { 3 } \pi$ and $\pi$ respectively.
Find the length of $P Q$ correct to 1 decimal place.
The line through $P$ and $Q$ meets the $x$-axis at $H ( h , 0 )$ and the $y$-axis at $K ( 0 , k )$.
Show that $h = \frac { 5 } { 9 } \pi$ and find the value of $k$.

CAIE P1 2018 June Q4

6 marks Moderate -0.8

4 The function f is such that $\mathrm { f } ( x ) = a + b \cos x$ for $0 \leqslant x \leqslant 2 \pi$. It is given that $\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5$ and $\mathrm { f } ( \pi ) = 11$.

Find the values of the constants $a$ and $b$. \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}
Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = k$ has no solution. \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632} The diagram shows a three-dimensional shape. The base $O A B$ is a horizontal triangle in which angle $A O B$ is $90 ^ { \circ }$. The side $O B C D$ is a rectangle and the side $O A D$ lies in a vertical plane. Unit vectors $\mathbf { i }$ and $\mathbf { j }$ are parallel to $O A$ and $O B$ respectively and the unit vector $\mathbf { k }$ is vertical. The position vectors of $A , B$ and $D$ are given by $\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }$ and $\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }$.

CAIE P1 2018 June Q10

8 marks Moderate -0.3

10

Solve the equation $2 \cos x + 3 \sin x = 0$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Sketch, on the same diagram, the graphs of $y = 2 \cos x$ and $y = - 3 \sin x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Use your answers to parts (i) and (ii) to find the set of values of $x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ for which $2 \cos x + 3 \sin x > 0$.

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 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 2004 November Q4

5 marks Moderate -0.8

4

Sketch and label, on the same diagram, the graphs of $y = 2 \sin x$ and $y = \cos 2 x$, for the interval $0 \leqslant x \leqslant \pi$.
Hence state the number of solutions of the equation $2 \sin x = \cos 2 x$ in the interval $0 \leqslant x \leqslant \pi$.

CAIE P1 2008 November Q5

8 marks Moderate -0.8

5 The function f is such that $\mathrm { f } ( x ) = a - b \cos x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$, where $a$ and $b$ are positive constants. The maximum value of $\mathrm { f } ( x )$ is 10 and the minimum value is - 2 .

Find the values of $a$ and $b$.
Solve the equation $\mathrm { f } ( x ) = 0$.
Sketch the graph of $y = \mathrm { f } ( x )$.

CAIE P1 2009 November Q2

4 marks Easy -1.2

2 The equation of a curve is $y = 3 \cos 2 x$. The equation of a line is $x + 2 y = \pi$. On the same diagram, sketch the curve and the line for $0 \leqslant x \leqslant \pi$.

CAIE P1 2010 November Q6