Questions - OCR - A-Level Maths

Find an equation for the tangent to $C$ at $P$. The tangent to $C$ at $P$ meets the coordinate axes at the points $Q$ and $R$.
Show that the area of triangle $O Q R$, where $O$ is the origin, is $\frac { 9 } { 3 - \mathrm { e } }$.

OCR C3 Q4

8 marks Moderate -0.3

4. The finite region $R$ is bounded by the curve with equation $y = \frac { 1 } { 2 x - 1 }$, the $x$-axis and the lines $x = 1$ and $x = 2$.

Find the exact area of $R$.
Show that the volume of the solid formed when $R$ is rotated through four right angles about the $x$-axis is $\frac { 1 } { 3 } \pi$.

OCR C3 Q5

8 marks Standard +0.3

5.
\includegraphics[max width=\textwidth, alt={}, center]{5dd332a5-56d9-407a-8ff6-fa59294b358d-2_520_787_246_479} The diagram shows the graph of $y = \mathrm { f } ( x )$. The graph has a minimum at $\left( \frac { \pi } { 2 } , - 1 \right)$, a maximum at $\left( \frac { 3 \pi } { 2 } , - 5 \right)$ and an asymptote with equation $x = \pi$.

Showing the coordinates of any stationary points, sketch the graph of $y = | \mathrm { f } ( x ) |$. Given that $$\mathrm { f } : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi$$
find the values of the constants $a$ and $b$,
find, to 2 decimal places, the $x$-coordinates of the points where the graph of $y = \mathrm { f } ( x )$ crosses the $x$-axis.

OCR C3 Q6

11 marks Standard +0.8

6. (i) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z }$$ (ii) Solve, for $0 \leq x < \pi$, the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7$$ giving your answers to 2 decimal places.

OCR C3 Q7

11 marks Moderate -0.3

7. The function $f$ is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R }$$

State the range of f .
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain. The function $g$ is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R }$$ Find, in terms of e,
the value of $\mathrm { gf } ( \ln 2 )$,
the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4$$

OCR C3 Q8

13 marks Standard +0.3

A curve has the equation $y = x ^ { 2 } - \sqrt { 4 + \ln x }$.
1. Show that the tangent to the curve at the point where $x = 1$ has the equation
$$7 x - 4 y = 11$$ The curve has a stationary point with $x$-coordinate $\alpha$.
Show that $0.3 < \alpha < 0.4$
Show that $\alpha$ is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with $x _ { 0 } = 0.35$, to find $\alpha$ correct to 5 decimal places.
You should show the result of each iteration.

OCR C3 Q1

5 marks Standard +0.3

(i) Solve the inequality

$$| x - 0.2 | < 0.03$$ (ii) Hence, find all integers $n$ such that $$\left| 0.95 ^ { n } - 0.2 \right| < 0.03$$

OCR C3 Q2

5 marks Standard +0.3

\includegraphics[max width=\textwidth, alt={}]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-1_474_823_685_482}

The diagram shows the curve with equation $y = x \sqrt { 2 - x } , 0 \leq x \leq 2$.
Find, in terms of $\pi$, the volume of the solid formed when the region bounded by the curve and the $x$-axis is rotated through $360 ^ { \circ }$ about the $x$-axis.

OCR C3 Q3

6 marks Standard +0.3

3. Solve, for $0 \leq y \leq 360$, the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$

OCR C3 Q4

7 marks Standard +0.3

A curve has the equation $x = y \sqrt { 1 - 2 y }$.
1. Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point $A$ on the curve has $y$-coordinate - 1 .
Show that the equation of tangent to the curve at $A$ can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where $p$ and $q$ are integers to be found.

OCR C3 Q5

7 marks Moderate -0.3

5. The function $f$ is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0$$

Solve the equation $\mathrm { f } ( x ) = 0$.
Sketch the curve $y = \mathrm { f } ( x )$. The function g is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where $a$ and $b$ are integers to be found.

OCR C3 Q6

8 marks Moderate -0.5

6. Find the value of each of the following integrals in exact, simplified form.

$\quad \int _ { - 1 } ^ { 0 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x$
$\int _ { 2 } ^ { 4 } \frac { 3 x ^ { 2 } - 2 } { x } \mathrm {~d} x$

OCR C3 Q7

10 marks Standard +0.3

7 $$f ( x ) = 2 + \cos x + 3 \sin x$$

Express $\mathrm { f } ( x )$ in the form $$\mathrm { f } ( x ) = a + b \cos ( x - c )$$ where $a , b$ and $c$ are constants, $b > 0$ and $0 < c < \frac { \pi } { 2 }$.
Solve the equation $\mathrm { f } ( x ) = 0$ for $x$ in the interval $0 \leq x \leq 2 \pi$.
Use Simpson's rule with four strips, each of width 0.5 , to find an approximate value for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$

OCR C3 Q8

11 marks Moderate -0.3

8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1$$

Express $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$.
Describe fully two transformations that would map the graph of $y = x ^ { 2 } , x \geq 0$ onto the graph of $y = \mathrm { f } ( x )$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.
Sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ on the same diagram and state the relationship between them.

OCR C3 Q9

13 marks Standard +0.3

9.
\includegraphics[max width=\textwidth, alt={}, center]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-3_501_1111_877_413} The diagram shows a graph of the temperature of a room, $T ^ { \circ } \mathrm { C }$, at time $t$ minutes.
The temperature is controlled by a thermostat such that when the temperature falls to $12 ^ { \circ } \mathrm { C }$, a heater is turned on until the temperature reaches $18 ^ { \circ } \mathrm { C }$. The room then cools until the temperature again falls to $12 ^ { \circ } \mathrm { C }$. For $t$ in the interval $10 \leq t \leq 60 , T$ is given by $$T = 5 + A \mathrm { e } ^ { - k t } ,$$ where $A$ and $k$ are constants.
Given that $T = 18$ when $t = 10$ and that $T = 12$ when $t = 60$,

show that $k = 0.0124$ to 3 significant figures and find the value of $A$,
find the rate at which the temperature of the room is decreasing when $t = 20$. The temperature again reaches $18 ^ { \circ } \mathrm { C }$ when $t = 70$ and the graph for $70 \leq t \leq 120$ is a translation of the graph for $10 \leq t \leq 60$.
Find the value of the constant $B$ such that for $70 \leq t \leq 120$ $$T = 5 + B \mathrm { e } ^ { - k t }$$

OCR C3 Q1

4 marks Moderate -0.3

Evaluate

$$\int _ { 2 } ^ { 6 } \sqrt { 3 x - 2 } \mathrm {~d} x$$

OCR C3 Q2

5 marks Moderate -0.8

Differentiate each of the following with respect to $x$ and simplify your answers.
1. $\frac { 6 } { \sqrt { 2 x - 7 } }$
2. $x ^ { 2 } \mathrm { e } ^ { - x }$
3. (i) Prove the identity
$$\sqrt { 2 } \cos ( x + 45 ) ^ { \circ } + 2 \cos ( x - 30 ) ^ { \circ } \equiv ( 1 + \sqrt { 3 } ) \cos x ^ { \circ }$$
Hence, find the exact value of $\cos 75 ^ { \circ }$ in terms of surds.

OCR C3 Q4

8 marks Standard +0.3

4. $\mathrm { f } ( x ) = x ^ { 2 } + 5 x - 2$ sec $x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.

Show that the equation $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, such that $1 < \alpha < 1.5$
Show that a suitable rearrangement of the equation $\mathrm { f } ( x ) = 0$ leads to the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { x _ { n } ^ { 2 } + 5 x _ { n } } \right)$$
Use the iterative formula in part (ii) with a starting value of 1.25 to find $\alpha$ correct to 3 decimal places. You should show the result of each iteration.

OCR C3 Q5

8 marks Standard +0.0

5. The function $f$ is defined by $$f ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 }$$

Find the exact value of $\mathrm { ff } ( 1 )$.
Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $x = 1$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.

OCR C3 Q6

9 marks Standard +0.8

6. (i) Sketch on the same diagram the graphs of $y = | x | - a$ and $y = | 3 x + 5 a |$, where $a$ is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes.
(ii) Solve the equation $$| x | - a = | 3 x + 5 a |$$

OCR C3 Q7

9 marks Standard +0.8

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The diagram shows the curve with equation $y = 2 x - \mathrm { e } ^ { \frac { 1 } { 2 } x }$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 4$.

Find the area of the shaded region, giving your answer in terms of e. The shaded region is rotated through four right angles about the $x$-axis.
Using Simpson's rule with two strips, estimate the volume of the solid formed.

OCR C3 Q8

10 marks Standard +0.8

8. (i) Sketch on the same diagram the graphs of $$y = \sin ^ { - 1 } x , - 1 \leq x \leq 1$$ and $$y = \cos ^ { - 1 } ( 2 x ) , \quad - \frac { 1 } { 2 } \leq x \leq \frac { 1 } { 2 }$$ Given that the graphs intersect at the point with coordinates $( a , b )$,
(ii) show that $\tan b = \frac { 1 } { 2 }$,
(iii) find the value of $a$ in the form $k \sqrt { 5 }$.

OCR C3 Q9

12 marks Standard +0.3

9. $\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , x \in \mathbb { R }$.

State the range of f . The curve $y = \mathrm { f } ( x )$ meets the $y$-axis at the point $P$ and the $x$-axis at the point $Q$.
Find the exact coordinates of $P$ and $Q$.
Show that the tangent to the curve at $P$ has the equation $$y = 3 e x + e - 2$$
Find to 3 significant figures the $x$-coordinate of the point where the tangent to the curve at $P$ meets the tangent to the curve at $Q$.

OCR C3 Q1

5 marks Standard +0.3

\includegraphics[max width=\textwidth, alt={}]{5e6a37a1-c51f-4637-aaae-48da6ab3eca0-1_305_606_219_539}

The diagram shows the curve with equation $y = \ln ( 2 + \cos x ) , x \geq 0$.
The smallest value of $x$ for which the curve meets the $x$-axis is $a$ as shown.

Find the value of $a$.
Use Simpson's rule with four strips of equal width to estimate the area of the region bounded by the curve in the interval $0 \leq x \leq a$ and the coordinate axes.

OCR C3 Q2

6 marks Standard +0.3

2. The functions $f$ and $g$ are defined by $$\begin{aligned} & f : x \rightarrow 2 - x ^ { 2 } , \quad x \in \mathbb { R } , \\ & g : x \rightarrow \frac { 3 x } { 2 x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \frac { 1 } { 2 } . \end{aligned}$$

Evaluate fg(2).
Solve the equation $\operatorname { gf } ( x ) = \frac { 1 } { 2 }$.

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