Questions - OCR C2 - A-Level Maths

Find the radius of the fourth circle to be drawn.
Show that when eight circles have been drawn, $w = 98.5 \mathrm {~mm}$ to 3 significant figures.
Find the total area of the design in square centimetres when ten circles have been drawn.

OCR C2 Q8

8. Given that for small values of $x$ $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 }$$ where $n$ is an integer and $n > 1$,

show that $n = 16$ and find the value of $a$,
use your value of $a$ and a suitable value of $x$ to estimate the value of (0.9985) ${ } ^ { 16 }$, giving your answer to 5 decimal places.

OCR C2 Q9

9.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-3_559_732_824_388} The diagram shows the curve with equation $y = \mathrm { f } ( x )$ which crosses the $x$-axis at the origin and at the points $A$ and $B$. Given that $$f ^ { \prime } ( x ) = 4 - 6 x - 3 x ^ { 2 }$$

find an expression for $y$ in terms of $x$,
show that $A$ has coordinates ( $- 4,0$ ) and find the coordinates of $B$,
find the total area of the two regions bounded by the curve and the $x$-axis.

OCR C2 Q1

The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by

$$u _ { n } = 2 ^ { n } + k n ,$$ where $k$ is a constant.
Given that $u _ { 1 } = u _ { 3 }$,

find the value of $k$,
find the value of $u _ { 5 }$.

OCR C2 Q2

2. Given that $$y = 2 x ^ { \frac { 3 } { 2 } } - 1 ,$$ find $$\int y ^ { 2 } \mathrm {~d} x .$$

OCR C2 Q3

(i) Sketch the curve $y = \sin x ^ { \circ }$ for $x$ in the interval $- 180 \leq x \leq 180$.
(ii) Sketch on the same diagram the curve $y = \sin ( x - 30 ) ^ { \circ }$ for $x$ in the interval $- 180 \leq x \leq 180$.
(iii) Use your diagram to solve the equation

$$\sin x ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ for $x$ in the interval $- 180 \leq x \leq 180$.

OCR C2 Q4

4. (i) Solve the inequality $$x ^ { 2 } - 13 x + 30 < 0$$ (ii) Hence find the set of values of $y$ such that $$2 ^ { 2 y } - 13 \left( 2 ^ { y } \right) + 30 < 0 .$$

OCR C2 Q5

\includegraphics[max width=\textwidth, alt={}]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-2_515_771_246_438}

The diagram shows the curve $y = \mathrm { f } ( x )$ where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and $k$ is a constant. The curve crosses the $x$-axis at the points $A , B$ and $C$.
Given that $A$ has coordinates $( - 4,0 )$,

show that $k = - 7$,
find the coordinates of $B$ and $C$.

OCR C2 Q6

6. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$

find an expression for $\mathrm { f } ( x )$. Given also that $$\mathrm { f } ( 2 ) = 2 \mathrm { f } ( 1 ) ,$$
find $\mathrm { f } ( 4 )$.

OCR C2 Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-3_376_892_221_427} The diagram shows a design painted on the wall at a karting track. The sign consists of triangle $A B C$ and two circular sectors of radius 2 metres and 1 metre with centres $A$ and $B$ respectively. Given that $A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}$ and $\angle A C B = 2.2$ radians,

find the size of $\angle A B C$ in radians to 3 significant figures,
show that $\angle B A C = 0.588$ radians to 3 significant figures,
find the area of triangle $A B C$,
find the area of the wall covered by the design.

OCR C2 Q8

8. The finite region $R$ is bounded by the curve $y = 1 + 3 \sqrt { x }$, the $x$-axis and the lines $x = 2$ and $x = 8$.

Use the trapezium rule with three intervals, each of width 2 , to estimate to 3 significant figures the area of $R$.
Use integration to find the exact area of $R$ in the form $a + b \sqrt { 2 }$.
Find the percentage error in the estimate made in part (a).

OCR C2 Q9

9. The first two terms of a geometric progression are 2 and $x$ respectively, where $x \neq 2$.

Find an expression for the third term in terms of $x$. The first and third terms of arithmetic progression are 2 and $x$ respectively.
Find an expression for the 11th term in terms of $x$. Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,
find the value of $x$,
find the sum of the first 50 terms of the arithmetic progression.

OCR C2 Q1

A geometric progression has first term 75 and second term - 15 .
1. Find the common ratio.
2. Find the sum to infinity.
3. Find the area of the finite region enclosed by the curve $y = 5 x - x ^ { 2 }$ and the $x$-axis.
4. During one day, a biological culure is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that $t$ hours after 8 a.m., the number of bacteria present, $N$, is given by
$$N = 20000 \times ( 1.06 ) ^ { t }$$ Using this model,
find the number of bacteria present at 11 a.m.,
find, to the nearest minute, the time when the initial number of bacteria will have doubled.

OCR C2 Q4

4.
\includegraphics[max width=\textwidth, alt={}, center]{fe47eac1-645a-46c6-a2b9-c4ad0bcaa538-1_433_844_1416_575} The diagram shows the curve with equation $y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0$.

Copy and complete the table below for points on the curve, giving the $y$ values to 2 decimal places.
$x$ 2 3 4 5 6
$y$ 2.89 6.36
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 6$.
Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.

OCR C2 Q5

5. (i) Given that $\sin \theta = 2 - \sqrt { 2 }$, find the value of $\cos ^ { 2 } \theta$ in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are integers.
(ii) Find, in terms of $\pi$, all values of $x$ in the interval $0 \leq x < \pi$ for which $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$

OCR C2 Q6

\includegraphics[max width=\textwidth, alt={}]{fe47eac1-645a-46c6-a2b9-c4ad0bcaa538-2_337_896_694_452}

The diagram shows triangle $A B C$ in which $A C = 8 \mathrm {~cm}$ and $\angle B A C = \angle B C A = 30 ^ { \circ }$.

Find the area of triangle $A B C$ in the form $k \sqrt { 3 }$. The point $M$ is the mid-point of $A C$ and the points $N$ and $O$ lie on $A B$ and $B C$ such that $M N$ and $M O$ are arcs of circles with centres $A$ and $C$ respectively.
Show that the area of the shaded region $B N M O$ is $\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }$.

OCR C2 Q7

7. (i) Expand $( 2 + x ) ^ { 4 }$ in ascending powers of $x$, simplifying each coefficient.
(ii) Find the integers $A , B$ and $C$ such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (iii) Find the real values of $x$ for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$

OCR C2 Q8

(i) The gradient of a curve is given by

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0$$ Find an equation for the curve given that it passes through the point $( 2,6 )$.
(ii) Show that $$\int _ { 2 } ^ { 3 } \left( 6 \sqrt { x } - \frac { 4 } { \sqrt { x } } \right) d x = k \sqrt { 3 }$$ where $k$ is an integer to be found.

OCR C2 Q9

9. The polynomial $\mathrm { f } ( x )$ is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where $k$ is a constant.
When $\mathrm { f } ( x )$ is divided by ( $x + 1$ ) the remainder is $r$.
When $\mathrm { f } ( x )$ is divided by $( x - 3 )$ the remainder is $3 r$.

Find the value of $k$.
Find the value of $r$.
Show that $( x - 5 )$ is a factor of $\mathrm { f } ( x )$.
Show that there is only one real solution to the equation $\mathrm { f } ( x ) = 0$.

OCR C2 Q1

Solve the equation

$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$

OCR C2 Q2

Find the coefficient of $x ^ { 2 }$ in the expansion of

$$( 1 + x ) ( 1 - x ) ^ { 6 }$$

OCR C2 Q3

(i) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where $k$ is a rational constant to be found.

OCR C2 Q4

4. The diagram shows triangle $P Q R$ in which $P Q = 7$ and $P R = 3 \sqrt { 5 }$.
Given that $\sin ( \angle Q P R ) = \frac { 2 } { 3 }$ and that $\angle Q P R$ is acute,

find the exact value of $\cos ( \angle Q P R )$ in its simplest form,
show that $Q R = 2 \sqrt { 6 }$,
find $\angle P Q R$ in degrees to 1 decimal place.

OCR C2 Q5

5. (i) Find $$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$ The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and the curve passes through the point $( 1,1 )$.
(ii) Show that the equation of the curve can be written in the form $$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$ where $a$ and $b$ are integers to be found.

OCR C2 Q6

6. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6$$ and that $x = - 3$ is a solution to the equation $\mathrm { f } ( x ) = 0$,

find the value of the constant $p$,
show that when $\mathrm { f } ( x )$ is divided by $( x - 2 )$ there is a remainder of 50 ,
find the other solutions to the equation $\mathrm { f } ( x ) = 0$, giving your answers to 2 decimal places.

\(x\)	2	3	4	5	6
\(y\)	2.89	6.36

Questions — OCR C2 (296 questions)

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