Questions - OCR - A-Level Maths

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
The point $P$ on the curve is the point at which the gradient takes its maximum value. Show that the tangent at $P$ passes through the point $( 0 , - 1 )$.

OCR C3 Specimen Q9

11 marks Standard +0.3

9
\includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_424_707_1260_724} The diagram shows the curve $y = \tan ^ { - 1 } x$ and its asymptotes $y = \pm a$.

State the exact value of $a$.
Find the value of $x$ for which $\tan ^ { - 1 } x = \frac { 1 } { 2 } a$. The equation of another curve is $y = 2 \tan ^ { - 1 } ( x - 1 )$.
Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of $a$.
Verify by calculation that the value of $x$ at the point of intersection of the two curves is 1.54 , correct to 2 decimal places. Another curve (which you are not asked to sketch) has equation $y = \left( \tan ^ { - 1 } x \right) ^ { 2 }$.
Use Simpson's rule, with 4 strips, to find an approximate value for $\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x$.

OCR C3 Q1

4 marks Moderate -0.5

Show that

$$\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x = \ln 3$$

OCR C3 Q2

5 marks Standard +0.3

Find the set of values of $x$ such that

$$| 3 x + 1 | \leq | x - 2 |$$

OCR C3 Q3

6 marks Standard +0.3

Find all values of $\theta$ in the interval $- 180 < \theta < 180$ for which

$$\tan ^ { 2 } \theta ^ { \circ } + \sec \theta ^ { \circ } = 1$$

OCR C3 Q4

6 marks Moderate -0.3

Solve each equation, giving your answers in exact form.
1. $\mathrm { e } ^ { 4 x - 3 } = 2$
2. $\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )$
3. (i) Prove, by counter-example, that the statement
  "cosec $\theta - \sin \theta > 0$ for all values of $\theta$ in the interval $0 < \theta < \pi$ " is false.
4. Find the values of $\theta$ in the interval $0 < \theta < \pi$ such that
$$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.

OCR C3 Q6

8 marks Standard +0.3

6. The curve $C$ has the equation $y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0$.

Show that the normal to $C$ at the point where $x = 3$ has the equation $$3 x + 5 y + 21 = 0$$
Find the $x$-coordinates of the stationary points of $C$.

OCR C3 Q7

11 marks Standard +0.3

7.
\includegraphics[max width=\textwidth, alt={}, center]{6cdbc2bc-8863-4003-a218-44552d75d137-2_556_777_246_468} The diagram shows the curve $y = \mathrm { f } ( x )$ which has a maximum point at ( $- 45,7$ ) and a minimum point at $( 135 , - 1 )$.

Showing the coordinates of any stationary points, sketch the curve with equation $y = 1 + 2 \mathrm { f } ( x )$. Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where $A$ is a constant,
show that $\mathrm { f } ( x )$ can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ }$$ where $R > 0$ and $0 < \alpha < 90$,
state the value of $A$,
find, to 1 decimal place, the $x$-coordinates of the points where the curve $y = \mathrm { f } ( x )$ crosses the $x$-axis.

OCR C3 Q8

12 marks Standard +0.3

8. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$

State the range of f .
Sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ on the same diagram.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3$$
Evaluate fg(-3).
Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x )$$

OCR C3 Q9

13 marks Standard +0.3

A curve has the equation $y = ( 2 x + 3 ) \mathrm { e } ^ { - x }$.
1. Find the exact coordinates of the stationary point of the curve.
The curve crosses the $y$-axis at the point $P$.
Find an equation for the normal to the curve at $P$. The normal to the curve at $P$ meets the curve again at $Q$.
Show that the $x$-coordinate of $Q$ lies between - 2 and - 1 .
Use the iterative formula $$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$ with $x _ { 0 } = - 1$, to find $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$. Give the value of $x _ { 4 }$ to 2 decimal places.
Show that your value for $x _ { 4 }$ is the $x$-coordinate of $Q$ correct to 2 decimal places.

OCR C3 Q1

6 marks Moderate -0.3

(i) Differentiate $x ^ { 3 } \ln x$ with respect to $x$.
(ii) Given that

$$x = \frac { y + 1 } { 3 - 2 y }$$ find and simplify an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $y$.

OCR C3 Q2

7 marks Moderate -0.5

2.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-1_638_677_749_443} The diagram shows the curves $y = 3 + 2 \mathrm { e } ^ { x }$ and $y = \mathrm { e } ^ { x + 2 }$ which cross the $y$-axis at the points $A$ and $B$ respectively.

Write down the coordinates of $A$ and $B$. The two curves intersect at the point $C$.
Find an expression for the $x$-coordinate of $C$ and show that the $y$-coordinate of $C$ is $\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }$.

OCR C3 Q3

8 marks Moderate -0.3

3. The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } . \end{aligned}$$

Evaluate $\mathrm { gf } ( 1 )$.
Find an expression for $\mathrm { g } ^ { - 1 } ( x )$.
Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

OCR C3 Q4

9 marks Moderate -0.3

(i) Use the identity for $\cos ( A + B )$ to prove that

$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for $\cos x \neq 0$, $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of $x$ in the interval $0 \leq x \leq 180 ^ { \circ }$ for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$

OCR C3 Q5

9 marks Standard +0.8

(i) Show that the equation

$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of $\pi$ the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for $x$ in the interval $0 \leq x \leq \pi$.

OCR C3 Q6

10 marks Standard +0.8

6.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation $y = \sqrt { \frac { x } { x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.

Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the $x$-axis.
Show that the volume of the solid formed is $\pi ( 3 - \ln 4 )$.

OCR C3 Q7

11 marks Standard +0.8

7. (i) Sketch on the same diagram the graphs of $y = 4 a ^ { 2 } - x ^ { 2 }$ and $y = | 2 x - a |$, where $a$ is a positive constant. Show, in terms of $a$, the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

OCR C3 Q8

12 marks Standard +0.3

A curve has the equation $y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
  [0pt]
2. Show that the curve has a stationary point in the interval [1.3,1.4].
The point $A$ on the curve has $x$-coordinate 2 .
Show that the tangent to the curve at $A$ passes through the origin. The tangent to the curve at $A$ intersects the curve again at the point $B$.
The $x$-coordinate of $B$ is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with $x _ { 0 } = - 1$.
Find $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 7 significant figures and hence state the $x$-coordinate of $B$ to 5 significant figures.

OCR C4 2006 January Q1

3 marks Easy -1.2

1 Simplify $\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }$.

OCR C4 2006 January Q2

5 marks Standard +0.3

2 Given that $\sin y = x y + x ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.

OCR C4 2006 January Q3

6 marks Moderate -0.3

Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.
Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder.

OCR C4 2006 January Q4

7 marks Standard +0.3

Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$.

OCR C4 2006 January Q5

8 marks Moderate -0.3

5 A curve is given parametrically by the equations $x = t ^ { 2 } , y = 2 t$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent to the curve at $\left( p ^ { 2 } , 2 p \right)$ is $$p y = x + p ^ { 2 } .$$
Find the coordinates of the point where the tangent at $( 9,6 )$ meets the tangent at $( 25 , - 10 )$.

OCR C4 2006 January Q6

9 marks Standard +0.8

Show that the substitution $x = \sin ^ { 2 } \theta$ transforms $\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$ to $\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$.
Hence find $\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$.

OCR C4 2006 January Q7

10 marks Standard +0.3

7 The expression $\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }$ is denoted by $\mathrm { f } ( x )$.

Express $\mathrm { f } ( x )$ in the form $\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$, where $A , B$ and $C$ are constants.
Given that $| x | < 1$, find the first 3 terms in the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$.

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