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HL Paper 1

Prove by mathematical induction that $\frac{d^{n}}{d x^{n}} (x^{2} e^{x}) = [x^{2} + 2 n x + n (n - 1)] e^{x}$ for $n \in ℤ^{+}$ .

[7]

Hence or otherwise, determine the Maclaurin series of $f (x) = x^{2} e^{x}$ in ascending powers of $x$ , up to and including the term in $x^{4}$ .

[3]

Hence or otherwise, determine the value of $\lim_{x \to 0} [\frac{{(x^{2} e^{x} - x^{2})}^{3}}{x^{9}}]$ .

[4]

Consider the function ${f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }$ .

Determine whether ${f_n}$ is an odd or even function, justifying your answer.

[2]

By using mathematical induction, prove that

${f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}$ where $m \in \mathbb{Z}$ .

[8]

Hence or otherwise, find an expression for the derivative of ${f_n}(x)$ with respect to $x$ .

[3]

Show that, for $n > 1$ , the equation of the tangent to the curve $y = {f_n}(x)$ at $x = \frac{\pi }{4}$ is $4x - 2y - \pi = 0$ .

[8]

The function $f$ is defined by $f\left( x \right) = {{\text{e}}^x}\,{\text{cos}}{\,^2}x$ , where 0 ≤ $x$ ≤ 5. The curve $y = f\left( x \right)$ is shown on the following graph which has local maximum points at A and C and touches the $x$ -axis at B and D.

Use integration by parts to show that $\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c$ , $c \in \mathbb{R}$ .

[5]

Hence, show that $\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c$ , $c \in \mathbb{R}$ .

[3]

Find the $x$ -coordinates of A and of C , giving your answers in the form $a + {\text{arctan}}\,b$ , where $a$ , $b \in \mathbb{R}$ .

[6]

Find the area enclosed by the curve and the $x$ -axis between B and D, as shaded on the diagram.

[5]

Show that ${\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x$ .

[2]

Show that ${\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}$ .

[4]

Hence or otherwise find $\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x$ in the form ${\text{ln}}\left( {a + \sqrt b } \right)$ where $a$ , $b \in \mathbb{Z}$ .

[9]

Consider the function $f$ defined by $f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}$ where $a$ is a positive constant.

The function $g$ is defined by $g(x) = x\sqrt {f(x)}$ for $\left| x \right| > a$ .

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = f(x)$ ;

[2]

a.i.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \frac{1}{{f(x)}}$ ;

[4]

a.ii.

Showing any $x$ and $y$ intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.

$y = \left| {\frac{1}{{f(x)}}} \right|$ .

[2]

a.iii.

Find $\int {f(x)\cos x{\text{d}}x}$ .

[5]

By finding $g'(x)$ explain why $g$ is an increasing function.

[4]

Consider the functions $f$ and $g$ defined on the domain $0 < x < 2\pi$ by $f\left( x \right) = 3\,{\text{cos}}\,2x$ and $g\left( x \right) = 4 - 11\,{\text{cos}}\,x$ .

The following diagram shows the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$

Find the $x$ -coordinates of the points of intersection of the two graphs.

[6]

Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .

[5]

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$ -coordinate of A on the graph of $g$ .

[6]

A continuous random variable $X$ has the probability density function

$f (x) = \{\begin{matrix} \frac{2}{(b - a) (c - a)} (x - a), & a \leq x \leq c \\ \frac{2}{(b - a) (b - c)} (b - x), & c < x \leq b \\ 0, & otherwise \end{matrix}$ .

The following diagram shows the graph of $y = f (x)$ for $a \leq x \leq b$ .

Given that $c \geq \frac{a + b}{2}$ , find an expression for the median of $X$ in terms of $a, b$ and $c$ .

Consider the function $f$ defined by $f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi$ .

The curvature at any point $(x,{\text{ }}y)$ on a graph is defined as $\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}$ .

Show that the function $f$ has a local maximum value when $x = \frac{{3\pi }}{4}$ .

[2]

Find the $x$ -coordinate of the point of inflexion of the graph of $f$ .

[2]

Sketch the graph of $f$ , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.

[3]

Find the area of the region enclosed by the graph of $f$ and the $x$ -axis.

[6]

Find the value of the curvature of the graph of $f$ at the local maximum point.

[3]

Find the value $\kappa$ for $x = \frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.

[2]

Let $f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0$ .

The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.

[5]

Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .

[5]

b.i.

The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.

[3]

b.ii.

Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.

[4]

By using the substitution $u = sec x$ or otherwise, find an expression for $\int_{0}^{\frac{π}{3}} {sec}^{n} x \tan x d x$ in terms of $n$ , where $n$ is a non-zero real number.

The continuous random variable $X$ has probability density function

$f (x) = \{\begin{cases} \frac{k}{\sqrt{4 - 3 x^{2}}}, & 0 \leq x \leq 1 \\ 0, & otherwise . \end{cases}$

Find the value of $k$ .

[4]

Find $E (X)$ .

[4]

A function $f$ is defined by $f (x) = x \sqrt{1 - x^{2}}$ where $- 1 \leq x \leq 1$ .

The graph of $y = f (x)$ is shown below.

Show that $f$ is an odd function.

[2]

The range of $f$ is $a \leq y \leq b$ , where $a, b \in ℝ$ .

Find the value of $a$ and the value of $b$ .

[6]

Consider the function defined by $f (x) = \frac{k x - 5}{x - k}$ , where $x \in ℝ \ \{k\}$ and $k^{2} \neq 5$ .

Consider the case where $k = 3$ .

State the equation of the vertical asymptote on the graph of $y = f (x)$ .

[1]

State the equation of the horizontal asymptote on the graph of $y = f (x)$ .

[1]

Use an algebraic method to determine whether $f$ is a self-inverse function.

[4]

Sketch the graph of $y = f (x)$ , stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

[3]

The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .

[6]

The curve $C$ is given by the equation $y = x\,{\text{tan}}\left( {\frac{{\pi xy}}{4}} \right)$ .

At the point (1, 1) , show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2 + \pi }}{{2 - \pi }}$ .

[5]

Hence find the equation of the normal to $C$ at the point (1, 1).

[2]

Consider the function $f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1$ .

Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .

[1]

a.i.

Factorize ${x^2} + 3x + 2$ .

[1]

a.ii.

Sketch the graph of $f(x)$ , indicating on it the equations of the asymptotes, the coordinates of the $y$ -intercept and the local maximum.

[5]

Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$ .

[4]

Sketch the graph of $y = f\left( {\left| x \right|} \right)$ .

[2]

Determine the area of the region enclosed between the graph of $y = f\left( {\left| x \right|} \right)$ , the $x$ -axis and the lines with equations $x = - 1$ and $x = 1$ .

[3]

Let $y = {\text{arccos}}\left( {\frac{x}{2}} \right)$

Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .

[2]

Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$ .

[7]

Let $f (x) = \sqrt{1 + x}$ for $x > - 1$ .

Show that $f'' (x) = - \frac{1}{4 \sqrt{{(1 + x)}^{3}}}$ .

[3]

Use mathematical induction to prove that $f^{(n)} (x) = {(- \frac{1}{4})}^{n - 1} \frac{(2 n - 3)!}{(n - 2)!} {(1 + x)}^{\frac{1}{2} - n}$ for $n \in ℤ, n \geq 2$ .

[9]

Let $g (x) = e^{m x}, m \in ℚ$ .

Consider the function $h$ defined by $h (x) = f (x) \times g (x)$ for $x > - 1$ .

It is given that the $x^{2}$ term in the Maclaurin series for $h (x)$ has a coefficient of $\frac{7}{4}$ .

Find the possible values of $m$ .

[8]

Consider $f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1$ .

For the graph of $y = f\left( x \right)$ ,

Find $f'\left( x \right)$ .

[2]

a.i.

Show that, if $f'\left( x \right) = 0$ , then $x = 2 - \sqrt 3$ .

[3]

a.ii.

find the coordinates of the $y$ -intercept.

[1]

b.i.

show that there are no $x$ -intercepts.

[2]

b.ii.

sketch the graph, showing clearly any asymptotic behaviour.

[2]

b.iii.

Show that $\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}$ .

[2]

The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .

[7]

Given that $\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10}$ and $\int_0^2 {f\left( x \right){\text{d}}x = 12}$ , find

$\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$ .

[4]

$\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$ .

[2]

Using the substitution $u = {\text{sin}}\,x$ , find $\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}}$ .

The function $f$ is defined by $f (x) = e^{x} \sin x$ , where $x \in ℝ$ .

The function $g$ is defined by $g (x) = e^{x} \cos x$ , where $x \in ℝ$ .

Find the Maclaurin series for $f (x)$ up to and including the $x^{3}$ term.

[4]

Hence, find an approximate value for $\int_{0}^{1} e^{x^{2}} \sin (x^{2}) d x$ .

[4]

Show that $g (x)$ satisfies the equation $g'' (x) = 2 (g' (x) - g (x))$ .

[4]

c.i.

Hence, deduce that $g^{(4)} (x) = 2 (g''' (x) - g'' (x))$ .

[1]

c.ii.

Using the result from part (c), find the Maclaurin series for $g (x)$ up to and including the $x^{4}$ term.

[5]

Hence, or otherwise, determine the value of $\lim_{x \to 0} \frac{e^{x} \cos x - 1 - x}{x^{3}}$ .

[3]

The graph of $y = f'\left( x \right)$ , 0 ≤ $x$ ≤ 5 is shown in the following diagram. The curve intercepts the $x$ -axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).

The shaded area enclosed by the curve $y = f'\left( x \right)$ , the $x$ -axis and the $y$ -axis is 0.5. Given that $f\left( 0 \right) = 3$ ,

The area enclosed by the curve $y = f'\left( x \right)$ and the $x$ -axis between $x = 1$ and $x = 4$ is 2.5 .

Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .

[1]

find the value of $f\left( 1 \right)$ .

[3]

find the value of $f\left( 4 \right)$ .

[2]

Sketch the curve $y = f\left( x \right)$ , 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

[3]

A function $f$ is defined by $f (x) = \frac{3}{x^{2} + 2}, x \in ℝ$ .

The region $R$ is bounded by the curve $y = f (x)$ , the $x$ -axis and the lines $x = 0$ and $x = \sqrt{6}$ . Let $A$ be the area of $R$ .

The line $x = k$ divides $R$ into two regions of equal area.

Let $m$ be the gradient of a tangent to the curve $y = f (x)$ .

Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

[4]

Show that $A = \frac{\sqrt{2} π}{2}$ .

[4]

Find the value of $k$ .

[4]

Show that $m = - \frac{6 x}{{(x^{2} + 2)}^{2}}$ .

[2]

Show that the maximum value of $m$ is $\frac{27}{32} \sqrt{\frac{2}{3}}$ .

[7]

Show that ${\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x$ where $r,\,x \in {\mathbb{R}^ + }$ .

The folium of Descartes is a curve defined by the equation ${x^3} + {y^3} - 3xy = 0$ , shown in the following diagram.

N17/5/MATHL/HP1/ENG/TZ0/07

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the $y$ -axis.

A particle moves along a straight line. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t,{\text{ }}t \geqslant 0$ . The first two times when the particle is at rest are denoted by ${t_1}$ and ${t_2}$ , where ${t_1} < {t_2}$ .

Find ${t_1}$ and ${t_2}$ .

[5]

Find the displacement of the particle when $t = {t_1}$

[2]

Use l’Hôpital’s rule to determine the value of

$\lim_{x \to 0} \frac{2 \sin x - \sin 2 x}{x^{3}}$ .

Write $2x - {x^2}$ in the form $a{\left( {x - h} \right)^2} + k$ , where $a{\text{, }}h{\text{, }}k \in \mathbb{R}$ .

[2]

Hence, find the value of $\int_{\frac{1}{2}}^{\frac{3}{2}} {\frac{1}{{\sqrt {2x - {x^2}} }}} {\text{d}}x$ .

[5]

Consider the expression $\frac{1}{\sqrt{1 + a x}} - \sqrt{1 - x}$ where $a \in ℚ, a \neq 0$ .

The binomial expansion of this expression, in ascending powers of $x$ , as far as the term in $x^{2}$ is $4 b x + b x^{2}$ , where $b \in ℚ$ .

Find the value of $a$ and the value of $b$ .

[6]

State the restriction which must be placed on $x$ for this expansion to be valid.

[1]

Using the substitution $x = \tan \theta$ show that $\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x = } \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta }$ .

[4]

Hence find the value of $\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x}$ .

[3]

A curve has equation $3x - 2{y^2}{{\text{e}}^{x - 1}} = 2$ .

Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .

[5]

Find the equations of the tangents to this curve at the points where the curve intersects the line $x = 1$ .

[4]

Find the value of $\int_{1}^{9} (\frac{3 \sqrt{x} - 5}{\sqrt{x}}) d x$ .

A function $f$ is defined by $f (x) = \frac{1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x \neq - 1, x \neq 3$ .

A function $g$ is defined by $g (x) = \frac{1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x > 3$ .

The inverse of $g$ is $g^{- 1}$ .

A function $h$ is defined by $h (x) = arctan \frac{x}{2}$ , where $x \in ℝ$ .

Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

[6]

Show that $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ .

[6]

b.i.

State the domain of $g^{- 1}$ .

[1]

b.ii.

Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .

[7]

Use the substitution $u = {x^{\frac{1}{2}}}$ to find $\int {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}}$ .

[4]

Hence find the value of $\frac{1}{2}\int\limits_1^9 {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}}$ , expressing your answer in the form arctan $q$ , where $q \in \mathbb{Q}$ .

[3]

A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ , the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .

Use l’Hôpital’s rule to determine the value of $\lim_{x \to 0} (\frac{2 x \cos (x^{2})}{5 \tan x})$ .

The acceleration, $a {ms}^{- 2}$ , of a particle moving in a horizontal line at time $t$ seconds, $t \geq 0$ , is given by $a = - (1 + v)$ where $v {ms}^{- 1}$ is the particle’s velocity and $v > - 1$ .

At $t = 0$ , the particle is at a fixed origin $O$ and has initial velocity $v_{0} {ms}^{- 1}$ .

Initially at $O$ , the particle moves in the positive direction until it reaches its maximum displacement from $O$ . The particle then returns to $O$ .

Let $s$ metres represent the particle’s displacement from $O$ and $s_{max}$ its maximum displacement from $O$ .

Let $v (T - k)$ represent the particle’s velocity $k$ seconds before it reaches $s_{max}$ , where

$v (T - k) = (1 + v_{0}) e^{- (T - k)} - 1$ .

Similarly, let $v (T + k)$ represent the particle’s velocity $k$ seconds after it reaches $s_{max}$ .

By solving an appropriate differential equation, show that the particle’s velocity at time $t$ is given by $v (t) = (1 + v_{0}) e^{- t} - 1$ .

[6]

Show that the time $T$ taken for the particle to reach $s_{max}$ satisfies the equation $e^{T} = 1 + v_{0}$ .

[2]

b.i.

By solving an appropriate differential equation and using the result from part (b) (i), find an expression for $s_{max}$ in terms of $v_{0}$ .

[5]

b.ii.

By using the result to part (b) (i), show that $v (T - k) = e^{k} - 1$ .

[2]

Deduce a similar expression for $v (T + k)$ in terms of $k$ .

[2]

Hence, show that $v (T - k) + v (T + k) \geq 0$ .

[3]

Use l’Hôpital’s rule to find $\lim_{x \to 0} (\frac{arctan 2 x}{\tan 3 x})$ .

Consider the curves ${C_1}$ and ${C_2}$ defined as follows

${C_1}\,{\text{:}}\,xy = 4$ , $x > 0$

${C_2}\,{\text{:}}\,{y^2} - {x^2} = 2$ , $x > 0$

Using implicit differentiation, or otherwise, find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ for each curve in terms of $x$ and $y$ .

[4]

Let P( $a$ , $b$ ) be the unique point where the curves ${C_1}$ and ${C_2}$ intersect.

Show that the tangent to ${C_1}$ at P is perpendicular to the tangent to ${C_2}$ at P.

[2]

Solve the differential equation $\frac{d y}{d x} = \frac{\ln 2 x}{x^{2}} - \frac{2 y}{x}, x > 0$ , given that $y = 4$ at $x = \frac{1}{2}$ .

Give your answer in the form $y = f (x)$ .

A particle moves in a straight line such that at time $t$ seconds $(t \geqslant 0)$ , its velocity $v$ , in ${\text{m}}{{\text{s}}^{ - 1}}$ , is given by $v = 10t{{\text{e}}^{ - 2t}}$ . Find the exact distance travelled by the particle in the first half-second.

Consider the function $q\left( x \right) = {x^5} - 10{x^2} + 15x - 6,{\text{ }}x \in \mathbb{R}$ .

Show that the graph of $y = q(x)$ is concave up for $x > 1$ .

[3]

e.i.

Sketch the graph of $y = q(x)$ showing clearly any intercepts with the axes.

[3]

e.ii.

Consider the curve $C$ defined by $y^{2} = \sin (x y), y \neq 0$ .

Show that $\frac{d y}{d x} = \frac{y \cos (x y)}{2 y - x \cos (x y)}$ .

[5]

Prove that, when $\frac{d y}{d x} = 0, y = \pm 1$ .

[5]

Hence find the coordinates of all points on $C$ , for $0 < x < 4 π$ , where $\frac{d y}{d x} = 0$ .

[5]

Find the coordinates of the points on the curve ${y^3} + 3x{y^2} - {x^3} = 27$ at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .

Find $\int {\arcsin x\,{\text{d}}x}$

The function $f$ is defined by $f\left( x \right) = {{\text{e}}^{{\text{sin}}\,x}}$ .

Find the first two derivatives of $f\left( x \right)$ and hence find the Maclaurin series for $f\left( x \right)$ up to and including the ${x^2}$ term.

[8]

Show that the coefficient of ${x^3}$ in the Maclaurin series for $f\left( x \right)$ is zero.

[4]

Using the Maclaurin series for ${\text{arctan}}\,x$ and ${{\text{e}}^{3x}} - 1$ , find the Maclaurin series for ${\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)$ up to and including the ${x^3}$ term.

[6]

Hence, or otherwise, find $\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{f\left( x \right) - 1}}{{{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)}}$ .

[3]

Find the equation of the tangent to the curve $y = e^{2 x} - 3 x$ at the point where $x = 0$ .

Given that $\int_0^{{\text{ln}}\,k} {{{\text{e}}^{2x}}} {\text{d}}x = 12$ , find the value of $k$ .

The lines $l_{1}$ and $l_{2}$ have the following vector equations where $λ, μ \in ℝ$ .

$l_{1} : r_{1} = (\begin{matrix} 3 \\ 2 \\ - 1 \end{matrix}) + λ (\begin{matrix} 2 \\ - 2 \\ 2 \end{matrix})$

$l_{2} : r_{2} = (\begin{matrix} 2 \\ 0 \\ 4 \end{matrix}) + μ (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$

By using the substitution $u = \sin x$ , find $\int \frac{\sin x \cos x}{\sin^{2} x - \sin x - 2} d x$ .

A right circular cone of radius $r$ is inscribed in a sphere with centre O and radius $R$ as shown in the following diagram. The perpendicular height of the cone is $h$ , X denotes the centre of its base and B a point where the cone touches the sphere.

Show that the volume of the cone may be expressed by $V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)$ .

[4]

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{{32\pi {R^3}}}{{81}}$ .

[4]

Consider the functions $f,\,\,g,$ defined for $x \in \mathbb{R}$ , given by $f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x$ and $g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x$ .

Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .