Consider the function \(f:x \to \sqrt {\frac{\pi }{4} - \arccos x} \).

(a)     Find the largest possible domain of f.

(b)     Determine an expression for the inverse function, \({f^{ - 1}}\), and write down its domain.

Find an expression for \({f^{ - 1}}(x)\).

Given that \(f(x)\) can be written in the form \(f(x) = A + \frac{B}{{2x - 1}}\), find the values of the constants \(A\) and \(B\).

Hence, write down \(\int {\frac{{3x - 2}}{{2x - 1}}} {\text{d}}x\).

Write \(\ln ({x^2} - 1) - 2\ln (x + 1) + \ln ({x^2} + x)\) as a single logarithm, in its simplest form.

Find the set of values of y for which this equation has real roots.

Hence determine the range of the function \(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}\).

Explain why f has no inverse.

Using the information shown in the diagram, find the values of a , b and c .

If g(x) = 3f(x − 2) ,

(i)     state the coordinates of the points where the graph of g intercepts the x-axis.

(ii)     Find the y-intercept of the graph of g .

Sketch the graph of \(y = f(x)\), indicating clearly any asymptotes and points of intersection with the \(x\) and \(y\) axes.

Find an expression for \({f^{ - 1}}(x)\).

Find all values of \(x\) for which \(f(x) = {f^{ - 1}}(x)\).

Solve the inequality \(\left| {f(x)} \right| < \frac{3}{2}\).

Solve the inequality \(f\left( {\left| x \right|} \right) < \frac{3}{2}\).

A function is defined by \(h(x) = 2{{\text{e}}^x} - \frac{1}{{{{\text{e}}^x}}},{\text{ }}x \in \mathbb{R}\) . Find an expression for \({h^{ - 1}}(x)\) .

The polynomial \(P(x) = {x^3} + a{x^2} + bx + 2\) is divisible by (x +1) and by (x − 2) .

Find the value of a and of b, where \(a,{\text{ }}b \in \mathbb{R}\) .

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

\(y = f(x)\);

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

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

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

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

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

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

Show that \(g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4\).

Find the range of \(g \circ f\).

Given that \(g \circ f\left( {\frac{{3\pi }}{{20}}} \right) = 7\), find the next value of \(x\), greater than \({\frac{{3\pi }}{{20}}}\), for which \(g \circ f(x) = 7\).

The graph of \(y = g \circ f(x)\) can be obtained by applying four transformations to the graph of \(y = \sin x\). State what the four transformations represent geometrically and give the order in which they are applied.

Find an expression for \(g \circ f(x)\), stating its domain.

Hence show that \(g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\).

Let \(y = g \circ f(x)\), find an exact value for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) at the point on the graph of \(y = g \circ f(x)\) where \(x = \frac{\pi }{6}\), expressing your answer in the form \(a + b\sqrt 3 ,{\text{ }}a,{\text{ }}b \in \mathbb{Z}\).

Show that the area bounded by the graph of \(y = g \circ f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = \frac{\pi }{6}\) is \(\ln \left( {1 + \sqrt 3 } \right)\).

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

Factorize \({x^2} + 3x + 2\).

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

Show that \(\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}\).

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

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

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 \(x = 1\).

For the polynomial equation \(p(x) = 0\), state

(i)     the sum of the roots;

(ii)     the product of the roots.

A new polynomial is defined by \(q(x) = p(x + 4)\).

Find the sum of the roots of the equation \(q(x) = 0\).

Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).

Prove by induction that \(\frac{{{{\text{d}}^n}y}}{{{\text{d}}{x^n}}} = n{3^{n - 1}}{{\text{e}}^{3x}} + x{3^n}{{\text{e}}^{3x}}\) for \(n \in {\mathbb{Z}^ + }\).

Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\).

Justify whether any such point is a maximum or a minimum.

Find the coordinates of any points of inflexion on the graph of \(y(x)\). Justify whether any such point is a point of inflexion.

Hence sketch the graph of \(y(x)\), indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.

The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such that \({\alpha ^2} + {\beta ^2} = 4\). Without solving the equation, find the possible values of the real number \(k\).

the degree of the polynomial;

the value of \({a_0}\).

Consider the function f , where \(f(x) = \arcsin (\ln x)\).

(a)     Find the domain of f .

(b)     Find \({f^{ - 1}}(x)\).

Show that \(\frac{1}{{\sqrt n  + \sqrt {n + 1} }} = \sqrt {n + 1}  - \sqrt n \) where \(n \ge 0,{\text{ }}n \in \mathbb{Z}\).

Hence show that \(\sqrt 2  - 1 < \frac{1}{{\sqrt 2 }}\).

Prove, by mathematical induction, that \(\sum\limits_{r = 1}^{r = n} {\frac{1}{{\sqrt r }} > \sqrt n } \) for \(n \ge 2,{\text{ }}n \in \mathbb{Z}\).

Let \(g(x) = {\log _5}\left| {2{{\log }_3}x} \right|\) . Find the product of the zeros of g .

Given that \(f\) is an even function, show that \(b = 0\).

Given that \(g\) is an odd function, find the value of \(r\).

The function \(h\) is both odd and even, with domain \(\mathbb{R}\).

Find \(h(x)\).

Given that \(q(x)\) has a factor \((x - 4)\), find the value of \(k\).

Hence or otherwise, factorize \(q(x)\) as a product of linear factors.

Write down the sum and the product of the roots of \(P(z) = 0\).

Show that \((z - 1)\) is a factor of \(P(z)\).

Find the value of \(b\) and the value of \(c\).

Hence find the complex roots of \(P(z) = 0\).

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

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

The function \(f\)  is defined as \(f(x) = \frac{{3x + 2}}{{x + 1}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne  - 1\).

Sketch the graph of \(y = f(x)\), clearly indicating and stating the equations of any asymptotes and the coordinates of any axes intercepts.

(i)     \(p =  - (\alpha  + \beta  + \gamma )\);

(ii)     \(q = \alpha \beta  + \beta \gamma  + \gamma \alpha \);

(iii)     \(c =  - \alpha \beta \gamma \).

It is now given that \(p =  - 6\) and \(q = 18\) for parts (b) and (c) below.

(i)     In the case that the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form an arithmetic sequence, show that one of the roots is \(2\).

(ii)     Hence determine the value of \(c\).

In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form a geometric sequence. Determine the value of \(c\).

Sketch the graph of \(y = h(x)\).

Find an expression for the composite function \(h \circ g(x)\) and state its domain.

Given that \(f(x) = h(x) + h \circ g(x)\),

(i)     find \(f'(x)\) in simplified form;

(ii)     show that \(f(x) = \frac{\pi }{2}\) for \(x > 0\).

Nigel states that \(f\) is an odd function and Tom argues that \(f\) is an even function.

(i)     State who is correct and justify your answer.

(ii)     Hence find the value of \(f(x)\) for \(x < 0\).

Write down the range of \(f\).

Find an expression for \({f^{ - 1}}(x)\).

Write down the domain and range of \({f^{ - 1}}\).

Show that \(f\) is an odd function.

Find \(f'(x)\).

Hence find the \(x\)-coordinates of any local maximum or minimum points.

Find the range of \(f\).

Sketch the graph of \(y = f(x)\) indicating clearly the coordinates of the \(x\)-intercepts and any local maximum or minimum points.

Find the area of the region enclosed by the graph of \(y = f(x)\) and the \(x\)-axis for \(x \ge 0\).

Show that \(\int_{ - 1}^1 {\left| {x\sqrt {1 - {x^2}} } \right|{\text{d}}x > \left| {\int_{ - 1}^1 {x\sqrt {1 - {x^2}} {\text{d}}x} } \right|} \).

Given that f and its derivative, \(f'\) , are continuous for all values in the domain of f , find the values of a and b .

Show that f is a one-to-one function.

Obtain expressions for the inverse function \({f^{ - 1}}\) and state their domains.

Determine whether or not \(f\)is continuous.

The graph of the function \(g\) is obtained by applying the following transformations to the graph of \(f\):

a reflection in the \(y\)–axis followed by a translation by the vector \(\left( \begin{array}{l}2\\0\end{array} \right)\).

Find \(g(x)\).

Sketch the graph of \(y = \frac{{1 - 3x}}{{x - 2}}\), showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Hence or otherwise, solve the inequality \(\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2\).

(i)     Find an expression for \(f'(x)\).

(ii)     Hence determine the coordinates of the point A, where \(f'(x) = 0\).

Find an expression for \(f''(x)\) and hence show the point A is a maximum.

Find the coordinates of B, the point of inflexion.

The graph of the function \(g\) is obtained from the graph of \(f\) by stretching it in the x-direction by a scale factor 2.

          (i)     Write down an expression for \(g(x)\).

          (ii)     State the coordinates of the maximum C of \(g\).

          (iii)     Determine the x-coordinates of D and E, the two points where \(f(x) = g(x)\).

Sketch the graphs of \(y = f(x)\) and \(y = g(x)\) on the same axes, showing clearly the points A, B, C, D and E.

Find an exact value for the area of the region bounded by the curve \(y = g(x)\), the x-axis and the line \(x = 1\).

state the value of \(a\) and the value of \(c\);

find the value of \(b\).

The equation \(5{x^3} + 48{x^2} + 100x + 2 = a\) has roots \({r_1}\), \({r_2}\) and \({r_3}\).

Given that \({r_1} + {r_2} + {r_3} + {r_1}{r_2}{r_3} = 0\), find the value of a.

Write down the numerical value of the sum and of the product of the roots of this equation.

The roots of this equation are three consecutive terms of an arithmetic sequence.

Taking the roots to be \(\alpha {\text{ , }}\alpha  \pm \beta \) , solve the equation.

Let f(x) = x⁴ + px³ + qx + 5 where p, q are constants.

The remainder when f(x) is divided by (x + 1) is 7, and the remainder when f(x) is divided by (x − 2) is 1. Find the value of p and the value of q.

The cubic polynomial \(3{x^3} + p{x^2} + qx - 2\) has a factor \((x + 2)\) and leaves a remainder 4 when divided by \((x + 1)\). Find the value of p and the value of q.

Without solving the equation, find the value of

(i)     \(\alpha  + \beta \);

(ii)     \(\alpha \beta \).

Another quadratic equation \({x^2} + px + q = 0,{\text{ }}p,{\text{ }}q \in \mathbb{Z}\) has roots \(\frac{2}{\alpha }\) and \(\frac{2}{\beta }\).

Find the value of \(p\) and the value of \(q\).

State the range of f and of g .

Find an expression for the composite function \(f \circ g(x)\) in the form \(\frac{{a{x^2} + bx + c}}{{3750}}\), where \(a,{\text{ }}b{\text{ and }}c \in \mathbb{Z}\) .

(i)     Find an expression for the inverse function \({f^{ - 1}}(x)\) .

(ii)     State the domain and range of \({f^{ - 1}}\) .

The domains of f and g are now restricted to {0, 1, 2, 3, 4} .

By considering the values of f and g on this new domain, determine which of f and g could be used to find a probability distribution for a discrete random variable X , stating your reasons clearly.

Using this probability distribution, calculate the mean of X .

Sketch the graphs on the same set of axes.

Given that the graphs enclose a region of area 18 square units, find the value of b.

(i)     Find \(\left( {g \circ f} \right)\left( x \right)\) and write down the domain of the function.

(ii)     Find \(\left( {f \circ g} \right)\left( x \right)\) and write down the domain of the function.

Find the coordinates of the point where the graph of \(y = f(x)\) and the graph of \(y = \left( {{g^{ - 1}} \circ f \circ g} \right)(x)\) intersect.

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

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 \(m \in \mathbb{Z}\).

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

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

A function f is defined by \(f(x) = \frac{{2x - 3}}{{x - 1}},{\text{ }}x \ne 1\).

(a)     Find an expression for \({f^{ - 1}}(x)\).

(b)     Solve the equation \(\left| {{f^{ - 1}}(x)} \right| = 1 + {f^{ - 1}}(x)\).

Find the value of p and the value of q .

The graph of f(x) is translated 3 units in the positive direction parallel to the x-axis. Determine the equation of the new graph.

Find an expression for \((f \circ f)(x)\) .

Let \({F_n}(x) = \frac{x}{{{2^n} - ({2^n} - 1)x}}\), where \(0 \leqslant x \leqslant 1\).

Use mathematical induction to show that for any \(n \in {\mathbb{Z}^ + }\)

\[\underbrace {(f \circ f \circ ... \circ f)}_{n{\text{ times}}}(x) = {F_n}(x)\] .

Show that \({F_{ - n}}(x)\) is an expression for the inverse of \({F_n}\) .

(i)     State \({F_n}(0){\text{ and }}{F_n}(1)\) .

(ii)     Show that \({F_n}(x) < x\) , given 0 < x < 1, \(n \in {\mathbb{Z}^ + }\) .

(iii)     For \(n \in {\mathbb{Z}^ + }\) , let \({A_n}\) be the area of the region enclosed by the graph of \(F_n^{ - 1}\) , the x-axis and the line x = 1. Find the area \({B_n}\) of the region enclosed by \({F_n}\) and \(F_n^{ - 1}\) in terms of \({A_n}\) .

State the set of values of \(a\) for which the function \(x \mapsto {\log _a}x\) exists, for all \(x \in {\mathbb{R}^ + }\).

Given that \({\log _x}y = 4{\log _y}x\), find all the possible expressions of \(y\) as a function of \(x\).

When the function \(q(x) = {x^3} + k{x^2} - 7x + 3\) is divided by (x + 1) the remainder is seven times the remainder that is found when the function is divided by (x + 2) .

Find the value of k .

Solve \({\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}\).

The roots of a quadratic equation \(2{x^2} + 4x - 1 = 0\) are \(\alpha \) and \(\beta \).

Without solving the equation,

(a)     find the value of \({\alpha ^2} + {\beta ^2}\);

(b)     find a quadratic equation with roots \({\alpha ^2}\) and \({\beta ^2}\).

When the polynomial \(3{x^3} + ax + b\) is divided by \((x - 2)\), the remainder is 2, and when divided by \((x + 1)\), it is 5. Find the value of a and the value of b.

Sketch on the same axes the curve \(y = \left| {\frac{7}{{x - 4}}} \right|\) and the line \(y = x + 2\), clearly indicating any axes intercepts and any asymptotes.

Find the exact solutions to the equation \(x + 2 = \left| {\frac{7}{{x - 4}}} \right|\).

Sketch the graph of \(y = {f^{ - 1}}(x)\) on the same axes.

State the range of \({f^{ - 1}}\).

Given that \(f(x) = \ln (ax + b),{\text{ }}x > 1\), find the value of \(a\) and the value of \(b\).

Find the set of values of x for which \(\left| {x - 1} \right| > \left| {2x - 1} \right|\).

Let \(f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1\).

If \(h = g \circ f\) , find

(a)     h(x) ;

(b)     \({h^{ - 1}}(x)\) , where \({h^{ - 1}}\) is the inverse of h.

Show that \(f(x) > 1\) for all x > 0.

Solve the equation \(f(x) = 4\).

(i)     Express each of the complex numbers \({z_1} = \sqrt 3  + {\text{i, }}{z_2} = - \sqrt 3  + {\text{i}}\) and \({z_3} = - 2{\text{i}}\) in modulus-argument form.

(ii)     Hence show that the points in the complex plane representing \({z_1}\), \({z_2}\) and \({z_3}\) form the vertices of an equilateral triangle.

(iii)     Show that \({\text{z}}_1^{3n} + z_2^{3n} = 2z_3^{3n}\) where \(n \in \mathbb{N}\).

(i)     State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them in modulus-argument form.

(ii)     If w is the solution to \({z^7} = 1\) with least positive argument, determine the argument of 1 + w. Express your answer in terms of \(\pi \).

(iii)     Show that \({z^2} - 2z\cos \left( {\frac{{2\pi }}{7}} \right) + 1\) is a factor of the polynomial \({z^7} - 1\). State the two other quadratic factors with real coefficients.

Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).

Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).

The same remainder is found when \(2{x^3} + k{x^2} + 6x + 32\) and \({x^4} - 6{x^2} - {k^2}x + 9\) are divided by \(x + 1\) . Find the possible values of k .

When \(3{x^5} - ax + b\) is divided by x −1 and x +1 the remainders are equal. Given that a , \(b \in \mathbb{R}\) , find

(a)     the value of a ;

(b)     the set of values of b .

(a)     Express the quadratic \(3{x^2} - 6x + 5\) in the form \(a{(x + b)^2} + c\), where a, b, c \( \in \mathbb{Z}\).

(b)     Describe a sequence of transformations that transforms the graph of \(y = {x^2}\) to the graph of \(y = 3{x^2} - 6x + 5\).

Find an expression for \(g(x)\).

State the equations of the asymptotes of the graph of \(g\).

Factorize \({z^3} + 1\) into a linear and quadratic factor.

Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\).

(i)     Show that \(\gamma \) is one of the cube roots of −1.

(ii)     Show that \({\gamma ^2} = \gamma - 1\).

(iii)     Hence find the value of \({(1 - \gamma )^6}\).

Given that the curve passes through the point \((a,{\text{ }}0)\), state the value of \(a\).

Use the substitution \(u = \ln x\) to find the area of the region \(R\).

(i)     Find the value of \({I_0}\).

(ii)     Prove that \({I_n} = \frac{1}{{\text{e}}} + n{I_{n - 1}},{\text{ }}n \in {\mathbb{Z}^ + }\).

(iii)     Hence find the value of \({I_1}\).

Find the volume of the solid formed when the region \(R\) is rotated through \(2\pi \) about the \(x\)-axis.

Given that \(A{x^3} + B{x^2} + x + 6\) is exactly divisible by \((x +1)(x − 2)\), find the value of A and the value of B .

Determine whether f is even, odd or neither even nor odd.

Show that \(f''(0) = 0\) .

John states that, because \(f''(0) = 0\) , the graph of f has a point of inflexion at the point (0, 2) . Explain briefly whether John’s statement is correct or not.

Sketch the graph of \(y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|\) for \(0 \leqslant x \leqslant 8\pi \).

Solve \(\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}\) for \(0 \leqslant x \leqslant 8\pi \).

The functions f and g are defined as:

\[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant 0\]

\[g(x) = \frac{1}{{x + 3}},{\text{ }}x \ne - 3.\]

(a)     Find \(h(x){\text{ where }}h(x) = g \circ f(x)\) .

(b)     State the domain of \({h^{ - 1}}(x)\) .

(c)     Find \({h^{ - 1}}(x)\) .

When \(f(x) = {x^4} + 3{x^3} + p{x^2} - 2x + q\) is divided by (x − 2) the remainder is 15, and (x + 3) is a factor of f(x) .

Find the values of p and q .

Solve the following equations:

(a)     \({\log _2}(x - 2) = {\log _4}({x^2} - 6x + 12)\);

(b)     \({x^{\ln x}} = {{\text{e}}^{{{(\ln x)}^3}}}\).

Find the inverse function \({f^{ - 1}}\), stating its domain.

Express \(g\left( x \right)\) in the form \(A + \frac{B}{{x - 2}}\) where A, B are constants.

Sketch the graph of \(y = g\left( x \right)\). State the equations of any asymptotes and the coordinates of any intercepts with the axes.

The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0.

State the domain and range of \(h \circ g\).

Express \(4{x^2} - 4x + 5\) in the form \(a{(x - h)^2} + k\) where a, h, \(k \in \mathbb{Q}\).

The graph of \(y = {x^2}\) is transformed onto the graph of \(y = 4{x^2} - 4x + 5\). Describe a sequence of transformations that does this, making the order of transformations clear.

Sketch the graph of \(y = f(x)\).

Find the range of f.

By using a suitable substitution show that \(\int {f(x){\text{d}}x = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \).

Prove that \(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}} \).

Express \(f(x)\) in the form \(A + \frac{B}{{x + 2}}\), where \(A\) and \(B \in \mathbb{Z}\).

Hence show that \(f'(x) > 0\) on D.

State the range of f.

(i)     Find an expression for \({f^{ - 1}}(x)\).

(ii)     Sketch the graph of \(y = f(x)\), showing the points of intersection with both axes.

(iii)     On the same diagram, sketch the graph of \(y = f'(x)\).

(i)     On a different diagram, sketch the graph of \(y = f(|x|)\) where \(x \in D\).

(ii)     Find all solutions of the equation \(f(|x|) = - \frac{1}{4}\).

(i)     Find \({f^{ - 1}}(x)\).

(ii)     State the domain of \({f^{ - 1}}\).

The function \(g\) is defined as \(g(x) = \ln x,{\text{ }}x \in {\mathbb{R}^ + }\).

The graph of \(y = g(x)\) and the graph of \(y = {f^{ - 1}}(x)\) intersect at the point \(P\).

Find the coordinates of \(P\).

The graph of \(y = g(x)\) intersects the \(x\)-axis at the point \(Q\).

Show that the equation of the tangent \(T\) to the graph of \(y = g(x)\) at the point \(Q\) is \(y = x - 1\).

A region \(R\) is bounded by the graphs of \(y = g(x)\), the tangent \(T\) and the line \(x = {\text{e}}\).

Find the area of the region \(R\).

A region \(R\) is bounded by the graphs of \(y = g(x)\), the tangent \(T\) and the line \(x = {\text{e}}\).

(i)     Show that \(g(x) \le x - 1,{\text{ }}x \in {\mathbb{R}^ + }\).

(ii)     By replacing \(x\) with \(\frac{1}{x}\) in part (e)(i), show that \(\frac{{x - 1}}{x} \le g(x),{\text{ }}x \in {\mathbb{R}^ + }\).

Sketch the graph of the function. You are not required to find the coordinates of the maximum.

Find the value of k .

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

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

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.

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

Shown below are the graphs of \(y = f(x)\) and \(y = g(x)\).

If \((f \circ g)(x) = 3\), find all possible values of x.

(i)     Find an expression for \(f'(x)\) .

(ii)     Given that the equation \(f'(x) = 0\) has two roots, state their values.

Sketch the graph of f , showing clearly the coordinates of the maximum and minimum.

Hence show that \({{\text{e}}^\pi } > {\pi ^{\text{e}}}\) .

State the two zeros of f .

Sketch the graph of f .

The region bounded by the graph, the x-axis and the y-axis is denoted by A and the region bounded by the graph and the x-axis is denoted by B . Show that the ratio of the area of A to the area of B is

\[\frac{{{e^\pi }\left( {{e^{\frac{\pi }{2}}} + 1} \right)}}{{{e^\pi } + 1}}.\]

Find the largest possible domain of the function \(g\) .

On the axes below, sketch the graph of \(y = g(x)\) . On the graph, indicate any asymptotes and local maxima or minima, and write down their equations and coordinates.

sketch the graph of \(f\);

show that \({\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x\);

find the volume of the solid formed when the graph of f is rotated through \(2\pi \) radians about the x-axis.

Sketch the graphs of \(y = \frac{x}{2} + 1\) and \(y = \left| {x - 2} \right|\) on the following axes.

Solve the equation \(\frac{x}{2} + 1 = \left| {x - 2} \right|\).

A function is defined as \(f(x) = k\sqrt x \), with \(k > 0\) and \(x \geqslant 0\) .

(a)     Sketch the graph of \(y = f(x)\) .

(b)     Show that f is a one-to-one function.

(c)     Find the inverse function, \({f^{ - 1}}(x)\) and state its domain.

(d)     If the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) intersect at the point (4, 4) find the value of k .

(e)     Consider the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) using the value of k found in part (d).

(i)     Find the area enclosed by the two graphs.

(ii)     The line x = c cuts the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) at the points P and Q respectively. Given that the tangent to \(y = f(x)\) at point P is parallel to the tangent to \(y = {f^{ - 1}}(x)\) at point Q find the value of c .

Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in \mathbb{R}\) . Show that

(i)     \({x^2} - {y^2} = - 5\) ;

(ii)     \(xy = 6\) .

Hence find the two square roots of \( - 5 + 12{\text{i}}\) .

For any complex number z , show that \({(z^*)^2} = ({z^2})^*\) .

Hence write down the two square roots of \( - 5 - 12{\text{i}}\) .

Explain why, of the four roots of the equation \(f(x) = 0\) , two are real and two are complex.

The curve passes through the point \(( - 1,\, - 18)\) . Find \(f(x)\) in the form

\(f(x) = (x - a)(x - b)({x^2} + cx + d),{\text{ where }}a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\) .

Find the two complex roots of the equation \(f(x) = 0\) in Cartesian form.

Draw the four roots on the complex plane (the Argand diagram).

Express each of the four roots of the equation in the form \(r{{\text{e}}^{{\text{i}}\theta }}\) .

The graph of \(y = \frac{{a + x}}{{b + cx}}\) is drawn below.

(a)     Find the value of a, the value of b and the value of c.

(b)     Using the values of a, b and c found in part (a), sketch the graph of \(y = \left| {\frac{{b + cx}}{{a + x}}} \right|\) on the axes below, showing clearly all intercepts and asymptotes.

The graph below shows \(y = f(x)\) , where \(f(x) = x + \ln x\) .

(a) On the graph below, sketch the curve \(y = {f^{ - 1}}(x)\) .

(b) Find the coordinates of the point of intersection of the graph of \(y = f(x)\) and the graph of \(y = {f^{ - 1}}(x)\) .

(i)     Solve the equation \(f'(x) = 0\) .

(ii)     Hence show the graph of \(f\) has a local maximum.

(iii)     Write down the range of the function \(f\) .

Show that there is a point of inflexion on the graph and determine its coordinates.

Sketch the graph of \(y = f(x)\) , indicating clearly the asymptote, x-intercept and the local maximum.

Now consider the functions \(g(x) = \frac{{\ln \left| x \right|}}{x}\) and \(h(x) = \frac{{\ln \left| x \right|}}{{\left| x \right|}}\) , where \(0 < x < {{\text{e}}^2}\) .

(i)     Sketch the graph of \(y = g(x)\) .

(ii)     Write down the range of \(g\) .

(iii)     Find the values of \(x\) such that \(h(x) > g(x)\) .

Sketch the graph of \(y = \frac{1}{{f(x)}}\).

Sketch the graph of  \(y = x{\text{ }}f(x)\) .

(i)     Sketch the graphs of \(y = \sin x\) and \(y = \sin 2x\) , on the same set of axes, for \(0 \leqslant x \leqslant \frac{\pi }{2}\) .

(ii)     Find the x-coordinates of the points of intersection of the graphs in the domain \(0 \leqslant x \leqslant \frac{\pi }{2}\) .

(iii)     Find the area enclosed by the graphs.

Find the value of \(\int_0^1 {\sqrt {\frac{x}{{4 - x}}} }{{\text{d}}x} \) using the substitution \(x = 4{\sin ^2}\theta \) .

The increasing function f satisfies \(f(0) = 0\) and \(f(a) = b\) , where \(a > 0\) and \(b > 0\) .

(i)     By reference to a sketch, show that \(\int_0^a {f(x){\text{d}}x = ab - \int_0^b {{f^{ - 1}}(x){\text{d}}x} } \) .

(ii)     Hence find the value of \(\int_0^2 {\arcsin \left( {\frac{x}{4}} \right){\text{d}}x} \) .

The diagram below shows a solid with volume V , obtained from a cube with edge \(a > 1\) when a smaller cube with edge \(\frac{1}{a}\) is removed.

Let \(x = a - \frac{1}{a}\)

(a)     Find V in terms of x .

(b)     Hence or otherwise, show that the only value of a for which V = 4x is \(a = \frac{{1 + \sqrt 5 }}{2}\) .

On the axes below, sketch the graph of \(y = \frac{1}{{f(x)}}\) , clearly showing the coordinates of the images of the points A, B and D, labelling them \({{\text{A}'}}\), \({{\text{B}'}}\), and \({{\text{D}'}}\) respectively, and the equations of any vertical asymptotes.

On the axes below, sketch the graph of the derivative \(y = f'(x)\) , clearly showing the coordinates of the images of the points  A and D, labelling them \({{\text{A}}}''\) and \({{\text{D}}}''\) respectively.

Draw the graph of y = f (x) on the blank grid below.

Hence state the value of

(i)     \(f'( - 3)\);

(ii)     \(f'(2.7)\);

(iii)     \(\int_{ - 3}^{ - 2} {f(x)dx} \).