Find the gradient of the line with equation 2x + 5y = 7

Circle your answer.

A curve has equation y = \frac{2}{\sqrt{x}}

Find \frac{dy}{dx}

Circle your answer.

When *θ* is small, find an approximation for \cos3\theta+\theta\sin2\theta in the form a + b\theta^2

What are the values of *a* and *b*?

Give your answers as decimals.

*a* =

*b*=

p(x)=2x^3+7x^2+2x−3

Use the factor theorem to show which of these expressions are factors of p(x):

Simplify the expression \frac{2x^3+7x^2+2x−3}{4x^2−1},\ x\ne\pm\frac{1}{2}

The diagram shows a sector *AOB* of a circle with centre *O* and radius *r* cm.

The angle *AOB* is *θ* radians

The sector has area 9 cm^{2} and perimeter 15 cm.

Show that *r* satisfies an equation of the form where *b* and *c* are real numbers.

What are the values of *b* and *c*?

*b* =

*c*=

Find the value of *θ*.

*θ*=

Sam goes on a diet. He assumes that his mass, *m* kg after *t* days, decreases at a rate
that is inversely proportional to the cube root of his mass.

Construct a differential equation involving *m*, *t* and a positive constant *k* to model this situation.

Find the values of *k* for which the equation (2k−3)x^2−kx+(k−1)=0 has equal roots.

*k*=

Given that u = 2^x, write down an expression for \frac{du}{dx}

Find the exact value of \int_0^1 2^x\sqrt{3+2^x}dx

Find \frac{dy}{dx}

*y* is increasing when 4x^2+bx+c<0
Find the values of the real numbers *b* and *c*.

*b* =

*c*=

Find the values of *x* for which *y* is increasing in the form a < x < b.

What are the values of *a* and *b*.

*a* =

*b*=

The function *f* is defined by

f(x)=4+3^{-x}\ ,\ x \in \R

Using set notation, state the range of *f*.

Put for answer in the form \left\{x:\ x>a\ ,\ x \in \R\right\}

*a*=

The inverse of *f* is f^{−1}

Using set notation, state the domain of f^{−1}

{Find an expression for f^{−1}(x)

The function *g* is defined by

g(x)=5−\sqrt{x}\ ,\ (x\in\R:x>0)

Find an expression for gf(x)

Solve the equation gf(x) = 2 , giving your answer in an exact form.

A circle with centre *C* has equation x^2+y^2+8x−12y=12

Find the coordinates of *C* and the radius of the circle.

*C* = (

The points *P* and *Q* lie on the circle.

The origin is the midpoint of the chord *PQ*.

Find the length of *PQ*. Give your answer in surd form.

A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram.

The shape of the cross-section of the sculpture can be modelled by the equation
x^2 + 2xy + 2y^2 = 10, where *x* and *y* are measured in metres.

The *x* and *y* axes are horizontal and vertical respectively.

Find the maximum vertical height above the platform of the sculpture in metres to 2 decimal places.

Express \cot 3\theta \cos 3\theta in the form a\cot 3\theta +b\cos 3\theta
where *a* and *b* are real numbers.

Determine the values of *a* and *b*.

*a* =

*b*=

An open-topped fish tank is to be made for an aquarium.

It will have a square horizontal base, rectangular vertical sides and a volume of 60 m^{3}.

The materials cost:

- £15 per m
^{2}for the base - £8 per m
^{2}for the sides.

Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value.

The height *x* metres, of a column of water in a fountain display satisfies the differential equation
\frac{dx}{dt}=\frac{8\sin 2t}{3\sqrt{x}}, where *t* is the time in seconds after the display begins.

Solve the differential equation, given that initially the column of water has zero height.

Express your answer in the form x=f(t)

*x*=

Find the maximum height of the column of water, giving your answer to the nearest cm.

Decide whether the statements below are:

- Always True
- Sometimes True
- Never True

Multiplying a rational number by a rational number gives an irrational number

Multiplying an irrational number by an irrational number gives an irrational number

Multiplying an irrational number by a rational number gives an irrational number

Determine

\lim _{n\rightarrow 0}\left(\frac{xe^x-\sin x}{x^2}\right)