 A-level Sample Test
Paper 1
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marks

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

\frac{2}{5}
\frac{5}{2}
- \frac{2}{5}
- \frac{5}{2}
[1 mark] ? marks

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

Find \frac{dy}{dx}

\frac{\sqrt{x}}{3}
\frac{1}{x\sqrt{x}}
−\frac{1}{x\sqrt{x}}
−\frac{1}{2x\sqrt{x}}
[1 mark] ? marks

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?

a =

b = [3 mark] ? marks

p(x)=2x^3+7x^2+2x−3

(a)

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

x-3
x-2
x+2
x+3
[2 marks] ? (b)

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

[4 marks] ? marks

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 cm2 and perimeter 15 cm.

(a)

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 = [4 marks] ? (b)

Find the value of θ.

θ = [4 marks] ?  marks

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.

[3 marks] ? marks

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

k = , [4 marks] ? marks
(a)

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

[1 mark]
(b)

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

[6 marks] ?  marks
A curve has equation y=\frac{2x+3}{4x^2+7}
(a)
(i)

Find \frac{dy}{dx}

[2 marks] ? (ii)

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

b =

c = [4 marks] ? (b)

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 = [2 marks] ? marks

The function f is defined by

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

(a)

Using set notation, state the range of f.

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

a = [2 marks]
(b)

The inverse of f is f^{−1}

(i)

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

{ ,\ x \in \R} [1 mark]
(ii)

Find an expression for f^{−1}(x)

[3 marks] ? (c)

The function g is defined by

g(x)=5−\sqrt{x}\ ,\ (x\in\R:x>0)

(i)

Find an expression for gf(x)

[1 mark] ? (ii)

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

[3 marks] ? marks

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

(a)

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

C = (, ) (b)

The points P and Q lie on the circle.

The origin is the midpoint of the chord PQ.

[5 marks] ? marks

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.

m [8 marks] ? marks

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 = [3 marks] ? marks

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 m3.

The materials cost:

• £15 per m2 for the base
• £8 per m2 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.

m [8 marks] ? marks

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.

(a)

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

x = [7 marks] ? (b)

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

cm [1 mark] ? marks

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

[3 marks] ? marks

Determine

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

[6 marks] ? Test check
Total: - of marks