4MA1 Mathematics A: January 2023 Paper 1HRHigher Tier Paper 1

The table shows information about the frame size, in cm, of 60 bicycles sold in a shop.

(a)Write down the modal class.

[1]

(b)Work out an estimate for the mean frame size.

cm[4]

The diagram shows a solid triangular prism.

6 cm

Work out the total surface area of the triangular prism. Give your answer correct to 3 significant figures.

cm²[3]

Here is a list of six numbers written in order of size.

Find the value of x, the value of y and the value of z

x =

y =

z = [3]

Divya and Yuan each pay for a holiday at a special offer price.

The amount that Divya pays is the same as the amount that Yuan pays.

Work out the value of k

k = [4]

C grams of chocolate is shared in the ratios 2 : 5 : 8 The difference between the largest share and the smallest share is 390 grams.

Work out the value of C

C = [3]

Solve the simultaneous equations

Show clear algebraic working.

x =

y = [3]

(a)Write 9.32 × 10^–5 as an ordinary number.

[1]

(b)Work out 3 × 10⁵– 6 × 10⁴ Give your answer in standard form.

[2]

(c)Work out (3 × 10⁵⁵) × (6 × 10⁶⁵) Give your answer in standard form.

[2]

(a)Factorise fully 18c³ d²– 21c²

[2]

(b)(i)Factorise y²– 3y – 18

[2]

(ii)Hence, solve y²– 3y – 18 = 0

[1]

The diagram shows an isosceles triangle ABC

AB = 7 cm AC = BC = ycm

The area of the triangle is 42 cm²

Work out the value of y

y = [4]

R and T are points on a circle, centre O

RT = 12 cm M is the midpoint of RT Angle ROM = 52°

Work out the area of the circle. Give your answer correct to 3 significant figures.

cm²[4]

The table shows information about the times, in minutes, that 80 patients had to wait to see a doctor.

(a)Complete the cumulative frequency table below.

[1]

(b)On the grid opposite, draw a cumulative frequency graph for your table.

[2]

(c)Use your graph to find an estimate for the median.

_minutes[1]

(d)Use your graph to find an estimate for the interquartile range.

_minutes[2]

Solve 2^–4x = 32

x =

[2]

Use algebra to show that 0.381 = 2155

[2]

T =

p = 0.51 correct to 2 significant figures. r = 6.3 correct to 2 significant figures.

Work out the upper bound for the value of T Show your working clearly.

[2]

(a)Complete the table of values for y = x³– 3x + 2

[2]

(b)On the grid, draw the graph of y = x³– 3x + 2 for values of x from –2 to 2

[2]

(c)By drawing a suitable straight line on the grid, use your graph to find an estimate for the solution of

Give your answer correct to one decimal place.

[3]

The function f is such that

(a)Find f  1 3  

[1]

(b)Find f ^–1(x)

f ^–1(x) = [2]

The function g is such that

(c)Express g(x) in the form a(x – b)²+ c

[3]

The area of triangle ABC is 12 cm²

Work out the perimeter of triangle ABC Give your answer correct to 3 significant figures.

cm[5]

Solve 3(x – 2 3) = x + 2 3

Give your answer in the form a + b 3 where a and b are integers. Show your working clearly.

x = [4]

P is inversely proportional to y² When y = 4, P = a

(a)Find a formula for P in terms of y and a

[3]

Given also that y is directly proportional to x and when x = a, P = 4a

(b)find a formula for P in terms of x and a

[3]

Here is a sketch of the curve y = a cos(x + b)° for 0  x  360

Given that 0 < b < 180

find the value of a and the value of b

a =

b = [2]

The diagram shows a triangular prism, ABCDEF, with a rectangular base ABCD

Work out the angle that BE makes with the plane ABCD Give your answer correct to one decimal place.

[4]

The diagram shows triangle OAB with OA extended to E

OA = a OB = b

M is the point on OB such that OM : MB = 4 : 1 N is the point on AB such that AN : NB = 3 : 2 OA : AE = 5 : 3

(a)Find an expression for ON in terms of a and b Give your answer in its simplest form.

ON = [2]

(b)Use a vector method to show that MNE is a straight line.[3]

G is the point on the curve with equation y = 8x²– 14x – 6 where the gradient is 10 The straight line Q passes through the point G and is perpendicular to the tangent at G

Find an equation for Q Give your answer in the form ax + by + c = 0 where a, b and c are integers.

[5]

An arithmetic sequence has first term 8 and common difference 11 The sequence has k terms, where k > 21

The sum of the last 20 terms of the sequence is 10 170

Find the value of k Show clear algebraic working.

k =

[5]