4MA1 Mathematics A: January 2019 Paper 1HRHigher Tier Paper 1

Show that 1 23 + 2 34 = 4 512[3]

There are 60 children in a club.

In the club, the ratio of the number of girls to the number of boys is 3 : 1

3

What fraction of the 60 children play a musical instrument?[4]

S is the point on PR such that angle RSQ = 90° RQ = 14 cm RS = 10 cm SP = 5 cm

Work out the length of PQ.

[4]

a, a, b and 40 are four numbers.

a is the least number. 40 is the greatest number.

The range of the four numbers is 14 The median of the four numbers is 30

Work out the value of a and the value of b.

[3]

The Shanghai Maglev Train takes 8 minutes to travel a distance of 30.5 kilometres.

Work out the average speed of the train. Give your answer in kilometres per hour.

[3]

The diagram shows the triangle PQR.

In the diagram, all the angles are in degrees.

RP = RQ

Find the value of y. Show clear algebraic working.

[4]

Solve the simultaneous equations

Show clear algebraic working.

[3]

N = 480 ´ 10⁹

(a)Write N as a number in standard form.[1]

(b)Write N as a product of powers of its prime factors. Show your working clearly.[3]

(c)Find the largest factor of N that is an odd number.[1]

The boys and the girls have some coins.

The mean number of coins that the boys have is 5.5 The girls have a total of 18 coins.

Work out the mean number of coins the 20 children have.

The numerators of the fractions form the sequence of whole numbers 1 2 3 4 … The denominators of the fractions form the sequence of odd numbers 1 3 5 7 …

(a)Write down an expression, in terms of n, for the nth term of this sequence of fractions.[2]

(b)Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1[3]

(a)Find dydx

[2]

The curve C has two turning points.

(b)Work out the x coordinates of the two turning points. Show your working clearly.[3]

(c)Show that the x-axis is a tangent to the curve C.[2]

euros, of 120 apartments in 2015

(a)Find an estimate for the number of these apartments with a value of 80 thousand euros or less in 2015[1]

The table gives information about the values, in thousands of euros, of the same 120 apartments in 2018

(b)On the grid opposite, draw a cumulative frequency diagram for this information.[2]

(c)Find an estimate for the increase in the median value for these apartments from 2015 to 2018

[2]

[2]

(b)Expand and simplify 4n(n – 3)(n + 5)[2]

(c)Factorise 4c²– 9d²[1]

(d)Simplify fully₂

[3]

7 of the beads are red. 3 of the beads are green. 2 of the beads are yellow.

Lucy takes at random a bead from the bag and keeps it. Then Julian takes at random a bead from the bag.

(a)Work out the probability that they each take a yellow bead.[2]

(b)Work out the probability that the beads they take are not the same colour.[3]

(b)Show that the ratio

is the same as the ratio

[3]

Show your working clearly.

B, C, D and E are points on a circle.

AB is the tangent at B to the circle. AB is parallel to ED. Angle ABE = 73°

Work out the size of angle DCE. Give a reason for each stage of your working.

M is the midpoint of the edge GH.

Find the size of the angle between the line MA and the plane ABCD. Give your answer correct to 1 decimal place.

The perimeter of the triangle is k cm.

Given that x = y − 1 find the value of k. Show your working clearly.

ABX and DCX are straight lines. → → AB = a BC = b → Find EX in terms of a and b. Give your answer in its simplest form.

(a)Find the value of p for which f⁻¹(p) = k

[3]

The function g is defined as g(x) = x² for x > 0

(b)Given that gf(a) = k for k > 1 find an expression for a in terms of k.

[3]