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TIME

INSTRUCTIONS TO CANDIDATES

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2 hours.

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Centre Number

Candidate Number

Write your Centre Number and Candidate Number in the spaces provided at the top of this page.You must answer the questions in the spaces provided.Do not write outside the boxed area on each page, on blank pages or tracing paper.Complete in black ink only. Do not write with a gel pen.All working should be clearly shown in the spaces provided. Marks may be awarded for partially correct solutions.Where rounding is necessary give answers correct to 2 decimal places unless stated otherwise.Answer all thirteen questions.

INFORMATION FOR CANDIDATES

The total mark for this paper is 100.Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question.You may use a calculator.The Formula Sheet is on pages 2 and 3.

General Certificate of Secondary EducationJanuary 2019

Further Mathematics

Unit 1 (With calculator)Pure Mathematics

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WEDNESDAY 16 JANUARY, AFTERNOON

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1 (a) Matrices P and Q are defined by

P = [ 2−1 −54 ] and Q = [45 −2–3 ] Express as a single matrix:

(i) P + 3Q

Answer __________________________ [1]

(ii) P2

Answer __________________________ [2]

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(b) Matrices A, B and C are defined by

A = [31 24], B = [6 3] and C = [26] Which one of the products

AB, AC, BC and CB

is impossible to form?

Answer __________________________ [1]

(c) Matrices D, E, F and G are defined by

D = [13 24 ], E = [45 23 ], F = [64 ] and G = [23 ] Which one of the following combinations

D + E, E − D, E + F or G – F

is impossible to form?

Answer __________________________ [1]

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2 Solve the equation

x2 + 8x − 7 = 0

by completing the square.

Give your answer in the form a ± √b, where a and b are whole numbers.

Answer __________________________ [4]

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3 (a) Find dydx if y = 7x

2 − 3x3

+ 5x4

4

Answer _____________________________________________ [3]

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(b) Find ∫ (9x3 + 37x4 − 5) dx

Answer _________________________________________________ [4]

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4 (a) The vectors a, b and c are defined by

a = [ 3−2 ], b = [ 1−4 ] and c = [−27 ] Find the vector x if

x + 2b = c + a

Answer __________________________ [2]

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(b) The vectors p and q are shown below.

p

q

On the grids below, draw diagrams to show the vectors

(i) p + 2q

[1]

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(ii) q − p

[1]

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5 (i) Solve the equation

cos θ = 0.3

for −180° ≤ θ ≤ 180°

Answer ___________________________________________ [2]

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(ii) Hence solve the equation

cos (x2 + 20°) = 0.3 for −360º ≤ x ≤ 360°

Answer ____________________________________________ [3]

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6 Matrices Y and Z are defined by

Y = [−35 21 ] and Z = [ 16−5 ] Find the matrix X such that YX = Z

Answer __________________________ [5]

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7 (a) Solve the equation

42x−3 = 6x+4

Answer __________________________ [5]

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(b) Given that

log35 = x and log32 = y

express log390 in terms of x and y.

Answer __________________________ [3]

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(c) Given that

log z = 3 log x − log y

express z in terms of x and y.

Answer __________________________ [2]

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8 Simplify fully the algebraic expressions

(a) x – 63x2 − 11x − 4

÷ 2x2 – 12x

2x − 8

Answer ___________________________________________ [4]

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(b) 2x + 2

− x – 72x2 + 3x − 2

Answer _________________________________________ [4]

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9 A curve is defined by the equation y = 3x2 − x + 4

The tangent to this curve at a point P is parallel to the line y = 11x + 5

(i) Find the coordinates of P.

Answer __________________________ [5]

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(ii) Find the equation of the normal to this curve at the point (−1, 8).

Answer ___________________________________________ [4]

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10 Andrea and Bill were at positions A and B on horizontal ground. The distance between A and B was 960 m.

Bill noticed that a pole at position P, 300 m from B, was directly in line with a tower T.

Andrea measured the angle TÂP as 63.44° and Bill measured the angle AB̂P as 56.75°, as shown in the diagram below.

T

A B

P

300 m

960 m

63.44°56.75°

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(i) Calculate the distance AP.

Answer __________________________ m [2]

(ii) Calculate the size of the angle PÂB.

Answer __________________________ ° [2]

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(iii) Write down the size of the angle TÂB.

Answer __________________________° [1]

(iv) Write down the size of the angle AT̂B.

Answer __________________________° [1]

(v) Calculate the distance AT.

Answer __________________________ m [2]

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Andrea observed lightning strike the tower and exactly 3.6 seconds later she heard thunder.

(vi) Calculate the speed of sound.

Answer __________________________ m/s [2]

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11 A curve is defined by the equation y = x3 − 2x2 − 8x

(i) Find the coordinates of the points where the curve meets the x-axis.

Answer _________________________________________ [3]

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(ii) Find the coordinates of the turning points of the curve.

Answer ____________________________________________________ [5]

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(iii) Using calculus, identify each turning point as either a maximum or a minimum point. You must show working to justify your answer.

Answer ________________________________________________________ [2]

(iv) Sketch the curve on the axes below.

x

y

[2]

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(v) Find the area enclosed by the curve and the positive x-axis.

Answer __________________________ [4]

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12 Brian wants to frame a rectangular family photograph.

The photograph has a width of x cm and a length of y cm.

The price for framing the photograph is made up of• the cost of the wooden frame at 12p per cm and• the cost of the glass at 5p per cm2.

The cost of the wooden frame is £18.00

(i) Show that x + y = 75

[1]

The cost of the glass is £67.50

(ii) Show that xy = 1350

[1]

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(iii) Hence, by solving these equations simultaneously, calculate the dimensions of Brian’s photograph.

Answer _____________ cm × _____________ cm [5]

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13 Fred plans to dig two plots in his garden. He wishes to have

• one small rectangular plot, 2 m wide and x m long for flowers, and• a larger rectangular plot, x m wide and y m long for shrubs,

as shown in the diagram below.

2

y

2

y

x

x

x

x

flowers

shrubs

Write down, in terms of x and y,

(i) an expression for the total area A of both plots,

Answer __________________________ [1]

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(ii) an expression for the total perimeter P around both plots.

Answer __________________________ [1]

Fred has enough topsoil to cover an area of 30 m2, which is the total area of both plots.

(iii) Express y in terms of x and hence show that the total perimeter is given by

P = 4x + 60x

[3]

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To cut down his work of trimming round the edges of the plots, Fred wishes to keep the total perimeter to a minimum.

(iv) Find the values of x and y which will minimise the total perimeter, showing that these values give a minimum.

Answer x = _____________ m, y = _____________ m [4]

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(v) Hence find the minimum total length round the edges of both plots that Fred will have to trim.

Answer _____________________ m [1]

THIS IS THE END OF THE QUESTION PAPER

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