Session 1Paper 1 Questions and Answers

Non Calculator

Harris Academy Supported Study

Question 1 (Unit 1 LO3 Differentiation)

Find the stationary points of the function

and determine their nature.

marks (7)

412)( 3 xxxf

Solution 1

Know to differentiate

Differentiate

Derivative equal to zero

Factorise and solve

Find y coordinates

Nature Table

State nature of TPs

............)( xf1

2

3

ans:

4

0123 2 x

123 2 x

2,20)2)(2(3 xxx

Max TP at (-2,20) Min TP at (2,-12)

5 )12,2()20,2(

6

7

x 2

)(xf

shape

2 00

Max TP at (-2,20) Min TP at (2,-12)

Question 2 (Unit 2 LO2 Integration)

Evaluate

marks (4)

dxx

xx

2

12

23 47

Solution 2

Prepare to integrate

Integrate

Substitute

Answer

2

1

247 dxxx1

2

3

ans:

4

2

1

12

472

xx

x

)47()142( 21

24

213

213

2

122

2

2

3 47dx

xx

x

x

x

Question 3 (Unit 2 LO3 Trigonometry)

If x is an acute angle such that 3

1tan x

find exact values for

marks (4,2)

(a)

(b)

x2sin

x2cos

Solution 3a

Construct a right angled triangle and use Pythagoras

Find sin x and cos x

Formula for sin2x

Substitute and answer

1

2

3

ans:

4

10

3cos

10

1sin xx

xxcossin2

10

3

10

12

5

3

X

1

3

10

6

10

Solution 3b

Formula for cos2x

Substitute and answer

1

2

ans:

xxx 22 sincos2cos

22

10

1

10

3

5

4

10

8

10

1

10

9

Note

usedbemayxorxx 22 sin211cos22cos

Question 4 (Unit 2 LO3 Trigonometry)

Solve

20032 xforxx ,cossin

marks (4)

Solution 4

ans: 23

32

23

,,,x

1 Use double angle formula

Factorise and form equations

Solve

Solve

xxx cossinsin 22

032 )sin(cos xx2

230 xorx sincos

23

2 orx3

23xsin

32

3 orx4

0xcos

Session 2Paper 1 Questions and Answers

Non Calculator

Harris Academy Supported Study

Question 5 (Unit 1 LO1 Straight Line)

The line with equation meets the

x and the y axes at the points A and B respectively

123 yx

(a) Determine the coordinates of A and B

marks (2,4)

(b) Find the equation of the perpendicular bisector of AB.

Solution 5a

ans: )4,0()0,12( BA

1 Coordinates of A

Coordinates of B

)0,12(A

)4,0(B2

Solution 5b

ans: 163 xy

1 Rearrange to y =…… and find gradient

Perpendicular gradient

Midpoint of AB

Find equation of line

431 xy

3m2

31m

)(, 2

402

012 3

)6(32 xy4

),( 26

Question 6 (Unit 1 LO2 Functions and graphs)

The diagram below shows part of the graph of .

The function has stationary points at and as shown

)(xgy)3,0(

marks (3)

)0,2(y

x

-3

2O

)(3 xgy Sketch the graph of the related function

Solution 6

Reflection in x-axis

Move 3 units up

Annotate graph

2

3

1

y

x

3

2O

y

x

6

(2,3)

O

Question 7 (Unit 1 Recurrence Relations)

A doctor administers 40ml of a drug to Mr Sick each week. Over the same period 80% of the drug in the bloodstream is removed.If the level in the bloodstream rises above 55ml the drug becomes toxic

(a) Write down a recurrence relation to model this situation.

(b) Find a limit and explain what it means in the context of the question.

marks (2,4)

Solution 7a

For correct multiplier

For recurrence relation

2.01

2

ans:

402.01 nn uu

402.01 nn uu

Solution 7b

a

bL

1

1

3

ans:

201

40

.L

toxicnotso5550

justify limit

use limit formula

calculate limit

explanation

50

4 toxicnotso5550

12.01 asexistslimit

2

Question 8 (Unit 2 LO1 Polynomials)

(a) For what value of k is

when k takes this value

marks (3,2)

2x

8223 kxxx

a factor of

?

(b) Hence fully factorise the expression 8223 kxxx

Solution 8a

Use synthetic division

Complete division

Calculate k

1

2

3

ans:

0204 k

5k

-2 1 -1 2k -8

-2 1 -1 2k -8

-2 6 -4k-12

1 -3 2k+6 -4k-20

5k

Solution 8b

Find quotient

Factorise fully

1

2

ans:

)43( 2 xx

-2 1 -1 -10 -8

-2 6 8

1 -3 -4 0

))(( 432 2 xxx

)4)(1)(2( xxx

)4)(1)(2( xxx

Session 3Paper 1 Questions and Answers

Non Calculator

Harris Academy Supported Study

Question 9 (Unit 2 LO1 Polynomials)

For what value of p does the equation

have equal roots?

01)1( 2 pxxp

marks (4)

Solution 9

Use the discriminant

Find values of a, b and c

Substitute and simplify

Calculate value of p

042 acb1

3

ans:

0142 )( pp

4 0)2)(2( pp

2p

2p

11 cpbpa2

0442 pp

Question 10 (Unit 2 LO4

Circle)

A block of wood of thickness t has to pass between two rollers

046422 yxyx

The equations of the two circles are

marks (4)

and

0246302222 yxyx

x

t

y

Find the maximum possible value of t

Solution 10

Find centre and radius of small circle

Find centre and radius of large circle

Calculate distance between centres

Calculate distance, t

3)3,2( rcentre1

2

3

ans:

4

10)15,11( rcentre

22 )315()211( d

15d

)310(15 t 2t

2t

Question 11 (Unit 1 LO3 Differentiation)

Part of the graph of

marks (5)

is shown in the diagram

The tangent to the curve at the point where x = 1is also shown

1 x

y

o

Find the equation of the tangent at the point where x = 1

)65( 2 xxxy

Solution 11

ans:

1 knowing to differentiate differentiate

gradient at x = 1

y coordinate

equation

.......dxdy

6103 2 xxdxdy2

16103 m

3

3 xy

26511 yxat ,4

)1(12 xy5

1 xatm dxdy

Question 12 (Unit 1 LO2 Functions and

graphs)

The functions 9)( 2 xxf

are defined on the set of real numbers.

marks (1,2,3)

and xxh 23)(

(a) Evaluate ))3(( fh

(b) Find an expression for ))(( xhf

(c) For what value(s) of x does )())(( xfxhf

Solution 12a

Evaluate h(f (3)) 3)0())3(( hfh1

1

2

ans:

)23( xf

9)23( 2 x

3

Solution 12b

Apply h

Apply f

ans: 9)23( 2 x

Solution 12c

ans:

1 Equation

Rearrange to ..... = 0

Factorise and solve

.99)23( 22 xx

09123 2 xx2

0)1)(3(3 xx3

31 orx

31 orx