Higher Maths Non Calculator Practice Practice …...Higher Maths Non Calculator Practice Practice Paper A 1. A sequence is defined by the recurrence relation 10 2 1, 3. nn u u u What

Higher Maths

Non Calculator Practice

Practice Paper A

1. A sequence is defined by the recurrence relation 1 0

2 1, 3.n n

u u u

What is the value of 2

?u

2. The line with equation 2 9 0kx y is parallel to the line with gradient 7.

What is the value of k?

3. A circle has equation 2 2 8 2 1 0.x y x y

What is the radius of this circle?

4. What is the derivative of 3 2

3

x

x

with respect to x?

5. Find 4

1.

2dx

x

6. If 2 12 37x x is written in the form 2( ) ,x p q find the value of q.

7. A sequence is generated by the recurrence relation 1

0 8 16.n n

u u

What is the limit of this sequence as ?n

8. A circle with centre ( 1, 5) passes through the point (2, 7).

What is the equation of the circle?

9. The vectors p and q with components

1

2

k

p and 3

2

k

q are perpendicular.


10. Identify the nature of the roots of the equation 22 8 3 0x x .

11. What is the value of 5 7

3 4cos tan ?

12. Given that 2

1

8log p , find the value of p.

13. Find 5(3 11)x dx

14. K and L are the points with coordinates (0, 1, 4) and (3, 2, 5) respectively.

If KM 3KL , find the coordinates of M.

15. 2

4( ) .

2 8h x

x x

For what values of x is ( )h x undefined?

16. Here are two statements about the graph

with equation ,xy a b shown opposite.

(1) 0 1;a

(2) y is always increasing

Which of these statements are true?

17. The diagram shows part of the graph of a cubic.

What is the equation of this graph?

8. Given that 4 4

log 2 log 5 ,y x express y in terms of x.

19. If . ( ) 18 p p q and 3,p find the value of .p q

y

x

xy a b

0

y

x 0

4 2 2

32

20. The diagram shows part of the

curve with equation 3

log ( ).y p x k

What is the value of p

21. Triangle PQR has vertices P( 3, 5),

Q(7, 3) and R( 1, 5), as shown.

(a) Find the equation of the median RM.

(b) Find the equation of the altitude AP.

(c) Find the coordinates of the point of intersection of RM and AP.

22. Find the stationary points on the curve given by 3 29 24 2y x x x and determine their

nature.

23. (a) Functions f and g are defined on suitable domains by

2( ) 2 5 and ( ) 1f x x g x x

Find ( ( )).f g x

(b) Sketch the curve with equation ( ( )).y f g x

24. (a) Show that π62sin 2cos 3 sin cos .x x x x

(b) Express 3 sin cosx x in the form sin( )k x a where π

.2

0 and 0k a

(c) Hence, or otherwise, solve π62sin 2cos 3,x x where π.0 2x

y

x 0

P

Q

R

y

x 0

Practice Paper B

1. Given that 4( ) 2 5 ,f x x x find (2).f

2. Find ( 3)(3 1) .x x dx

3. P and Q have coordinates (2, 3, 2) and (1, 0, 5).

What is the distance between P and Q?

4. If 2 8 3x x is expressed in the form 2( ) ,x p q what is the value of q?

5. Here are two statements about the equation 23 5 1 0.x x

(1) The roots are equal.

(2) The roots are rational.

Which of these statements is true?

6. Find all the values of x in the interval 0 2x for which 3

2cos .x

7. S is the point with coordinates (2, 1, 1) , T (4, 1, 5) and U(5, 2, 7) .

Find the ratio in which T divides SU.

8. Given that 4sin(3 2),y x find .dy

dx

9. The angle between the line shown in

the diagram and the x-axis is 3.

What is the gradient of the line?

10. Given that 2

3log 9 ,

a what is the value of a?

11. What is the maximum value of π59 4sin ?x

y

x 0 3

y

2cos( )y px q

12. Find 4(3 11) .x dx

13. The graph shown in the diagram has equation of the form 2cos( ) .y px q

What are the values of p and q?

14. Given that 2

1( )

16h x

x

, what is the largest possible domain for ?h

15. Vector t has components

4

0

3

. u is a unit vector such that ku t , where 0.k

Find the value of k.

16. The diagram shows the graph of ( ).y f x

Sketch the graph of 2 ( )?y f x

17. The equation of the parabola shown is of the form ( 4).y kx x


18. Simplify 3 3

2log log ( 1).x x

19. What is the solution to 2 4 5 0 ?x x

y

x 0

5

1

x 0

2

y

y

x 0 4

( 4)y kx x ( 1,10)

20. If 210v t and the rate of change of v with respect to t at , 0t k k is 160, find

the value of k.

21. A circle with equation 2 2 6 2 9 0x y x y has centre 1

C .

(a) Write down the coordinates of the centre 1

C and find the length of the

radius of this circle.

A second circle with equation 2 2( 3) ( 7) 36x y has centre2

C .

(b) (i) Find the distance between the centres 1

C and 2

C .

(ii) Hence find the minimum distance between the circumferences of the two circles.

22. A is the point with coordinates (1, 1, 2), B(3, 0, 3) and C( 2, 3, 4).

(a) Express AB and AC in component form.

(b) Find the size of angle BAC.

23. Solve 2sin2 5cosx x for 0 2x .

y

x 0

y

x 0

24. The diagram shows part of the quartic with equation ( ).y g x

There are stationary points at 2, 0 and .x x x a

On separate diagrams sketch the graph of

(a) ( ).y g x

(b) ( 3).y g x

25. Find the values of x for which the function 2 3( ) 5 24 3f x x x x is decreasing.

26. P is the point with coordinates ( 1, 6) and Q is(3, 10) .

Find the locus of points which are equidistant from both P and Q.

y

x a

0

Practice Paper C

1. A sequence is defined by the recurrence relation

1 0

2 5, 6n n

u u u

What is the value of 2

?u

2. Here are two statements about the line with equation 3 4 8 0.x y

(1) This line is parallel to a line with gradient 3

4 .

(2) This line cuts the y-axis at the point (0, 8) .


3. Functions f and g are defined on suitable domains by

( ) 3 5 and ( ) 2 .f x x g x x

Find an expression for ( ( )).f g x

4. A curve has equation 3 2 5y x x .

What is the gradient of the tangent at the point where 2 ?x

5. A circle with centre ( 2, 1) passes through the point (5, 2).


6. Find 3

2.dx

x

7. 3 2( ) 2 7.g x x x x

What is the remainder when ( )g x is divided by ( 1) ?x

8. Vectors u and v are shown in the diagram below.

QR 3 ST

Find PQ in terms of u and v.

P

R

Q

S

T

u

3u

v

9. P and Q are the points with coordinates ( 1, 0, 5) and (2, 3, 3) respectively.

If PR 2PQ , find the coordinates of R.

10. What is the exact value of 5

4 4sin cos

?

11. Find 5cos(2 1) .x dx

12. Given that 2 2 2

log 3log log 8y x , express y in terms of x.

13. Given that 4sin ,y x find .dy

dx

14. If 25 6x x is written in the form 2( ) ,p x q what is the value of p?

15. Solve 2 1

3tan x for

2x

16. The diagram shows the graph with equation log ( ).by x a

What are the values of a and b?

17. What is the nature of the roots of the quadratic equation 2 10 25 ?x x

18. The diagram shows part of the graph of cubic with equation ( ).y g x

The graph has turning points at 1 and 2.x x

Sketch the graph of 𝑦 = 𝑔′(𝑥).

x

y

0 2 1

x 0

log ( )b

y x a

4 3

(1, 1)

y

19. Solve 2 8 15 0.x x

20. The diagram shows part of the curve ( ).y f x

The curve passes through the points K (0, 3) and L (2, 27).

Which of the following represents the equation of the curve?

A 2 3y x B 13xy C 3xy e D 3 24xy

21. A function f is defined by 3 2( ) 2 3 ,f x x x where x is a real number.

(a) Find the coordinates of the points where the curve with equation ( )y f x crosses the x and y-axes.

(b) Find the stationary points on the curve ( )y f x and determine their

nature.

(c) (i) Sketch the curve ( ).y f x

(ii) Hence solve 3 22 3 .x x

22. Two sequences are generated by the recurrence relations

1

1

0 4 8 4

2n n

n n

u u

v kv

The two sequences approach the same limit as .n

(a) Evaluate this limit.

(b) Hence determine the value of k.

23. Given that 4

5sin a and

2

5sin ,b where

π π,

2 20 and 0a b find the exact values of :

(a) sin( );a b

(b) tan( ).a b

24. In the triangle opposite 2 units a b

Find . ( ) a a b c

a b

c

K

x 0

L y

Practice Paper D

1. The midpoint of the line joining G( 1, 3, 7) to H(5, 1, )p is M( , 1, 4).q


2. Given that 5

1( ) ,

3f x

x find ( ).f x

3. If 2 12 7x x is written in the form 2( ) ,x a r find the value of r.

4. A straight line passes through the points (4, 3) and (0, 1).

What is the equation of the line?

5. Functions f and g are defined on the set of real numbers by

2( ) 1 and ( ) 3 5f x x g x x

What is the value of ( ( 1)) ?g f

6. The vectors with components

4

7

3

and

5

2

t

are perpendicular.

What is the value of t?

7. The diagram shows a right-angled triangle

with sides 1, 3 and 10.

What is the value of cos2 ?x

8. Find

0

2

2

6x dx

9. For what value of k does the equation 22 4 0x x k have equal roots?

10. DE and EF have components

5

2

3

and

2

1

1

respectively.

Given that D has coordinates ( 2, 0, 2) , what are the coordinates of F ?

1

3

10

x

11. What is the maximum value of π7

98 3sin ?x

12. Find 3(2 5) .x dx

13. How many solutions does the equation( 7 cos 3)(4tan 9) 0x x have in

the interval 0 2x

14. Given that ( ) 4sin3 ,f x x find 6.f

15. The diagram shows the line ST with equation 2 0.x y

The angle between ST and the positive direction of the x-axis is

Find an expression for

A 1 1

2tan B 1 1

2tan C 1tan 2 D 1tan 2

16. What is the value of 2

2

log 32?

log 8

17. The diagram shows a sketch of the curve with equation

( 2)( 2)( )y k x x x a

What are the values of a and k?

18. Here are two statements about the function 2( ) 4.f x x

(1) The largest possible domain is 2 2.x

(2) The range is ( ) 0.f x


19. Given that

0, for 3

( ) 0, for 3

0, for 3

x

f x x

x

Sketch a curve to represent ( ) ?y f x

20. If 25 ,x a find an expression for x.

y

x 0 2 5

21. A( 2, 4),B(10, 4) and C(4, 8) are the vertices of triangle ABC shown in the

diagram.

(a) Write down the equation of the altitude from C.

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

(c) Find the point of intersection of the lines found in (a) and (b).

22. P is the point (4, 1, 2), Q is (5, 2, 0) and R is (7, 4, 4).

(a) Show that P, Q and R are collinear.

(b) Find the ratio in which Q divides PR.

23. Find the equation of the tangent to the curve with equation

4

yx

at the point where 2.x

24. (a) Given that 2( ) 3 2 10f x x x and ( 2)x is a factor of ( ),f x find a

formula for ( ).f x

(b) Hence factorise ( )f x fully.

(c) Solve ( ) 0.f x

y

x 0

4y

x

2

y

x 0

A B

C

25. The graph illustrates the law .by ax

The straight line joins the points (0, 4) and (1, 0).

Find the values of a and b.

2log y

2log x

4

1

0

Practice Paper E

1. K and L have position vectors

2

0

1

and

1

3

1

respectively.

What is the magnitude of KL ?

2. If 3( ) 4 7,f x x x find ( 2).f

3. Find 1

24x x dx

4. A function f is defined on the set of real numbers by ( ) 4 5.f x x

Find an expression for ( ( )).f f x

5. Evaluate 2

.4 3

4 2 sin cos

6. A circle with centre ( 3, 4) passes through the point ( 2, 2).


7. 3 2( ) 2 5 4.f x x x x

What is the remainder when ( )f x is divided by ( 2) ?x

8. The diagram shows the part of the graph of the cubic ( ).y f x

Sketch the graph of 4 ( )?y f x

x 0

y

4

4

1

9. The graph shown in the diagram has equation 2sin( ).y p qx


10. A sequence is generated by the recurrence relation 1

7 2 .n n

u u

If 2

5,u what is the value of 0

?u

11. For what value of k does the equation 2 6 1 0kx x have equal roots?

12. Find 4(2 7) .x dx

13. Given that 2( ) 6f x x and (1) 5,f find a formula for ( )f x in terms of x.

14. What are the coordinates of the centre of the circle with equation

2 23 3 6 18 5 0 ?x y x y

15. The diagram shows part of the graph of a cubic function.

What is the equation of this graph?

x 0

3 1 6

y

x 0

4

y

16. The diagram shows part of the graph of the cubic ( ).y f x

There are stationary points at 0 and 3.x x

Sketch the graph of ( ) ?y f x

17. If 24 8 1x x is expressed in the form 24( ) ,x p q what is the value of q?

18. If 2 2

3log log 5 3,t find the value of t.

19. If 34p x find the rate of change of p with respect to x when 2.x

20. Find the solutions for 28 2 0 ?x x

21. A line joins the points P( 4, 3) and Q(2, 7) .

Find the equation of the perpendicular bisector of PQ.

22. Show that the line with equation 2 10y x is a tangent to the circle with

equation 2 2 2 4 15 0x y x y and find the coordinates of the point of

contact of the tangent and circle.

23. The diagram shows a right-angled triangle with height 2 units, base 1 unit and an angle of p.

(a) Find the exact values of:

(i) cos ;p

(ii) cos2 .p

(b) By writing 3 2 ,p p p find the exact value of cos3 .p

P y

x

Q

0

2

1

p

y

x 0 3

24. A function f is defined by 3 2( ) 2 4 1,f x x x x where 0 3.x

Find the maximum and minimum values of f.

25. (a) Express 2 2 cos 2 2 sinx x in the form cos( ) ,k x a where 0k

and 0 360.a

(b) Find:

(i) the maximum value of 3 2 2 sin 2 2 cosx x ;

(ii) a value of x where this maximum value occurs in the interval 0 360.x

PRACTICE PAPER F

1. If ( ) (3 1)( 4),f x x x find ( ).f x

2. Vectors p is given by 3 i j k and q is 2 2 i j k.

What are the components of 2 ?p q

3. A circle has equation 2 2 2 8 2 0.x y x y

What is the radius of this circle?

4. The line with equation 4 1 0kx y is parallel to the line with gradient 3.


5. What is the derivative of 22 6

,2

x

x

with respect to x?

6. Find 5 2 .x dx

7. A circle centre (3, 5) passes through the point ( 1, 4).


8. What is the value of π2 11

3 6sin cos ?

9. Determine the coordinates of the stationary point and it’s nature for the curve with equation

𝑦 = 4 − (𝑥 − 9)2

10. Here are two statements about the equation

25 3 1 0x x

(1) The roots are unequal;

(2) The roots are irrational.


11. What is the minimum value of π611 3cos 2 ?x

( 2)( 5)y k x x

12. If 5sin(7 2 ),y x find .dy

dx

13. Find the radius of the circle with equation 2 2 8 3.x y x

14. 2

1( ) .

25g x

x

For what value(s) of x is ( )g x undefined?

15. The diagram shows part of the graph of the cubic ( ).y f x

There are three roots at 4, 2 and as shown.x x x m There are two stationary points lying

between the roots.

Sketch the graph of ( ) ?y f x

16. The equation of the parabola shown

is of the form ( 2)( 5).y k x x


17. What is the maximum value of 2 (3 1)(3 1)?y x x

18. Given that . 5 and .( ) 54, a b a a b find .a

19. If 6 6 6

log 2log log 12y x express y in terms of x.

20. Sketch the graph of 5

1log ?x

y

y

x 0 2

(4, 4)

y

x 0 4 2 m

( )y f x

5

21. (a) (i) Show that ( 1)x is a factor of 3 2( ) 2 3 5 6.f x x x x

(ii) Hence factorise ( )f x fully.

(b) Given that 2

0

6 6 5 6, 0,

p

x x dx p find the value of p.

22. (a) Write 2 6 13x x in the form 2( ) .x a b

(b) (i) Sketch the graph of 2 6 13.y x x

(ii) State the range of values of y.

(c) Write down the maximum value of 2

1.

6 13x x

23. The diagram shows part of the curve with equation log ( ).b

y x a

The curve passes through the points P ( 4, 0) and Q (4, 2).

Find the values of a and b.

24. (a) Write 2cos x in terms of cos2 .x

(b) Find 24cos .x dx

25. In the triangle shown

2 and 2 3. p q r

Find . ( ) p p q r

P

Q

y

x 0

log ( )b

y x a

p

q

r

2

3

ANSWERS PAPER A

1. 15 11. 3

2

2. 14 12. 3

3. 18 units 13. 61

18(3 11)x c

4. 22 2

3 3x x 14. (9, 4, 7)

5. 3

1

6c

x 15. 2 and 4

6. 1 16. Only statement 2 is correct

7. 80 17. 2( 2)( 2)( 4)y x x x

8. 2 2( 1) ( 5) 13x y 18. 16

5y

x

9. -2 19. -9

10. 2 real and distinct roots 20. 2 21. (a) 3 2 0x y

(b) 2 0x y

(c) (1, 1)

22. Maximum turning point at (2, 18)

Minimum turning point at (4, 14)

23. (a) 2 22( 1) 5 or 2 4 7x x x

(b) 24. (a) Proof

(b) 62sin x

(c) 5

2 6,

y

x 0

7

PAPER B 1. 59 11. 13

2. 3 24 3x x x c 12. 51

15(3 11)x c

3. 19 units 13. 𝑝 = 4, 𝑞 = 3

4. -13 14. 4, 4

5. 2 real and distinct roots 15. 1

5

6. 7

6 6 and

16.

7. 2 : 1 17. 2

8. 12cos(3 2)x 18. 2

3log

1

x

x

9. 3 19. 5 1x

10. 27 20. 8 21. (a) Centre ( 3, 1) and radius 1 unit

(b) (i) 10 units (ii) 3 units

22. (a)

2 3

AB 1 and AC 4

1 2

(b) Angle BAC 2

90 or

23. , 2 2

24. (a) (b)

25. : 2 4,x x x x

26. 4 9 0x y

y

x 0 a 2

( )y g x y

3 1

( 3)y g x

x 0 3a

PAPER C

1. 9 11. 5

2sin(2 1)x c

2. Only statement 1 is correct 12. 38y x

3. ( ( )) 11 3f g x x 13. 34sin cosx x

4. 10 14. 14

5. 2 2( 2) ( 1) 58x y 15. 5

6

6. 2

33x c 16. 𝑎 = 4, 𝑏 =

7. 3 17. 2 real and distinct roots

8. 4 3u v 18.

9. (5, 6, 1) 19. 3 or 5x x

10. 0 20. 13xy

21. (a) 3

2(0, 0) and , 0

(b) Maximum turning point at (0, 0)

Minimum turning point at (1, 1)

(c) (i)

(ii) 3

2x

22. (a) Limit : 14

(b) 6

7k

23. (a) 2

sin( )5

a b

(b) tan( ) 2a b

24. 8

y

x 0

PAPER D 1. 𝑝 = 1, 𝑞 = 2 11. 11

2. 6

5

3x 12. 41

8(2 5)x c

3. -29 13. 2

4. 1 0x y 14. 0

5. 1 15. 1tan 2

6. 2 16. 5

3

7. 4

5 17. 𝑎 = −5, 𝑘 =

1

4

8. 16 18. Only statement 2 is correct

9. 2 19.

10. (1, 3, 0) 20. 2

log 5a

x

21. (a) 4x

(b) 3 2 9 0x y

(c) 3

24,

22. (a) Proof e.g. show that QR 2PQ

(b) 1 : 2

23. 4 0x y

24. (a) 3 2( ) 10 8f x x x x

(b) ( ) ( 2)( 1)( 4)f x x x x

(c) { 4, 1, 2}

25. 16 and 4a b

PAPER E

1. 22 11. 9

2. 8 12. 51

10(2 7)x c

3. 5

144

5x x c 13. 3( ) 2 3f x x

4. ( ( )) 16 25f f x x 14. (1, 3)

5. -2 15. 22( 1) ( 3)y x x

6. 2 2( 3) ( 4) 5x y 16.

7. -6 17. -5

8. 18. 3 40

9. 𝑝 = 2, 𝑞 = 6 19. 3

4

10. 3 20. 4 or 2x x

21. 3 5 7 0x y

22. Point of contact ( 3, 4)

23. (a) (i) 1

5 (ii)

3

5

(b) 11

5 5

24. Maximum value 1, minimum value 7

25. (a) 4cos( 315)x

(b) Maximum value 7 at 315x

PAPER F 1. 6 11x 11. 8

2.

5

0

4

12. 10cos(7 2 )x

3. 19 13. 19

4. 12 14. 5 and 5

5. 23 x 15.

6. 7

55

7x c 16. 2

7. 2 2( 3) ( 5) 17x y 17. 3

8. 3 18. 270

9. maximum at (9, 4) 19. 212y x

10. Both statements are correct 20. 21. (a) (i) Show that ( 1) 0f (ii) ( 1)( 2)(2 3)x x x

(b) 3

2p

22. (a) 2( 3) 4x

(b) (i) (ii) 4y

(c) 1

4

23. 5, 3a b

24. (a) 1 1 1

2 2 2(1 cos2 ) or cos2x x (b) 2 sin2x x c

25. 8

y

x 0

13

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Higher Maths Non Calculator Practice Practice …...Higher Maths Non Calculator Practice Practice Paper A 1. A sequence is defined by the recurrence relation 10 2 1, 3. nn u u u What