KS3 maths worksheets › Weekly checks › Weekly check: Year 9 part 3 Year 9 · Key Stage 3 · free KS3 maths weekly check Weekly check: Year 9 part 3
20 mixed questions from 7 Year 9 topics, 53 marks. It covers surface area and volume to quadratic, cubic, reciprocal and circle graphs, the last third of the year in the order schools usually teach it.
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The questions
1 ABC is a triangle. The side BC is extended to the point D.
Angle BAC = a and angle ABC = b .
Prove that angle ACD = a + b .
Give a reason for each stage of your working.
A B a b D C Diagram NOT accurately drawn 3 marks Show answer Angle ACB = 180 - a - b (angles in a triangle add up to 180°), so angle ACD = 180 - (180 - a - b) = a + b (angles on a straight line add up to 180°)
See the working A n g l e s i n a t r i a n g l e a d d u p t o 1 8 0 °
A C B = 1 8 0 - a - b A n g l e s o n a s t r a i g h t l i n e a d d u p t o 1 8 0 ° A C D = 1 8 0 - A C B A C D = 1 8 0 - ( 1 8 0 - a - b ) A C D = a + b
A n g l e A C D = a + b
2 Reflect the shape in the mirror line.
2 marks Show answer Triangle with corners on the other side of the line, each the same distance from the line
See the working e a c h c o r n e r : g o a t r i g h t a n g l e s t o t h e m i r r o r
s a m e d i s t a n c e o n t h e o t h e r s i d e
d i a g o n a l s q u a r e s o n a 4 5 ° l i n e
j o i n t h e n e w c o r n e r s
i m a g e d r a w n , s a m e s i z e , f l i p p e d o v e r t h e l i n e
3 Rectangle A has vertices at (1, 1), (3, 1), (3, 2) and (1, 2).
Rectangle B has vertices at (2, 2), (6, 2), (6, 4) and (2, 4).
Describe fully the single transformation that maps rectangle A onto rectangle B .
2 marks Show answer Enlargement, scale factor 2, centre (0, 0).
See the working C o m p a r e s i d e l e n g t h s
A : w i d t h 2 , h e i g h t 1
B : w i d t h 4 , h e i g h t 2
S c a l e f a c t o r = 4 ÷ 2 S c a l e f a c t o r = 2
E a c h v e r t e x d o u b l e s , s o c e n t r e ( 0 , 0 )
( 1 , 1 ) → ( 2 , 2 )
( 3 , 1 ) → ( 6 , 2 )
( 3 , 2 ) → ( 6 , 4 )
E n l a r g e m e n t , s c a l e f a c t o r 2 , c e n t r e ( 0 , 0 )
4 A graph shows Kelly's total pay plotted against the number of hours she works.
The graph is a straight line through (0, 0) and through the point (8, 96), where pay is in pounds.
Work out Kelly's hourly rate of pay. Work out Kelly's total pay for working 11 hours. 0 1 2 3 4 5 6 7 8 9 10 11 12 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Hours worked Total pay (£)
3 marks Show answer a) £12 per hourb) £132
See the working a )
8 h o u r s → £ 9 6
£ 1 2 p e r h o u r
b )
£ 1 3 2
5 Triangle P has vertices at (2, 1), (5, 1) and (5, 3).
Triangle Q has vertices at (2, −1), (5, −1) and (5, −3).
Describe fully the single transformation that maps triangle P onto triangle Q .
2 marks Show answer Reflection in the x-axis.
See the working C o m p a r e m a t c h i n g v e r t i c e s
( 2 , 1 ) → ( 2 , - 1 )
( 5 , 1 ) → ( 5 , - 1 )
( 5 , 3 ) → ( 5 , - 3 )
x s t a y s t h e s a m e , y c h a n g e s s i g n
( x , y ) → ( x , - y )
R e f l e c t i o n i n t h e x - a x i s
6 A prism has a cross-section with an area of 24 cm2 . The prism is 15 cm long. Work out the volume of the prism. Another prism has a volume of 540 cm3 . Its cross-section has an area of 72 cm2 . Work out the length of this prism. 3 marks Show answer a) 360 cm³b) 7.5 cm
See the working a )
V = a r e a o f c r o s s - s e c t i o n × l e n g t h
3 6 0 c m ³
b )
7 . 5 c m
7 A cube has a volume of 64 cm3 .
Work out the total surface area of the cube.
3 marks Show answer 96 cm²
See the working S i d e l e n g t h = c u b e r o o t o f 6 4
s = ∛ 6 4 s = 4 A r e a o f o n e f a c e = 4 × 4 A r e a o f o n e f a c e = 1 6 A c u b e h a s 6 f a c e s S A = 6 × 1 6 S A = 9 6
9 6 c m ²
8 Find the equation of the straight line that is perpendicular to y = 2x - 3 and passes through the point (4, 1).
3 marks Show answer y = -x 2
See the working P e r p e n d i c u l a r : n e g a t i v e r e c i p r o c a l
G r a d i e n t o f y = 2 x - 3 i s 2
m = - 1 2 y = - 1 2 x + c S u b s t i t u t e ( 4 , 1 ) 1 = - 1 2 × 4 + c 1 = - 2 + c + 2 c = 3 + 2
y = - 1 2 x + 3
9 The diagram shows five sketch graphs, labelled A , B , C , D and E .
Match each equation below to the letter of its graph.
y = -x 2 y = x 3 y = 5 x 3 marks Show answer a) Cb) Dc) E
See the working a )
U p s i d e - d o w n p a r a b o l a
C
b )
S - s h a p e d c u b i c t h r o u g h t h e o r i g i n
D
c )
R e c i p r o c a l c u r v e , t w o b r a n c h e s
E
10 Triangle T and the point P (1, 1) are drawn on the grid.
Enlarge triangle T by scale factor 3, centre P.
0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 x y T P 3 marks Show answer Triangle with vertices (4, 4), (10, 4), (4, 7)
See the working f r o m P ( 1 , 1 ) t o e a c h c o r n e r , t h e n × 3
( 2 , 2 ) : 1 r i g h t , 1 u p → 3 r i g h t , 3 u p → ( 4 , 4 )
( 4 , 2 ) : 3 r i g h t , 1 u p → 9 r i g h t , 3 u p → ( 1 0 , 4 )
( 2 , 3 ) : 1 r i g h t , 2 u p → 3 r i g h t , 6 u p → ( 4 , 7 )
c h e c k : r a y s f r o m P g o t h r o u g h e a c h p a i r
i m a g e a t ( 4 , 4 ) , ( 1 0 , 4 ) , ( 4 , 7 )
11 Triangle PQR is a right-angled triangle with the right angle at Q.
PQ = 7 cm and QR = 9 cm.
Work out the length of PR.
Give your answer correct to 1 decimal place.
P Q R 7 cm 9 cm Diagram NOT accurately drawn 2 marks Show answer 11.4 cm
See the working P R i s t h e h y p o t e n u s e
P R 2 = P Q 2 + Q R 2 P R 2 = 7 2 + 9 2 P R 2 = 4 9 + 8 1 P R 2 = 1 3 0 P R = √ 1 3 0 P R = 1 1 . 4 0 1 7 . . .
R o u n d t o 1 d p
1 1 . 4 c m
12 AB and CD are parallel lines. The line PQ crosses them at P and Q.
One of the angles at P is 62°, as shown.
Write down the value of x . Give a reason for your answer. Write down the value of z . Give a reason for your answer. Work out the value of y . Give a reason for your answer. 62° x y z A B C D P Q Diagram NOT accurately drawn 3 marks Show answer a) x = 62 (corresponding angles are equal)b) z = 62 (alternate angles are equal)c) y = 118 (angles on a straight line add up to 180°)
See the working a )
x = 6 2
C o r r e s p o n d i n g a n g l e s a r e e q u a l
b )
z = x = 6 2
A l t e r n a t e a n g l e s a r e e q u a l
c )
y = 1 8 0 - 6 2
y = 1 1 8
A n g l e s o n a s t r a i g h t l i n e a d d u p t o 1 8 0 °
13 Here is a table of values for y = x 3 − 4x + 1
Complete the table. On the grid, draw the graph of y = x 3 − 4x + 1 for values of x from −2 to 2 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 4 5 O x y
4 marks Show answer a) 4, -2, 1b) smooth curve through (-2, 1), (-1, 4), (0, 1), (1, -2), (2, 1)
See the working a )
x = - 1 : ( - 1 ) 3 - 4 × ( - 1 ) + 1 = - 1 + 4 + 1 = 4
x = 1 : 1 - 4 + 1 = - 2
x = 2 : 8 - 8 + 1 = 1
4 , - 2 , 1
b ) p l o t t h e f i v e p o i n t s
s m o o t h c u r v e : u p , o v e r t h e t o p , d o w n , t h e n u p
S - s h a p e d c u r v e d r a w n
14 Triangle ABC is a right-angled triangle with the right angle at B.
AB = 6 cm and BC = 8 cm.
Work out the length of AC.
A B C 6 cm 8 cm Diagram NOT accurately drawn 2 marks Show answer 10 cm
See the working A C i s t h e h y p o t e n u s e
A C 2 = A B 2 + B C 2 A C 2 = 6 2 + 8 2 A C 2 = 3 6 + 6 4 A C 2 = 1 0 0 A C = √ 1 0 0 A C = 1 0
1 0 c m
15 Triangle A is drawn on the grid.
Translate triangle A by the vector 5−3 . Label the image B.
−5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 O x y A 2 marks Show answer Triangle B with vertices (1, -2), (3, -2), (1, 1)
See the working t o p n u m b e r 5 : 5 r i g h t
b o t t o m n u m b e r - 3 : 3 d o w n
( - 4 , 1 ) , ( - 2 , 1 ) , ( - 4 , 4 )
m o v e e a c h c o r n e r 5 r i g h t , 3 d o w n
( 1 , - 2 ) , ( 3 , - 2 ) , ( 1 , 1 )
B h a s c o r n e r s ( 1 , - 2 ) , ( 3 , - 2 ) , ( 1 , 1 )
16 The distance-time graph shows Marek's car journey from home to a garden centre and back.
For how many minutes does Marek stay at the garden centre? Work out his speed, in km/h, on the way to the garden centre. Work out his speed, in km/h, on the way home. 09:00 09:30 10:00 10:30 11:00 11:30 12:00 0 10 20 30 40 50 Time Distance from home (km) 3 marks Show answer a) 30 minutesb) 40 km/hc) 30 km/h
See the working a )
H o r i z o n t a l l i n e = s t o p p e d
3 0 m i n u t e s
b )
4 0 k m f r o m 0 9 : 0 0 t o 1 0 : 0 0
S p e e d = d i s t a n c e t i m e S p e e d = 4 0 1 S p e e d = 4 0
4 0 k m / h
c )
4 0 k m f r o m 1 0 : 3 0 t o 1 1 : 5 0
T i m e = 8 0 m i n u t e s T i m e = 8 0 6 0 h o u r s T i m e = 4 3 h o u r s S p e e d = 4 0 ÷ 4 3 S p e e d = 4 0 × 3 4 S p e e d = 3 0
3 0 k m / h
17 In triangle ABC, AB = AC.
The point D lies on BC so that AD bisects angle BAC.
Prove that triangle ABD is congruent to triangle ACD.
A B C D Diagram NOT accurately drawn 3 marks Show answer Triangle ABD is congruent to triangle ACD by SAS
See the working T r i a n g l e s A B D a n d A C D
A B = A C ( g i v e n ) ∠ B A D = ∠ C A D ( A D b i s e c t s ∠ B A C ) A D = A D ( c o m m o n s i d e )
T h e a n g l e i s b e t w e e n t h e t w o s i d e s
A B D ≡ A C D ( S A S )
18 The diagram shows a right-angled triangle PQR.
Write down the name of the side that is the hypotenuse. Show that the three lengths fit Pythagoras' theorem. P Q R 5 cm 12 cm 13 cm Diagram NOT accurately drawn 2 marks Show answer a) PRb) 5² + 12² = 25 + 144 = 169 and 13² = 169
See the working a )
O p p o s i t e t h e r i g h t a n g l e
P R
b )
5 2 + 1 2 2 = 2 5 + 1 4 4
= 1 6 9
1 3 2 = 1 6 9
5 2 + 1 2 2 = 1 3 2
19 Triangle B is an enlargement of triangle A.
Write down the scale factor of the enlargement. Find the coordinates of the centre of the enlargement. 3 marks Show answer a) 2b) (1, 5)
See the working a ) c o m p a r e m a t c h i n g s i d e s
A : 1 s q u a r e , B : 2 s q u a r e s
2
b ) j o i n e a c h c o r n e r o f B t o t h e m a t c h i n g c o r n e r o f A
e x t e n d t h e l i n e s u n t i l t h e y m e e t
t h e y m e e t a t ( 1 , 5 )
c h e c k ( 3 , 4 ) : 2 r i g h t , 1 d o w n f r o m ( 1 , 5 )
× 2 : 4 r i g h t , 2 d o w n → ( 5 , 3 ) , a c o r n e r o f B
( 1 , 5 )
20 A rectangle is 9 cm long and 6 cm wide.
Work out the length of a diagonal of the rectangle.
Give your answer correct to 1 decimal place.
2 marks Show answer 10.8 cm
See the working D i a g o n a l i s t h e h y p o t e n u s e
d 2 = 9 2 + 6 2 d 2 = 8 1 + 3 6 d 2 = 1 1 7 d = √ 1 1 7 d = 1 0 . 8 1 6 6 . . .
R o u n d t o 1 d p
1 0 . 8 c m
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Every question here is original and written by Robinson Tuition, drawn from the Year 9 topic worksheets. It follows the Key Stage 3 programme of study.
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