5. In the rectangle ABCD, the diagonal AC is 8cm and the height BC is 4cm.

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1 SURDS. Simplify (a) 47 (b) ( ) 6 (,). Epress 0 4 as a surd in its simplest form.. Epress with a rational denominator 4. Epress as a fraction with a rational denominator 4 (). In the rectangle ABCD, the diagonal AC is 8cm and the height BC is 4cm. D C 8 cm 4 cm A B (a) Calculate the length of the rectangle, giving your answer as a surd in its simplest form. () (b) Calculate the area of triangle ABC. 6. Find the length of the diagonal, AB, of this rectangle leaving your answer as a surd in its simplest form. 7 B (4) A 0 marks Pegasys 0

2 INDICES m. Simplify m (). Simplify the epression below, giving your answer with a positive power. 8 m m. Epress p ( p p ) in its simplest form. 4. Simplify a a a (). Epress in its simplest form 4 y y y 6. Evaluate Simplify, epressing your answer with positive indices. 8. Simplify ( y 4 6 ) ( y ) 8 k ( k ) 9. Epress a ( a a ) in its simplest form. 0. Epress a ( a ) in its simplest form. a 0 marks Pegasys 0

3 SCIENTIFIC NOTATION / SIGNIFICANT FIGURES. Write the numbers in each of these sentences in standard form. (a) The mass of the moon is about kg (b) The relative density of hydrogen is Write the numbers in each of these sentences in full. (a) The number of seconds in a decade is about 0 8 () (b) The size of a molecule of water is roughly 0 (). Calculate each of the following, giving your answers in standard form. (a) (4 0 0 ) ( 0 ) (b) ( 0 ) ( ) (,, ) 4. The Earth is 9 million miles from the sun, which is one astronomical unit (AU). The distance from the sun to Jupiter is AU. Calculate the distance in miles from the sun to Jupiter and give your answer in standard form.. A company s profit for the year was 0 8. Calculate the profit made per day, giving your answer to the nearest. 6. Use your calculator to find the following. Answer correct to significant figures (a) 84 (96 7) (b) 0 ( 9) 8 ( 4) (d) 00 04³ (,,, ) 0 marks Pegasys 0

4 ALGEBRAIC EXPRESSIONS with BRACKETS Multiply out the brackets and simplify in each question.. (a) ( + 7) + (b) 6y (y + ) 7(s ) (). (a) ( + ) (b) m (8 m) y (w y) (). (a) 9(a + ) + 7(a + 7) (b) 7(y 8) (y 6) (, ) 4. (a) ( + 4)( + 7) (b) (y 9)(y ) (s + )(s ) (d) (a + )(a + 9) (e) (w 8)(w + ) (f) (4 ) (,,,,, ). (a) ( + )( + ) (b) ( )(² ) () 0 marks Pegasys 0

5 FACTORISING an ALGEBRAIC EXPRESSION Factorise each epression in the following:. (a) y + y (b) 4 49 s 0 (). (a) (b) 0 4 k + k 6 (6). (a) a + 7a (b) 7w w (6) 4. (a) (b) m 6m 9 6 (6). (a) 8 (b) a +ab + b () marks Pegasys 0

6 COMPLETING the SQUARE. Write each of the following quadratic epressions in the form a ( b) c : (a) 6 (b) 4 8 (d) 6 (,,, ). Show that the function f ( ) 6 7 can be written in the form f ( ) ( p) q and write down the values of p and q. Hence state the minimum value of the function and the corresponding value of. (4) marks Pegasys 0

7 ALGEBRAIC FRACTIONS. Simplify: (a) 9 7 (b) w w 0 (d) 6 (4). Simplify: (a) ( ) ( )( ) (b) 6 ( ) (,, ). Simplify: (a) m m m 4 m (b) 4 4 (d) (,,, ) 4. Epress each of the following in its simplest form. (a) 7 9k k (b) 9 a a (d) 4 y y (,,, ) marks Pegasys 0

8 DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS. The line CD passes through the points (0, ) and (6, 0) C y 0 D Calculate the gradient of CD. (). A line passes through the points A(, 4) and B(8, ). Find the gradient of the line AB.. Prove that the points A(0, ), B(4, 4) and C(6, ) all lie on the same straight line. () 4. The points S(k, ), T(0, ) and U(, ) are collinear. Find the value of k. (4). Calculate the gradient of a line which is parallel to the line passing through F(, 7) and G(8, ). 6. The line which passes through (4, ) and (7, ) is parallel to the line through (, y) and (, ). Find the value of y. (4) 7. What is the gradient of the line perpendicular to the line with equation y =? () 8. The line which passes through (, ) and (6, 4) is perpendicular to the line through Pegasys 0

9 (4, b) and (, ). Find the value of b. () 0 marks WORKING with the LENGTH of ARC and AREA of a SECTOR of a CIRCLE Give your answers correct to significant figures where necessary.. (a) Find the length of the minor arc AB in this circle. () O (b) Calculate the area if the minor sector AOB. 8 cm () 0 o A B. (a) Find the length of the major arc PQ in this circle. () O 8 o cm (b) Calculate the area of the major sector POQ. () P Q. The length of arc CD is 88 cm. 4. The area of sector OPQ is 00 cm. Calculate the circumference Calculate the size of angle, o, to of the circle. the nearest degree. O o O cm o C D P Q. Ornamental paving slabs are in the shape of part of a sector of a circle. Calculate the area of the slab shown. (4) Pegasys 0 cm 0 cm

10 8 marks WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID Give your answers correct to significant figures where necessary.. A cone has a base diameter of 6cm and a height of 7cm. 7cm Calculate the volume of the cone, giving your answer correct to sig figs. [Volume of cone = r h ] () 6cm. A lead sinker is in the shape of a cone with a hemispherical base. The total height of the sinker is cm and the diameter of the base is 0cm () Calculate the volume of lead required to make the sinker. [Volume of sphere = 4 r ] cm 0cm. (a) Calculate the volume of the largest sphere which will fit inside a cube of side cm. (b) Calculate the volume of wasted space between the two. [Answer to nearest cm ] (, ) 4. A pyramid has a square base of side 6cm and a vertical height of 9cm. Calculate the volume of the pyramid correct to significant figures. (4) 6 marks Pegasys 0

11 ANSWERS SURDS. (a) (b).. 4. ( ) 6. (a) 4 (b) 8 INDICES. m.. p m 4. 6a. y y 8. k² 9. a 0. a a 4 SCIENTIFIC NOTATION / SIGNIFICANT FIGURES. (a) (b) (a) (b) (a) (b) (a) (b) 7 4 (d) 80 Pegasys 0

12 ALGEBRAIC EXPRESSIONS with BRACKETS. (a) + (b) 6y 7s 7. (a) 4 + (b) 4m m² y w 0y³. (a) a + 94 (b) 8y 6 4. (a) ² (b) y² y + 7 s² + 0s 4 (d) a² + a + 4 (e) 6w² w 8 (f) 6² (a) ³ (b) ³ 7² FACTORISING an ALGEBRAIC EXPRESSION. (a) y(y + ) (b) ( 7)( + 7) (s )(s + ) (a) ( + )( + ) (b) ( )( + ) (k+ 6)(k ). (a) (4a )(a + 4) (b) (7w 9)(w + ) (4 )( ) 4. (a) 4( + )( + ) (b) (m )(m + ) ( 4)(+ ). (a) (² + 9)( )( + ) (b) (a + b)(a + b). COMPLETING the SQUARE. (a) ( ) (b) ( ). p = 8; q = 7. Minimum value = 7 when = 8 0 ( 4) (d) 0 ( ) Pegasys 0

13 ALGEBRAIC FRACTIONS. (a). (a) ( ) (b) w (d) (b) ( ). (a) 9m 0 (b) 7m 4 0 ( ) (d) ( )( ) 4. (a) (b) a (d) DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS k = 6.. Proof [gradients 9 6. y = 7 7. ] 8. 6 WORKING with the LENGTH of ARC and AREA of a SECTOR of a CIRCLE. (a) 8 cm (b) 7 6cm². (a) 9 0cm (b) 7 4cm². cm o. 78cm² WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID. 40cm³. 44cm³. (a) 770cm³ (b) 60cm³ 4. 0cm³ Pegasys 0