CFE National 5 Resource

Pegasys Educational Publishing CFE National 5 Resource Unit Expressions and Formulae Homework Exercises Homework exercises covering all the Unit topics + Answers + Marking Schemes Pegasys 0

National 5 Homework Expressions and Formulae SURDS. Simplify (a) 47 5 (b) ( ) 6 (,). Express 5 0 45 as a surd in its simplest form. (). Express with a rational denominator 5 () 4. Express as a fraction with a rational denominator 5 4 () 5. 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 5 (4) A 0 marks Pegasys 0

National 5 Homework Expressions and Formulae INDICES 5 m. Simplify m (). Simplify the expression below, giving your answer with a positive power. 5 8 m m (). Express p ( p p ) in its simplest form. () 4. Simplify a a a () 5. Express in its simplest form 4 y y y () 6. Evaluate 4 6 () 7. Simplify, expressing your answer with positive indices. 8. Simplify ( x 4 6 y ) ( x y ) () 8 k ( k ) () 9. Express a ( a a ) in its simplest form. () 0. Express a ( a ) in its simplest form. () a 0 marks Pegasys 0

National 5 Homework Expressions and Formulae SCIENTIFIC NOTATION / SIGNIFICANT FIGURES. Write the numbers in each of these sentences in standard form. (a) The mass of the moon is about 79 50 000 000 000 000 000 000 kg () (b) The relative density of hydrogen is 0 000 089 9 (). 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) 4 0 80 5 ( 0 ) (4 50 0 6 ) (,, ) 4. The Earth is 9 million miles from the sun, which is one astronomical unit (AU). The distance from the sun to Jupiter is 5 AU. Calculate the distance in miles from the sun to Jupiter and give your answer in standard form. () 5. 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 57) (b) 0 ( 59) 58 ( 4) (d) 500 045³ (,,, ) 0 marks Pegasys 0

National 5 Homework Expressions and Formulae ALGEBRAIC EXPRESSIONS with BRACKETS Multiply out the brackets and simplify in each question.. (a) (x + 7) + x (b) 6y 5(y + ) 7(s ) (). (a) x (x + ) (b) m (8 m) y (w 5y) (). (a) 9(a + 5) + 7(a + 7) (b) 7(y 8) 5(y 6) (, ) 4. (a) (x + 4)(x + 7) (b) (y 9)(y ) (s + )(s ) (d) (a + 5)(a + 9) (e) (w 8)(w + ) (f) (4x ) (,,,,, ) 5. (a) (x + )(x + ) () (b) (x )(x² x ) () 0 marks Pegasys 0

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

National 5 Homework Expressions and Formulae COMPLETING the SQUARE. Write each of the following quadratic expressions in the form a ( x b) c : (a) x 6x (b) x 5x 4 x 8x (d) x 6x (,,, ). Show that the function f ( x) x 6x 7 can be written in the form f ( x) ( x p) q and write down the values of p and q. Hence state the minimum value of the function and the corresponding value of x. (4) marks Pegasys 0

National 5 Homework Expressions and Formulae ALGEBRAIC FRACTIONS. Simplify: (a) 9 57 (b) w w 5x 0 (d) x 6x (4). Simplify: (a) (x ) (x )(x ) (b) x 5x 6 ( x ) x x x 6 4x 4 (,, ). Simplify: (a) m m m 4 m (b) 5 4 5 4 x x (d) x x x x (,,, ) 4. Express each of the following in its simplest form. (a) 7 9k k (b) x 5 9x a a (d) x 4x y y (,,, ) 5 marks Pegasys 0

National 5 Homework Expressions and Formulae DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS. The line CD passes through the points (0, 5) and (6, 0) C y 0 x 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(, 5) are collinear. Find the value of k. (4) 5. 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 = x 5? () 8. The line which passes through (, ) and (6, 4) is perpendicular to the line through (4, b) and (, ). Find the value of b. () 0 marks Pegasys 0

National 5 Homework Expressions and Formulae 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 5 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, x o, to of the circle. () the nearest degree. () O 5 o O cm x o C D P Q 5. 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 0 cm

8 marks National 5 Homework Expressions and Formulae 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 (5) 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 5cm. (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. Pegasys 0

Calculate the volume of the pyramid correct to significant figures. (4) 6 marks ANSWERS National 5 Homework Expressions and Formulae SURDS. (a) (b). 5. 4. 5(4 6. 4 ) 5 6 5. (a) 4 (b) 8 National 5 Homework Expressions and Formulae INDICES. m.. p 5 m 4. 6a 5. y 7 6. 8 5 x 7. y 8. k² 9. a 0. a a 4 National 5 Homework Expressions and Formulae SCIENTIFIC NOTATION / SIGNIFICANT FIGURES. (a) 7 95 0 (b) 8 99 0 5. (a) 0 000 000 (b) 0 00. (a) 6 0 9 (b) 5 5 0 5 4 8 0 5 4. 4 86 0 8 5. 8 767 6. (a) 5 (b) 7 45 (d) 850 Pegasys 0

National 5 Homework Expressions and Formulae ALGEBRAIC EXPRESSIONS with BRACKETS. (a) 5x + (b) 6y 5 7s 7. (a) x 4 + x (b) 4m m² y w 0y³. (a) a + 94 (b) 8y 6 4. (a) x² + x + 8 (b) y² y + 7 s² + 0s 4 (d) a² + a + 45 (e) 6w² w 8 (f) 6x² 4x + 9 5. (a) x³ + x + x + (b) x³ 7x² + 4x + 4 National 5 Homework Expressions and Formulae FACTORISING an ALGEBRAIC EXPRESSION. (a) y(y + 5) (b) (x 7)(x + 7) 5(s )(s + ) (a) (x + 5)(x + 5) (b) (x )(x + ) (k+ 6)(k ). (a) (4a )(a + 4) (b) (7w 9)(w + ) (4x )(x ) 4. (a) 4(x + )(x + ) (b) (m )(m + ) ( 4x)(+ x) 5. (a) x(x² + 9)(x )(x + ) (b) (a + b)(a + b) National 5 Homework Expressions and Formulae. COMPLETING the SQUARE. (a) ( ) x (b) ( x 5) 55. p = 8; q = 57. Minimum value = 57 when x = 8 0 ( x 4) (d) 0 ( x ) Pegasys 0

National 5 Homework Expressions and Formulae ALGEBRAIC FRACTIONS. (a). (a) (x ) (b) w x (d) (b) ( x ) x x x. (a) 9m 0 (b) 7m 4 0 x x( x ) (d) x x ( x )( x ) 4. (a) (b) 5x a (d) x National 5 Homework Expressions and Formulae DETERMINING the GRADIENT of a STRAIGHT LINE given TWO POINTS. 5. 6 4. k = 6 5.. Proof [gradients 9 6. y = 7 7. ] 8. 6 National 5 Homework Expressions and Formulae 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. 5 cm 4. 80 o 5. 785cm² National 5 Homework Expressions and Formulae WORKING with the VOLUME of a SOLID SPHERE, CONE, PYRAMID Pegasys 0

. 40cm³. 445cm³. (a) 770cm³ (b) 60cm³ 4. 0cm³ National 5 Surds Homework Marking Scheme - EF. (a) 7 simplifying 47 answer (b) 6 6 multiplying out brackets simplifying to answer [4 marks]. 5 5 5 simplifying 45 and 0 5 answer [ marks]. knowing to multiply by 5 5 6 answer [ marks] 4. 5 4 4 4 (4 5(4 ) )(4 ) knowing to multiply by multiplying 4 4 5(4 ) simplifying [ marks] 5. (a) (b) x 8 4 using Pythagoras x 48 x 48 finding x x 6 x 4 simplifying [ marks] A = ½ bh = ½ 4 4 substituting = 8 answer [ marks] 6. ( 7) ( 5) using Pythagoras 8 0 48 + squaring surds 4 simplifying [4 marks] Total: 0 marks Pegasys 0

National 5 Indices Homework Marking Scheme - EF. answer [ mark].. m simplifying m writing with positive power [ marks] 5 0 p p multiplying brackets p 5 simplifying [ marks] 4. 6 number a³ multiplying letter 6a answer [ marks] 5. y 5 simplifying numerator y 7 answer [ marks] 6. 4 6 knowing how to find fractional power 8 answer [ marks] 7. 5 x y x y 5 simplifying writing with positive powers [ marks] 8. k 6 simplifying bracket k² answer [ marks] 9. 4 a a a 4 0 multiplying brackets simplifying [ marks] 0. 4 a a 4 0 multiplying brackets a [ marks] simplifying Total: 0 marks Pegasys 0

National 5 Scientific Notation/Significant Figures Homework Marking Scheme - EF. (a) 7 95 0 each part (b) 8 99 0 6 each part [4 marks]. (a) 0 000 000 answer (b) 0 00 answer [ marks]. (a) 60 000 000 answer 6 0 9 scientific notation [ marks] (b) 55 000 answer 5 5 0 5 scientific notation [ marks] 4 800 000 answer 4 8 0 5 scientific notation [ marks] 4. 9 000 000 5 correct calculation 4 86 0 8 answer in scientific notation [ marks] 5. 0 8 65 correct calculation 8 767 answer correctly rounded [ marks] 6. (a) 5 (b) 7 45 (d) 850 each answer [ marks] Total: 0 marks Pegasys 0

National 5 Algebraic Expressions with Brackets Homework Marking Scheme - EF. (a) 5x + (b) 6y 5 7s 7 [ marks]. (a) x 4 + x (b) 4m m² y²w 0y³ [ marks]. (a) a + 94 (b) 8y 6 [ marks] 4. (a) x + x + 8 (b) y y + 7 s² + 0s 4 (d) a + a + 45 (e) 6w² w 8 (f) 6x² 4x + 9 [6 marks] 5. (a) x³ + x x² + (b) x x² x 4x² + 6x + 4 x³ 7x² + 4x + 4 [5 marks] Total: 0 marks Pegasys 0

National 5 Factorising and Algebraic Expression Homework Marking Scheme - EF. (a) y (y + 5) (b) (x 7) (x + 7) each bracket 5(s ) (s + ) each bracket [5 marks]. (a) (x + 5)(x + 5) each bracket (b) (x ) (x + ) each bracket (x + 6) (x ) each bracket [6 marks]. (a) (4a )(a + 4) each bracket (b) (7w 9)(w + ) each bracket (4x )(x ) each bracket [6 marks] 4. (a) 4(x + )(x + ) common factor (b) (m )(m + ) common factor brackets ( 4x)(+ x) common factor brackets [6 marks] 5. (a) x(x² + 9)( x² 9) common factor brackets x(x² + 9)(x )(x + ) difference of squares (b) (a + b)(a + b) each bracket [5 marks] Correct brackets with signs round the wrong way gains mark. Total: 8 marks Pegasys 0

National 5 Completing the Square Homework Marking Scheme - EF. (a) ( x )... (b) ( x 5) 55 0.. ( x 4) (d) 0. ( x ) [8 marks]. p = 8; q = 57. Minimum value = 57 when x = 8 [4 marks] Total: marks Pegasys 0

National 5 Algebraic Fractions Homework Marking Scheme - EF. (a) (b) w x x (d). (a) (x ) (b) ( x )( x ) x ( x ) ( x )( x ) x ( x )( x ) x each answer [4 marks] answer factorising numerator cancelling to answer factorising numerator factorising denominator cancelling to answer [6 marks]. (a) (b) (d) 9m 0 answer 5( m 4) ( m ) 0 correct denominator 7 4 m 0 correct numerator 4( x ) x x( x ) correct denominator x x( x ) correct numerator ( x )( x ) ( x )( x ) ( x )( x ) correct denominator x x x 4 ( x )( x ) x x ( x )( x ) correct numerator simplified numerator [8 marks} 4. (a) answer (b) 6x 45x multiplying 5x simplifying a a multiplying a simplifying Pegasys 0

y (d) inversion 4x simplifying [7 marks] x Total: 5 marks Pegasys 0

National 5 Determining the Gradient of a Straight Line Homework Marking Scheme - EF. 5 m answer [ mark] 6. 4 m working 8 answer [ marks]. 4 m AB 4 0 gradient 4 m AB 6 4 gradient m AB = m BC with B a common point so A, B and C are collinear. conclusion [ marks] 4. m ST 0 k 0 k gradient 5 m TU 0 4 gradient 0 k 4 equating gradients k = 6 answer [4 marks] 5. 7 9 m working 8 answer [ marks] 6. m 4 7 4 y y m 5 y 4 5 gradient gradient equating gradients y 7 answer [4 marks] Pegasys 0

7. m perp answer [ mark] 4 8. m ; m perp both gradients 6 b 6 equating to perpendicular gradient b 6 answer [ marks] Total: 0 marks Pegasys 0

National 5 Arcs and Sectors Homework Marking Scheme - EF. (a) 0 46 = 8 cm 60 correct fraction substitution answer (b) 0 60 48 = 7 6cm² correct fraction substitution answer [6 marks]. (a) 40 = 9 0cm 60 correct fraction substitution answer (b) 60 4 5 = 7 4cm² correct fraction substitution answer [6 marks] 60. 8 8 = 5 cm working 5 answer [ marks] x 4. 444 00 relevant working 60 x = 80 o answer [ marks] 5. 90 60 4 5 = 96cm² substitution answer 90 45 = 77cm² 60 substitution and answer Shaded area = 96 77 = 785cm² answer [4 marks] Total: 0 marks Pegasys 0

National 5 Volumes of Solids Homework Marking Scheme - EF. V 8 7 correct substitution 40cm³ answer correctly rounded units [ marks]. V 5 7 correct height 8cm³ correct answer V 5 correct substitution 6cm³ correct answer Total: 445 cm³ answer correctly rounded [5 marks]. (a) V = 5³ = 80 cm³ [accept 75] dimension of 5 answer (b) 4 V 7 5 substitution 770cm³ answer Wasted space = 60cm³ answer correctly rounded [5 marks] 4. V A h correct formula V 6 69 substitution 0cm³ answer correctly rounded [ marks] Total: 6 marks Pegasys 0