dfm 1 Test 1 Question 1 Factorise the following: 6π₯ + 24 ………………………… Question 2 Factorise the following: π₯2 + 2π₯ ………………………… Question 3 Factorise the following quadratic: π₯2 + 10π₯ + 21 ………………………… Question 4 Factorise the following quadratic: π2 − 7π + 10 ………………………… Question 5 Factorise the following quadratic: π2 + 2π − 3 ………………………… www.drfrostmaths.com - Test 1 dfm 2 Question 6 Make π₯ the subject of 10π¦ = 5π§+10π€ 9π₯ π₯ = ………………………… Question 7 The ratio (π¦ + π₯): (π¦ − π₯) is equivalent to π: 1. Find an expression for π¦ in terms of π and π₯. π¦ = ………………………… Question 8 Make π₯ the subject of the formula where π₯ is positive: 2π₯ − 4π§π₯ = 5π’ π₯ = ………………………… Question 9 Make π₯ the subject of the formula where π₯ is positive: (π₯ − 3π¦) = 4π₯π§ + 3π’ π₯ = ………………………… Question 10 Expand and simplify −7 + 6(5π₯ − 8) ………………………… Question 11 Expand www.drfrostmaths.com - Test 1 dfm 3 9π¦(4π¦ + 7) ………………………… Question 12 Expand and simplify 2(3π₯ − 4) + 4(5π₯ − 2) ………………………… Question 13 Expand and simplify 2(5π₯ − 1) − 2(π₯ + 5) ………………………… Question 14 Expand and simplify −4 − (4π₯ + 3) ………………………… Question 15 Expand and simplify 2π₯(π₯ + 1) + π₯(2π₯ + 3) ………………………… Question 16 [AQA GCSE Nov 2013 1H Q15a] Expand and simplify (2π₯ + 1)(3π₯ − 4) www.drfrostmaths.com - Test 1 dfm 4 ………………………… (2 marks) Question 17 Expand and simplify (π₯ − 10)(π₯ + 10) ………………………… Question 18 Expand and simplify (π₯ + 3)(π₯ − 5) ………………………… Question 19 Expand and simplify: (π₯ − 4 )2 ………………………… Question 20 Expand and simplify (2π₯ + 3)(π₯ + 6) ………………………… Question 21 Given that Angle π΄ : Angle π΅ = 2: 7 determine the value of π¦ in the diagram. www.drfrostmaths.com - Test 1 dfm 5 Give your answer correct to 1 decimal place. π¦ = ………………………… cm Question 22 The distance between the top of a building and a point π on the ground is 13 m and the angle of elevation is 59° from the point π to the top of the building. Find the height of the building. Give your answer correct to 1 decimal place. β = ………………………… m Question 23 Determine the area of the triangle in the diagram. Give your answer correct to 1 decimal place. ………………………… cm 2 Question 24 Determine whether it is possible to construct the triangle with the lengths and angles given in the diagram below. www.drfrostmaths.com - Test 1 dfm 6 The triangle can be constructed. [ ] The triangle can not be constructed. [ ] Question 25 πππ is an isosceles triangle where ππ = ππ . Work out the length marked π§ on the diagram. Give your answer correct to 1 decimal place. π§ = ………………………… cm Question 26 The diagram below shows the isosceles triangle πππ . www.drfrostmaths.com - Test 1 dfm 7 Find the area of triangle πππ . Give your answer to 1 decimal place. ………………………… cm 2 Question 27 The diagram below shows the isosceles triangle πππ . The area of triangle πππ is 270 cm 2 . Find the perimeter of triangle πππ . ………………………… cm Question 28 From point π·, Anna walks 80 m due south to point πΈ. From πΈ, she then walks 140 m due east to point πΉ. Work out the length of π· πΉ. Round your answer to 1 decimal place. www.drfrostmaths.com - Test 1 dfm 8 ………………………… m Question 29 Tim makes a framework from metal rods. The framework is in the shape of the right-angled triangle πππ shown in the diagram The rods that Tim uses to make the framework have a weight of 1 kg per metre. Work out the total weight of the framework. Give your answer, in kg, correct to 1 decimal place. ………………………… kg www.drfrostmaths.com - Test 1 dfm Mark scheme Question 1 6(π₯ + 4) Question 2 π₯(π₯ + 2) Question 3 (π₯ + 7)(π₯ + 3) We find two numbers that add together to give 10 and multiply together to give 21. These are 7 and 3. Using these, we factorise as follows: π₯2 + 10π₯ + 21 = (π₯ + 7)(π₯ + 3) Question 4 (π − 2)(π − 5) We find two numbers that add together to give 7 and multiply together to give 10. These are 2 and 5. Using these, we factorise as follows, being very careful to keep track of minus signs: π2 − 7π + 10 = (π − 2)(π − 5) Question 5 (π − 1)(π + 3) We find two numbers that have difference 2 and multiply together to give 3. These are 1 and 3. Using these, we factorise as follows, being very careful to keep track of minus signs: π2 + 2π − 3 = (π − 1)(π + 3) Question 6 5π§+10π€ 90π¦ β Make π₯ the subject. www.drfrostmaths.com - Test 1 9 dfm 10 5π§+10π€ 10π¦ = 9π₯ × 9π₯ ↓ ↓ × 9π₯ 90π₯π¦ = 5π§ + 10π€ ÷ 90π¦ ↓ ↓ ÷ 90π¦ 5π§+10π€ π₯ = 90π¦ Question 7 π₯(π+1) π−1 Question 8 5π’ 2−4π§ You need to factorise by π₯ first, and then divide by the bracket. 2π₯ − 4π§π₯ = 5π’ π₯(2 − 4π§) = 5π’ ÷ (2−4π§) ↓ ↓ ÷ (2−4π§) 5π’ π₯ = 2−4π§ 5π’ ∴ π₯ = 2−4π§ Question 9 3π’+3π¦ 1−4π§ You need to expand the bracket, put π₯ on the left hand-side, factorise by π₯, and then divide by the bracket. (π₯ − 3π¦) = 4π₯π§ + 3π’ π₯ − 3π¦ = 4π₯π§ + 3π’ π₯ − 4π₯π§ = 3π’ + 3π¦ π₯(1 − 4π§) = 3π’ + 3π¦ ÷ (1−4π§) ↓ ↓ ÷ (1−4π§) 3π’+3π¦ π₯ = 1−4π§ ∴ π₯= 3π’+3π¦ 1−4π§ Question 10 30π₯ − 55 β Expand the bracket. www.drfrostmaths.com - Test 1 dfm 11 −7 + 6(5π₯ − 8) = −76 × 5π₯ + 6 × −8 = −7 + 30π₯ − 48 β Collect like terms and simplify. −7 + 30π₯ − 48 = 30π₯ − 7 − 48 = 30π₯ − 55 Question 11 36π¦2 + 63π¦ β Multiply all terms in the bracket by 9π¦, and then simplify. 9π¦(4π¦ + 7) = 9π¦ × 4π¦ + 9π¦ × 7 = 36π¦2 + 63π¦ Question 12 26π₯ − 16 β Expand each bracket. 2(3π₯ − 4) + 4(5π₯ − 2) = 6π₯ − 8 + 20π₯ − 8 β Collect like terms and simplify. 6π₯ − 8 + 20π₯ − 8 = 6π₯ + 20π₯ − 8 − 8 = 26π₯ − 16 Question 13 8π₯ − 12 β Expand each bracket. 2(5π₯ − 1) − 2(π₯ + 5) = 10π₯ − 2 − 2π₯ − 10 β Collect like terms and simplify. 10π₯ − 2 − 2π₯ − 10 = 10π₯ − 2π₯ − 2 − 10 = 8π₯ − 12 Question 14 −4π₯ − 7 www.drfrostmaths.com - Test 1 dfm 12 β Expand the bracket. = −4 − (4π₯ + 3) = −4 − 1 (4π₯ + 3) = −4 − 4π₯ − 3 β Collect like terms and simplify. = −4 − 4π₯ − 3 = −4π₯ − 4 − 3 = −4π₯ − 7 Question 15 4π₯2 + 5π₯ β Expand each bracket. 2π₯(π₯ + 1) + π₯(2π₯ + 3) = 2π₯2 + 2π₯ + 2π₯2 + 3π₯ β Collect like terms and simplify. 2π₯2 + 2π₯ + 2π₯2 + 3π₯ = 2π₯2 + 2π₯2 + 2π₯ + 3π₯ = 4π₯2 + 5π₯ Question 16 6π₯2 − 5π₯ − 4 Question 17 π₯2 − 100 β Multiply each term in the first bracket by each term in the second bracket. (π₯ − 10)(π₯ + 10) = π₯ × π₯ + π₯ × 10 − 10 × π₯ − 10 × 10 = π₯2 + 10π₯ − 10π₯ − 100 β Simplify. www.drfrostmaths.com - Test 1 dfm 13 = π₯2 − 100 Question 18 π₯2 − 2π₯ − 15 β Multiply each term in the first bracket by each term in the second bracket. (π₯ + 3)(π₯ − 5) = π₯×π₯−π₯×5+3×π₯−3×5 = π₯2 − 5π₯ + 3π₯ − 15 β Simplify. = π₯2 − 2π₯ − 15 Question 19 π₯2 − 8π₯ + 16 β Multiply each term in the first bracket by each term in the second bracket. (π₯ − 4 )2 = (π₯ + −4)(π₯ + −4) = π₯ × π₯ + π₯ × −4 + −4 × π₯ + −4 × −4 = π₯2 − 4π₯ − 4π₯ + 16 β Simplify. = π₯2 − 8π₯ + 16 Question 20 2π₯2 + 15π₯ + 18 β Multiply each term in the first bracket by each term in the second bracket. (2π₯ + 3)(π₯ + 6) = 2π₯ × π₯ + 2π₯ × 6 + 3 × π₯ + 3 × 6 = 2π₯2 + 12π₯ + 3π₯ + 18 β Simplify. = 2π₯2 + 15π₯ + 18 Question 21 π¦ =32.2cm www.drfrostmaths.com - Test 1 dfm 14 β Find the angle π΄. π΄ = 90 × π΄ = 20 2 2+7 β Label the sides. β Decide the trigonometric ratio to use. SOH CAH TOA Therefore we use tan β Write an equation and solve. tan(π) = tan(20) = π¦ = π π» 11 π¦ 11 tan(20) = 32.2 cm Question 22 β =11.1 m β Draw a diagram. β Draw a triangle and label the sides. www.drfrostmaths.com - Test 1 dfm 15 β Decide the trigonometric ratio to use. SOH CAH TOA Therefore we use sin β Write an equation and solve. sin(π) = π π» β 13 sin(59) = β = 13 sin(59) = 11.1 m Question 23 32.8cm 2 β Label the sides. β Decide the trigonometric ratio to use to find the unknown short edge length. SOH CAH TOA Therefore we use tan β Find the length of the unknown short edge. tanπ = π΄ = π π΄ π tanπ 9 tan51 = = 7.288 … β Find the area. www.drfrostmaths.com - Test 1 dfm 16 Area = = = height × base 2 7.288… × 9 2 32.8 cm2 Question 24 The triangle can not be constructed. β Find the values of π΄π΅2 , π΅πΆ2 and π΄πΆ2 . π΄π΅2 = 1.52 = 2.25 π΅πΆ2 = 10.52 = 110.25 π΄πΆ2 = 112 = 121 β Compare the value of π΄π΅2 +π΅πΆ2 with the value of π΄πΆ2 . (Could not display math) Therefore, by the converse of Pythagoras' theorem, the triangle with the given lengths and angle can not be constructed. Question 25 π§ =4cm The isosceles triangle can be split along its height to give a right-angle triangle on both sides. Using Pythagoras' theorem, π§2 = = = π§= π§= 3.72 + 1.52 13.69 + 2.25 15.94 √15.94 4 cm www.drfrostmaths.com - Test 1 dfm 17 Question 26 145.2cm 2 β Use the line of symmetry of the isosceles triangle to draw a right-angled triangle, with the right angle at the midpoint of ππ. β Use Pythagoras' Theorem to find the missing side in this triangle. ππ 2 = 18 2 − 9.5 2 = 233.75 π π = 15.2889 … ππ β Find the area of triangle πππ. π΄πππ = 1 2 × 19 × 15.2889 … = 145.2444 … ππ 2 β Round the answer to one decimal place. π΄πππ = 145.2 ππ 2 Question 27 75cm β Find the base of triangle πππ . 1 2 × ππ × 22.5 ×2 ↓ ππ × 22.5 ÷ 22.5 ↓ ππ = 270 ↓ ×2 = 540 ↓ ÷ 22.5 = 24 ππ www.drfrostmaths.com - Test 1 dfm 18 β Use the line of symmetry of the isosceles triangle to draw a right-angled triangle, with the right angle at the midpoint, π, of ππ. β Use Pythagoras' theorem to find the hypotenuse in this triangle. π₯ 2 = 12 2 + 22.5 2 = 650.25 π₯ = 25.5 ππ β Add together all the sides of triangle πππ . πππππππ‘ππ = 24 + 2 × 25.5 = 75 ππ Question 28 161.2m β Draw a diagram showing Anna's journey. β Use Pythagoras' theorem to find the length of π·πΉ. π·πΉ 2 = 80 2 + 140 2 = 26000 π·πΉ = 161.2452 … π β Round the answer to 1 decimal place. www.drfrostmaths.com - Test 1 dfm 19 π·πΉ = 161.2 π Question 29 14.2kg β Use Pythagoras' theorem to find the length ππ. π π 2 = 2.9 2 + 5.32 = 36.5 π π = 6.0415 … β Find the total length of all the rods in the framework. π‘ππ‘ππ πππππ‘β = ππ + ππ + ππ = 2.9 + 5.3 + 6.0415 … = 14.2415 … β Multiply the total length the weight per metre. π‘ππ‘ππ π€πππβπ‘ = 14.2415 … × 1 = 14.2415 … β Round this value to one decimal place. = 14.2 ππ www.drfrostmaths.com - Test 1
