GCSE Higher Tier Practice Questions

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1 GCSE Higher Tier Practice Questions GCSE Higher Tier Practice Questions Contents: Topic Page Solving Linear, Quadratic and Simultaneous Equations Algebra: Factorising Practice and Fractions Changing Subjects of Formulae & Solving Inequalities 4 4 Quadratic Functions: Completing the Square and Sketching 5 5 Forming and Solving Quadratic Equations 5 6 Proportionalit or Variation 6 7 Number: Estimating and Factorising 6 8 Constructions, Loci and Bearings 7 9 Vectors 8 0 Probabilit 9 Calculations in Right-Angled Triangles 0 Calculations Needing Sine and Cosine Rules 0A Upper and Lower Bounds and Standard Form Answers (C) E J Tricke and M E B George 004

2 GCSE Higher Tier Practice Questions Solving Linear, Quadratic and Simultaneous Equations *Alwas tr to CHECK our answers* Eercise. Solve: = 9. 9 =. 7 = 4 5 = = ( ) = = = 5 0 Eercise. Solve the following sets of simultaneous equations: a + b = 8 7 = 6.. 5a b = 4 = p + 5q = 0 p q = 5 a 5b = 8 5a b = 4 = + = = = 97 = = 5 Eercise. Solve: = = = = = = = = 0 Eercise.4 Solve, to sig fig, using the b ± b ac formula 4 = a. 9 + = = = = 0 5. = = = 8 Eercise.5 Solve b factorising or b using the formula = 8. + = = 5 = = Eercise.6 Simultaneous equations - one linear and one quadratic: *Epect to reach a quadratic equation in one letter which will factorise (but ou ma use the quadratic formula instead of factorising if ou prefer it) *Check our answers in the original equations + = 4 = + = = 6 + = 4 = 5 = + = = 4 = + = = = = 8 = 4 9 = 44 + = 4 = (C) E J Tricke and M E B George 004

3 GCSE Higher Tier Practice Questions Algebra: Factorising Practice and Fractions Eercise. Factorise full: a b + 5a b 6. 9abc b c n 4n a a 4 0. n 8n b 4b 6. 5b 0b 05 Eercise. Factorise full:. b + c + b + c. h h b + b. 6am bm an + bn 6a + b + a + b 5. 8pr qr 4 ps + qs m p Without a calculator, evaluate: Eercise. Simplif: m n.. 5mn Eercise.4 Epress as a single fraction, full simplified: p q pq a + 5b 5. + a 4a 8a b a a 4b Eercise.5 Solve and check our solutions (all quadratics reached correctl do factorise) m + m. = 4. = 4 m + 5 m. = a a = a a + (C) E J Tricke and M E B George 004

4 GCSE Higher Tier Practice Questions 4 5. a 4 7 = a (C) E J Tricke and M E B George 004

5 GCSE Higher Tier Practice Questions 5 Changing Subjects of Formulae & Solving Inequalities Eercise. Solve:. In each case, make the subject of the formula: (a) a + b = c (b) q = r p n = m (d) l = t m a (e) a ( + b) = c( + d) (f) = s n (g) a b = n c (h) a + b + c = 0. In each case, make the subject of the formula: a + b n a (a) = (b) = c + d b a + = b (d) a = m n (e) p (g) (i) = 4q (f) ( ) a = l + n T = π (h) a + b = c g = + (j) 4 = m z. A = π r + π rh Find h, if A = 704 and r = 7 (give our answer to sig. fig.) 4 V r (give our answer to sig. fig.) = π Find r, if V = A = a + b + c Find a, if A = 9.0, b = and c = (give answer to sig. fig.) 6. T = π l g T = 6.95 and l =. Find g to sig. fig. + b 7. = + d and d = 5 c a Find, if a = 7, b =, c = 6 r 8. In each case, make the subject of or R the formula: (a) V = π r h (b) = + s r t 4 V = π r (d) A = 4π r NR (e) r = a + 4b (f) R = A + 00 Eercise. Solve the following and show our answer clearl on a number line:. 0 + < > < < 7 8. < < 5 Eercise. List all the INTEGERS that satisf the following inequalities:. 5 + <. <. < 4 9 < < 7 6. < 7 n 5 (Each time, check our answers carefull) Eercise.4 = Use this graph to solve the inequalit 6 < 4, giving our estimated solutions to d.p (C) E J Tricke and M E B George 004

6 GCSE Higher Tier Practice Questions 6 4 Quadratic Functions: Completing the Square and Sketching. f ( ) 6 + (a) Re-write f() in the form f ( ) ( a) + b (b) Hence state the minimum value of f() and the value of for which this minimum occurs. Sketch the curve =f(). f ( ) (a) Re-write f() in the form f ( ) ( + c) d (b) Hence state the minimum value of f() and the value of for which this minimum occurs. Sketch the curve =f(). f ( ) + + (a) Re-write f() in the form f ( ) ( + e) + f (b) Hence state the minimum value of f() and the value of for which this minimum occurs. Sketch the curve =f() f ( ) 8 f ( ) A + B (b) Hence state the maimum value of f() and the value of for which this maimum occurs. Sketch the graph of =f() (a) Re-write f() in the form ( ) 5. f ( ) 0 f ( ) C D (b) Hence state the maimum value of f() and the value of for which this maimum occurs. Sketch the graph of =f() (a)re-write f() in the form ( ) 6. f ( ) 8 4 (a) Re-write f() in the form ( ) f ( ) a + b (b) Hence state the maimum value of f() and the value of for which this maimum occurs. Sketch the graph of =f() 7. f ( ) 6 (a) Sketch =f() (b) B considering f() in the form f ( ) ( a) b, or otherwise, state the horizontal and vertical transformations that would map = 6 onto the graph of = 5 Forming and Solving Quadratic Equations. Show that the information given about the lengths in this right-angled triangle satisfies the equation: 4 + = 0. Hence find.(two solutions). All measurements are in centimetres Both of these quadrilaterals are rectangles and the inner one is cut out of the outer one. The REMAINING white area is 6cm. Show that this information can give the equation + 8 = 65 and solve for.. All measurements in cm The shaded right-angled triangle is removed from the rectangle. The remaining area is 90 cm. Show clearl that this gives the equation + = 90 and solve it. Not drawn to scale. The curve is = 8 (a) State the coordinates of D. C (0,4) (b) B completing the square A B in the form = ( a) b or D M otherwise, give the coordinates of M, the minimum point of the graph. B factorising = 8, find the coordinates of A and B. (d) Find the equation of the line CB and the coordinates of C. 5. f ( ) 6 + (a)rearrange f() in the form f ( ) ( c) d (b)hence solve, in surd form, 6 + = 0 Now solve 6 + = 0 using the formula b ± b 4ac = leaving our answers in a full simplified surd form. Show that these two answers are identical to our answers from (b). (C) E J Tricke and M E B George 004

7 6 Proportionalit or Variation Eercise 6.. p varies directl as q. Fill in the following table: p q The area of a circle is proportional to the square of the radius. What happens if: (a) the radius is doubled? (b) the radius is trebled? the radius is made 0 times as big?. m varies as the square root of a. When a = 4, m =.6 (a) Find the equation linking m and a. (b) Find: m when a = 6 a when m = 4 m when a = 9 No calculators in this question! P is propotional to h. When h = 5, P = 45 (a) Find P when h = 0 and when h = 00 (b) Find h when P = 80 and when P = It is known that is proportional to. Fill in the following table: also find: (a) when = 50 (b) when = If a stone is dropped from the top of a building, the time t it takes to reach the ground is proportional to the square root of the height h of the building. 7 Number: Estimating and Factorising (NO calculators to be used!). Given that = Find: (a) 8 0 (b) V = π r h Estimate V when r = 9.8 and h = 0. (give our answer to sig. fig.) A stone was timed and took 5 seconds to drop 00 feet. If the stone onl dropped 50 feet, how long would it take to reach the ground? Eercise 6.. t is inversel proportional to v. Complete the following table: v t It is known that is inversel proportional to. Fill in the following table: No calculators in this question! H is inversel proportional to the square root of T. When T = 6, H =.5 (a) Find: H when T = 00 and when T = 0.0 (b) Find: T when H = and when H = 0. The resistance R in a fied length of wire varies inversel as the square of the diameter d of the wire. If the diameter is 5mm, the resistance is 0.04 ohms. Find the resistance if the diameter is 4mm. 5. (Give our answers to sig. fig.) The air pressure available from a biccle pump is inversel proportional to the square of the diameter of the pump. If a pressure of 6 units is available from a diameter of 5mm, (a) Find the pressure available from a pump with a diameter of mm. (b) Find the diameter of a pump whose pressure is 0 units. l. T = π Estimate T when g = 9.8 g and l = 55. (tr sf for this estimate) Epress the following numbers as products of their prime factors: {e.g. = } (a) 9 (b) Hence find their highest common factor and their lowest common multiple in factor form. 5. Find an estimate to sig. fig. for:

8 8 Constructions, Loci and Bearings In questions, and 6 use onl a pencil, a ruler and a pair of compasses on PLAIN paper. Do NOT erase an construction lines or arcs.. Construct the sketched triangle accuratel: AB = 8cm angle B = 90 angle A = 0 Measure AC to the nearest mm. 5.. (a) Construct rectangle ABCD. (b) Construct the locus of all points equidistant from AD and DC. Construct the locus of all points 6cm from A, inside ABCD. (d) Shade the region inside ABCD where the points are nearer DC than AD but less than 6cm from A. (e) Calculate the area of this shaded region, to sig. fig.. {not drawn to scale CALCULATE our answers!} The bearing of A from B is. The bearing of C from B is. BA = 8cm CA = 4cm. Find the bearing of C from A and the distance BC (each to sig. fig.) 6. A B Draw a line AB 8cm long and construct accuratel the locus of all points 4cm from an point on AB. Find the area within this locus to the nearest cm. 7. B A ship sails from port P on a bearing of 06 to Q. The distance PQ is km. At Q the ship changes direction and sails to R on a bearing of 75. QR = 9km. Find the direct distance PR and the bearing of R from P, giving our answers to sig. fig. A C not to scale B s bearing from A is 076. B s bearing from C is 40. AB = BC. Find the bearing of A from C D (a) Evaluate the area of the quadrilateral ABCD. (b) Find the lengths of AB, BC, CD and AD to sig. fig. Calculate the angle A to sig. fig.

9 9 Vectors Eercise 9.. Eercise 9.. O is the centre of a regular heagon, ABCDEF. AB = BC = n Find, in terms of or n or both, the following vectors: AC, AD, CD, FA, FB, AE (a) Write a, b, c, p, q, r, s, and t each as column vectors. (b) Give, in simplified surd form, a b c q t As ou can see, p = a Write q, r, s, and t each as a multiple of a or b or c.. {Tip: When ou have a vector question in an eam or test, NEVER attempt to solve it without a DIAGRAM even if there is no grid drawn for ou!} O is the origin and S is the point (0,). 6 SP =. T is the midpoint of SP. (a) Find, as a column vector, TO (b) Find OT R is on the line = 0. If OR = OT, find the possible coordinates of R.. In each set of vectors below, find the one vector that is not parallel to all the other vectors in its row: 4 4 (a),,, 5 0 5, (b),,,, Given that in each case p and q ARE parallel, find k: (a) p = 6a + 8b q = ka + 6b (b) p = a + kb q = 9a b. This diagram shows a rough sketch of quadrilaterals, ABEF and BCDE. (a) If a = b, what can ou conclude about A, B and C? (b) If a = b = e = d, what tpe of figure is ABCDEF? If g = c, what tpe of figure is EBCD? (d) If d + c = e + g, name the four points that are vertices of a parallelogram.. ABCD is a parallelogram. AB = s AD = t (a) Find, in terms of s or t or both: (i) BD (ii) BP (iii) BQ (iv) AQ (v) QC

10 (b) Show clearl, giving reasons, that APCQ is a parallelogram. 0 Probabilit If ou are asked to find an probabilit, remember that the answer: CANNOT EXCEED ONE i. e. 0 Pr obabilit CANNOT BE NEGATIVE. Three coins are tossed at the same time. List all the possible outcomes. Find the probabilit of obtaining: (a) Heads (b) Heads and one Tail no Heads (d) at least one Head.. Cards with the numbers to 0 are placed in a hat. Find the probabilit of selecting: (a) an even number (b) a number less than 4 a square number (d) a prime number less than 40 (e) a prime number greater than 90.. Two dice, one red and one blue, are thrown simultaneousl. Show all the possible outcomes on a sample space (probabilit space). Find the probabilit of obtaining: (a) a total of 0 (b) a total of a total of less than 6 (d) the same number on both dice (e) a total more than 9. (f) Which is the most likel total? In each of the following, state whether the events A and B are mutuall eclusive: (a) Two coins are tossed Event A : Heads (i) Event B : Head, Tail Event A : Heads (ii) Event B : at leasthead (b) A card is drawn from a pack Event A : a spade Event B : an ace A counter is drawn from a bo of red, blue and white counters Event A : a red counter Event B : not a blue counter 5. In a European car factor, 85% of the cars manufactured are left-hand drive, the rest (obviousl) are right-hand drive. The probabilit that a car needs its steering adjusted before leaving the factor is 0.. The percentage of cars that are left-hand drive and do NOT need adjustment to their steering is 75. Show that steering adjustment and right / left hand drive are NOT independent. 6. What is the probabilit of selecting a King from a full pack of cards? What is the probabilit of selecting a red card from a full pack? Show that the event selecting a King and the event selecting a red card are independent. 7. A bag contains 6 red marbles and 4 blue marbles. A marble is drawn at random and not replaced. Two further draws are made, again without replacement. Find the probabilit of drawing: (a) red marbles (b) blue marbles no red marbles (d) at least one red marble. 8. Sall goes to school in the mornings. The probabilit that her alarm clock works is 0.9. If her alarm clock has worked the probabilit that she is on time for registration is If her alarm clock has failed the probabilit that she is late for registration is Draw a tree diagram to help ou work out the probabilit that: (a) Her alarm clock worked and she is on time (b) She is on time, regardless of her clock. 9. A bo contains milk chocolates and plain chocolates. Two are selected one after the other, WITHOUT replacement. Find, in terms of and, the probabilit of choosing: (a) a milk chocolate on the first choice (b) milk chocolates on both the first and second choices one of each sort of chocolate (d) two plain chocolates.

11 Calculations in Right-Angled Triangles Eercise. Find each angle to sig.fig... Eercise. Find each side to sig.fig Calculations Needing Sine and Cosine Rules a b c b + c a Remember: = = and a = b + c bc CosA or CosA = SinA SinB SinC bc In each case, find the side lettered, or the angle marked, to sig. fig. (Triangles NOT drawn to scale)... {two possible answers!}

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14 ANSWERS Solving Equations Eercise. = 5 = 4 = 4 = 4 5 = 6 = 4 7 = 8 = 0 Eercise. a = 4 = b = = p = 5 = 0 4 q = 0 = 9 a = 4 = 5 6 b = = = 9 7 = 4 Eercise. = or = 6 or = or 8 4 = 4 or 6 5 = or 4 6 = or 7 = 6 (repeated) 8 = 4 or 9 Eercise.4 = 7.54 or. 46 = or. 57 = 8.9 or =.07 or = 8.7 or = 0.9 or = 7.07 or Eercise.5 = or 0. 8 = 6 or = 5 or 4 = or 5 5 = or Eercise.6 = 5 = or = = 5 = 5 = or = = 5 = 5 = or = = 5 = 5 = 4 or = = 5 = 5 = 5 or = = 5 = 5 = 6 or = = 5 = 5 = 7 or = = 5 = 5 = 8 or = = 5 = 5 = 9 or = = 5 Algebra Practice Eercise. ( + 5) ( + ) (5 4 ( + ) ) 5 a b(a + 5b) 6 bc(a b) 7 ( + 8)( ) 8 ( n )( n ) 9 ( a + 4)( a 6) 0 ( n )( n ) ( + )( + 5) ( + )( + 5) ( )( + ) 4 5 ( )( + ) 5 ( b + )( b 7) 6 5( b + )( b 7) Eercise. ( + )( b + c) ( )( h b) ( m n)(a b) 4 ( a + b)( + ) 5 ( r s)(4 p q) 6 ( + )( + ) 7 ( + )( + ) 8 ( + )( ) 9 ( + 5)( ) 0 ( + 4)( 7) ( 6 + 5)( ) ( 6 + )( + ) ( + )( ) 4 ( + )( ) 5 ( )( ) 6 ( m 7)(m + 7) 7 ( 6 p )(6 + p) Eercise. 4m 5n Eercise.4 9q 8p p q b + 6a 5a b 6a 6a + 5 8a 0b a 4 4ab ( )( + ) ( 5) + +

15 GCSE Higher Tier Practice Questions ( + )( + ) 7 7 ( + )( ) + 8 ( )( + ) ( ) 0 ( + )( ) Eercise.5 = or 4 m = 4 a = or 4 a = 4 or + 5 a = 5 or Changing Subjects & Inequalities Eercise. c b (a) = (b) = p( r + q) a m = (d) = m( l t) n cd ab (e) = (f) = ns + a a c n + b c b (g) = (h) = a + c a Remember: cd ab ab cd a c c a d b (a) = (b) a c = ± b a (d) (e) (f) = ± = ± p ± = m n a l a q + n n = b + a T (g) g = π c b (h) = a z (i) = z + (j) m = 6 h = 9.0 (sf ) 4 r =.6 (sf ) 5 a = 0 (sf ) 6 g = 9.8 (sf ) 7 = 9 8 V st (a) r = ± (b) r = π h t s V r = (d) 4 π (e) (f) a + b r = ± 00A R = 00 N Eercise. < < 6 5 < < < < 4 r = ± A 4π Eercise. -, -, 0,,, -, 0,,,, 4, 5, 6, 7, 8 4, 5, 6, 7, 8, 9, 0,, 4 -, -4, +, , -4, -5, -6, +, +4, +5, +6 6, Eercise.4 Approimatel:.8 <.4 and +.4 < Quadratic Functions (a) f ( ) ( ) + 4 (b) min is 4 when = (0,) =f() (,4) (b) min is - when = - =f() (0, 7) (a) f ( ) ( + ) (, ) + (b) min is when = - =f() (0,) (,) (a) f ( ) ( + ) f ( ) (b) ma is 7 when = -4 4 (a) ( ) ( 4,7) (0,)

16 GCSE Higher Tier Practice Questions 6 f ( ) 5 5 (b) ma is 5 when = 5 5 (a) ( ) f ( ) 4 + (b) ma is -4 when = - 6 (a) ( ) (, 4) (, 9) (5,5) (0, 8) (6,0) ( ) 9, so for this curve to map onto =, it needs to move or to 9 the left and 9 up. 5 Quadratic Equations B Pthagoras: ( 4) + + ) = (4 + ) ( ) = = 0 ( )( ) = 0 = or = ( 4 + )( + ) ( )( ) = 6 (4 + + ) ( 5 + 6) = = = 0 ( ) = 0 ( + )( 5) = 0 = 5 ( 6 )(4) = ()( ) = = = 0 ( + 45)( ) = 0 cannot be negative, so = 4 (a) D is at (0,-8), (b) = ( ) 9 so M is at (,-9) A is at(-,0) and B is at (4,0) (d) CB has equation = + 4 and C is at (-,7) 5 (a) 6 + ( ) 8 (b) = ± + 6 ± 6 = + 6 ± = + 6 ± 4 = = ± 6 Proportionalit or Variation Eercise 6. Missing p values: 0.,0,6,80 Missing q vlaues: 5,0,50 (a) becomes 4 as big (b) becomes 9 as big becomes 00 as big (a) m = 6. 8 a (b)m = 7., a = 5, m = (a) P = 80, P = 000 (b) h = ± 0, h = ± 5 Missing values: 0,40 Missing value: (a) = 50 (b) = 5 6 time =.8 seconds (sf) Eercise 6. Missing v values: 40, 80 Missing t values: 0, 0, Missing values: 0.4, 0.5, Missing value: 00 (a) H =.5, H = 50 (b) T = 5, T = Resistance = ohms or ohms 5 (a) pressure = 69.4 units (sf) (b) diameter =.4 mm (sf) 7 Number: Estimating and Factorising (a) 8960 (b) V 4000 T 4 4(a) 6 (b) 7 HCF = = 4 LCM = 6 7 {=4784 is unnecessar} 5 appro. 0 8 Constructions, Loci and Bearings AC = 9. cm Area = 0.cm PR = 7.9km, bearing = 40 4 bearing = 98 5 bearing = 64, BC = 9.8cm

17 GCSE Higher Tier Practice Questions 7 6 Area = 4 cm 7 (a) Area = square units (b) AB = 5.8 units, BC = 5 CD = 7. AD = Vectors Eercise 9. (a) a = b = 4 c = p = q = 6 4 r = s = t = 4 (b) a = 5 b = c = q = t = 4 q = b r = - c s = -a t = b (a) TO = 4 (b) OT = 5 R is either (-5,0) or (+5,0) 4 4 (a) (b) (a) k = (b) k = -8 Eercise 9. AC = + n AD = n CD = n - FA = n FB = n AE = n - (a) A, B and C would be collinear. (b) It would be all one parallelogram. It would be a trapezium. (d)vertices would be E, C, B, F (a) (i) BD = t s (ii) BP = t s (iii) (iv) BQ = t s AQ = t + s (v) QC = s + t (b) AP = s + t AP = QC and AP is parallel to QC APCQ is a parallelogram 0 Probabilit (a) (b) (d) 8 7 (a) (b) (e) 50 (a) 8 5 (d) 5 (b) 6 (d) 6 (e) 6 (f) 7 is the most likel total 4 (a) (i) mutuall eclusive (ii) not mutuall eclusive (b) not mutuall eclusive not mutuall eclusive 5 P(LHD) P(no adjustment) = = P(King) = = 5 P(Red) = P(Red) P(King) = 6 P(Red King) = = (a) 6 (b) 0 9 (d) (a)p(alarm ok & on time) = (b) P(on time) = (a) P(Milk st ) = + (b) ( ) P(M,M)= ( + )( + ) P(one of each)= ( + )( + ) (d) P(Plain,Plain)= ( ) ( + )( + ) Calculations in Right-Angled Triangles Eercise. = 47.7 = 67.4 = = = = = 0 {all to s.f.} Eercise. = 5.0 cm = 7.68 cm = 6.4 cm 4 = 0.4 cm 5 = 9 cm 6 = 0.8 cm 7 = 6.9 cm Sine and Cosine Rules = 7.45 cm = cm = = 54 or 6 5 = 8. cm 6 = 5.06 cm 7 =.5 8 = = 6.68 cm