Created by T. Madas MIXED SURD QUESTIONS. Created by T. Madas

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1 MIXED SURD QUESTIONS

2 Question 1 (**) Write each of the following expressions a single simplified surd a) b) 21 7 C1A, 2 6, 3 7 Question 2 (**) Write each of the following surd expressions as simple as possible a) ( 7 + 2)( 1+ 7 ) b) C1M, , 4

3 Question 3 (**) A rectangle has area 2 12cm and length cm Find its width in the form a + b 7, where a and b are integers Question 4 (**) Write each of the following surd expressions as simple as possible a) b) ( 2 + 3)( 4 12 ) 3 6, 2

4 Question 5 (**) Write each of the following surd expressions as simple as possible a) b) ( 7 + 3)( 2 7 3) 4 3, Question 6 (**) Write each of the following surd expressions as simple as possible a) ( 5 + 2)( 3 5) 14 b) 18 ( 2 ) , 2 2

5 Question 7 (**+) Write each of the following surd expressions as simple as possible a) ( 4 5) 2 b) C1A, , 3 3 Question 8 (**+) Write each of the following expressions a single simplified surd a) b) , 7 5

6 Question 9 (**+) Write each of the following surd expressions as simple as possible a) b) , Question 10 (**+) Write each of the following expressions a single simplified surd a) b) ( 2 + 5)( 5 20 ) 7 7, 5

7 Question 11 (**+) Write each of the following surd expressions as simple as possible a) ( 3 8) 2 b) , 3 7 Question 12 (**+) Write each of the following surd expressions as simple as possible a) b) , 8 2 3

8 Question 13 (**+) Write the following expression in the form k 3, where k is an integer ( ) Question 14 (**+) Write each of the following surd expressions as simple as possible a) b) , 3+ 7

9 Question 15 (**+) Write each of the following surd expressions as simple as possible a) ( 2 + 3) 2 b) ( ) , 20 3 Question 16 (**+) The area of a triangle is ( 3+ 3) 2 cm Given the base of the triangle is of the triangle 3 cm, find in exact simplified surd form the height h =

10 Question 17 (**+) Write each of the following surd expressions as simple as possible a) b) , Question 18 (**+) A rectangular room has an area of m The length of the room is m Find the width of the room, giving the answer as an exact surd in its simplest form C1F, 4 7

11 Question 19 (**+) Write each of the following surd expressions as simple as possible a) b) , Question 20 (**+) 2z z 8 6 = 2 Solve the above equation giving your answer in the form k 2, where k is an integer C1I, z = 3 2

12 Question 21 (**+) Write each of the following surd expressions as simple as possible a) ( 2 6 3) 2 3 b) , Question 22 (**+) Express f ( x ), x 0 in the form ax + b 2 2 ( ) ( 2) ( 1 2 ) f x x + + x 5x + 5

13 Question 23 (***) Write each of the following surd expressions as simple as possible a) b) ( ) 2 2, 3 Question 24 (***) Write each of the following surd expressions as simple as possible a) b) , 3 + 2

14 Question 25 (***) Write each of the following surd expressions as simple as possible a) ( 1+ 2 ) 3 b) C1E, , 11 3 Question 26 (***) Write each of the following surd expressions as simple as possible a) ( ) 2 ( )( ) b) 3 48, 9 7 3

15 Question 27 (***) Write each of the following surd expressions as simple as possible a) b) , Question 28 (***) Show that the above surd expression simplifies to a positive integer, stating its value C1L, 2

16 Question 29 (***) Write each of the following surd expressions as simple as possible a) ( 3 2 ) 2 b) C1J, 5 2 6, 14 3 Question 30 (***) a) Simplify the following expression, writing the final answer in the form a + b 3, where a and b are integers b) Solve the equation 1 x x =, x 0 16 C1N, 2 3, x = ± 4

17 Question 31 (***) If x = 3 show that ( x 2 )( x x 3 ) = 8 3 proof Question 32 (***) Simplify fully ( )( ) C1D, 70 3

18 Question 33 (***) Write each of the following surd expressions as simple as possible a) b) C1C, 2, 2 2 Question 34 (***) x y x + y Write the above expression as a single fraction in its simplest form 2x 2 x y

19 Question 35 (***) 2 + y = 2 y Solve the above equation giving the answer in the form a + b 2, where a and b are integers C1K, y = Question 36 (***) a) Simplify the following expression, writing the final answer in the form a + b 3, where a and b are integers b) Solve the equation 2 2 x+ = 4 2 C1O, , x = 1 2

20 Question 37 (***) Write each of the following surd expressions as simple as possible a) ( )( 1+ 2 ) 2 b) , 2 Question 38 (***) 98 ( ) 2 Write the above surd expression in the form a + b 2, where a and b are integers C1G,

21 Question 39 (***+) a) Simplify fully each of the following expressions, writing the final answer in terms of 3 i ii b) Solve the equation ( ) x = 8 Detailed workings must be shown in this question MP1-J, 7 3, 3, x = 1

22 Question 40 (***+) It is given that that for some constants a, b and c a 2 + b 3 + c Find the value of a, the value of b and the value of c C1H, a = 1, b = 2, c = 1

23 Question 41 (***+) A right angled triangle ABC is shown in the figure below The lengths of AC and BC are 6 cm and 2 3 cm, respectively A 6 cm B 2 3 cm C Find the area of the triangle ABC in the form k 2, where k is an integer k = 6

24 Question 42 (***+) a) Simplify fully each of the following expressions, writing the final answer in terms of 2 i ii ( 2 + 3)( ) b) Solve the equation 27 3 t 3 3 t 1 = SYN-A, 8 2, 7 2, t = 1 4

25 Question 43 (***+) ( ) f x x 8x, x > 0 Show clearly that f ( ) 3 = k 3, where k is a constant MP1-K, proof

26 Question 44 (***+) Write each of the following surd expressions as simple as possible a) Give the answer in the form a + b 7, where a and b are integers b) Give the answer in the form k, where k is an integer C1Y, , 6

27 Question 45 (***+) A cylinder has a radius of cm and a height of ( 2 + 1) cm Show, by detailed working, that the volume of this cylinder is exactly 3 π ( ) cm SYN-B, proof

28 Question 46 (***+) a) Solve the equation 3 3 y = 9 b) Express 525, in the form a b c, where a, b and c are prime numbers C1W, y = 3,

29 Question 47 (***+) a) If x is a real number solve the following indicial equation x x x = 0 b) Express 98 8, 1+ 2 in the form a + b 2, where a and b are integers MP1-N, x = 2,

30 Question 48 (***+) Show clearly, without approximating and without using any calculating aid that > 7 proof Question 49 (***+) 3 p =, q = and r = 4 Prove that p + q + r = pqr C1Y, proof

31 Question 50 (***+) a) Simplify fully each of the following expressions, writing the final answer as a single simplified surd i ( 2 + 3)( 2 3 3) ii b) Solve the equation w 8w = 0, w 0 C1V, 3, 3, w = 1 4

32 Question 51 (***+) ( 2 + 3) ( 1 3) Write the above surd expression in the form a + b 3, where a and b are integers C1Q, Question 52 (***+) ( x ) 3 3 = x + 3 Solve the above equation giving the answer in the form a + b 3, where a and b are integers C1P, x =

33 Question 53 (****) Show clearly that ( 1+ 3) 4 = C1X, proof

34 Question 54 (****) The two rectangles shown in the figure below are similar A D E H 3 B 12 2 C F G It is further given that in suitable units AB = 3, BC = 12 2 and FG = Find the exact length of EF C1R, EF =

35 Question 55 (****) Solve the following system of simultaneous equations ( x y ) = y = 3x C1U, ( 2,3 2 ),( 2, 3 2 )

36 Question 56 (****+) The positive constants p and q satisfy the following equation p 2 p q = 2 p + p 3p + q Show by a detailed method that p + 2 p q = p MP1-Z, proof

37 Question 57 (****+) Four circles are touching in such a way so that their centres form the corners of a square ABCD These four circles are circumscribed by a larger circle This is shown in the figure below A D B C Show that the ratio of the total area of the four smaller circles to the area of the larger circle is given by :1 SYN-I, proof

38 Question 58 (****+) If x = 3 120, show clearly that x = x 2 3 MP1-V, proof Question 59 (****+) If x = , show clearly that x = x 2 3 MP1-Y, proof

39 Question 60 (****+) Show clearly, without approximating and without using any calculating aid that a) > b) 3 3 > 2 c) 2 1 > 3 2 proof Question 61 (****+) It is given that if k is a non zero constant then ( k ) Determine the value of k k = 3

40 Question 62 (****+) non calculator f ( x) 2 x + x x + x 1, x R, x 1 Find without the use of a calculator the value of 5 ( 6 ) f 12 Detailed workings must be shown in this question f = 2 SYN-H, ( )

41 Question 63 (****+) Show that can be expressed in the form a + b 2, where a and b are integers to be found SP-E, = Question 64 (*****) Show clearly without the use of any calculating aid that = a b, where a and b are integers to be found a = b = 2

42 Question 65 (*****) Determine, in exact simplified surd form, the solution pair ( a, b ) of the following simultaneous equations 2 x 3 y 5 + = and ( ) x ( ) y = 10 6 Detailed workings must be shown in this question SYN-F, x = 3 + 2, y = 3 2

43 Question 66 (*****) 3 3 z = Verify that z is a solution of the equation z 3 3z 8 = 0 proof Question 67 (*****) Find the rational solution of the following equation x 2 3 3x = , x Q SP-O, x = 2 3

44 Question 68 (*****) Show clearly without the use of any calculating aid that = a + b c, where a, b and c are integers to be found a = 4, b = 3, c = 2 Question 69 (*****) Show clearly without the use of any calculating aid that = k, where k is an integer to be found k = 3

45 Question 70 (*****) Determine, in exact simplified surd form, the solution pair ( a, b ) of the following simultaneous equations ( a ) b ( ) = and 6 a 3 b = 2 3 Detailed workings must be shown in this question MP1-X, a = 1+ 2, b = 2

46 Question 71 (****+) non calculator Evaluate ( ) f + 2 ( ) 4 ( 2)( 1)( 3) f x x x x + x, x R You must show detailed workings in this question SPX-D, 1 ( ) f = 9 2

47 Question 72 (*****) 2 1 f p p p ( ) ( ) 2, p R, p 0 Given that 1 p + < 2, find f ( p ) in its simplest form p SP-G, proof

48 Question 73 (*****) x x x x x + 2 +, It is given that the above nested radical converges to a limit L, L R Determine the range of possible values of x SP-T, x 9 4 Question 74 (*****) Show clearly that = 1+ 3 proof

49 Question 75 (*****) Show that can be written in the form 3 a + 3 b + 1, where a and b are integers to be found MP1-T, Question 76 (*****) Show clearly that = + You may not use verification in this question C1T, proof

50 Question 77 (*****) Show clearly that = a + b 3, where a and b are constants to be found a = 1, b = 2 3 Question 78 (*****) Show that can be written in the form ( ) 1 2 to be found 10 a 5 + b 7, where a and b are positive integers MP1-W, ( )

51 Question 79 (*****) A = 3 xy + 2yz + 2xz Given that ( ) 1 x = 3, ( ) 1 y = 3 and ( ) 2 z = , show clearly that 3 A = 3 6 C1T, proof Question 80 (*****) Find an exact simplified value for

52 Question 81 (*****) Rationalize the denominator of the following surd = + Show detailed workings in this question MP1-S, proof

53 Question 82 (*****) Solve the following quadratic equation ( ) x x = Give one of the roots in the form p + q 3 and the other root in the form r 3, where p, q and r are integers C1S, x = 3, x = Question 83 (*****) Express , in the form p q, where p and q are integers 3 2

54 Question 84 (*****) Solve the following simultaneous equations, to find in exact form where appropriate, the value or values of x and k x 2 9x + 10 = 0 and k = x x + 6 = , k = 2 2 SP-A, x ( )

55 Question 85 (*****) Find in exact simplified form ( + )

56 Question 86 (*****) ( ) f a 2 a 2 a 1 a , a R, a 1 a 2a 1 Show that f ( a ) can be simplified to a linear polynomial in a a SP-V, f ( a) = a + 1

57 Question 87 (*****) ( ) h x 1 2 x + x 1, x R, x 1 Show that h( x ) can be expressed in the form f ( x) g ( x), where f ( x ) and g ( x ) are linear functions to be found SP-Z, h( x) x + 1 x 1 = 2 2

58 Question 88 (*****) Using a detailed method show that = SP-S, proof

59 Question 89 (*****) f x y x y x y x y, y x y x y x y x ( x y) Simplify f (, ) x y in a form not involving square roots SP-C, f ( x y) x x y y, ± y y x x

60 Question 90 (*****) By using the substitution 3 10 ± 6 3 = u ± v, where u Q, v Q, simplify fully the following cubic radical expression SP-B, 2

61 Question 91 (*****) By using the substitution 3 20 ± 14 2 = u ± v, where u Q, v Q, simplify fully the following cubic radical expression SPX-F, 2 2

62 Question 92 (*****) 2 2x f ( x) x + 1, x R Factorize f ( x ) into a product of 2 simple linear factors SP-M, ( x )( x )

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