Indices Exam Questions PDF

Created by T. Madas INDICES Exam Questions Created by T. Madas  Created by T. Madas Question 1 (**) a) Evaluate the following indicial expression, giving the final answer as an exact simplified fraction. 3 − 12...

Created by T. Madas INDICES Exam Questions Created by T. Madas  Created by T. Madas Question 1 (**) a) Evaluate the following indicial expression, giving the final answer as an exact simplified fraction. 3 − 12 42 + 4. b) Simplify fully the following expression 12 y −5. 3 y −2 4 C1R , 17 , 4 y −3 = 3 2 y Created by T. Madas  Created by T. Madas Question 2 (**) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions. i. 2−4 + 8−1. 3 ii. ( ) 81 16 4. b) Simplify fully the following expression 2 ( 4 xy 2 ). ( 2 x )3 2 y4 C1A , 3 , 27 , 2 y 4 x −1 = 16 8 x Created by T. Madas  Created by T. Madas Question 3 (**+) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions where appropriate. 3 2 i. 64 2 + 64 3. − 12 ii. ( ) 25 16. b) Simplify fully the following expression 3 3a 4 × (10a ). 2 3 ( 5a ) 528 , 4 , 24a 5 Created by T. Madas  Created by T. Madas Question 4 (**+) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions where appropriate. 1 1 i. 16 2 + 16 4. −2 ii. ( 23 ). b) Simplify fully the following expression 5 x2 × x. 6 , 9 , x3 4 Created by T. Madas  Created by T. Madas Question 5 (**+) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions. i. 2−5 − 8−2. 3 ii. ( ) 4 9 2. b) Solve the equation − 13 y = 8. C1G , 1 , 8 , 1 64 27 512 Question 6 (**+) Evaluate the following indicial expression, giving the final answers as an exact simplified fraction. − 23  36 12 + 16 14   .   C1D , 1 4 Created by T. Madas  Created by T. Madas Question 7 (**+) a) Evaluate the following indicial expression, giving the final answers as an exact simplified fraction. 5 −1 4× 42 + 8 3. The answer to this part of the question must be fully supported by a detailed method, justifying each step in the evaluation. b) Simplify fully the following expression 4 ( 2 pq 2 ) × 5 p q6. MP1-D , 257 , 80 p 5q11 2 Created by T. Madas  Created by T. Madas Question 8 (***) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions. 1 − 13 i. 83 + 8. ii. 8−4 × 211. b) Simplify fully 9 x6 y 4. 2 3 2 ( 3x y ) 1 C1F , 5 , 1 , 2 2 3xy 4 Created by T. Madas  Created by T. Madas Question 9 (***) a) Simplify the following expression, writing the final answer in the form a + b 3 , where a and b are integers 3− 3. 3+ 3 b) Solve the equation x x −1 = , x ≠ 0. 16 C1N , 2 − 3 , x = ±4 Created by T. Madas  Created by T. Madas Question 10 (***) a) Evaluate the following indicial expression, giving the final answers as an exact simplified fraction. − 32 ( ) 61 4. b) Simplify fully the following expression 4 ( 12 x3 y 2 z ). 2 6 2 (4x z ) 8 8 8 , 3 x8 y8 z −8 = 3 x y 125 4 4 z8 Created by T. Madas  Created by T. Madas Question 11 (***) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions, where appropriate. 6 i.. 2−2 3 ii. ( ) 17 9 2. b) Solve the equation 3 z 2 = 27. C1Y , 24 , 64 , z = 9 27 Created by T. Madas  Created by T. Madas Question 12 (***) a) Simplify the following expression, writing the final answer in the form a + b 3 , where a and b are integers 2 3 −1. 2− 3 b) Solve the equation 2 x+ 2 = 4 2. C1O , 4 + 3 3 , x = 1 2 Question 13 (***) a) Evaluate the following indicial expression, giving the final answers as an exact simplified fraction. 27 −4 × 311. b) Solve the equation − 12 t = 1. 4 1 , t = 16 3 Created by T. Madas  Created by T. Madas Question 14 (***) − 12 1 6x − x2 = 5. 1 a) Show that the substitution y = x 2 transforms the above indicial equation into the quadratic equation y2 + 5 y − 6 = 0. b) Solve the quadratic equation and hence find the root of the indicial equation. x =1 Question 15 (***) The points ( 2,14 ) and ( 6,126 ) lie on the curve with equation y = ax n , x ∈ » where a and n are non zero constants. Find the value of a and the value of n. a= 7 , n=2 2 Created by T. Madas  Created by T. Madas Question 16 (***) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions. 1 − 12 i. 42 + 9. ii. 325 × 8−10. b) Simplify fully the following expression 3a 3bc × 6a 2b 2c3. 2abc 4 7 , 1 , 3a 2b 3 32 Created by T. Madas  Created by T. Madas Question 17 (***) 1 − 13 t 3 = 2 + 15t. 1 a) Show that the substitution x = t 3 transforms the above indicial equation into the quadratic equation x 2 − 2 x − 15 = 0. b) Solve the quadratic equation and hence find the two solutions of the indicial equation. t = −27, t = 125 Created by T. Madas  Created by T. Madas Question 18 (***) a) Evaluate the following indicial expressions, giving the final answers as exact simplified fractions where appropriate. 5 3 i. 8 3 − 16 4. 3 ii. ( 2.25)− 2. b) Simplify fully the following expression 4 − 12  2a 12 b3  × 4a 6b2     ( ). 11 8b 24 , 8 , 8a −1b11 = 27 a Question 19 (***) Given that the curve with equation 1 y = ax − x 3 , x ≥ 0 , 1  passes though the point with coordinates  , 0  , find the value of the constant a. 8  a=4 Created by T. Madas  Created by T. Madas Question 20 (***) a) Evaluate the following indicial expressions, giving the answers as integers. 4  36 12 + 16 14  3. i.     1 −2. ii. ( ) 4 b) Simplify fully the following expression 1  k 32 × 8k −3  3.     − 12 2 16 , 16 , 2k = k Created by T. Madas  Created by T. Madas Question 21 (***+) a) Simplify fully each of the following expressions, writing the final answer in terms of 3. i. 108 + 3. 6+ 3 ii.. 2 +1 b) Solve the equation 3 (5 − x ) 2 = 8. 7 3, 3 , x =1 Created by T. Madas  Created by T. Madas Question 22 (***+) a) Evaluate the following indicial expressions, giving the answers as integers. −3 i. ( 13 ). 86 ii.. 323 b) Simplify fully the following expression 1  6a 7b 2 × 9b  3  .  2 a  108 , 8 , 3a 2b Created by T. Madas  Created by T. Madas Question 23 (***+) a) Simplify fully each of the following expressions, writing the final answer as a single simplified surd. i. ( 2 + 3 )( 2 ) 3 −3. 6 +3 2 ii.. 6+ 2 b) Solve the equation 1 8w 2 − w−1 = 0 , w ≠ 0. C1V , 3, 3 , w= 1 4 Created by T. Madas  Created by T. Madas Question 24 (***+) 1 − 13 t 3 = 2 + 15t , t ≠ 0. 1 Use the substitution x = t 3 to solve the above indicial equation. t = −27, t = 125 Question 25 (***+) An exponential curve has equation y = ab x , x ∈ » , where a and b are non zero constants. ( ) ( ) The points A 1 ,1 , B ( 2,8 ) and C − 1 , k lie on this curve. 2 2 a) Find the values of a and b. b) Find the value of k. a= 1 , b=4 , k = 1 2 4 Created by T. Madas  Created by T. Madas Question 26 (***+) − 23  1 1 3 1 1  125 3 × 25 2 + 16 4 × 64 3 + .  −1   49 2  Evaluate the above indicial expression, giving the final answer as a simplified fraction. You may not use any calculating aid in the above question, and detailed workings must support the answer. SYN-D , 1 16 Created by T. Madas  Created by T. Madas Question 27 (***+) a) Simplify fully each of the following expressions, writing the final answer in terms of 2. i. 98 + 2. ii. ( )( 2 +3 2−3 2.) b) Solve the equation 27t =3 3. 3t −1 SYN-A , 8 2 , −7 2 , t = 1 4 Created by T. Madas  Created by T. Madas Question 28 (***+) a) Evaluate the following indicial expression, giving the final answers as an exact simplified fraction. 1 − 43 16 2 + 16. b) Solve the equation − 23 x = 64. c) Simplify fully 2  x 32 + 2 x − 32 .     4 SYN-V , 33 , x = 1 , x3 + 4 + 3 8 512 x Created by T. Madas  Created by T. Madas Question 29 (***+) An exponential curve has equation y = ab x , x ∈ » where a and b are non zero constants. The points A (1,7 ) and B ( 3,175) lie on this curve. Given that b > 0 , find the values of a and b. a = 1.4 , b = 5 Question 30 (***+) Solve the following simultaneous equations without using a calculator 8 y = 42 x +1 27 2 y = 9 x −3 SYN-W , ( − 53 , − 149 ) Created by T. Madas  Created by T. Madas Question 31 (***+) a) Solve the equation 3 3y =. 9 b) Express 525 , in the form a b c , where a , b and c are prime numbers. C1W , y = − 3 , 5 3 7 2 Question 32 (***+) Find the exact coordinates of the point of intersection between the curves with equations y = 8x and y = 4 × 2− x. ( 12 , 2 2 ) Created by T. Madas  Created by T. Madas Question 33 (***+) a) If x is a real number solve the following indicial equation 2 x  x 2 − 2 x  1 − 12  =0.   b) Express 98 − 8 , 1+ 2 in the form a + b 2 , where a and b are integers. MP1-N , x = 2 , 10 − 5 2 Created by T. Madas  Created by T. Madas Question 34 (***+) Find, without the use of any calculating aid, the solution of the equation 1 × 42 x = 6464. 2 MP1-A , x = 96.25 Question 35 (***+) 4 x − 2 x+ 2 = 32. a) Show that the substitution y = 2 x transforms the above indicial equation into the quadratic equation y 2 − 4 y − 32 = 0. b) Solve the quadratic equation and hence find the root of the indicial equation. x=3 Created by T. Madas  Created by T. Madas Question 36 (****) Solve the following simultaneous equations without using a calculator. 4 x − 3 y = 11 3 3 9 y +3 = 27 x ( 12 , −3) Question 37 (****) Given that the curve with equation 1 − 32 y = kx 2 − x , x≥0,  5  passes though the point with coordinates  3, 3  , show clearly that k = 16.  3  9 proof Created by T. Madas  Created by T. Madas Question 38 (****) The indicial equation 2 x+1 + 23− x = 17 , x ∈ » , is to be solved by a suitable substitution. a) Show clearly that the substitution y = 2 x transforms the above indicial equation into the quadratic equation 2 y 2 − 17 y + 8 = 0. b) Solve the quadratic equation by factorization and hence determine the two solutions of the indicial equation. C1M , x = −1, 3 Question 39 (****) Solve the equation − 12 ( 25 x 2 ) = 2, x ≠ 0. x=± 1 10 Created by T. Madas  Created by T. Madas Question 40 (****) Given that 1 − 12 1 − 12 a = x2 + x and b = x2 − x , show clearly that 2  1 a 2b 2 + 4 ≡  x + .  x MP1-G , proof Created by T. Madas  Created by T. Madas Question 41 (****) 22 p − 2 − 2 p −2 − 3 = 0 , p ∈ » , a) Show clearly that the substitution x = 2 p transforms the above indicial equation into the quadratic equation x 2 − x − 12 = 0. b) Solve the quadratic equation and hence determine the value of p. p=2 Created by T. Madas  Created by T. Madas Question 42 (****) ( ) 100 x − 10001 10 x−1 + 100 = 0. a) Show that the substitution y = 10 x transforms the above indicial equation into the quadratic equation 10 y 2 − 10001 y + 1000 = 0. b) Solve the quadratic equation and hence find the two solutions of the indicial equation. C1Z , x = −1, x=3 Question 43 (****) Determine the value of k. 2288 + 2285 = 2k. 9 You must show full workings. MP1-I , k = 285 Created by T. Madas  Created by T. Madas Question 44 (****+) Solve the following indicial equation 6 x + 2 × 21− x = 8. 3 You must show full workings. C1S , x = −3 Question 45 (****+) Show clearly that − 12 ( ) 1 + x x2 −1 ≡ 1. 1 2 x + ( x 2 − 1) 2 x −1 SYN-M , proof Created by T. Madas  Created by T. Madas Question 46 (****+) Solve the following exponential equation 16 + 8 x +1 − 4 x+1 − 2 x +5 = 0 , x ∈ ». SP-J , x = ±1 Question 47 (****+) Determine the value of k. 2399 − 2395 = 32k. 15 You must show full workings. SYN-M , k = 79 Created by T. Madas  Created by T. Madas Question 48 (****+) Solve the following equation 2  −3  2 1 x 2 x5 3  1  − x ×  1  3 + ×  x  2 x −    x   (  + x x3 x ) 7 =4.    x x =1 Question 49 (****+) Solve the following indicial equation 2n = 1. 2 n × 26 You must show full workings. SYN-K , n = 9 Created by T. Madas  Created by T. Madas Question 50 (****+) Show with a detailed method that 3 16 − 3 2 3 3 = k 4 4 where k is a constant to be found. SYN-L , k = 1 2 Question 51 (*****) 8 f ( x) ≡ − x, x ≠ 0. x2 Show that f  −2 3  = 3 3 2. 4   You must show detailed workings. SPX-L , proof Created by T. Madas  Created by T. Madas Question 52 (*****) A = 3 xy + 2 yz + 2 xz. 2 1 1 2 ( ) Given that x = 4 3 3 ( ) , y= 4 3 3 ( ) and z = 3 4 3 , show clearly that A = 3 3 6 C1T , proof Created by T. Madas  Created by T. Madas Question 53 (*****) A curve has equation f ( x ) ≡ 3ax + b , x ∈ » , where a and b are non zero constants. Find the value of a and the value of b , given further that f ( 2) = 3 − 3 and f ( 3) = 2 3. SPX-E , a = 1 , b = − 3 2 Created by T. Madas  Created by T. Madas Question 54 (*****) A curve has equation f ( x ) ≡ 2ax + b , x ∈ » , where a and b are non zero constants. Find the value of a and the value of b , given further that f ( 2) = 5 and f ( −2 ) = 4. 2 MP1-S , a = − 1 , b = 2 2 Created by T. Madas  Created by T. Madas Question 55 (*****) A curve has equation f ( x ) ≡ 4ax+b , x ∈ » , where a and b are non zero constants. Find the value of a and the value of b , given further that ( ) f 2 =134 3 4 and ( ) f 3 = 1 2. 2 2 SYN-C , a = 1 , b = −1 2 Created by T. Madas  Created by T. Madas Question 56 (*****) Find the solutions for the following equation. x 2 + 2 x −8 ( 2x 2 − 7x + 4 ) = 1. V , SYN-U , x = −4, 1 , 2, 3 2 Created by T. Madas  Created by T. Madas Question 57 (*****) Find a solution for the following equation. 4 x +5 ( 2 x2 − 5x − 10) =1. 2 V , SYN-Z , x = − 3 2 Created by T. Madas  Created by T. Madas Question 58 (*****) Determine the value of x and the value of y in the following equation 153 x −2 × 61−2 y = 6.25 , ( 3x − 2 ) ∈ ». SP-R , x = 4 , y = 3 3 2 Question 59 (*****) 2m +1 + 2m = 3n + 2 − 3n. Given that m and n are positive integers, find the value of m and the value of n. SP-N , m = 3 , n = 1 Created by T. Madas  Created by T. Madas Question 60 (*****) Find the term independent of x in the expansion of 10  x +1 x −1   2 − .  3 1 1   x − x3 +1 x − x 2  SP-U , 210 Created by T. Madas  Created by T. Madas Question 61 (*****) Simplify the following expression. x6 − yz 4 x4 + x z 2 y Give the answer in the form x 2 − x r y s z t , where r , s and t are constants. 1 SPX-Q , x 2 − x −1 y 2 z 2 Created by T. Madas