FIS 0444 Mathematics 1 Unit 1 Exercises & Answers PDF

FIS 0444 Mathematics 1 Unit 1- Algebra Indices a m  a n = a m+ n (a m ) n = a mn = (a n ) m (ab) m = a m b m a am...

FIS 0444 Mathematics 1 Unit 1- Algebra Indices a m  a n = a m+ n (a m ) n = a mn = (a n ) m (ab) m = a m b m a am m am (a p a q ) m = a pm a qm   = m n = a m  a −n b b a m −m m ap  a pm a b  q  = qm   =  b  b b a Surds Theorem ( ) (b ) m m 1  1n  ( b) 1 m = b = b n n = n bn = b b =  b  = n m n n m m b n n   n ab = n a  n b n m a = mn a a na n = b nb tn a tm = n a m Logarithms Theorem a y = x, y = log a x loga xy = loga x + loga y log a x c = c log a x x loga 1 = 0 loga a = 1 loga = loga x − loga y y a loga x = x log b N 1 log a N = loga b = log b a logb a Polynomials P( x) = Q( x) D( x) + r ( x), where Q(x)= quotient; D(x)= divisor and r(x)= remainder. -1- FIS 0444 Mathematics 1 Unit 1- Algebra Exercise 1 g. 50 + 8 + 32 Determine whether each given real number is a 1. natural number, an integer, a rational number, or 9. Simplify the following. an irrational number. a. h. a. 999 5 a 6 3 1− a b. − 5 b. 10 i. 2 a+ b c. − 6 5 2 a− b 3 c. 2 d. 25 2+ 2 e. 3 d. 5 3+ 2 2. Solve each of the following equation. 2 e. 1+ 2 a. 3 = 27 x 3 −1 3 b. 10 x = 0.01 f. 3 3−4 c. (2 x ) 2 = 2 5 6 −2 3 d. 2 x−3 = 4 x+1 g. 2 1 1 − e. 2 x  2 x +1 = 5 −2 5+2 2 f. 2 −92 +8 = 0 2x x 10. Express with rational denominators: g. 32 x − 10  3 x + 9 = 0 a. 1 3+ 2− 5 4n + 2 5  25 n b. 1 3. Simplify 6 n −1 5 3 + 2 +1 4. Show that 5n + 2 − 5n +1 − 5n is divisible by 19 for all 5− 3 positive integer values of n. 11. If x = , find the value of 8 x − x 2. 5+ 3 5. Show that 4 m+1 + 4 m−1 − 4 m is divisible by 13 for 12. Without using a calculator, evaluate each of the all positive integers of m. following. a. lg1000 56 n  94 n  152 n b. 6. If 2n = k 8n , where k is a positive log 2 128 3 c. log5 0.2 integer, find the value of k. 13. Express each of the following as a single 25m  10m +1 logarithm. 7. Simplify. 2m −1  52 + 3m a. lg 24 – lg 6 – lg 2 b. 3 lg x – 2 lg ( xy) + lg y 8. Simplify c. 1 log2 25 + log2 3 − 2 log2 15 a. 18 + 8 2 b. 50 + 32 − 72 d. log a p 2 + 2 log a q − 2 c. ( 2 − 3) 2 10 d. (4 3 − 3 2 ) 2 x −1 14. Solve the equation 3 − 3 − 2x = 0. 3 e. 27  12 f. 32  15  24 15. If h 3 x −1 = k 1−2 y and h 2 x −2 = k 4 y −3 , show that -2- FIS 0444 Mathematics 1 Unit 1- Algebra 16xy − 8 y − 11x + 5 = 0. a. 3 log 2 + 2 log5 − log 20 b. 2 log a − logb − log c 16. Evaluate (log3 m) (logm 81). 1 17. Solve the following equations. c. log x + log y − 3 log z a. logx 8 = 1.5 2 b. d. 3 logc a − 2 logc b + 1 6 x +1 = 18 2 c. log3 ( x + 2) = log9 (6x + 4) 5. log c a Show that log bc a =. 1 + log c b Exercise 2 6. If x = log y z, y = log z x and z = logx y ,show that 1. Evaluate 1 xyz = 1. a. 50 b. 36 2 c. − 1 d. − 3 7. Given that log10 ( x − y + 1) = 0 and 2 4 49 81 1 −3 − 3 1 + log10 ( xy) = 0 ,show that x = y =. 1 1 2 10 e.   f.   2 4 1 1 8. 1 1 Show that log16 ( xy) = log4 x + log4 y. 1 1 9 82 2 2 2 g. 12  3 2 2 h. 1 2 2 9. Solve each of the following equations for x, giving 1 1 i. your answers correct to two decimal places. 27  3 4 4 a. 2 x−1 = 3 x +1 2. Rationalise the denominator for each of the b. 5 x − 2 = 2 x +3 following surds. c. 32 x −1 = 5 x 10 2 d. 61− x = 2 3 x +1 a. b. 5 3 10. Solve each of the following equations for x, giving 3 1+ 2 your answers correct to two decimal places. c. d. 2 5− 3 3 −1 a. 4 2 x − 5(4 x ) + 6 = 0 3+2 2 1 1 b. 2 2 x+1 − 11(2 x ) + 5 = 0 e. f. + 5− 3 2 +1 2 −1 11. Solve the following equations. 3. Evaluate each of the following expression. a. 52 x − 6(5 x+1 ) + 125 = 0 a. log3 1 b. 4 y − 5(2 y +1 ) + 16 = 0 81 b. log 2 0.25 12. Evaluate x if c. log 4 2 log 2 (1 + x) + log 2 (5 − x) − log 2 ( x − 2) = 3. d. (log 4 2)(log 2 16) 13. Solve the simultaneous equations: 1 e. 5 log8 12 − log8 3 log x y + log y x = ; xy = 64. 2 1 f. log12 4 + 2 log36 2 + log36 3 14. Solve the simultaneous equations: log 2 ( x − 2 y ) = 5; log 2 x − log 4 y = 4. 4. Express each of the following as a single expression. -3- FIS 0444 Mathematics 1 Unit 1- Algebra Exercise 3 and d such that 1. Find the degree of each of the following x 4 + ax3 + 5x 2 + x + 3  ( x 2 + 4)(x 2 − x + b) + cx + d polynomials.. Hence, with these values of a, b, c and d , write a. 3x 2 + 12x − 1 b. 5x 3 + 8 down the quotient and the remainder when the c. x15 − 1 d. 4x + 7 x 4 polynomial x 4 + ax3 + 5x 2 + x + 3 is divided by e. x 7 − x 4 + 2x 2 − 9 f. 2 + 6x3 − 9x 5 x2 + 4 2. Given f ( x) = x 4 − 3x 3 + 2 x 2 + 5x − 1, 7. Given that g ( x) = 4 x 5 − 3 x 3 + 2 x + 1 , x 4 + x 2 + x + 1  ( x 2 + A)(x 2 − 1) + Bx + C , h( x) = 2 x 3 + x 2 − 1 , determine the numerical values of A, B and C. evaluate Hence, deduce the remainder when x 4 + x 2 + x + 1 a. f (2) b. g (0) is divided by ( x + 1)(x − 1). c. 1 d. f (−1) + g (2) h  Exercise 4 2 1. If we divide the polynomial P by the factor x − c e. g (4) − 2h(1) f. f (−3) + 3g (−1) − 5h(5) and we obtain the equation P( x) = ( x − c)Q( x) + R( x) , then we say that x − c is the divisor, Q(x) is the _____________ , and 3. Given f ( x) = 5x 3 − 4 x 2 + 3x + 2 , R (x) is the ______________. g ( x) = x 4 + x 2 − 7 , 2. a. If we divide the polynomial P (x) by the h( x) = 3x 5 + 7 x 2 + 5x − 4 , factor x − c and we obtain a remainder find 0, then we know that x − c is a a. f ( x) + g ( x) b. g ( x) − 2h( x) _____________ of P. c. 3 f ( x) − 2 g ( x) d. 2 g ( x) + 3h( x) b. If we divide the polynomial P (x) by the e. 3 f ( x) − g ( x) + h( x) f. f ( x) + 2 g ( x) − 3h( x) factor x − c and we obtain a remainder of k, then we know that P(c) = ______. 3. For the function f ( x) = x 4 − 6 x3 + x 2 + 24x − 20 , 4. Find the coefficients of the term indicated in square brackets in the expansions of the polynomials use long division to determine which of the below. following are factors of f (x). a. ( x − 3)(3x 3 + 2 x 2 − x + 4) [x2 ] a. x +1 b. x−2 b. ( x 2 + x + 1)(x 2 + 4 x − 3) [x3 ] x+5 c. c. ( x 2 − 5x + 2)(x 4 − 3x 2 + 1) [x4 ] d. (2 x 3 − x 2 + 1)(4 x 3 + x + 3) [x4 ] 4. For the function g ( x) = x3 − 2 x 2 − 11x + 12 , use e. ( x 2 + 3x − 5) 2 [x2 ] the long division method to determine which of the following are factors of g (x). f. x( x 2 + 1)(x 3 + 2 x − 1) [x4 ] a. x−4 5. Find the quotient and the remainder when b. x−3 c. x −1 a. x 2 + 6 x + 5 is divided by x + 1. b. 2 x 3 + x 2 + 5 x − 4 is divided by 2 x − 1. In question 5 to 7, a polynomial P (x) and a divisor c. 4 x 4 − 3x 2 + x + 2 is divided by x − 2. d (x) are given. Use long division method to find the d. x 3 + 6 x − 2 is divided by x 2 + x + 1. quotient Q(x) and the remainder R (x) when P (x) is e. x 4 + 2 x 2 − 1 is divided by x 2 ( x − 1) 2. divided by d (x) , and express P (x) in the form f. 4 x 4 + 3x 2 + x + 2 is divided by ( x − 1)(x + 1). d ( x).Q( x) + R( x) 6. Find the numerical values of the constants a, b, c -4- FIS 0444 Mathematics 1 Unit 1- Algebra 5. P( x) = x 3 − 8, factor of p( x) − 5 , find the remainder when q(x) d ( x) = x + 2. is divided by x − 2. 17. Find a polynomial of degree 3 that has zeros -1, 6. P( x) = x 3 + 6 x 2 − 25x + 18, 1 ,3. d ( x) = x + 9. 18. Find a polynomial of degree 4 that has zeros -1, 1 ,3, 5. 7. P( x) = x 4 − 2 x 2 + 3, d ( x) = x + 2. 19. Find a polynomial of degree 3 that has zeros 1, -2 and 3 in which the coefficient of x 2 is 3. 8. The x 3 + x 2 + ax − 24 gives a expression remainder of -10 when divided by x + 1. Find the value of the constant a. Exercise 5 Decompose the following into partial fractions. 1. 2 9. If 2 x 3 + 7 x 2 − 5x + k leaves a remainder of 3 when divided by 2 x + 1 ,find the value of the ( x − 1)( x + 1) constant k. 2. 5 ( x − 1)( x + 4 ) 10. The expression 2 x 3 + px2 + qx + 5 leaves a 3. 12 remainder of -3 when divided by x − 1 and a x −9 2 remainder of 7 when divided by x − 2. Find the 4. x + 14 values of the constants p and q. x − 2x − 8 2 5. x 11. When x 4 + px3 + qx2 + rx + 6 is divided by x + 1, x − 1 and x + 2 , the remainders are 6, 12 8 x − 10 x + 3 2 and 6 respectively. Find the values of p, q, and r. 6. 9 x2 − 9 x + 6 2 x3 − x 2 − 8 x + 4 12. By using the factor theorem, show that 3x − 4 is a 7. 2x factor of f (x) ,where f ( x)  12x 3 − 4 x 2 − 13x − 4. 4 x + 12 x + 9 2 Hence, factorise f (x) completely. 8. 4 x2 − x − 2 x 4 + 2 x3 13. If p( x) = 2 x 3 + x 2 − 8x − 4 , show that x − 2 is a 9. −10 x 2 + 27 x − 14 ( x − 1) ( x + 2 ) 3 factor of p (x). Hence, factorise p (x) into its linear factors. 10. x−3 14. By using the factor theorem, find one of the x3 + 3x factors of the polynomial 11. 2 x3 + 7 x + 5 g ( x)  2 x − 9 x + 7 x + 6. ( x2 + x + 2)( x2 + 1) 3 2 hence, factorise g (x) completely. 12. x 4 + x3 + x 2 − x + 1 x ( x 2 + 1) 2 15. When a polynomial f ( x)  x 3 + ax2 − 20x + 24 is divided by x − 1 , the remainder is 7. Find this value of a, find the three linear factors of f (x). 16. Polynomials p (x) and q(x) are given by relationship q( x) = p( x)(x 5 − 2 x + 2). If x − 2 is a -5- FIS 0444 Mathematics 1 Unit 1- Algebra Answers: 12. a. 3 Exercise 1 b. 7 1. a. N c. -1 b. Q 13 a. lg 2 c. Z b. x d. N lg y e. Irrational number 2. a. 2 c. − log 2 15 b. -2 d. pq 5 2 loga c. a 2 14. 0.6309 d. -5 16. 4 e. -1 17. a. 4 f. 0 or 3 b. .783 g. 0 or 2 c. 0 or 2 3. 125 6. 15 Exercise 2 4 1. a. 1 b. 6 7. 1 1 5 c. d. 8. a. 7 27 5 2 e. 8 f. 8 b. 3 2 g. 6 h. 6 c. 5−2 6 i. 3 d. 66 − 24 6 2. a. 2 5 b. 6 e. 18 3 f. 2 5 2 15 + 3 ( 3 + 1)(1 + 2 ) c. d. g. 11 2 17 2 9. a. 5 3 ( 3 + 2 2 )( 5 + 3 ) e. f. 2 2 3 2 b. 3. a. -4 2 5 b. -2 c. 2− 2 1 c. d. 5( 3 − 2 ) 2 e. 1 d. 2 ( 3 + 1)( 2 + 1) 3 2 e. f. 1 2 (4 − 3 3 )( 6 + 2 3 ) f. 2 6 4. a. 1 or lg10 g. 5 +6 a2 h. a(1 + a ) b. lg bc 1− a x y i. (2 a + b ) 2 c. lg z3 4a − b a 3c 10. a. 3 2 + 2 3 + 30 d. logc 2 b 12 5. log c a b. 2− 6+2 1 + log c b 4 9. a. -4.419 11. 1 b. 5.782 -6- FIS 0444 Mathematics 1 Unit 1- Algebra c. 1.870 5. ( x + 2)(x 2 − 2 x + 4) − 16 d. 0.284 10. a. 0.50 or 0.793 6. ( x + 9)(x 2 − 3x + 2) b. -1 or 2.322 7. ( x + 2)(x 3 − 2 x 2 + 2 x − 4) + 11 11. a. 1 or 2 8. -14 b. 1 or 3 9. -1 12. 3 10. p = 3; q = −13 13. x = 4, y = 16; x = 16, y = 4 11. p = 3; r = 0; q = 2 14. x = 64, y = 16 12. (3x − 4)(2 x + 1) 2 Exercise 3 13. ( x − 2)(x + 2)(2 x + 1) 1. a. 2 b. 3 c. 15 d. 4 14. ( x − 2)(x − 3)(2 x + 1) e. 7 f. 5 15. x − 2, x − 2, x + 6 2. a. 9 b. 1 16. 150 c. 1 d. 109 17. x 3 − 3x 2 − x + 3 − 2 18. x 4 − 8x 3 + 14x 2 + 8x − 15 e. 3909 f. -1212 19. 3 3 15 3. a. x + 5 x − 3x + 3x − 5 4 3 2 − x + 3x + 2 x−9 2 2 b. − 6 x 5 + x 4 − 13x 2 − 10x + 1 c. − 2 x 4 + 15x 3 − 14x 2 + 9 x + 20 Exercise 5 d 9 x 5 + 2 x 4 + 23x 2 + 15x − 26 1. 1 1 − e. 3x 5 − x 4 + 15x 3 − 6 x 2 + 14x + 9 x −1 x + 1 f. − 9 x 5 + 2 x 4 + 5x 3 − 23x 2 − 12x 2. 1 1 − 4. a. -7 x −1 x + 4 b. 5 3. 2 2 c. -1 − x −3 x +3 d. 2 4. 3 2 e. -1 − f. 3 x−4 x+2 5. a. x + 5; 0 5. 1 3 − + b. x 2 + x + 3; − 1 2 ( 2 x − 1) 2 ( 4 x − 3) c. 4 x 3 + 8x 2 + 13x + 27; 56 6. 2 + 3 − 1 d. x − 1; 6 x − 1 x − 2 x + 2 2x −1 7. 1 3 e. 1; 2 x 3 + x 2 − 1 − 2 x + 3 ( 2 x + 3) 2 f. 4 x 2 + 7; x + 9 6. x 2 − x + 1; 5x − 1 8. 2 1 2 − 3− 7. x+3 x x x+2 9. 4 4 2 1 − + + Exercise 4 x + 2 x − 1 ( x − 1) ( x − 1)3 2 1. quotient; remainder 10. x +1 1 2. a. factor − b. k x2 + 3 x 3. a. No 11. 2x − 5 5 + 2 b. Yes x + x + 2 x +1 2 c. No 12. 1 x+2 1 − + 4. a. Yes x + 1 ( x 2 + 1) 2 2 x b. Not c. Yes -7-