Prime Review Online Algebra 01 Solutions PDF

## PRIME REVIEW ONLINE ### MECHANICAL ENGINEERING REVIEW ### ALGEBRA 01 #### MODULE 1.1 SOLUTIONS 1. When rounded-off to four significant figures, 102.4886 becomes: - A. 102.4 - B. 102.4889 - C. 102.48 - D. 102.5 **Solution:** Four Significant of 102.4886 = 102.5 **Ans. D** 2. Which of...

## PRIME REVIEW ONLINE ### MECHANICAL ENGINEERING REVIEW ### ALGEBRA 01 #### MODULE 1.1 SOLUTIONS 1. When rounded-off to four significant figures, 102.4886 becomes: - A. 102.4 - B. 102.4889 - C. 102.48 - D. 102.5 **Solution:** Four Significant of 102.4886 = 102.5 **Ans. D** 2. Which of the following statements is correct? - A. 0.0010 has one significant figures - B. 200 has two significant figures - C. 100.00 has five significant figures - D. 0.003280 has three significant figures **Solution:** - A. 0.0010 has one significant figures - Has 2 significant figures - B. 200 has two significant figures - Has 1 significant figure - C. 100.00 has five significant figures - Has 5 significant figures - D. 0.003280 has three significant figures - Has 4 significant figures **Ans. C** 3. The most significant digit of the number 0.2015 is - A. 0 - B. 1 - C. 2 - D. 5 **Solution:** 0.2015 ↓ Most Significant digit **Ans. 2 C.** 4. Solve for the value of x of the equation: $x^{10} = 10$ → so; the exponents: $x = \sqrt[10]{10}$ - A. 0.618 - B. 1.258 - C. 1.852 - D. 0.453 **Solution:** subst: $x = \sqrt[10]{10}$ $x = [10]^{\frac{1}{10}} = 10^{\frac{1}{10}} = 1.258$ **Ans. B** 5. If $9^x = 27^y$ and $8^z = 16^y$, then what is the value of $x : y : z$? - A. 1:3:6 - B. 3:1:2 - C. 6:2:1 - D. 6:4:3 **Solution:** $9^x = 27^y$ $3^2 = 3^3$ eqn. (1) $8^z = 16^y$ $2^3 = 2^4$ eqn. (2) equate exponents eqn. (1): $2x = 3y$ $y = \frac{2x}{3}$ equate exponents eqn. (2) and subs't y: $3y = 4z$ $z = \frac{3y}{4}$ $z = \frac{2x}{3} \times \frac{3}{4}$ $z = \frac{x}{2}$ thus: $x : y : z$ $x : \frac{2x}{3} : \frac{x}{2}$ $6 : 4 : 3$ **Ans. D** ### MATHEMATICS & BASIC ENGINEERING SCIENCES ### PRIME'S SUPPLEMENTARY ENHANCERS (ALGEBRA 01) 6. If $3x = 4y$, then $\frac{3x^2}{4y^2}$ =? - A. 3/4 - B. 4/3 - C. 27/64 - D. 9/16 **Solution:** $\frac{3x}{4y} = 1$ $\frac{3x^2}{4y^2} = \frac{3^2 x^2}{4^2y^2} = (\frac{3x}{4y})^2 = 1^2 = 1$ **Ans. B** 7. If $log_{10}x = \frac{1}{n}$, then the value of x is: - A. 10 - B. 10⁻¹ - C. 10ⁿ - D. 10 **Solution:** $log_{10}x = \frac{1}{n}$ $x = 10^{\frac{1}{n}}$ - property of logarithm **Ans. B** 8. In a morning walk, three persons step off together. Their steps measure 75 cm, 80 cm and 90 cm respectively. What is the minimum distance each should walk so it can cover the same distance in complete steps? - A. 3600 cm - B. 4500 cm - C. 7200 cm - D. 540,000 cm **Solution:** Person A: 75 cm Person B: 80 cm Person C: 90 cm Minimum distance each can cover means we need the LCM of the three: A: 75 = 5² * 3 B: 80 = 5 * 2⁴ C: 90 = 5 * 3² * 2 thus: LCM = 2⁴ * 3² * 5² = 3600 cm **Ans. A** 9. Determine the absolute value of the complex number 3 + 4i. - A. 4 - B. 8 - C. 5 - D. 6 **Solution:** Abs value = $\sqrt {a^2 + b^2}$ = $\sqrt {3^2 + 4^2}$ = 5 Alternate Solution: By: Calculator Mode 2: CMPLX |3 + 4i| = 5 **Ans. C** 10. Find the value of x : $\sqrt(20 - x) = x$ - A. 4, -5 - B. -4, 5 - C. -4, -5 - D. No Solution **Solution:** $(\sqrt(20-x))=(x)^2$ $20 - x = x^2$ $x^2 + x - 20 = 0$ → $(x + 5)(x - 4) = 0$ equate each factor to zero and solve the value of x $(x + 5) = 0$; x = -5 checking for (x = 5) $\sqrt(20 - x) = x$ $\sqrt(20 + 5) = 5$ $\sqrt(25) = ±5$ $(x - 4) = 0$; x = 4 checking for (x = 4) $\sqrt(20 - x) = x$ $\sqrt(20 - 4) = 4$ $\sqrt(16) = ±4$ therefore: x = -5 and x = 4 are extraneous roots **Ans. D. No Solution** 11. Find the value of x to satisfy the following equations: x + y = 7; x² + 2y² = 34 - A. 3 - B. 4 - C. 2 - D. 5 **Solution:** x + y = 7 eq. (1) and x² + 2y² = 34 eqn. (2) Subs. y from eqn. (1) to eqn. (2): x² + 2(7 - x)² = 34 Use Calculator: Shift Solve x = 4 **Ans. B.** 12. What are all the ordered triples of real numbers (x, y, z) which satisfy the following equations? $(x + y)(x + y + z) = 384$ $(y + z)(x + y + z) = 288$ $(x + z)(x + y + z) = 480$ - A. I only - B. II only - C. I and II - D. none of these **Solution** Add all the given equations: $(x + y)(x + y + z) = 384$ → eqn. (1) $(y + z)(x + y + z) = 288$ → eqn. (2) $(x + z)(x + y + z) = 480$ → eqn. (3) $(2x + 2y + 2z)(x + y + z) = 384 + 288 + 480$ $2(x + y + z) = 1152$ $x + y + z = \sqrt(576) = ± 24$ From eqn. (1): $(24 - x)(± 24) = 288$ $24 - x = ± 12$ $x = ± 12$ From eq. 2: $(24 - y)(± 24) = 480$ $24 - y = ± 20$ $y = ± 4$ From eq. 3: $(24 - z)(± 24) = 384$ $24 - z = ± 16$ $z = ± 8$ thus: I and II are correct Alternate Solution: Subst. x, y, and z of I and II to the given equations. **Ans. C.** 13. Solve for the constant "m" in $x^2 - 4x + 3m = 0$ if one root exceeds by the other by 2. - A. 1 - B. 3 - C. 2 - D. 5 **Solution:** Sum of roots: $r_1 + r_2 = 4$ note: $r_1 = r_2 + 2$ (given condition) $r_2 + (r2 + 2) = 4$ $r_2 = 1$ from eqn. 1. $r_1 = 3$ Product of roots: ($r_1 = 1; r_2 = 3$) $(1)(3) = 3m$ **Ans. A. m = 1** 14. Which of the following is a prime number? - A. 377 - B. 313 - C. 357 - D. 333 **Solution:** 377 = 13 x 29 → composite 313 = prime 357 = 3 x 119 → composite 333 = 3 x 111 → composite **Ans. B.** 15. If $y = x + \frac{1}{x}$, x cannot be equal to 0, then the equation $(x^2 - 3x + 1)(x^2 - 5x + 1) = 6x^2$ reduces to: - A. $y^2 - 8y + 9 = 0$ - B. $y^2 - 8y + 7 = 0$ - C. $y^2 + 8y + 9 = 0$ - D. $y^2 - 8y + 7 = 0$ **Solution:** $y = x + \frac{1}{x}$ → eqn. 1 $(x^2 - 3x + 1)(x^2 - 5x + 1) = 6x^2$ → eqn. 2 From eqn. 1: $(y - 3)(y - 5) = 6$ $y^2 - 8y + 15 = 6$ $y^2 - 8y + 9 = 0$ **Ans. A** 16. Solve the value of x: $x^2 + \frac{x}{4} + \frac{1}{4} = \frac{17}{4}$ - A. 8 - B. 1/8 - C. -8 - D. All of these **Solution** $4(x^2) + 4 = 17x$ By Factoring: $4(x^2) - 17(x) + 4 = 0$ $[4(x) - 1](x - 4) = 0$ Equate each factor to zero: $4(x) - 1 = 0$ $x = \frac{1}{4}$ $x - 4 = 0$ $x = 4$ x = 4, $\frac{1}{4}$ thus: $x = ± 8$ Alternate Solutions: Use Calculator By Shift Solve all the choices ② By using CALC to check the eqn. Ex. Try choices A, B, and C. $x^2 + \frac{x}{4} - \frac{17}{4}$ CALC x 8, -8, $\frac{1}{8}$ = all of these equated to 0 thus; x = 8, -8, $\frac{1}{8}$ and $-\frac{1}{8}$ are values of x **Ans. D.** 17. If a, b and c be three real numbers such that $a + 2b + 4c = 0$. Then, the equation $ax^2 + bx + c = 0$. - A. has its roots lying within 1 < x < 0 - B. has both the roots complex - C. has its roots lying within 2 < x < 6 - D. has one of its roots equal to 1½ **Solution:** a + 2b + 4c = 0 eqn. 1 $ax^2 + bx + c = 0$ → eqn. 2 From eqn. 1, $-\frac{a}{4} + -\frac{2b}{4} + c = 0$ → $-\frac{a}{4} + -\frac{b}{2} + c = 0$ By comparison of eqn. 2 and eqn. 3 x = 2 **Ans. D.** 18. Two or more equations are equal if and only if they have the same - A. solution set - B. degree - C. order - D. variable set **Ans. A.** 19. Solve for x: $\sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1$ - A. 13/25 - B. 17/25 - C. 16/25 - D. 22/25 **Solution:** $\sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1$ $\sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1$ Use Calcu: Shift Solve thus; $x = 0.64 = \frac{16}{25}$ Alternate Solution: CALC can also be used **Ans. C.** 20. Which of the following statement is NOT correct? - A. π⁰ = 1 - B. e¹ = 1 - C. 0⁰ = 1 - D. ln 1 = 0 **Solution:** π⁰ = 1 correct e¹ = 1 correct 0⁰ indeterminate ln 1 = 0 correct **Ans. C.** 21. The equation whose roots are twice the roots of the equation $x^2 - 3x + 3 = 0$ is: - A. $x^2 - 3x + 6 = 0$ - B. $x^2 - 6x + 12 = 0$ - C. $x^2 - 4x + 8 = 0$ - D. $x^2 - 8x + 6 = 0$ **Solution:** $x^2 - 3x + 3 = 0$ By quadratic formula: x = $\frac{3 ± \sqrt({3^2 - 4(1)(3)})}{2(1)} = \frac{3 ± \sqrt(9 - 12)}{2} = \frac{3 ± \sqrt(-3)}{2} = \frac{3 ± \sqrt3i}{2}$ Twice the roots will be : $2[x] = 3 + \sqrt3i$ Sum: $\frac{3}{2} + \frac{3 - \sqrt3i}{2}$ = 6 Product: $\frac{3}{2} \times \frac{3 - \sqrt3i}{2}$ = $\frac{9 + 3}{4}$ = 12 thus: the eqn. is $x^2 - 6x + 12 = 0$ Alternate Solution: Subst, any arbitrary value on y and a to solve for x and see if the same value is also get when y and a are substituted from the choices. i.e. say: a = 3 and y = 4.4 then: $log_a x = 5(4.4) + log_ay$ x = $1.381 * 10^{22}$ Try: D. $y^{a^n}$ $(4.4)^{(3)^{4.4}}$ = $1.381 * 10^{22}$ **Ans. D.** 22. Solve the value of x in the equation: 38.5x = 6.5(x - 2) - A. -2.10 - B. -5.01 - C. 3.8 - D. -3.1 **Solution:** 38.5x = 6.5*2 Use Calculator: Shift Solve x = - 2.10 **Ans. A.** 23. Simplify: log x + log x³ + log x³ - A. log (x + x² + x) - B. log 9x - C. 9 log x - D. 15 log x **Solution:** log x + log x² + log x³ log x(x²)(x³) = log x⁹ 9 log x Alternate Solution: let x, say 2.2 So, log x + log x³ + lóg x³ CALC x ? 2.2 = 3.081804127 then, look for the same value in the choices when x is subs. Try choice C. 9 log x CALC x ? 2.2 = 3.081804127 Since they are equal therefore this is the answer. **Ans. C.** 24. Given the equation $2x^2 + x - 10 = 0$. Find the product of roots. - A. 5 - B. -1/2 - C. -5 - D. ½ **Solution:** 2x² + x - 10 = 0 Alternate Solution: by: Calculator Ax² + Bx + C = 0 2x² + x - 10 = 0 Mode 5(eqn) 3(quaratic) Product of roots: $\frac{C}{A} = -\frac{10}{2} = -5$ x = 2, 5/2 get the product: 2(-5/2) = -5 **Ans. C.** 25. Given: $log_a x = 5y + log_a y$. Find the value of x. - A. x = a⁵x - B. x = 5ay - C. x = 5a⁵ - D. x = ya⁵ **Solution:** $log_a x = 5y + log_a y$ $log_a x = log_a y + 5y$ $log_a (\frac{x}{y}) = 5y$ $\frac{x}{y} = a^{5y}$ $x = ya^{5y}$ **Ans. D.** 26. If the sum of the roots of a quadratic equation is 12 and the product is 36, then what is difference of the roots? - A. 1 - B. 0 - C. 3 - D. 6 **Solution:** Sum = 12 Product = 36 then; the eqn. is $x^2 - 12x + 36 = 0$ By Mode 5 (egn) 3 (quadratic), x = 6 Then, the difference 6 - 6 = 0 **Ans. B.** 27. What is the logarithm of a negative number. - A. zero - B. rational - C. irrational - D. complex **Solution:** log(negative no) = complex **Ans. D.** 28. Solve the value of x: $x^{-3} - 9x^{-3/2} + 8 = 0$. - A. 1/8 - B. - 1 - C. 8 - D. 4 **Solution** $x^{-3} - 9x^{-3/2} + 8 = 0$ $(x^{-3/2})^2 - 9x^{-3/2} + 8 = 0$ By quadratic formula/factorization: x = 8, 1 thus: x = 1, 1/4 Alternate Solution: By Calculator By Shift Solve: x = 1/4 ② By CALC using trial and error of choices as x. **Ans. D.** 29. Solve the value of x: 1.4 = $(\frac{0.0613}{x})^{-3.2}$ - A. 0.5471 - B. 0.04751 - C. 0.7541 - D. 0.4571 **Solution:** 1.4 = $(\frac{0.0613}{x})^{-3.2}$ Use Calcuator: Shift Solve: x = 0.04751 **Ans. B.** 30. The surface area of A of a hollow cone is given by A = πrL.. Determine, correct to 1 decimal place, the surface area when r = 3.0 cm and L = 8.5 cm. - A. 80.11 cm² - B. 80.1 cm² - C. 80.1106 cm² - D. 80.0 cm³ **Solution:** A = πrL = π(3.0 cm)(8.5 cm) A = 80.1 cm² **Ans. B.** 31. Solve for x if $8^{(x)} = 2^{(y + 2)}$ and $16^{(3x - y)} = 4^y$. - A. 1 - B. 2 - C. 3 - D. 4 **Solution** $8^ x = 2^4$ eqn. 1 $16^ {3x-y} = 4^y$ eqn. 2 from eqn. 1 and equate exponents: $2^3 = 2^4$ $3x = y + 2$ eqn. 3 from eqn. 2 and equate exponents: $2^4(3x - y) = 2^2 y$ eqn. 4 from eqn. 3 and eqn. 4: $3x = y + 2$ → $3x - y = 2$ $4(3x - y) = 2y$ → $12x - 6y = 0$ By: Calculator: Mode 5 (equation) 1(axn + byn = cn) x = 2, y = 4 **Ans. B.** 32. What are the roots of the cubic equation $x^3 - 8x - 3 = 0$? - A. x = 7.90, -3, -0.38 - B. x = -3, -2, 2 - C. x = -3, -0.38, 2 - D. x = -2.62, -0.38, 3 **Solution:** $x^3 - 8x - 3 = 0$ By: Calcu: Mode 5 (equation) 4(ax³ + bx² + cx + d = 0) x = 3, -0.382, -2.62 **Ans. D.** 33. What is most nearly the value of x that satisfies the expression $17.3 = e^{1.1x}$? - A. 0.17 - B. 2.6 - C. 5.8 - D. 15 **Solution:** $17.3 = e^{1.1x}$ Use Calcu: Shift Solve, x = 2.6 **Ans. B** 34. What is the solution of the equation $50x^2 + 5(x - 2)^2 = -1$, where x is a real-valued variable? - A. - 6.12 or - 3.88 - B. - 0.52 or 0.700 - C. 7.55 - D. no solution **Solution:** $50x^2 + 5(x - 2)^2 = -1$ No solution because no real value of x will result in a negative no. when squared. **Ans. D.** 35. Which of the following is a happy number ? - A. 143 - B. 69 - C. 88 - D. 100 **Solution:** Happy Number: 100: $1^2 + 0^2 + 0^2 = 1$(Results in 1) **Ans. D.** 36. How many real solutions does the equation: $x^7 +14x^3 + 16x + 30x - 560 = 0$ have? - A. 1 - B. 3 - C. 5 - D. 7 **Solution:** $x^7 +14x^3 + 16x + 30x - 560 = 0$ By Descartes Rule of sign: *Positive The number of positive real roots a polynomial equation f (x) = 0 with real coefficient is equal to the number of variation in signs of f (x) or less than that number by an even positive integer! f(x) = $x^7 + 14x^3 + 16x + 30x - 560 = 0$ * Negative f(-x) = $x^7 - 14x^3 + 16x - 30x - 560$ none NO change of sign so NO negative real root Thus, number of real root(s) = 1 **Ans. A.** 37. What is the discriminant of the equation: 4x² - 8x + 5 = 0 ? - A. 8 - B. -16 - C. 16 - D. -8 **Solution** 4x² - 8x + 5 D = b² - 4ac = (-8)² - 4(4)(5) D = -16 **Ans. B.** 38. How many positive real root(s) in the given equation $4x^3 - 2x^2 + x - 3 = 0$? - A. 2 - B. 1 - C. 3 - D. 0 **Solution:** $4x^3 - 2x^2 + x - 3 = 0$ Mode 5(equation) 4(cubic) x = 1, -0.25 + 0.83i, 0.25 - 0.83i only one **Ans. B.** 39. What is the natural logarithm of e? - A. $\frac{1}{xy}$ - B. xy - C. $2.718xy$ - D. $\frac{2.718}{xy}$ **Solution:** By property of log: ln e = $xy\ln e = xy$ **Ans. B.** 40. What is the value of $(0.001)^\frac{2}{3}$? - A. antilog [log 0.001] - B. antilog [log 0.001]² - C. antilog [log(0.001)]$\frac{2}{3}$ - D. antilog [log 0.001]$\frac{2}{3}$ **Solution:** $(0.001)^\frac{2}{3} = x$ then, $log x = log (0.001)^\frac{2}{3}$ $log x = \frac{2}{3}log (0.001)$ $x = antilog[\frac{2}{3} log (0.001)]$ **Ans. D.** 41. Factor the expression $3x^3 - 3x^2 - 18x$ - A. 3x(x - 3)(x + 2) - B. 3x(x + 3)(x - 2) - C. 3x(x + 3)(x + 2). - D. 3x(x - 3)(x - 2) **Solution:** $3x^3 - 3x^2 - 18x = 3x(x^2 - x - 6)$ = 3x(x - 3)(x + 2) Alternate Solution: Subst. any arbitrary value to x, say x = 1.1 then determine the value. $3x^3 - 3x^2 - 18x$ CALC x? 1.1 = - 19.437 then check the choices for the same value when x = 1.1 Try A. (3x)(x - 3)(x + 2) CALC x ? 1.1 = -19.187 **Ans. A.** 42. Which of the following best describe 8 + 0i? - A. irrational number - B. real number - C. imaginary - D. surd **Solution** 8 + 0i Real No. **Ans. B.** 43. Simplify the given fraction $\frac{3x - 1}{x + 3} \times \frac{x^2 - 1}{x^2 + 3x + 2} \times \frac{x + 2}{2x - 3}$ - A. $\frac{1}{2x - 3}$ - B. $\frac{1}{x + 1}$ - C. $\frac{1}{x + 1}$ - D. $\frac{1}{x - 1}$ **Solution** $\frac{3x - 1}{x + 3} \times \frac{x^2 - 1}{x^2 + 3x + 2} \times \frac{x + 2}{2x - 3}$ = $\frac{3x - 1}{x + 3} \times \frac{(x + 1)(x - 1)}{(x + 3)(x + 1)} \times \frac{x + 2}{2x - 3}$ = $\frac{(3x-1)(x+1)(x-1)(x+2)}{(x+1)(x-1)(x+3)(2x-3)}$ = $\frac{(3x^2 + 5x + (-2) - (x^2 + 2x - 3) - (x - 1))}{x^2 + 3x + 2}$ = $\frac{5x^2 + 13x + 1 - (x^2 + 2x - 3)}{(x + 1)(x + 2)}$ = $\frac{5x^2 + 13x + 1 - x^2 - 2x + 3}{(x + 1)(x + 2)}$ = $\frac{4x^2 + 11x + 4}{(x + 1)(x + 2)}$ = $\frac{4x^2 + 11x + 4}{(x + 1)(x + 2)} \times \frac{(x - 1)}{(x - 1)}$ = $\frac{(4x^2 + 11x + 4)(x - 1)}{(x+1)(x+2)(x-1)}$ = $\frac{(4x^3 + 7x^2 + (4) - (4x^2 + 11x + 4))}{(x + 1)(x + 2)(x - 1)}$ = $\frac{4x^3 + 7x^2 + 4 - 4x^2 - 11x - 4}{(x + 1)(x + 2)(x - 1)}$ = $\frac{4x^3 + 3x^2 - 11x}{(x + 1)(x + 2)(x - 1)}$ = $\frac{(x + 1)(x + 2)(4x - 1)}{(x + 1)(x + 2)(x - 1)}$ = $\frac{4x - 1}{x - 1}$ Alternate Solution: Substitute any arbitrary value to x, i.e. x = 2.2 and determine the value of the expression $\frac{3x - 1}{x + 1} \times \frac{x^2 - 1}{x^2 + 3x + 2} \times \frac{x + 2}{2x - 3}$ CALC x ? 2.2 = 5/6 Check the choices by substituting the value x as 2.2 Try D. $\frac{1}{x - 1}$ CALC x 2.2 = 5/6 Since the values are the same they are equal, **Ans. D.** 44. Find the equation of whose roots are the reciprocal of the roots of $2x^2