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    LEARNING UNIT 1-Exercise

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    Xuan Xian

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    LEARNING UNIT 1-Exercise

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    Xuan Xian
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    LEARNING UNIT 1-exercise

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    LEARNING UNIT 1-Exercise

    Uploaded by

    Xuan Xian

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
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    LEARNING UNIT 1 (BASIC ALGEBRAIC OPERATIONS)

    EXERCISE

    1. Given 𝑃(𝑥) = 2𝑥 2 + 4𝑥 + 1 and 𝑄(𝑥) = 2𝑥 + 3. Determine the following.


    (a) 𝑃(𝑥) + 𝑄(𝑥)
    Ans: 2𝑥 2 + 6𝑥 + 4

    (b) 𝑄(𝑥) − 𝑃(𝑥)


    Ans: −2𝑥 2 − 2𝑥 + 2

    (c) 𝑃(𝑥)𝑄(𝑥)
    Ans: 4𝑥 3 + 14𝑥 2 + 14𝑥 + 3

    2. Given point 𝑋(4,3) and point 𝑌(−2,0).


    (a) Find the midpoint and distance between them. Ans: (1, 3⁄2)

    (b) Sketch the point on the Cartesian Plane.


    (c) Determine the slope of the line obtained with the given pairs of points.
    Ans: 1⁄2

    (d) Find the Point-Slope Form and Slope-Intercept Form of the equation of
    the line.
    1 1
    Ans: Point-Slope Form, 𝑦 − 3 = (𝑥 − 4), Slope-Intercept Form, 𝑦 = 𝑥 + 1
    2 2

    (e) Find the y-intercept and x-intercept of the line.


    Ans: y-intercept, (0,1) and x-intercept, (−2, 0)

    3. 2𝑥 1 1
    Solve 𝑥 2 −1 + 𝑥+1 = 𝑥−1. Ans: No solution

    4. A chemist wants to make 500ml of 5% HCL from 10% solution. What is the
    volume of water and the volume of 10% acid needed?
    Ans: 250 ml water and 250 ml of 10% acid

    5. Solve
    (a) 4𝑥 2 − 12 = 4𝑥 + 36 (using factoring method)
    Ans: 𝑥 = 4 or 𝑥 = −3
    2
    (b) 9𝑥 = 24 (using extracting square roots)
    2√6
    Ans: 𝑥 = ±
    3

    2
    (c) 2𝑥 − 3𝑥 − 5 = 0 (using completing the square)
    Ans: 𝑥 = 5⁄2 or 𝑥 = −1

    (d) 5𝑥 2 + 3𝑥 − 7 = 0 (using quadratic formula)


    −3±√149
    Ans: 𝑥 =
    10
    6. Determine the nature or type of solutions of the following equations.
    (a) −𝑥 2 − 6𝑥 + 6 = 0
    Ans: Two distinct real solutions, graph has two x-intercepts

    (b) 𝑥 2 − 4𝑥 + 4 = 0
    Ans: One repeated real solution, graph has one x-intercept

    (c) 2𝑥 2 + 8 = 0
    Ans: No real solutions, graph has no x-intercept (has two complex solution)

    7. Write the following quotient of complex numbers in standard form.


    (a) 4 + 3𝑖
    2 − 5𝑖
    7 26
    Ans: + 𝑖
    29 29

    (b) 2𝑖
    −3 + 4𝑖
    8 6
    Ans: − 𝑖
    25 25

    8. Solve |𝑥 2 − 3| = 5 − 2𝑥
    Ans: 𝑥 = 2 or 𝑥 = −4 , 𝑥 = 1 ± 𝑖

    9. Solve the following inequalities. Write using interval notation or solution set
    and present the inequality on a real number line.
    (a) 3𝑥 − 9 > 0
    Ans: (3, ∞)

    (b) 0 ≤ −4𝑥 + 3 < 5


    Ans: (− 1⁄2 , 3⁄4]

    10. Solve 2𝑥 3 −3𝑥 2 − 32𝑥 > −48 using the following methods.
    Ans: (−4, 3⁄2) ∪ (4, ∞)

    (a) Real number line


    (b) Graphical method
    (c) Table of sign

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