MCR3U Final Exam Review: Free Practice Exam

Free MCR3U Practice Exam

21 questions (23 parts), up to 3 from each of the 7 units, taken from the free tests on Martian Lab. Do them in one sitting, on paper, then open the answers.

1. Functions & Transformations

   For the following relations, fill in the table.

   Image	Domain	Range	Solve	Is this relation a function?
   A	D =	-	Find $x$ if $f(x)=1$
   B	D =	R =	$f(2) =$
   C	-	R =	$f^{-1}(0) =$
   D	D =	-	-

   

   Show solution

   Image	Domain	Range	Solve	Is this relation a function?
   A	D = $\{x \in \mathbb{R},\ x > -4\}$	-	$\{x \in \mathbb{R},\ 1 \leq x \leq 3\}$	Yes
   B	D = $\{x \in \mathbb{R},\ -4 \leq x \leq 5\}$	R = $\{-1, 2\}$	$f(2) = 2$	Yes
   C	-	R = $\{y \in \mathbb{R},\ -3 \leq y \leq 4\}$	$f^{-1}(0) =-2, 2, or 4$	Yes
   D	D = $\{1, 2, 4\}$	-	-	No

   

2. Functions & Transformations

   State the domain of $f(x) = \dfrac{3}{-x-2}$. D = _____

   Show answer

   $\{\chi \in \mathbb{R},\ \chi \neq -2\}$

3. Functions & Transformations

   State the domain of $f^{-1}(x)$, if $f(x) = -3(x-9)^2 - 5$. D = _____

   Show answer

   Range of $f(x)$ is $\{y \in \mathbb{R},\ y \leq -5\}$ Therefore, the domain of $f^{-1}(x)$ is $\{\chi \in \mathbb{R},\ \chi \leq -5\}$

4. Rational Expressions

   Factor fully. $x^{2} + 2xy + y^{2} - z^{2} - 2z - 1$

   Show answer

   $= x^{2} + 2xy + y^{2} - (z^{2} + 2z + 1)$ $= \left(x + y\right)^{2} - \left(z + 1\right)^{2}$ $= \left(x + y - z - 1\right)\left(x + y + z + 1\right)$

5. Rational Expressions

   Simplify the following, show all your work. State restrictions if necessary. $\dfrac{4x}{6x^2+7x-3} - \dfrac{5x}{2x^2+15x+18}$

   Show solution

   $= \frac{4x}{(2x+3)(3x-1)} - \frac{5x}{(x+6)(2x+3)}$ $= \frac{4x(x+6) - 5x(3x-1)}{(2x+3)(3x-1)(x+6)}$ $= \frac{4x^2 - 24x - 15x^2 + 5x}{(2x+3)(3x-1)(x+6)}$ $= \frac{-11x^2 - 21x}{(2x+3)(3x-1)(x+6)}$ $x \neq \dfrac{-3}{2}, \dfrac{1}{3}, -6$

6. Rational Expressions

   Simplify and state any restrictions on the variable. $\frac{x-2}{6x^2-7x-5} \div \frac{-3x^2+6x}{3x^2+7x-20} + \frac{2x+3}{4x^2-12x+9}$

   Show solution

   $= \frac{(x-2)}{(2x+1)(3x-5)} \div \frac{-3x(x-2)}{(x+4)(3x-5)} + \frac{2x+3}{(2x-3)^2}$ $= \frac{(x-2)}{(2x+1)(3x-5)} \times \frac{(x+4)(3x-5)}{-3x(x-2)} + \frac{2x+3}{(2x-3)^2}$ $= \frac{(x+4)}{-3x(2x+1)} + \frac{2x+3}{(2x-3)^2}$ $= \frac{(x+4)(2x-3)^2 - 3x(2x+1)(2x+3)}{-3x(2x+1)(2x-3)^2}$ $= \frac{(x+4)\!\left(4x^2-12x+9\right) - 3x\!\left(4x^2+8x+3\right)}{-3x(2x+1)(2x-3)^2}$ $= \frac{4x^3 + 4x^2 - 39x + 36 - 12x^3 - 24x^2 - 9x}{-3x(2x+1)(2x-3)^2}$ $= \frac{-8x^3 - 20x^2 - 48x + 36}{-3x(2x+1)(2x-3)^2},\; x \neq 0, 2, -4, \frac{5}{3}, \frac{-1}{2}, \frac{3}{2}$

7. Quadratic Functions

   Express $y = x^2 - 5x$ in vertex form

   Show answer

   $y = \left(x - \dfrac{5}{2}\right)^2 - \dfrac{25}{4}$

8. Quadratic Functions

   What is the equation of the axis of symmetry of the function $y = 2x^2 - 7x - 24$

   Show answer

   $x = \dfrac{7}{4}$

9. Quadratic Functions

   Solve the linear-quadratic system. Express solution(s) in exact value. $f(x) = 3x^2 + 17x + 16$ $g(x) = -x - 2$

   Show solution

   $3x^2 + 17x + 16 = -x - 2$ $3x^2 + 18x + 18 = 0$ $x^2 + 6x + 6 = 0$ $x = \frac{-6 \pm \sqrt{36 - 24}}{2}$ $x = -3 \pm \sqrt{3}$ $x = -3 + \sqrt{3}, \quad y = 1 - \sqrt{3}$ $x = -3 - \sqrt{3}, \quad y = 1 + \sqrt{3}$ $\therefore$ Points of intersections are $(-3+\sqrt{3},\; 1-\sqrt{3})$ and $(-3-\sqrt{3},\; 1+\sqrt{3})$

10. Exponential Functions

    Evaluate the expression $\dfrac{x^{-3}}{y^n}$ for $x = 4, y = 7, n = -2$.

    1. $\dfrac{49}{64}$
    2. $\dfrac{64}{49}$
    3. $\dfrac{7}{8}$
    4. 3136

    Show answer

    a) $\dfrac{49}{64}$

11. Exponential Functions

    Which of the following is equivalent to the expression $\left(a^3 b^4\right)^{-2}\!\left(a^{-3}b^{-5}\right)^{-4}$?

    1. $a^6 b^{12}$
    2. $a^{12}b^{15}$
    3. $\dfrac{1}{a^6 b^{12}}$
    4. $\dfrac{1}{a^6 b^7}$

    Show answer

    a) $a^6 b^{12}$

12. Exponential Functions

    Which of the following is equivalent to the expression $\left(\dfrac{\sqrt[3]{z^7}}{\sqrt{z^5}}\right)^4$?

    1. $\dfrac{1}{\sqrt{z^3}}$
    2. $z$
    3. $\sqrt[35]{z^4}$
    4. $\dfrac{1}{\sqrt[3]{z^2}}$

    Show answer

    d) $\dfrac{1}{\sqrt[3]{z^2}}$

13. Trigonometric Ratios

    Based on the figure, which of the following represents the length of $AB$?

    

    1. 12
    2. 13
    3. 14
    4. 15

    Show answer

    b) 13

14. Trigonometric Ratios

    Based on figure 1, which of the following represents the trigonometry ratio of $\cot(A)$?

    

    1. $\dfrac{12}{5}$
    2. $\dfrac{12}{15}$
    3. $\dfrac{5}{13}$
    4. $\dfrac{14}{12}$

    Show answer

    a) $\dfrac{12}{5}$

15. Trigonometric Ratios

    Based on figure 1, which of the following represents the measurement of angle $B$ to the nearest degree?

    

    1. $23^\circ$
    2. $21^\circ$
    3. $59^\circ$
    4. $67^\circ$

    Show answer

    d) $67^\circ$

16. Trigonometric Functions

    Complete the following table.

    Function	Period	Amplitude	Phase shift	Vertical displacement	Range
    $y = 3\cos(3x + 180^\circ) - 1$
    $y = \sin\frac{3}{4}(2x + 180^\circ) - 1$
    $y = \frac{1}{3}\cos\!\left(4x - \frac{90^\circ}{2}\right) + 3$
    $y = \underline{\quad} \cos \underline{\quad\quad}$	$90^\circ$	$\dfrac{1}{2}$	$-60^\circ$	$-3$
    $y = \underline{\quad} \sin \underline{\quad\quad}$	$720^\circ$	$3$	$60^\circ$	N/A

    Show solution

    Function	Period	Amplitude	Phase shift	Vertical displacement	Range
    $y = 3\cos(3x + 180^\circ) - 1$	$120^\circ$	$3$	$-60^\circ$	$-1$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -4 \leq y \leq 2 \end{array}\right.$
    $y = \sin\frac{3}{4}(2x + 180^\circ) - 1$	$240^\circ$	$1$	$-90^\circ$	$-1$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -2 \leq y \leq 0 \end{array}\right.$
    $y = \frac{1}{3}\cos\!\left(4x - \frac{90^\circ}{2}\right) + 3$	$90^\circ$	$\dfrac{1}{3}$	$+11.25^\circ$	$+3$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ \frac{8}{3} \leq y \leq \frac{10}{3} \end{array}\right.$
    $y = \frac{1}{2}\cos\left[2(x+60)\right]^\circ - 3$	$90^\circ$	$\dfrac{1}{2}$	$-60^\circ$	$-3$	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -\frac{7}{2} \leq y \leq -\frac{5}{2} \end{array}\right.$
    $y = 3\sin\!\left[\frac{1}{2}(x-60)\right]^\circ$	$720^\circ$	$3$	$60^\circ$	N/A	$\left\{\begin{array}{l} y \in \mathbb{R} \\ -3 \leq y \leq 3 \end{array}\right.$

17. Trigonometric Functions

    How many x intercepts will the graph of the equation $y = -2\sin\theta - 1$ from $-360^\circ \leq \theta \leq 360^\circ$?

    Show solution

    $0 = -2\sin\theta - 1$ $1 = -2\sin\theta$ $-\tfrac{1}{2} = \sin\theta$ $\therefore\ -360^\circ \leq \theta \leq 360^\circ$ $\therefore\ \theta_1 = -30^\circ \; \theta_2 = 330^\circ$ $\theta_3 = 210^\circ \; \theta_4 = -150^\circ$ There are 4 x-intercepts.

18. Trigonometric Functions

    A Ferris wheel at Charlotte's Amusement Park has a diameter of $18\,\mathrm{m}$ and travels at a rate of 6 revolutions every 2 min. At the bottom of the ride, the passenger is $2\,\mathrm{m}$ above the ground.

    1. Sketch the graph of the rider's height above the ground, in meters, verses the time, in seconds, for 2 minutes. Assume the rider starts at the highest position at $t = 0$. Be sure to show the 5 major points of each period, the central axis, and the appropriate labels.
       Show solution

       A correct sketch has amplitude $9$, midline $h=11$, period $20\,\mathrm{s}$, and starts at the maximum point. Major points for one period are $(0,20)$, $(5,11)$, $(10,2)$, $(15,11)$, $(20,20)$. This pattern repeats every $20\,\mathrm{s}$ for $0 \leq t \leq 120$.
    2. Write the equation of the height that represents the height $h$ above the ground at time $t$ as a cosine function.
       Show answer

       $h(t) = 9\cos 18t + 11$
    3. How high above the ground will the rider be at $t = 78\,\mathrm{s}$?
       Show solution

       $h(78) = 9\cos 18 \cdot 78 + 11$ $h(78) = 18.281152\,\mathrm{m}$ $h(78) \approx 18.28\,\mathrm{m}$ $\therefore$ the rider will be approximately $18.28\,\mathrm{m}$ above the ground at $t=78\,\mathrm{s}$.

19. Discrete Functions

    The fifth term in the sequence defined by $t_1 = 3$, $t_n = t_{n-1} - 4$ is

    1. -5
    2. -9
    3. -13
    4. -17

    Show answer

    c) -13

20. Discrete Functions

    The sum of the first eight terms of the arithmetic series where $a = -6$ and $d = \dfrac{1}{4}$ is

    1. -41
    2. -36.75
    3. -32.25
    4. -4.25

    Show answer

    a) -41

21. Discrete Functions

    What is the 15th term of the arithmetic sequence that has $t_2 = -8$ and $t_6 =20$?

    1. -7
    2. 83
    3. 81
    4. -71

    Show answer

    b) 83

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