Multiple Choice

1. A line passes through the points $P(-3, 7)$ and $Q(x, 1)$. If the slope of the line is $m = -\frac{2}{3}$, what is the value of $x$?

   $9$

   $-6$

   $3$

   $6$

2. The points $A(1, 5)$, $B(4, k)$, and $C(10, -7)$ are collinear. What is the value of $k$?

   $2$

   $1$

   $3$

   $-1$

3. A company's profit, $P$, in thousands of dollars, is a linear function of the number of items sold, $n$. When $100$ items are sold, the profit is $\$15,000$. When $300$ items are sold, the profit is $\$27,000$. What is the average increase in profit for every $50$ items sold?

   $\$150$

   $\$3,000$

   $\$60$

   $\$12,000$

4. A straight path passes through the origin $(0,0)$, point $A(2, y_A)$, and point $B(x_B, 10)$. The slope of the path is $m = \frac{3}{4}$. What is the sum of $y_A$ and $x_B$?

   $\frac{89}{6}$

   $\frac{43}{5}$

   $\frac{46}{3}$

   $\frac{13}{2}$

5. A line passes through the points $P(a, 2a+1)$ and $Q(2a, a-3)$. What is the slope of this line in terms of $a$?

   $\frac{-a + 2}{a}$

   $\frac{-a - 4}{a}$

   $\frac{3a - 2}{a}$

   $-a - 4$

6. The slope of the line passing through points $P(k, 7)$ and $Q(-3, 1)$ is $m = -\frac{3}{2}$. What is the value of $k$?

   $k = 7$

   $k = -7$

   $k = 1$

   $k = -11$

7. Points $A(2, 5)$, $B(x, 11)$, and $C(-1, -4)$ are collinear. What is the value of $x$?

   $x = 2$

   $x = 0$

   $x = 6$

   $x = 4$

8. A small plane is descending. At $1:00$ PM, its altitude is $6000$ feet. At $1:15$ PM, its altitude is $4500$ feet. What is the average rate of change of altitude, in feet per minute, during this time interval?

   $400$ feet per minute

   $-400$ feet per minute

   $100$ feet per minute

   $-100$ feet per minute

9. The vertices of a quadrilateral are $P(1, 3)$, $Q(4, y)$, $R(7, 3)$, and $S(4, 0)$. If $PQRS$ is a parallelogram, what is the value of $y$?

   $y = 6$

   $y = 3$

   $y = 0$

   $y = 9$

10. A line passes through the point $P(2, -3)$ and has a slope of $m = \frac{1}{2}$. If another point on this line is $Q(x, 0)$, what is the value of $x$?

    $x = 8$

    $x = -4$

    $x = 2$

    $x = 5$

11. A line passes through the points $(k, 5)$ and $(4, -1)$ and has a gradient (slope) of $-2$. What is the value of $k$?

    $2$

    $1$

    $-3$

    $7$

12. A cyclist travels at a constant speed. At $10:00$ AM, they have cycled $40$ km from their starting point. By $1:30$ PM on the same day, they have cycled $131$ km from their starting point. If the 'distance cycled' is plotted on the y-axis and 'time elapsed from a reference point' (in hours) is on the x-axis, what is the gradient (slope) of the line representing the cyclist's journey?

    $42 \text{ km/h}$

    $37 \text{ km/h}$

    $30 \text{ km/h}$

    $26 \text{ km/h}$

13. The points $P(-3, -2)$, $Q(x, 4)$, and $R(9, 10)$ lie on the same straight line. What is the value of $x$?

    $-1$

    $0$

    $6$

    $3$

14. Consider a quadrilateral with vertices $A(1, 1)$, $B(5, 3)$, $C(x, 7)$, and $D(3, 9)$. If side $BC$ is parallel to side $AD$, what is the value of $x$?

    $1$

    $6$

    $4$

    $7$

15. A line passes through the points $(m, 7)$ and $(2m + 1, -5)$. If the gradient (slope) of this line is $-4$, what is the value of $m$?

    $0$

    $3$

    $-1$

    $2$