Question

(Theoretical Probability MC)

A bag consists of 5 marbles. There is 1 red marble, 1 orange marble, 1 green marble, 1 yellow marble and 1 blue marble. Which table shows the sample space for choosing 2 marbles from the bag with replacement?


Red Orange Green Yellow Blue
Red Red Red Red Red Red
Orange Orange Orange Orange Orange Orange
Green Green Green Green Green Green
Yellow Yellow Yellow Yellow Yellow Yellow
Blue Blue Blue Blue Blue Blue

Red Orange Green Yellow Blue
Red Red, Red Red, Orange Red, Green Red, Yellow Red, Blue
Red Red, Red Red, Orange Red, Green Red, Yellow Red, Blue

Red Orange Green Yellow Blue
Red Red, Red Orange, Red Green, Red Yellow, Red Blue, Red
Orange Red, Orange Orange, Orange Green, Orange Yellow, Orange Blue, Orange
Green Red, Green Orange, Green Green, Green Yellow, Green Blue, Green
Yellow Red, Yellow Orange, Yellow Green, Yellow Yellow, Yellow Blue, Yellow
Blue Red, Blue Orange, Blue Green, Blue Yellow, Blue Blue, Blue

Red Orange Green Yellow Blue
Red Red Orange, Red Green, Red Yellow, Red Blue, Red
Orange Red, Orange Orange Green, Orange Yellow, Orange Blue, Orange
Green Red, Green Orange, Green Green Yellow, Green Blue, Green
Yellow Red, Yellow Orange, Yellow Green, Yellow Yellow Blue, Yellow
Blue Red, Blue Orange, Blue Green, Blue Yellow, Blue Blue


Question 2: Polygon JKLM is drawn with vertices J(−4, −3), K(−4, −6), L(−1, −6), M(−1, −3). Determine the image coordinates of K′ if the pre-image is reflected across y = −4.

K′(−4, −4)
K′(4, −6)
K′(−4, 6)
K′(−4, −2)

Triangle UVW has vertices at U(−2, 0), V(−3, 1), W(−3, 3). Determine the vertices of image U′V′W′ if the preimage is rotated 90° clockwise.

U′(0, −2), V′(−1, −3), W′(−3, −3)
U′(0, 2), V′(1, 3), W′(3, 3)
U′(2, 0), V′(3, −1), W′(3, −3)
U′(−2, 0), V′(−3, 1), W′(3, −3)

Triangle ABC with vertices at A(4, 3), B(3, −2), C(−3, 1) is dilated using a scale factor of 3.5 to create triangle A′B′C′. Determine the vertex of point C′.

C′(−10.5, 1)
C′(−10.5, 3.5)
C′(−3, 3.5)
C′(−10.5, −3.5)

Answer 1

Red Orange Green Yellow Blue Red Red, Red Orange, Red Green, Red Yellow, Red Blue Orange Red, Orange Orange, Orange Green, Orange Yellow, Orange Blue Green Red, Green Orange, Green Green, Green Yellow, Green Blue Yellow Red, Yellow Orange, Yellow Green, Yellow Yellow, Yellow Blue Blue Red, Blue Orange, Blue Green, Blue Yellow, Blue Blue





K′(−4, −2)

U′(−2, 0), V′(−3, −1), W′(−3, −3)

C′(-10.5, 1)

Answer 2

Answer 1: The sample space for choosing 2 marbles from the bag with replacement consists of all possible pairs of marbles that can be drawn. Since there are 5 marbles in the bag, there are 5 choices for the first marble and 5 choices for the second marble. Therefore, there are 5 x 5 = 25 possible pairs of marbles that can be drawn. Each pair can be represented as an ordered pair of the form (marble 1, marble 2). The correct table that shows the sample space for choosing 2 marbles from the bag with replacement is:

mathematica

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Red Orange Green Yellow Blue

Red (Red, Red) (Red, Orange) (Red, Green) (Red, Yellow) (Red, Blue)

Orange (Orange, Red) (Orange, Orange) (Orange, Green) (Orange, Yellow) (Orange, Blue)

Green (Green, Red) (Green, Orange) (Green, Green) (Green, Yellow) (Green, Blue)

Yellow (Yellow, Red) (Yellow, Orange) (Yellow, Green) (Yellow, Yellow) (Yellow, Blue)

Blue (Blue, Red) (Blue, Orange) (Blue, Green) (Blue, Yellow) (Blue, Blue)

Therefore, the correct answer is option (D).

Answer 2: To reflect a point across the line y = -4, we can find the distance between the point and the line and then add twice this distance to the y-coordinate of the point.

The y-coordinate of point K is -6, and the distance between point K and the line y = -4 is |-6 - (-4)| = 2. Therefore, the image of point K, K', will be 2 units above the line y = -4. Adding twice this distance to the y-coordinate of point K gives:

K'(−4, -6 + 2(2)) = K'(−4, −2)

Therefore, the correct answer is option (D).

Answer 3: To rotate a point (x, y) 90° clockwise, we can apply the transformation (x', y') = (y, -x).

Using this transformation on the vertices of triangle UVW gives:

U′(0, -2), V′(1, -3), W′(3, -3)

Therefore, the correct answer is option (A).

Answer 4: To dilate a point (x, y) by a scale factor of 3.5 with center of dilation at the origin, we can apply the transformation (x', y') = (3.5x, 3.5y).

Using this transformation on the vertices of triangle ABC gives:

A′(14, 10.5), B′(10.5, -7), C′(-10.5, 3.5)

Therefore, the correct answer is option (A).