Question

1. If (x + 4) is a factor of x³ + 2x²− 7x + k, what is the value of k?

2. If the 4th term of GP is 8 and the 7th term is 1, find the 2nd term,
3. An urn contains 3 white balls and 1 black ball. Determine the probability of drawing two white balls in succession from the urn without replacing the ball after each drawing.
4. Simplify {cos /[sin + 1]} + tan
5. Points A and B are 100 m apart and are of the same elevation as the foot of the tower. The angles of elevation of the top of the tower from the points A and B are 21 degrees and 32 degrees respectively. How far is A from the tower in meters?
6. How many triangles are determined by the vertices of a regular hexagon?
7. The product of two numbers is increased by 4 is P. If one number is D, find the other number.

Answer 1

The value of k is -60, the 2nd term in the GP is 16, the probability of drawing two white balls in succession is 1/2, Point A is approximately 75.7 meters from the tower, a regular hexagon determines 20 triangles, and the second number in the product is (P-4)/D.

Step-by-step explanation:

1. If (x + 4) is a factor of x³ + 2x²− 7x + k, we can use the Factor theorem which states that if (x-a) is a factor of a polynomial, the polynomial will equal zero when x = a. Thus, substituting x = -4 into the polynomial, we get (-4)³ + 2(-4)² − 7(-4) + k = 0. Solving for k, gives us k = -60.

2. For a geometric progression (GP), the nth term of can be represented by ar^(n-1). Given the 4th term is 8 and the 7th term is 1, we can set up two equations: ar^3 = 8 and ar^6 = 1. Dividing the second equation by the square of the first gives: r^3 = 1/8. So, r = 0.5 (cube root of 1/8). Substituting r = 0.5 into the first equation, we get a = 8/0.5^3 = 32. Therefore, the 2nd term is ar = 32 * 0.5 = 16.

3. The probability of drawing two white balls in succession without replacing the ball after each drawing can be found by multiplying the probability of drawing one white ball and the probability of drawing a second white ball given that one white ball has been removed. There are 3 white balls out of 4, so, P(White1) = 3/4. After removing one white ball, there are 2 white balls out of 3, so, P(White2|White1) = 2/3. Therefore, the probability is (3/4)*(2/3) = 1/2.

4. Unable to provide a accurate response as the question about cos, sin and tan is incomplete.

5. To solve the problem of elevation, use the tangent of the given angles. From point A, tan(21) = (Height of Tower / Distance from A). From B, tan(32) = (Height of Tower / (100-Distance from A)). By equating the two and solving for Distance from A, we find it to be approximately 75.7 meters.

6. A regular hexagon has 6 vertices. The number of triangles determined by those vertices is given by the combination 6C3 = 6! / (3!(6-3)!) = 20.

7. If product of two numbers increased by 4 is P and one of the numbers is D, then the other number is (P-4) / D.

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