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Q. For which value of k does the equation x^2 - kx + 9 = 0 have roots that are both positive?

A. k < 6
B. k > 6
C. k = 6
D. k = 0

Solution

For both roots to be positive, k must be greater than 6, as the sum of the roots must be positive.

Correct Answer: B — k > 6

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Q. For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?

A. k < 4
B. k = 4
C. k > 4
D. k ≤ 4

Solution

The discriminant must be negative: k^2 - 16 < 0, hence k > 4.

Correct Answer: C — k > 4

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Q. For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x = -1?

A. 0
B. 1
C. -1
D. 2

Solution

f'(x) = 2kx + 2; f'(-1) = -2k + 2 must exist for any k.

Correct Answer: A — 0

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Q. For which value of m is the function f(x) = { 3x + m, x < 1; 2, x = 1; mx + 1, x > 1 continuous at x = 1?

A. -1
B. 0
C. 1
D. 2

Solution

Setting 3 + m = 2 and 2 = m + 1 gives m = 1 for continuity.

Correct Answer: B — 0

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Q. For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?

A. -1
B. 0
C. 1
D. 2

Solution

Setting the two pieces equal at x = 0 gives us p = -1.

Correct Answer: B — 0

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Q. From a group of 10 people, how many ways can a committee of 4 be formed?

A. 210
B. 120
C. 150
D. 180

Solution

The number of ways to choose 4 people from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. From a point A, the angle of elevation of the top of a building is 45 degrees. If the height of the building is 20 meters, how far is point A from the base of the building?

A. 10 m
B. 20 m
C. 30 m
D. 40 m

Solution

Using tan(45°) = height/distance, we have 1 = 20/distance. Therefore, distance = 20 m.

Correct Answer: B — 20 m

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Q. From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the height of the hill is 50 meters, how far is the point from the base of the hill?

A. 50 m
B. 75 m
C. 100 m
D. 125 m

Solution

Using tan(30°) = height/distance, we have 1/√3 = 50/distance. Therefore, distance = 50√3 ≈ 86.6 m.

Correct Answer: C — 100 m

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Q. From a point on the ground, the angle of elevation of the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?

A. 50 m
B. 60 m
C. 70 m
D. 80 m

Solution

Using tan(30°) = height/100, we have 1/√3 = height/100. Therefore, height = 100/√3 ≈ 57.74 m.

Correct Answer: A — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a 40 m high building is 45 degrees. How far is the point from the base of the building?

A. 40 m
B. 20 m
C. 30 m
D. 50 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 40/1 = 40 m.

Correct Answer: A — 40 m

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Q. From a point on the ground, the angle of elevation to the top of a 75 m high building is 45 degrees. How far is the point from the building?

A. 75 m
B. 50 m
C. 100 m
D. 25 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 75/1 = 75 m.

Correct Answer: A — 75 m

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Q. From a point on the ground, the angle of elevation to the top of a building is 45 degrees. If the building is 50 meters tall, how far is the point from the base of the building?

A. 50 m
B. 25 m
C. 100 m
D. 75 m

Solution

Distance = height / tan(angle) = 50 / 1 = 50 m.

Correct Answer: A — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 100 meters, what is the height of the hill?

A. 100√3 m
B. 50 m
C. 100 m
D. 50√3 m

Solution

Using tan(30°) = height/distance, we have height = distance * tan(30°) = 100 * (1/√3) = 100/√3 = 50 m.

Correct Answer: B — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 20 m, how far is the point from the base of the hill?

A. 20 m
B. 10 m
C. 30 m
D. 40 m

Solution

Using tan(45°) = height/distance, we have 1 = 20/distance. Therefore, distance = 20 m.

Correct Answer: A — 20 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 100 m, how far is the point from the base of the hill?

A. 100 m
B. 50 m
C. 200 m
D. 150 m

Solution

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 100/1 = 100 m.

Correct Answer: A — 100 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the point is 100 meters away from the base of the hill, what is the height of the hill?

A. 100 meters
B. 50 meters
C. 200 meters
D. 150 meters

Solution

Height = distance * tan(angle) = 100 * tan(45°) = 100 meters.

Correct Answer: A — 100 meters

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 50 m, how far is the point from the base of the hill?

A. 25 m
B. 50 m
C. 70 m
D. 100 m

Solution

Using tan(45°) = height/distance, we have 1 = 50/distance. Therefore, distance = 50 m.

Correct Answer: B — 50 m

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Q. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the height of the hill is 40 m, how far is the point from the base of the hill?

A. 20 m
B. 40 m
C. 60 m
D. 80 m

Solution

Using tan(45°) = height/distance, we have 1 = 40/distance. Therefore, distance = 40 m.

Correct Answer: B — 40 m

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Q. From a point on the ground, the angle of elevation to the top of a tower is 60 degrees. If the tower is 30 m high, how far is the point from the base of the tower?

A. 15 m
B. 30 m
C. 20 m
D. 10 m

Solution

Using tan(60°) = height/distance, we have √3 = 30/distance. Therefore, distance = 30/√3 m.

Correct Answer: A — 15 m

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Q. From a set of 8 different books, how many ways can you choose 3 books?

A. 56
B. 84
C. 28
D. 70

Solution

The number of ways to choose 3 books from 8 is given by 8C3 = 56.

Correct Answer: B — 84

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Q. From the top of a 20-meter high building, the angle of depression to a car parked on the ground is 60 degrees. How far is the car from the base of the building?

A. 10√3 meters
B. 20 meters
C. 30 meters
D. 40 meters

Solution

Distance = height / tan(angle) = 20 / tan(60°) = 20 / √3 = 10√3 meters.

Correct Answer: A — 10√3 meters

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Q. From the top of a 50 m high building, the angle of depression to a point on the ground is 45 degrees. How far is the point from the base of the building?

A. 25 m
B. 50 m
C. 70 m
D. 100 m

Solution

Using tan(45°) = height/distance, we have 1 = 50/distance. Therefore, distance = 50 m.

Correct Answer: B — 50 m

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Q. From the top of a 50-meter high building, the angle of depression to a point on the ground is 45 degrees. How far is the point from the base of the building?

A. 50 meters
B. 100 meters
C. 25 meters
D. 70 meters

Solution

Distance = height / tan(angle) = 50 / tan(45°) = 50 meters.

Correct Answer: A — 50 meters

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Q. From the top of a 60 m high building, the angle of depression to a point on the ground is 30 degrees. How far is the point from the base of the building?

A. 60√3 m
B. 30√3 m
C. 60 m
D. 30 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 60/√3 = 60√3 m.

Correct Answer: A — 60√3 m

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Q. From the top of a 60 m high cliff, the angle of depression to a boat in the sea is 30 degrees. How far is the boat from the base of the cliff?

A. 60√3 m
B. 30√3 m
C. 60 m
D. 30 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 60/√3 = 60√3 m.

Correct Answer: A — 60√3 m

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Q. Given A = 3i + 4j and B = 0i + 0j, find A · B.

A. 0
B. 12
C. 7
D. 3

Solution

A · B = (3)(0) + (4)(0) = 0.

Correct Answer: A — 0

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Q. Given vectors A = (2, -1, 3) and B = (4, 0, -2), find A × B.

A. (-1, -10, 4)
B. (1, 10, -4)
C. (10, -1, 4)
D. (10, 1, -4)

Solution

A × B = |i  j  k|\n|2 -1  3|\n|4  0 -2| = (-1, -10, 4)

Correct Answer: A — (-1, -10, 4)

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Q. Given vectors A = (x, y, z) and B = (1, 2, 3), if A · B = 14, what is the value of x + 2y + 3z?

A. 14
B. 10
C. 8
D. 6

Solution

A · B = x*1 + y*2 + z*3 = 14, thus x + 2y + 3z = 14.

Correct Answer: A — 14

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Q. Given vectors P = (4, 0, -3) and Q = (1, 2, 1), find the scalar product P · Q.

A. -1
B. 5
C. 10
D. 2

Solution

P · Q = 4*1 + 0*2 + (-3)*1 = 4 + 0 - 3 = 1.

Correct Answer: B — 5

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Q. Given vectors P = (4, 1, 0) and Q = (0, 2, 3), find the scalar product P · Q.

A. 0
B. 6
C. 8
D. 10

Solution

P · Q = 4*0 + 1*2 + 0*3 = 0 + 2 + 0 = 2.

Correct Answer: B — 6

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Engineering & Architecture Admissions

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