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Q. Calculate the scalar product of A = (1, 1, 1) and B = (2, 2, 2).

A. 3
B. 4
C. 5
D. 6

Solution

A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: D — 6

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Q. Calculate the scalar product of the vectors (1, 0, 0) and (0, 1, 0).

A. 0
B. 1
C. 2
D. 3

Solution

Scalar product = 1*0 + 0*1 + 0*0 = 0.

Correct Answer: A — 0

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Q. Calculate the scalar product of the vectors (1, 2, 3) and (4, 5, 6).

A. 32
B. 33
C. 34
D. 35

Solution

Scalar product = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: A — 32

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Q. Calculate the scalar product of the vectors (2, 3, 4) and (4, 3, 2).

A. 28
B. 29
C. 30
D. 31

Solution

Scalar product = 2*4 + 3*3 + 4*2 = 8 + 9 + 8 = 25.

Correct Answer: A — 28

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Q. Calculate the scalar product of the vectors (3, 0, -3) and (1, 2, 1).

A. 0
B. 1
C. 2
D. 3

Solution

Scalar product = 3*1 + 0*2 + (-3)*1 = 3 + 0 - 3 = 0.

Correct Answer: A — 0

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Q. Calculate the scalar product of the vectors A = (1, 2, 3) and B = (4, 5, 6).

A. 32
B. 30
C. 28
D. 26

Solution

A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. Calculate the scalar product of the vectors A = (4, -1, 2) and B = (2, 3, 1).

A. 10
B. 8
C. 6
D. 12

Solution

A · B = 4*2 + (-1)*3 + 2*1 = 8 - 3 + 2 = 7.

Correct Answer: A — 10

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Q. Calculate the scalar product of the vectors K = (0, 1, 2) and L = (3, 4, 5).

A. 10
B. 11
C. 12
D. 13

Solution

K · L = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

Correct Answer: A — 10

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Q. Calculate the variance of the data set {2, 4, 4, 4, 5, 5, 7, 9}.

A. 4
B. 6
C. 5
D. 3

Solution

Mean = 5, Variance = [(2-5)² + (4-5)² + (4-5)² + (4-5)² + (5-5)² + (5-5)² + (7-5)² + (9-5)²] / 8 = 4.

Correct Answer: B — 6

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Q. Calculate the variance of the data set {4, 8, 6, 5, 3}.

A. 2.5
B. 3.2
C. 1.5
D. 4.0

Solution

Mean = (4+8+6+5+3)/5 = 5.2. Variance = [(4-5.2)² + (8-5.2)² + (6-5.2)² + (5-5.2)² + (3-5.2)²]/5 = 2.5.

Correct Answer: A — 2.5

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Q. Calculate the vector product of A = (3, 2, 1) and B = (1, 0, 2).

A. (4, 5, -2)
B. (2, 5, -3)
C. (2, -5, 3)
D. (5, -2, 3)

Solution

A × B = |i  j  k|\n|3  2  1|\n|1  0  2| = (4, 5, -2)

Correct Answer: A — (4, 5, -2)

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Q. Calculate the weighted mean of the following data: (10, 2), (20, 3), (30, 5).

A. 20
B. 25
C. 30
D. 35

Solution

Weighted Mean = (10*2 + 20*3 + 30*5) / (2 + 3 + 5) = 220 / 10 = 22.

Correct Answer: B — 25

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Q. Calculate the weighted mean of the following data: (2, 3), (4, 5), (6, 7) where the first number is the weight.

A. 4.5
B. 5
C. 5.5
D. 6

Solution

Weighted mean = (2*3 + 4*5 + 6*7) / (2 + 4 + 6) = (6 + 20 + 42) / 12 = 68 / 12 = 5.67.

Correct Answer: C — 5.5

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Q. Calculate the weighted mean of the numbers 10, 20, and 30 with weights 1, 2, and 3 respectively.

A. 20
B. 25
C. 30
D. 15

Solution

Weighted mean = (10*1 + 20*2 + 30*3) / (1 + 2 + 3) = 140 / 6 = 23.33.

Correct Answer: B — 25

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Q. Calculate the weighted mean of the scores 70, 80, and 90 with weights 1, 2, and 3 respectively.

A. 80
B. 85
C. 90
D. 75

Solution

Weighted mean = (70*1 + 80*2 + 90*3) / (1 + 2 + 3) = 85.

Correct Answer: B — 85

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Q. Calculate ∫ from 0 to 1 of (1 - x^2) dx.

A. 1/3
B. 1/2
C. 2/3
D. 1

Solution

The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

Correct Answer: B — 1/2

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Q. Calculate ∫ from 0 to 1 of (1/x) dx.

A. 0
B. 1
C. ln(1)
D. ln(2)

Solution

The integral evaluates to [ln(x)] from 0 to 1 = ln(1) - ln(0) which diverges.

Correct Answer: C — ln(1)

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Q. Calculate ∫ from 0 to 1 of (2x^2 + 3x + 1) dx.

A. 1
B. 2
C. 3
D. 4

Solution

The integral evaluates to [2x^3/3 + (3/2)x^2 + x] from 0 to 1 = (2/3 + 3/2 + 1) = 3.

Correct Answer: C — 3

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Q. Calculate ∫ from 0 to 1 of (4x^3 - 2x^2 + x) dx.

A. 1/4
B. 1/3
C. 1/2
D. 1

Solution

The integral evaluates to [x^4 - (2/3)x^3 + (1/2)x^2] from 0 to 1 = (1 - 2/3 + 1/2) = 1/6.

Correct Answer: C — 1/2

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Q. Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.

A. 0
B. 1
C. 2
D. 3

Solution

The integral evaluates to [x^4 - x^3 + x^2 - x] from 0 to 1 = (1 - 1 + 1 - 1) = 0.

Correct Answer: B — 1

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Q. Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.

A. 1
B. 2
C. 3
D. 4

Solution

The integral evaluates to [x^4 - (4/3)x^3 + x] from 0 to 1 = 1 - (4/3) + 1 = 2/3.

Correct Answer: A — 1

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Q. Calculate ∫ from 0 to 1 of (6x^2 - 4x + 1) dx.

A. 1
B. 2
C. 3
D. 4

Solution

The integral evaluates to [2x^3 - 2x^2 + x] from 0 to 1 = (2 - 2 + 1) = 1.

Correct Answer: A — 1

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Q. Calculate ∫ from 0 to 1 of (x^2 * e^x) dx.

A. 1/e
B. 2/e
C. 3/e
D. 4/e

Solution

Using integration by parts, the integral evaluates to (2/e - 1/e) = 1/e.

Correct Answer: B — 2/e

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Q. Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.

A. 1
B. 2
C. 3
D. 4

Solution

The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.

Correct Answer: C — 3

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Q. Calculate ∫ from 0 to 1 of (x^2 + 2x + 1) dx.

A. 1
B. 2
C. 3
D. 4

Solution

The integral evaluates to [x^3/3 + x^2 + x] from 0 to 1 = (1/3 + 1 + 1) - (0) = 7/3.

Correct Answer: C — 3

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Q. Calculate ∫ from 0 to 1 of (x^2 + 4x + 4) dx.

A. 3
B. 4
C. 5
D. 6

Solution

The integral evaluates to [x^3/3 + 2x^2 + 4x] from 0 to 1 = (1/3 + 2 + 4) = 25/3.

Correct Answer: C — 5

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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.

A. 0
B. 1
C. 1/3
D. 2/3

Solution

The integral evaluates to [x^5/5 - 2x^3/3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = (15/15 - 10/15 + 3/15) = 8/15.

Correct Answer: D — 2/3

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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.

A. 0
B. 1/5
C. 1/3
D. 1/2

Solution

The integral evaluates to [x^5/5 - (2/4)x^4 + (1/3)x^3] from 0 to 1 = (1/5 - 1/2 + 1/3) = 1/30.

Correct Answer: B — 1/5

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Q. Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.

A. 2
B. 4
C. 6
D. 8

Solution

The integral evaluates to [x^4/4 - x^3 + 4x] from 0 to 2 = (4 - 8 + 8) - 0 = 4.

Correct Answer: C — 6

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Q. Calculate ∫ from 0 to π of sin(x) dx.

A. 0
B. 1
C. 2
D. 3

Solution

The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.

Correct Answer: C — 2

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