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Q. Find the minimum value of the function f(x) = x^4 - 8x^2 + 16.

A. 0
B. 2
C. 4
D. 8

Solution

f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0, which is the minimum value.

Correct Answer: A — 0

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Q. Find the particular solution of dy/dx = 2x with the initial condition y(0) = 1.

A. y = x^2 + 1
B. y = x^2 - 1
C. y = 2x + 1
D. y = 2x - 1

Solution

Integrating gives y = x^2 + C. Using the initial condition y(0) = 1, we find C = 1.

Correct Answer: A — y = x^2 + 1

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Q. Find the particular solution of dy/dx = x + y, given y(0) = 1.

A. y = e^x + 1
B. y = e^x - 1
C. y = x + 1
D. y = x + e^x

Solution

The general solution is y = e^x + C. Using the initial condition y(0) = 1, we find C = 1.

Correct Answer: A — y = e^x + 1

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Q. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.

A. (1, 4)
B. (2, 3)
C. (3, 0)
D. (0, 0)

Solution

f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).

Correct Answer: C — (3, 0)

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Q. Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.

A. (1, 3)
B. (2, 2)
C. (3, 1)
D. (0, 6)

Solution

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. The point of inflection is at (2, f(2)) = (2, 2).

Correct Answer: A — (1, 3)

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Q. Find the point of intersection of the lines y = 2x + 1 and y = -x + 4.

A. (1, 3)
B. (2, 5)
C. (3, 7)
D. (4, 9)

Solution

Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).

Correct Answer: A — (1, 3)

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Q. Find the point of intersection of the lines y = x + 1 and y = -x + 5.

A. (2, 3)
B. (3, 2)
C. (1, 2)
D. (0, 1)

Solution

Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).

Correct Answer: A — (2, 3)

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Q. Find the projection of vector A = (2, 3) onto vector B = (1, 1).

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

Solution

Projection of A onto B = (A · B) / |B|^2 * B; A · B = 2*1 + 3*1 = 5; |B|^2 = 1^2 + 1^2 = 2; Projection = (5/2)(1, 1) = (2.5, 2.5).

Correct Answer: A — 1

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Q. Find the projection of vector A = (3, 4) onto vector B = (1, 2).

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

Solution

Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).

Correct Answer: B — 2

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Q. Find the range of the data set: 10, 15, 20, 25, 30.

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

Solution

Range = Maximum - Minimum = 30 - 10 = 20.

Correct Answer: A — 15

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Q. Find the range of the data set: 12, 15, 20, 22, 30.

A. 10
B. 15
C. 18
D. 20

Solution

Range = Maximum - Minimum = 30 - 12 = 18.

Correct Answer: C — 18

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Q. Find the range of the data set: 12, 15, 22, 30, 5.

A. 25
B. 22
C. 30
D. 15

Solution

Range = max - min = 30 - 5 = 25.

Correct Answer: A — 25

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Q. Find the range of the data set: 8, 12, 15, 20, 25.

A. 12
B. 15
C. 17
D. 20

Solution

Range = Maximum - Minimum = 25 - 8 = 17.

Correct Answer: A — 12

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Q. Find the real part of the complex number z = 2 + 3i.

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

Solution

The real part of z = 2 + 3i is 2.

Correct Answer: A — 2

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Q. Find the real part of the complex number z = 2e^(iπ/3).

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

Solution

The real part is 2 * cos(π/3) = 2 * 1/2 = 1.

Correct Answer: B — 2

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Q. Find the real part of the complex number z = 3 + 4i.

A. 3
B. 4
C. 7
D. 0

Solution

The real part of z is 3.

Correct Answer: A — 3

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Q. Find the real part of the complex number z = 4 + 3i.

A. 4
B. 3
C. 7
D. 0

Solution

The real part of z = 4 + 3i is 4.

Correct Answer: A — 4

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Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).

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

Solution

The real part is 4 * cos(π/3) = 4 * 1/2 = 2.

Correct Answer: A — 2

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Q. Find the real part of the complex number z = 5 - 2i.

A. 5
B. -2
C. 2
D. 0

Solution

The real part of z = 5 - 2i is 5.

Correct Answer: A — 5

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Q. Find the roots of the equation x^2 + 4x + 4 = 0.

A. -2
B. 2
C. 0
D. 4

Solution

The equation factors to (x + 2)^2 = 0, giving a double root x = -2.

Correct Answer: A — -2

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Q. Find the roots of the equation x^2 + 5x + 6 = 0.

A. -2, -3
B. -1, -6
C. -3, -2
D. 0, -6

Solution

The roots are x = -2 and x = -3.

Correct Answer: C — -3, -2

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Q. Find the roots of the quadratic equation x^2 + 4x + 4 = 0.

A. {-2}
B. {2, -2}
C. {-4, 0}
D. {0, 4}

Solution

The equation factors to (x + 2)(x + 2) = 0, giving a double root x = -2.

Correct Answer: A — {-2}

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

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

Solution

Scalar product = 3*1 + (-2)*4 + 5*(-1) = 3 - 8 - 5 = -10.

Correct Answer: A — -1

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Q. Find the scalar product of the vectors (4, 5) and (1, 2).

A. 14
B. 13
C. 12
D. 11

Solution

Scalar product = 4*1 + 5*2 = 4 + 10 = 14.

Correct Answer: A — 14

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Q. Find the scalar product of the vectors (7, 8, 9) and (0, 1, 2).

A. 26
B. 27
C. 28
D. 29

Solution

Scalar product = 7*0 + 8*1 + 9*2 = 0 + 8 + 18 = 26.

Correct Answer: A — 26

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

A. -1
B. 5
C. 10
D. 11

Solution

A · B = 2*4 + 3*(-1) = 8 - 3 = 5.

Correct Answer: C — 10

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Q. Find the scalar product of the vectors A = 5i + 12j and B = 3i - 4j.

A. -33
B. 33
C. 39
D. 45

Solution

A · B = (5)(3) + (12)(-4) = 15 - 48 = -33.

Correct Answer: A — -33

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Q. Find the scalar product of the vectors G = (2, -3, 1) and H = (4, 0, -2).

A. -2
B. 0
C. 2
D. 8

Solution

G · H = 2*4 + (-3)*0 + 1*(-2) = 8 + 0 - 2 = 6.

Correct Answer: A — -2

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Q. Find the scalar product of the vectors G = (5, -3, 2) and H = (1, 1, 1).

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

Solution

G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.

Correct Answer: D — 3

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