Q. Find the minimum value of the function f(x) = x^4 - 8x^2 + 16.
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
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B.
y = x^2 - 1
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C.
y = 2x + 1
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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.
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A.
y = e^x + 1
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B.
y = e^x - 1
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C.
y = x + 1
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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)
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B.
(2, 3)
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C.
(3, 0)
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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.
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A.
(1, 3)
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B.
(2, 2)
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C.
(3, 1)
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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.
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A.
(1, 3)
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B.
(2, 5)
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C.
(3, 7)
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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)
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B.
(3, 2)
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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).
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).
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.
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.
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.
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.
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.
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).
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.
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.
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)).
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.
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.
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
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B.
-1, -6
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C.
-3, -2
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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}
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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).
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).
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).
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).
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).
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.
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).
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).
Solution
G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.
Correct Answer: D — 3
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