Q. What is the condition for the lines represented by the equation 5x^2 + 4xy + 3y^2 = 0 to be parallel?
Solution
The condition for parallel lines is that the determinant of the coefficients must be zero.
Correct Answer: C — 0
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Q. What is the condition for two lines ax + by + c1 = 0 and ax + by + c2 = 0 to be parallel?
-
A.
c1 = c2
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B.
a/b = c1/c2
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C.
a/b = c2/c1
-
D.
a = 0
Solution
Two lines are parallel if their coefficients of x and y are proportional, which means c1 must equal c2.
Correct Answer: A — c1 = c2
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Q. What is the condition for two lines to be parallel?
-
A.
m1 = m2
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B.
m1 + m2 = 0
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C.
m1 * m2 = -1
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D.
m1 - m2 = 0
Solution
Two lines are parallel if their slopes are equal, i.e., m1 = m2.
Correct Answer: A — m1 = m2
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Q. What is the condition for two lines to be perpendicular?
-
A.
m1 * m2 = -1
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B.
m1 + m2 = 0
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C.
m1 - m2 = 1
-
D.
m1 * m2 = 1
Solution
Two lines are perpendicular if the product of their slopes m1 and m2 is -1.
Correct Answer: A — m1 * m2 = -1
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Q. What is the conjugate of the complex number z = 2 + 5i?
-
A.
2 - 5i
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B.
2 + 5i
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C.
-2 + 5i
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D.
-2 - 5i
Solution
The conjugate of z = 2 + 5i is z̅ = 2 - 5i.
Correct Answer: A — 2 - 5i
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Q. What is the conjugate of the complex number z = 5 - 2i?
-
A.
5 + 2i
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B.
5 - 2i
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C.
2 - 5i
-
D.
2 + 5i
Solution
The conjugate of z = 5 - 2i is 5 + 2i.
Correct Answer: A — 5 + 2i
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Q. What is the conjugate of the complex number z = 5 - 6i?
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A.
5 + 6i
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B.
5 - 6i
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C.
6 - 5i
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D.
6 + 5i
Solution
The conjugate of z = 5 - 6i is 5 + 6i.
Correct Answer: A — 5 + 6i
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Q. What is the critical point of f(x) = x^3 - 3x^2 + 4?
Solution
Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2.
Correct Answer: B — 2
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Q. What is the critical point of f(x) = x^3 - 6x^2 + 9x?
Solution
Setting f'(x) = 0 gives critical points at x = 1, 2, and 3.
Correct Answer: C — 2
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Q. What is the cross product of the vectors (1, 0, 0) and (0, 1, 0)?
-
A.
(0, 0, 1)
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B.
(1, 1, 0)
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C.
(0, 0, 0)
-
D.
(1, 0, 0)
Solution
Cross product = (1, 0, 0) × (0, 1, 0) = (0, 0, 1).
Correct Answer: A — (0, 0, 1)
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Q. What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?
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A.
(-3, 6, -3)
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B.
(-3, 6, 3)
-
C.
(3, -6, 3)
-
D.
(3, 6, -3)
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
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Q. What is the cross product of u = (1, 2, 3) and v = (4, 5, 6)?
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A.
(-3, 6, -3)
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B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, 2, 3)
Solution
u × v = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
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Q. What is the cross product of vectors (1, 2, 3) and (4, 5, 6)?
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A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, 2, 3)
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
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Q. What is the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6)?
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A.
(-3, 6, -3)
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B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, -2, 1)
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
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Q. What is the derivative of f(x) = 3x^3 - 5x + 2?
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A.
9x^2 - 5
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B.
3x^2 - 5
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C.
9x^2 + 5
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D.
3x^2 + 5
Solution
f'(x) = 9x^2 - 5.
Correct Answer: A — 9x^2 - 5
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Q. What is the derivative of f(x) = 5x^4 - 3x + 2?
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A.
20x^3 - 3
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B.
20x^3 + 3
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C.
15x^3 - 3
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D.
5x^3 - 3
Solution
The derivative f'(x) = d/dx(5x^4 - 3x + 2) = 20x^3 - 3.
Correct Answer: A — 20x^3 - 3
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Q. What is the derivative of f(x) = e^x?
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A.
e^x
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B.
x*e^x
-
C.
1
-
D.
0
Solution
The derivative of e^x is e^x.
Correct Answer: A — e^x
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Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
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A.
0
-
B.
1
-
C.
2
-
D.
undefined
Solution
f'(x) = (2x)/(x^2 + 1), thus f'(0) = 0.
Correct Answer: A — 0
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Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 1?
Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 1.
Correct Answer: B — 1
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Q. What is the derivative of f(x) = ln(x^2 + 1)?
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. What is the derivative of f(x) = sin(x) + cos(x)?
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A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
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C.
sin(x) + cos(x)
-
D.
-sin(x) + cos(x)
Solution
Using the derivatives of sine and cosine, f'(x) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. What is the derivative of f(x) = sin(x^2)?
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A.
2x cos(x^2)
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B.
cos(x^2)
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C.
2x sin(x^2)
-
D.
sin(x^2)
Solution
Using the chain rule, f'(x) = cos(x^2) * 2x = 2x cos(x^2).
Correct Answer: A — 2x cos(x^2)
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Q. What is the derivative of f(x) = tan(x)?
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A.
sec^2(x)
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B.
csc^2(x)
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C.
sec(x)
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D.
tan^2(x)
Solution
The derivative of tan(x) is sec^2(x).
Correct Answer: A — sec^2(x)
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Q. What is the derivative of f(x) = x^2 * e^x?
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A.
e^x(2x + x^2)
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B.
e^x(2x)
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C.
e^x(x^2 + 2)
-
D.
e^x(x^2 + 1)
Solution
Using the product rule, f'(x) = e^x * (x^2 + 2x).
Correct Answer: A — e^x(2x + x^2)
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Q. What is the derivative of f(x) = x^2 + 2x + 1?
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A.
2x + 2
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B.
2x + 1
-
C.
x + 2
-
D.
2x
Solution
f'(x) = 2x + 2.
Correct Answer: A — 2x + 2
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Q. What is the derivative of f(x) = x^2 at x = 3?
Solution
f'(x) = 2x; f'(3) = 2*3 = 6.
Correct Answer: B — 6
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Q. What is the derivative of f(x) = x^3 - 4x^2 + 6x?
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A.
3x^2 - 8x + 6
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B.
3x^2 + 8x + 6
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C.
2x^2 - 4x + 6
-
D.
3x^2 - 4x + 6
Solution
The derivative f'(x) = d/dx(x^3 - 4x^2 + 6x) = 3x^2 - 8x + 6.
Correct Answer: A — 3x^2 - 8x + 6
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Q. What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5?
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A.
4x^3 - 12x^2 + 12x - 24
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B.
3x^2 - 12x + 6
-
C.
4x^3 - 12x^2 + 6
-
D.
12x^2 - 12
Solution
The derivative f'(x) = 4x^3 - 12x^2 + 12x - 24.
Correct Answer: A — 4x^3 - 12x^2 + 12x - 24
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Q. What is the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1?
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A.
4x^3 - 12x^2 + 12x - 4
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B.
3x^2 - 12x + 6
-
C.
4x^3 - 12x^2 + 6
-
D.
12x^2 - 12
Solution
The derivative is f'(x) = 4x^3 - 12x^2 + 12x - 4.
Correct Answer: A — 4x^3 - 12x^2 + 12x - 4
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Q. What is the derivative of f(x) = x^4?
-
A.
4x^3
-
B.
3x^4
-
C.
2x^4
-
D.
x^3
Solution
f'(x) = 4x^3.
Correct Answer: A — 4x^3
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