Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).
Solution
Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.
Correct Answer: A — 3
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Q. Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
Solution
Scalar triple product = A · (B × C) = 0, as vectors are coplanar.
Correct Answer: A — 0
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Q. Find the second derivative of f(x) = e^x at x = 0.
Solution
f''(x) = e^x, thus f''(0) = e^0 = 1.
Correct Answer: B — 1
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Q. Find the second derivative of f(x) = ln(x^2 + 1).
-
A.
-2/(x^2 + 1)^2
-
B.
2/(x^2 + 1)^2
-
C.
0
-
D.
-1/(x^2 + 1)
Solution
First derivative f'(x) = (2x)/(x^2 + 1). Second derivative f''(x) = (2(x^2 + 1) - 4x^2)/(x^2 + 1)^2 = -2/(x^2 + 1)^2.
Correct Answer: A — -2/(x^2 + 1)^2
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Q. Find the second derivative of f(x) = x^3 - 6x^2 + 9x.
Solution
f'(x) = 3x^2 - 12x + 9; f''(x) = 6x - 12. At x = 2, f''(2) = 6(2) - 12 = 0.
Correct Answer: A — 6
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Q. Find the second derivative of f(x) = x^4 - 4x^3 + 6x^2.
-
A.
12x - 24
-
B.
12x^2 - 24
-
C.
24x - 12
-
D.
24x^2 - 12
Solution
First derivative f'(x) = 4x^3 - 12x^2 + 12. Second derivative f''(x) = 12x^2 - 24.
Correct Answer: A — 12x - 24
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Q. Find the slope of the line passing through the points (2, 3) and (4, 7).
Solution
The slope m is given by (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.
Correct Answer: A — 2
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Q. Find the slope of the line represented by the equation 2x - 3y + 6 = 0.
-
A.
2/3
-
B.
-2/3
-
C.
3/2
-
D.
-3/2
Solution
Rearranging gives y = (2/3)x + 2, so slope = 2/3.
Correct Answer: B — -2/3
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Q. Find the slope of the line that passes through the points (0, 0) and (5, 5).
Solution
The slope m = (5 - 0) / (5 - 0) = 1.
Correct Answer: B — 1
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Q. Find the slope of the tangent line to the curve y = sin(x) at x = π/4.
-
A.
1
-
B.
√2/2
-
C.
√3/2
-
D.
0
Solution
The derivative f'(x) = cos(x). At x = π/4, f'(π/4) = cos(π/4) = √2/2.
Correct Answer: B — √2/2
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Q. Find the slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0.
-
A.
-1, -2
-
B.
-3, -1
-
C.
1, 2
-
D.
2, 3
Solution
The slopes can be found by solving the quadratic equation for m in terms of x and y.
Correct Answer: B — -3, -1
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Q. Find the slopes of the lines represented by the equation 6x^2 - 5xy + y^2 = 0.
-
A.
-1/6, 5
-
B.
1/6, -5
-
C.
5/6, -1
-
D.
1, -1
Solution
The slopes can be calculated from the quadratic equation, yielding slopes of 5/6 and -1.
Correct Answer: C — 5/6, -1
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Q. Find the solution for the inequality 2(x - 1) ≥ 3.
-
A.
x ≥ 2.5
-
B.
x ≤ 2.5
-
C.
x ≥ 1.5
-
D.
x ≤ 1.5
Solution
2(x - 1) ≥ 3 => x - 1 ≥ 1.5 => x ≥ 2.5.
Correct Answer: C — x ≥ 1.5
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Q. Find the solution for the inequality 2x + 3 ≤ 5.
-
A.
x ≤ 1
-
B.
x ≥ 1
-
C.
x ≤ 2
-
D.
x ≥ 2
Solution
2x + 3 ≤ 5 => 2x ≤ 2 => x ≤ 1.
Correct Answer: A — x ≤ 1
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Q. Find the solution for the inequality 2x + 5 > 3x - 1.
-
A.
x < 6
-
B.
x > 6
-
C.
x < 4
-
D.
x > 4
Solution
2x + 5 > 3x - 1 => 5 + 1 > 3x - 2x => 6 > x => x < 6.
Correct Answer: D — x > 4
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Q. Find the solution for the inequality 3x + 2 ≥ 11.
-
A.
x ≥ 3
-
B.
x ≤ 3
-
C.
x ≥ 2
-
D.
x ≤ 2
Solution
3x + 2 ≥ 11 => 3x ≥ 9 => x ≥ 3.
Correct Answer: A — x ≥ 3
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Q. Find the solution for the inequality 3x - 5 ≥ 4.
-
A.
x ≥ 3
-
B.
x ≤ 3
-
C.
x ≥ 2
-
D.
x ≤ 2
Solution
3x - 5 ≥ 4 => 3x ≥ 9 => x ≥ 3.
Correct Answer: A — x ≥ 3
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Q. Find the solution for the inequality 6x - 4 ≤ 2x + 8.
-
A.
x ≤ 3
-
B.
x ≥ 3
-
C.
x ≤ 2
-
D.
x ≥ 2
Solution
6x - 4 ≤ 2x + 8 => 4x ≤ 12 => x ≥ 3.
Correct Answer: B — x ≥ 3
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Q. Find the solution for the inequality: -x + 4 ≤ 2.
-
A.
x ≥ 2
-
B.
x ≤ 2
-
C.
x ≥ 4
-
D.
x ≤ 4
Solution
-x + 4 ≤ 2 => -x ≤ -2 => x ≥ 2.
Correct Answer: B — x ≤ 2
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Q. Find the solution for the inequality: 8x - 3 > 5.
-
A.
x > 1
-
B.
x < 1
-
C.
x > 0.5
-
D.
x < 0.5
Solution
8x - 3 > 5 => 8x > 8 => x > 1.
Correct Answer: A — x > 1
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Q. Find the solution of the differential equation y' = 2y + 3.
-
A.
y = Ce^(2x) - 3/2
-
B.
y = Ce^(-2x) + 3/2
-
C.
y = 3/2 - Ce^(2x)
-
D.
y = 3/2 + Ce^(-2x)
Solution
This is a linear first-order equation. The general solution is y = 3/2 + Ce^(-2x).
Correct Answer: D — y = 3/2 + Ce^(-2x)
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Q. Find the solution of the differential equation y'' + 4y = 0.
-
A.
y = C1 cos(2x) + C2 sin(2x)
-
B.
y = C1 e^(2x) + C2 e^(-2x)
-
C.
y = C1 e^(x) + C2 e^(-x)
-
D.
y = C1 sin(2x) + C2 cos(2x)
Solution
This is a second-order linear homogeneous differential equation. The characteristic equation has roots ±2i.
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. Find the solution of the first-order linear differential equation dy/dx + y = e^x.
-
A.
y = e^x + Ce^(-x)
-
B.
y = e^x - Ce^(-x)
-
C.
y = e^(-x) + Ce^x
-
D.
y = e^(-x) - Ce^x
Solution
Using an integrating factor e^x, we solve to get y = e^x + Ce^(-x).
Correct Answer: A — y = e^x + Ce^(-x)
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Q. Find the solution set for the inequality -2x + 3 > 1.
-
A.
x < 1
-
B.
x > 1
-
C.
x < 2
-
D.
x > 2
Solution
Subtract 3 from both sides: -2x > -2. Then divide by -2 (reverse the inequality): x < 1.
Correct Answer: B — x > 1
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Q. Find the solution set for the inequality -2x + 4 < 0.
-
A.
x > 2
-
B.
x < 2
-
C.
x > 4
-
D.
x < 4
Solution
Subtract 4 from both sides: -2x < -4. Then divide by -2 (reverse the inequality): x > 2.
Correct Answer: A — x > 2
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Q. Find the solution set for the inequality -2x + 5 > 1.
-
A.
x < 2
-
B.
x > 2
-
C.
x < 3
-
D.
x > 3
Solution
-2x + 5 > 1 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
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Q. Find the solution set for the inequality -3x + 1 ≤ 4.
-
A.
x ≥ -1
-
B.
x ≤ -1
-
C.
x ≥ 1
-
D.
x ≤ 1
Solution
-3x + 1 ≤ 4 => -3x ≤ 3 => x ≥ -1.
Correct Answer: B — x ≤ -1
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Q. Find the solution set for the inequality -4x + 1 < 3.
-
A.
x < -0.5
-
B.
x > -0.5
-
C.
x < 0.5
-
D.
x > 0.5
Solution
-4x + 1 < 3 => -4x < 2 => x > -0.5.
Correct Answer: A — x < -0.5
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Q. Find the solution set for the inequality -x + 4 < 2.
-
A.
x > 2
-
B.
x < 2
-
C.
x > 4
-
D.
x < 4
Solution
-x + 4 < 2 => -x < -2 => x > 2.
Correct Answer: D — x < 4
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Q. Find the solution set for the inequality -x + 5 < 2.
-
A.
x > 3
-
B.
x < 3
-
C.
x ≥ 3
-
D.
x ≤ 3
Solution
-x + 5 < 2 => -x < -3 => x > 3.
Correct Answer: A — x > 3
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