Q. Find the angle between the vectors (1, 0, 0) and (0, 1, 0).
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
180 degrees
Solution
The angle θ = cos⁻¹((u · v) / (|u| |v|)) = cos⁻¹(0) = 90 degrees.
Correct Answer: B — 90 degrees
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Q. Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).
-
A.
0°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*0 + 2*2 = 6. |A| = √(1^2 + 2^2 + 2^2) = 3, |B| = √(2^2 + 0^2 + 2^2) = 2√2. cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45°.
Correct Answer: C — 60°
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Q. Find the angle between the vectors A = (1, 2, 2) and B = (2, 1, 1).
-
A.
60°
-
B.
45°
-
C.
30°
-
D.
90°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*1 + 2*1 = 6; |A| = √(1^2 + 2^2 + 2^2) = 3; |B| = √(2^2 + 1^2 + 1^2) = √6. Thus, cos(θ) = 6 / (3√6) = 1/√6, θ = 45°.
Correct Answer: B — 45°
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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1) if A · B = |A||B|cos(θ).
-
A.
60°
-
B.
45°
-
C.
90°
-
D.
30°
Solution
A · B = 3*1 + (-2)*1 + 1*1 = 3 - 2 + 1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. cos(θ) = 2/(√14 * √3). θ = 60°.
Correct Answer: A — 60°
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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1).
-
A.
60°
-
B.
45°
-
C.
90°
-
D.
30°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 3*1 + (-2)*1 + 1*1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. θ = cos^(-1)(2/(√14 * √3)).
Correct Answer: A — 60°
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Q. Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.
Solution
The area between the curves is given by ∫(from -2 to 2) (4 - x^2) dx = [4x - x^3/3] from -2 to 2 = 16/3.
Correct Answer: B — 16/3
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Q. Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Solution
The area between the curves y = x^2 and y = 4 is given by ∫(from 0 to 2) (4 - x^2) dx = [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 4/3.
Correct Answer: A — 4
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Q. Find the area between the curves y = x^3 and y = x from x = 0 to x = 1.
-
A.
1/4
-
B.
1/3
-
C.
1/2
-
D.
1/6
Solution
The area between the curves is given by ∫(from 0 to 1) (x - x^3) dx = [x^2/2 - x^4/4] from 0 to 1 = (1/2 - 1/4) = 1/4.
Correct Answer: B — 1/3
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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3).
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: A — 6
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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.
Solution
Area = 0.5 * |AB × AC| = 0, as points are collinear.
Correct Answer: A — 0
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Q. Find the area of the triangle with vertices (0,0), (4,0), and (4,3).
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: B — 12
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Q. Find the area under the curve y = e^x from x = 0 to x = 1.
Solution
The area is given by the integral from 0 to 1 of e^x dx. This evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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Q. Find the area under the curve y = x^2 + 2x from x = 0 to x = 3.
Solution
The area under the curve is given by ∫(from 0 to 3) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 3 = (27/3 + 9) = 18.
Correct Answer: C — 15
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Q. Find the area under the curve y = x^2 from x = 0 to x = 2.
Solution
The area under the curve y = x^2 from 0 to 2 is given by the integral ∫(from 0 to 2) x^2 dx = [x^3/3] from 0 to 2 = (2^3/3) - (0^3/3) = 8/3.
Correct Answer: C — 8/3
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Q. Find the area under the curve y = x^2 from x = 0 to x = 3.
Solution
Area = ∫ from 0 to 3 of x^2 dx = [1/3 * x^3] from 0 to 3 = 9.
Correct Answer: A — 9
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Q. Find the area under the curve y = x^2 from x = 1 to x = 3.
-
A.
8/3
-
B.
10/3
-
C.
9/3
-
D.
7/3
Solution
The area is given by the integral ∫ (x^2) dx from 1 to 3. This evaluates to [x^3/3] from 1 to 3 = (27/3 - 1/3) = 26/3.
Correct Answer: B — 10/3
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Q. Find the area under the curve y = x^4 from x = 0 to x = 1.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The area under the curve y = x^4 from 0 to 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Find the area under the curve y = x^4 from x = 0 to x = 2.
Solution
The area is given by the integral from 0 to 2 of x^4 dx. This evaluates to [x^5/5] from 0 to 2 = (32/5) = 16.
Correct Answer: C — 16
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Q. Find the argument of the complex number z = -1 - i.
-
A.
-3π/4
-
B.
3π/4
-
C.
π/4
-
D.
-π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.
Correct Answer: A — -3π/4
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Q. Find the arithmetic mean of the first five prime numbers.
Solution
First five primes: 2, 3, 5, 7, 11. Mean = (2 + 3 + 5 + 7 + 11) / 5 = 28 / 5 = 5.6.
Correct Answer: C — 7
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Q. Find the arithmetic mean of the numbers 12, 15, 18, 21, and 24.
Solution
Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18.
Correct Answer: A — 18
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Q. Find the coefficient of x^0 in the expansion of (2x + 3)^4.
Solution
The coefficient of x^0 is C(4, 0) * (2x)^0 * 3^4 = 1 * 81 = 81.
Correct Answer: A — 81
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Q. Find the coefficient of x^1 in the expansion of (x + 2)^5.
Solution
The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.
Correct Answer: B — 20
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Q. Find the coefficient of x^2 in the expansion of (3x - 4)^6.
-
A.
540
-
B.
720
-
C.
480
-
D.
360
Solution
The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.
Correct Answer: B — 720
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^6.
-
A.
-540
-
B.
-720
-
C.
540
-
D.
720
Solution
The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Correct Answer: A — -540
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Q. Find the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution
The coefficient of x^3 is C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Correct Answer: B — 15
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Q. Find the coefficient of x^3 in the expansion of (x + 2)^6.
-
A.
80
-
B.
120
-
C.
160
-
D.
240
Solution
The coefficient of x^3 is C(6,3) * (2)^3 = 20 * 8 = 160.
Correct Answer: B — 120
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^5.
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: A — -10
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^6.
-
A.
-20
-
B.
-15
-
C.
-10
-
D.
-6
Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: A — -20
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Q. Find the coefficient of x^3 in the expansion of (x - 3)^5.
-
A.
-135
-
B.
-90
-
C.
-60
-
D.
-45
Solution
The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.
Correct Answer: A — -135
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