Q. Evaluate ∫_0^1 (x^3 - 3x^2 + 3x - 1) dx.
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Solution
The integral evaluates to [x^4/4 - x^3 + (3/2)x^2 - x] from 0 to 1 = 0.
Correct Answer: A — 0
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Q. Evaluate ∫_0^1 (x^4 - 2x^2 + 1) dx.
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Solution
∫_0^1 (x^4 - 2x^2 + 1) dx = [x^5/5 - (2/3)x^3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = 1/15.
Correct Answer: B — 1
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Q. Evaluate ∫_0^1 (x^4) dx.
A.
1/5
B.
1/4
C.
1/3
D.
1/2
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Solution
The integral evaluates to [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Evaluate ∫_0^π/2 cos^2(x) dx.
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Solution
∫_0^π/2 cos^2(x) dx = π/4.
Correct Answer: A — π/4
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Q. Evaluate ∫_0^π/2 sin^2(x) dx.
A.
π/4
B.
π/2
C.
π/3
D.
π/6
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Solution
Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Correct Answer: A — π/4
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Q. Evaluate ∫_1^2 (3x^2 - 4) dx.
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Solution
The integral evaluates to [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Correct Answer: A — 1
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Q. Evaluate ∫_1^2 (3x^2 - 4x + 1) dx.
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Solution
∫_1^2 (3x^2 - 4x + 1) dx = [x^3 - 2x^2 + x] from 1 to 2 = (8 - 8 + 2) - (1 - 2 + 1) = 1.
Correct Answer: B — 1
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Q. Evaluate ∫_1^3 (2x + 1) dx.
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Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 10
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Q. Evaluate: sin^(-1)(0) + cos^(-1)(0).
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Solution
sin^(-1)(0) = 0 and cos^(-1)(0) = π/2, thus the sum is 0 + π/2 = π/2.
Correct Answer: B — π/2
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Q. Evaluate: sin^(-1)(1) + cos^(-1)(0).
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Solution
sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.
Correct Answer: A — π/2
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Q. Find the 10th term of the sequence defined by a_n = 3n + 2.
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Solution
a_10 = 3(10) + 2 = 30 + 2 = 32.
Correct Answer: A — 32
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Q. Find the 10th term of the sequence defined by a_n = 3n^2 + 2n.
A.
320
B.
302
C.
290
D.
310
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Solution
a_10 = 3(10^2) + 2(10) = 300 + 20 = 320.
Correct Answer: B — 302
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Q. Find the angle between the lines represented by the equation 2x^2 - 3xy + y^2 = 0.
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
90 degrees
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Solution
The angle between the lines can be found using the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the lines. The slopes can be found from the equation.
Correct Answer: B — 45 degrees
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Q. Find the angle between the lines y = 2x + 1 and y = -0.5x + 3.
A.
60 degrees
B.
45 degrees
C.
90 degrees
D.
30 degrees
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Solution
The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.
Correct Answer: A — 60 degrees
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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
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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°
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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°
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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°
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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°
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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.
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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.
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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
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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).
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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.
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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), (0,3).
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Solution
Area = 0.5 * base * height = 0.5 * 4 * 3 = 6.
Correct Answer: A — 6
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Q. Find the area of the triangle with vertices (0,0), (4,0), and (4,3).
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Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: B — 12
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Q. Find the area of the triangle with vertices at (0,0), (4,0), and (0,3).
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Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: A — 6
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Q. Find the area under the curve y = e^x from x = 0 to x = 1.
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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.
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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.
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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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