Q. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)
-
A.
√14
-
B.
√6
-
C.
√9
-
D.
√12
Solution
Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.
Correct Answer: A — √14
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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)
Solution
Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.
Correct Answer: B — √14
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Q. Calculate the gravitational potential energy of a 2 kg mass at a height of 5 m. (g = 9.8 m/s²)
-
A.
98 J
-
B.
19.6 J
-
C.
39.2 J
-
D.
49 J
Solution
Potential Energy (PE) = m * g * h = 2 kg * 9.8 m/s² * 5 m = 98 J
Correct Answer: C — 39.2 J
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Q. Calculate the integral ∫ (x^2 + 2x + 1) dx.
-
A.
(1/3)x^3 + x^2 + x + C
-
B.
(1/3)x^3 + x^2 + C
-
C.
(1/3)x^3 + 2x^2 + C
-
D.
(1/3)x^3 + x^2 + x
Solution
The integral of x^2 is (1/3)x^3, the integral of 2x is x^2, and the integral of 1 is x. Thus, ∫ (x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + x + C
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Q. Calculate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + 2x + C
-
C.
x^2 + x + C
-
D.
(1/3)x^3 + (1/2)x^2 + C
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
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Q. Calculate the integral ∫ (x^3 - 4x) dx.
-
A.
(1/4)x^4 - 2x^2 + C
-
B.
(1/4)x^4 - 2x^2
-
C.
(1/4)x^4 - 4x^2 + C
-
D.
(1/4)x^4 - 2x^2 + 1
Solution
The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Correct Answer: A — (1/4)x^4 - 2x^2 + C
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Q. Calculate the integral ∫ cos^2(x) dx.
-
A.
(1/2)x + (1/4)sin(2x) + C
-
B.
(1/2)x + C
-
C.
(1/2)x - (1/4)sin(2x) + C
-
D.
(1/2)x + (1/2)sin(2x) + C
Solution
Using the identity cos^2(x) = (1 + cos(2x))/2, we find that ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.
Correct Answer: A — (1/2)x + (1/4)sin(2x) + C
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Q. Calculate the integral ∫ from 0 to π of sin(x) dx.
Solution
The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer: C — 2
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Q. Calculate the integral ∫(2 to 3) (x^3) dx. (2023)
Solution
∫(2 to 3) (x^3) dx = [x^4/4] from 2 to 3 = (81/4 - 16/4) = 65/4 = 16.25.
Correct Answer: C — 8
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Q. Calculate the interquartile range (IQR) for the data set: 1, 3, 7, 8, 9, 10.
Solution
Q1 = 3, Q3 = 9; IQR = Q3 - Q1 = 9 - 3 = 6.
Correct Answer: A — 4
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Q. Calculate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Infinity
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer: B — 1/2
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Q. Calculate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.
Correct Answer: B — 1
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Q. Calculate the limit: lim (x -> 0) (ln(1 + x)/x) (2023)
-
A.
1
-
B.
0
-
C.
Undefined
-
D.
Infinity
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.
Correct Answer: A — 1
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Q. Calculate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
3
-
B.
1
-
C.
0
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 3, so the limit is 3.
Correct Answer: A — 3
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Q. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)
-
A.
0
-
B.
1
-
C.
5
-
D.
Undefined
Solution
Using the limit property lim (x -> 0) (tan(kx)/x) = k, we have lim (x -> 0) (tan(5x)/x) = 5.
Correct Answer: C — 5
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Q. Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Using the fact that sin(x) ~ x as x approaches 0, we find that lim (x -> 0) (x^3)/(sin(x)) = 0.
Correct Answer: A — 0
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2 = (x + 1)/(x - 1). Thus, lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
3
-
D.
Undefined
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer: C — 3
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Q. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
Solution
Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.
Correct Answer: D — 4
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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.
Correct Answer: D — Undefined
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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Correct Answer: A — 3/5
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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)
Solution
Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.
Correct Answer: B — 5/2
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Q. Calculate the mean absolute deviation for the data set: 1, 2, 3, 4, 5.
Solution
Mean = 3. Mean Absolute Deviation = (|1-3| + |2-3| + |3-3| + |4-3| + |5-3|)/5 = (2 + 1 + 0 + 1 + 2)/5 = 1.5.
Correct Answer: B — 1.5
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Q. Calculate the mean of the following data: 5, 10, 15, 20.
-
A.
10
-
B.
12.5
-
C.
15
-
D.
17.5
Solution
Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.
Correct Answer: B — 12.5
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Q. Calculate the mean of the following numbers: 10, 20, 30, 40, 50.
Solution
Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30.
Correct Answer: A — 30
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Q. Calculate the mean of the following numbers: 4, 8, 12, 16, 20.
Solution
Mean = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12.
Correct Answer: C — 14
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Q. Calculate the molality of a solution if the boiling point elevation is 1.024 °C. (K_b for water = 0.512 °C kg/mol)
-
A.
1 mol/kg
-
B.
2 mol/kg
-
C.
0.5 mol/kg
-
D.
0.25 mol/kg
Solution
Molality = ΔT_b / (i * K_b) = 1.024 / (2 * 0.512) = 1 mol/kg
Correct Answer: B — 2 mol/kg
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Q. Calculate the moment of inertia of a hollow sphere of mass M and radius R about an axis through its center.
-
A.
2/5 MR^2
-
B.
3/5 MR^2
-
C.
2/3 MR^2
-
D.
MR^2
Solution
The moment of inertia of a hollow sphere about an axis through its center is I = 2/5 MR^2.
Correct Answer: B — 3/5 MR^2
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