Q. Calculate the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
-
A.
ad - bc
-
B.
ab + cd
-
C.
ac - bd
-
D.
bc - ad
Solution
The determinant is calculated as \( ad - bc \).
Correct Answer:
A
— ad - bc
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Q. Calculate the determinant | 1 0 0 | | 0 1 0 | | 0 0 1 |.
Solution
The determinant of the identity matrix is 1.
Correct Answer:
B
— 1
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Q. Calculate the determinant | 2 3 | | 4 5 | + | 1 1 | | 1 1 |.
Solution
The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.
Correct Answer:
B
— 1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 2 | | 0 1 3 |.
Solution
The determinant evaluates to 0 as the rows are linearly dependent.
Correct Answer:
A
— -1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.
Solution
Using the determinant formula, we find that the determinant evaluates to 0.
Correct Answer:
A
— -1
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Q. Calculate the distance between the points (1, 1) and (4, 5).
Solution
Using the distance formula: d = √[(4 - 1)² + (5 - 1)²] = √[9 + 16] = √25 = 5.
Correct Answer:
B
— 5
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Q. Calculate the distance between the points (1, 2) and (1, 5).
Solution
Using the distance formula: d = √[(1 - 1)² + (5 - 2)²] = √[0 + 9] = √9 = 3.
Correct Answer:
A
— 3
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Q. Calculate the distance between the points (6, 8) and (2, 3).
Solution
Using the distance formula: d = √[(2 - 6)² + (3 - 8)²] = √[16 + 25] = √41.
Correct Answer:
B
— 6
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Q. Calculate the distance between the points (6, 8) and (6, 2).
Solution
Using the distance formula: d = √((6 - 6)² + (2 - 8)²) = √(0 + 36) = √36 = 6.
Correct Answer:
A
— 6
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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 integral ∫(2 to 5) (4x - 1) dx. (2023)
Solution
∫(2 to 5) (4x - 1) dx = [2x^2 - x] from 2 to 5 = (50 - 5) - (8 - 2) = 40.
Correct Answer:
A
— 20
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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^2 sin(1/x))
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Since |sin(1/x)| ≤ 1, we have |x^2 sin(1/x)| ≤ |x^2|. Thus, lim (x -> 0) x^2 sin(1/x) = 0.
Correct Answer:
A
— 0
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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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