Q. Find the odd one out: 2, 3, 5, 7, 9, 11 (2021)
Solution
All numbers except 9 are prime numbers.
Correct Answer:
D
— 9
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Q. Find the odd one out: Dog, Cat, Rabbit, Car (2023)
-
A.
Dog
-
B.
Cat
-
C.
Rabbit
-
D.
Car
Solution
Car is an inanimate object; the others are animals.
Correct Answer:
D
— Car
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Q. Find the odd one out: January, February, March, Sunday (2023)
-
A.
January
-
B.
February
-
C.
March
-
D.
Sunday
Solution
Sunday is a day of the week; the others are months.
Correct Answer:
D
— Sunday
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Q. Find the particular solution of dy/dx = 2x with the initial condition y(0) = 1.
-
A.
y = x^2 + 1
-
B.
y = x^2 - 1
-
C.
y = 2x + 1
-
D.
y = 2x - 1
Solution
Integrating gives y = x^2 + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer:
A
— y = x^2 + 1
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Q. Find the particular solution of dy/dx = 2y with the initial condition y(0) = 1.
-
A.
y = e^(2x)
-
B.
y = e^(2x) + 1
-
C.
y = 1 + e^(2x)
-
D.
y = e^(2x) - 1
Solution
The general solution is y = Ce^(2x). Using the initial condition y(0) = 1 gives C = 1, so y = e^(2x).
Correct Answer:
A
— y = e^(2x)
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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(x)
-
D.
y = 2e^(x)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(2x)
-
D.
y = 2e^(x/4)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the particular solution of dy/dx = x + y, given y(0) = 1.
-
A.
y = e^x + 1
-
B.
y = e^x - 1
-
C.
y = x + 1
-
D.
y = x + e^x
Solution
The general solution is y = e^x + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer:
A
— y = e^x + 1
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Q. Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)
-
A.
(1, 4)
-
B.
(2, 3)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. f(2) = 3.
Correct Answer:
C
— (3, 0)
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Q. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.
-
A.
(1, 4)
-
B.
(2, 3)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).
Correct Answer:
C
— (3, 0)
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Q. Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
-
A.
(1, 3)
-
B.
(2, 2)
-
C.
(3, 1)
-
D.
(0, 6)
Solution
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. The point of inflection is at (2, f(2)) = (2, 2).
Correct Answer:
A
— (1, 3)
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Q. Find the point of intersection of the lines 2x + 3y = 6 and x - y = 1. (2020)
-
A.
(0, 2)
-
B.
(2, 0)
-
C.
(1, 1)
-
D.
(3, 0)
Solution
Solving the equations simultaneously, we find the intersection point is (1, 1).
Correct Answer:
C
— (1, 1)
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Q. Find the point of intersection of the lines 2x + y = 10 and x - y = 1. (2020)
-
A.
(3, 4)
-
B.
(4, 2)
-
C.
(2, 6)
-
D.
(5, 0)
Solution
Solving the equations simultaneously, we find the intersection point is (3, 4).
Correct Answer:
A
— (3, 4)
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Q. Find the point of intersection of the lines y = 2x + 1 and y = -x + 4.
-
A.
(1, 3)
-
B.
(2, 5)
-
C.
(3, 7)
-
D.
(4, 9)
Solution
Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).
Correct Answer:
A
— (1, 3)
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Q. Find the point of intersection of the lines y = x + 1 and y = -x + 5.
-
A.
(2, 3)
-
B.
(3, 2)
-
C.
(1, 2)
-
D.
(0, 1)
Solution
Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).
Correct Answer:
A
— (2, 3)
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Q. Find the point of intersection of the lines y = x + 2 and y = -x + 4. (2023)
-
A.
(1, 3)
-
B.
(2, 4)
-
C.
(3, 5)
-
D.
(0, 2)
Solution
Setting x + 2 = -x + 4 gives 2x = 2, so x = 1. Substituting x back gives y = 3. Thus, the point is (1, 3).
Correct Answer:
A
— (1, 3)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 that has a horizontal tangent. (2023)
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 2)
-
D.
(3, 4)
Solution
To find horizontal tangents, set the derivative y' = 3x^2 - 6x = 0. This gives x = 0 and x = 2. The point (1, 2) has a horizontal tangent.
Correct Answer:
B
— (1, 2)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 where the tangent is horizontal. (2023)
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 2)
-
D.
(3, 4)
Solution
To find horizontal tangents, set dy/dx = 0. dy/dx = 3x^2 - 6x = 0 gives x = 0 and x = 2. At x = 1, y = 2.
Correct Answer:
B
— (1, 2)
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Q. Find the projection of vector A = (2, 3) onto vector B = (1, 1).
Solution
Projection of A onto B = (A · B) / |B|^2 * B; A · B = 2*1 + 3*1 = 5; |B|^2 = 1^2 + 1^2 = 2; Projection = (5/2)(1, 1) = (2.5, 2.5).
Correct Answer:
A
— 1
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Q. Find the projection of vector A = (3, 4) onto vector B = (1, 2).
Solution
Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).
Correct Answer:
B
— 2
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Q. Find the range of the data set: 10, 15, 20, 25, 30.
Solution
Range = Maximum - Minimum = 30 - 10 = 20.
Correct Answer:
A
— 15
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Q. Find the range of the data set: 12, 15, 20, 22, 30.
Solution
Range = Maximum - Minimum = 30 - 12 = 18.
Correct Answer:
C
— 18
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Q. Find the range of the data set: 12, 15, 22, 30, 5.
Solution
Range = max - min = 30 - 5 = 25.
Correct Answer:
A
— 25
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Q. Find the range of the data set: 8, 12, 15, 20, 25.
Solution
Range = Maximum - Minimum = 25 - 8 = 17.
Correct Answer:
A
— 12
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Q. Find the real part of the complex number 4 + 5i. (2023)
Solution
The real part of the complex number 4 + 5i is 4.
Correct Answer:
A
— 4
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Q. Find the real part of the complex number z = 2 + 3i.
Solution
The real part of z = 2 + 3i is 2.
Correct Answer:
A
— 2
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Q. Find the real part of the complex number z = 2e^(iπ/3).
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer:
B
— 2
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Q. Find the real part of the complex number z = 3 + 4i.
Solution
The real part of z is 3.
Correct Answer:
A
— 3
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Q. Find the real part of the complex number z = 4 + 3i.
Solution
The real part of z = 4 + 3i is 4.
Correct Answer:
A
— 4
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Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).
Solution
The real part is 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer:
A
— 2
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