Mathematics
Q. In how many ways can 4 different fruits be selected from 10 fruits?
-
A.
210
-
B.
120
-
C.
240
-
D.
300
Solution
The number of ways to choose 4 fruits from 10 is C(10, 4) = 210.
Correct Answer: A — 210
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Q. In how many ways can 4 different prizes be distributed among 10 students? (2023)
-
A.
5040
-
B.
10000
-
C.
2400
-
D.
120
Solution
The number of ways to distribute 4 different prizes among 10 students is 10P4 = 5040.
Correct Answer: A — 5040
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Q. In how many ways can 4 prizes be distributed among 10 students?
-
A.
210
-
B.
5040
-
C.
10000
-
D.
1001
Solution
The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.
Correct Answer: B — 5040
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Q. In how many ways can 4 students be seated in a row of 6 chairs? (2021)
-
A.
360
-
B.
720
-
C.
120
-
D.
240
Solution
Choose 4 chairs from 6 (6C4) and arrange 4 students (4!). Total = 15 * 24 = 360.
Correct Answer: B — 720
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Q. In how many ways can 5 different books be arranged on a shelf? (2021)
-
A.
60
-
B.
120
-
C.
240
-
D.
720
Solution
The number of arrangements of 5 distinct books is 5! = 120.
Correct Answer: D — 720
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Q. In how many ways can 5 different colored balls be arranged in a line?
-
A.
60
-
B.
120
-
C.
240
-
D.
720
Solution
The number of ways to arrange 5 different colored balls is 5! = 120.
Correct Answer: D — 720
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Q. In how many ways can 6 different colored balls be arranged in a row? (2018)
-
A.
720
-
B.
600
-
C.
480
-
D.
360
Solution
The number of arrangements of 6 distinct balls is 6! = 720.
Correct Answer: A — 720
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Q. In how many ways can 6 people be seated in a row?
-
A.
720
-
B.
600
-
C.
480
-
D.
360
Solution
The number of arrangements of 6 people is 6! = 720.
Correct Answer: A — 720
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Q. In how many ways can 7 people be divided into 3 groups if one group must have 3 people?
-
A.
210
-
B.
300
-
C.
420
-
D.
560
Solution
Choose 3 people from 7: C(7, 3) = 35. The remaining 4 can be divided into 2 groups in 2 ways. Total = 35 * 2 = 70.
Correct Answer: B — 300
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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^0?
Solution
The coefficient of x^0 is given by 4C4 * (2x)^0 * (3)^4 = 1 * 81 = 81.
Correct Answer: A — 81
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Q. In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)
-
A.
-300
-
B.
-600
-
C.
600
-
D.
300
Solution
The coefficient of x^2 is C(5,2) * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.
Correct Answer: A — -300
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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)
-
A.
-150
-
B.
-300
-
C.
300
-
D.
150
Solution
The coefficient of x^2 is C(6,2) * (2)^2 * (-5)^4 = 15 * 4 * 625 = -37500.
Correct Answer: A — -150
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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)
-
A.
-600
-
B.
-720
-
C.
720
-
D.
600
Solution
The coefficient of x^3 is C(6,3)(2)^3(-5)^3 = 20 * 8 * -125 = -20000.
Correct Answer: B — -720
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Q. In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)
-
A.
540
-
B.
810
-
C.
729
-
D.
486
Solution
The coefficient of x^4 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: A — 540
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Q. In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)
-
A.
-144
-
B.
-216
-
C.
216
-
D.
144
Solution
The coefficient of x^2 is C(4,2)(3)^2(-4)^2 = 6 * 9 * 16 = -864.
Correct Answer: B — -216
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Q. In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)
Solution
The coefficient of x^3 is C(4,3) * (3)^1 = 4 * 3 = 12.
Correct Answer: A — 36
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Q. In the expansion of (x - 1)^5, what is the coefficient of x^3?
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: A — -10
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Q. Solve the differential equation dy/dx = 2x + 1.
-
A.
y = x^2 + x + C
-
B.
y = x^2 + 2x + C
-
C.
y = 2x^2 + x + C
-
D.
y = x^2 + C
Solution
Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.
Correct Answer: A — y = x^2 + x + C
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Q. Solve the differential equation dy/dx = 2y + 3. (2023)
-
A.
y = Ce^(2x) - 3/2
-
B.
y = Ce^(-2x) + 3/2
-
C.
y = 3e^(2x)
-
D.
y = 2e^(2x) + C
Solution
Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.
Correct Answer: A — y = Ce^(2x) - 3/2
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Q. Solve the differential equation dy/dx = y/x. (2023)
-
A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the differential equation y' = 5 - 2y.
-
A.
y = 5/2 + Ce^(-2x)
-
B.
y = 5 + Ce^(-2x)
-
C.
y = 2 + Ce^(2x)
-
D.
y = 5/2 - Ce^(-2x)
Solution
This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).
Correct Answer: A — y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation y' = 5y + 3.
-
A.
y = (3/5) + Ce^(5x)
-
B.
y = (5/3) + Ce^(5x)
-
C.
y = Ce^(5x) - 3
-
D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the solution y = (3/5) + Ce^(5x).
Correct Answer: A — y = (3/5) + Ce^(5x)
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Q. Solve the differential equation y'' - 3y' + 2y = 0.
-
A.
y = C1e^(2x) + C2e^(x)
-
B.
y = C1e^(x) + C2e^(2x)
-
C.
y = C1e^(-x) + C2e^(-2x)
-
D.
y = C1e^(3x) + C2e^(x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).
Correct Answer: B — y = C1e^(x) + C2e^(2x)
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Q. Solve the equation y' = 6y + 12.
-
A.
y = 2 - Ce^(-6x)
-
B.
y = Ce^(6x) - 2
-
C.
y = 2 + Ce^(6x)
-
D.
y = 6Ce^(-x)
Solution
This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).
Correct Answer: A — y = 2 - Ce^(-6x)
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Q. Solve the first-order differential equation dy/dx = y/x.
-
A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is a separable equation. Separating variables and integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
-
A.
y = x + C/x
-
B.
y = Cx - x
-
C.
y = Cx + x
-
D.
y = C/x + x
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.
Correct Answer: A — y = x + C/x
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Q. What are the roots of the equation 3x² - 12x + 12 = 0? (2019)
Solution
Dividing the equation by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.
Correct Answer: B — 4
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Q. What are the roots of the equation x² - 5x + 6 = 0? (2021)
-
A.
1 and 6
-
B.
2 and 3
-
C.
3 and 2
-
D.
0 and 5
Solution
The roots can be found using the factorization method: (x - 2)(x - 3) = 0, hence the roots are 2 and 3.
Correct Answer: B — 2 and 3
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Q. What is the 3rd term in the expansion of (2x + 3)^4?
-
A.
108x^2
-
B.
216x^2
-
C.
324x^2
-
D.
432x^2
Solution
The 3rd term is given by C(4,2) * (2x)^2 * (3)^2 = 6 * 4x^2 * 9 = 216x^2.
Correct Answer: B — 216x^2
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Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)
-
A.
112
-
B.
128
-
C.
256
-
D.
64
Solution
The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.
Correct Answer: A — 112
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