Undergraduate
Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.
A.
540
B.
720
C.
810
D.
960
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Solution
The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.
Correct Answer: A — 540
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)
A.
80
B.
100
C.
120
D.
160
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Solution
The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.
Correct Answer: A — 80
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Q. Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)
A.
240
B.
360
C.
480
D.
600
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Solution
The coefficient of x^4 is C(5,4)(3)^4(2)^1 = 5 * 81 * 2 = 810.
Correct Answer: B — 360
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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)
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Solution
The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.
Correct Answer: A — -64
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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)
A.
(3, 4, 5)
B.
(2, 3, 4)
C.
(4, 5, 6)
D.
(5, 6, 7)
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Solution
Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).
Correct Answer: A — (3, 4, 5)
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Q. Find the derivative of f(x) = sin(x) + cos(x).
A.
cos(x) - sin(x)
B.
-sin(x) - cos(x)
C.
sin(x) + cos(x)
D.
-cos(x) + sin(x)
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Solution
The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. Find the derivative of f(x) = x^5 + 3x^3 - 2x.
A.
5x^4 + 9x^2 - 2
B.
5x^4 + 6x^2 - 2
C.
3x^2 + 5x^4 - 2
D.
5x^4 + 3x^2 - 2
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Solution
The derivative f'(x) = d/dx(x^5 + 3x^3 - 2x) = 5x^4 + 9x^2 - 2.
Correct Answer: A — 5x^4 + 9x^2 - 2
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Q. Find the determinant of F = [[4, 5], [6, 7]]. (2020)
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Solution
Det(F) = (4*7) - (5*6) = 28 - 30 = -2.
Correct Answer: A — -2
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Q. Find the determinant of the matrix \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2021)
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Solution
The determinant is \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer: A — 10
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Q. Find the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10. (2023)
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Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.
Correct Answer: A — 2
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Q. Find the general solution of the equation y' = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = C/x^3
D.
y = Cx^3
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Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer: A — y = Ce^(x^3)
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Q. Find the integral of (2x + 1)^3 dx. (2019)
A.
(1/4)(2x + 1)^4 + C
B.
(1/3)(2x + 1)^4 + C
C.
(1/5)(2x + 1)^4 + C
D.
(1/2)(2x + 1)^4 + C
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Solution
Using substitution, the integral is (1/4)(2x + 1)^4 + C.
Correct Answer: A — (1/4)(2x + 1)^4 + C
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Q. Find the integral of cos(2x)dx. (2023)
A.
(1/2)sin(2x) + C
B.
sin(2x) + C
C.
(1/2)cos(2x) + C
D.
2sin(2x) + C
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Solution
The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. Find the integral of cos(x). (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
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Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. Find the integral of e^x dx. (2022)
A.
e^x + C
B.
e^x
C.
x e^x + C
D.
ln(e^x) + C
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Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. Find the integral of sin(x). (2020)
A.
-cos(x) + C
B.
cos(x) + C
C.
sin(x) + C
D.
-sin(x) + C
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Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. Find the integral of sin(x)dx. (2020)
A.
-cos(x) + C
B.
cos(x) + C
C.
sin(x) + C
D.
-sin(x) + C
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Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. Find the integral of x^5 dx. (2020)
A.
(1/6)x^6 + C
B.
(1/5)x^6 + C
C.
(1/4)x^6 + C
D.
(1/7)x^6 + C
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Solution
The integral is (1/6)x^6 + C.
Correct Answer: B — (1/5)x^6 + C
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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6. (2023)
A.
√49
B.
√36
C.
√45
D.
√50
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Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
Correct Answer: A — √49
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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)
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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 slope of the tangent line to f(x) = 2x^3 - 3x^2 + 4 at x = 1. (2021)
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Solution
f'(x) = 6x^2 - 6. f'(1) = 6(1)^2 - 6 = 0.
Correct Answer: B — 2
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Q. Find the solution of the differential equation dy/dx = y^2.
A.
y = 1/(C - x)
B.
y = C/(x - 1)
C.
y = Cx
D.
y = e^(x)
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Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)
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Solution
The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.
Correct Answer: A — -15
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Q. Find the value of 5! (5 factorial). (2019)
A.
120
B.
100
C.
150
D.
90
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Solution
5! = 5 × 4 × 3 × 2 × 1 = 120.
Correct Answer: A — 120
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Q. Find the value of k for which the equation x² + kx + 16 = 0 has equal roots. (2022)
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Solution
For equal roots, the discriminant must be zero: k² - 4*1*16 = 0, thus k² = 64, k = ±8. The value of k can be -8.
Correct Answer: A — -8
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Q. Find the value of k if the equation x² + kx + 16 = 0 has no real roots. (2022)
A.
k < 8
B.
k > 8
C.
k < 0
D.
k > 0
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Solution
For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, which gives k > 8.
Correct Answer: B — k > 8
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Q. For a monatomic ideal gas, the ratio of specific heats (γ) is approximately: (2019)
A.
1.5
B.
1.67
C.
1.4
D.
2
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Solution
For a monatomic ideal gas, γ = C_p/C_v = 5/3 = 1.67.
Correct Answer: B — 1.67
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Q. For a process at constant volume, which of the following is true? (2023)
A.
Work done is zero
B.
Heat added equals change in internal energy
C.
Both A and B
D.
None of the above
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Solution
At constant volume, no work is done (W=0), and the heat added equals the change in internal energy (Q=ΔU).
Correct Answer: C — Both A and B
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Q. For a reaction with a rate constant of 0.02 M⁻¹s⁻¹ and initial concentration of 0.5 M, what is the time taken to reach 0.25 M in a second-order reaction? (2023)
A.
25 s
B.
50 s
C.
10 s
D.
20 s
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Solution
Using t = 1 / (k[A₀]) * (1/[A] - 1/[A₀]), t = 1 / (0.02 * 0.5) * (1/0.25 - 1/0.5) = 25 s.
Correct Answer: A — 25 s
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Q. For a reaction with an activation energy of 50 kJ/mol, what is the rate constant at 300 K if R = 8.314 J/(mol·K)? (2022)
A.
0.001 M/s
B.
0.01 M/s
C.
0.1 M/s
D.
1 M/s
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Solution
Using the Arrhenius equation, k = Ae^(-Ea/RT). Calculate k using the given values.
Correct Answer: C — 0.1 M/s
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