MHT-CET
Q. What is the general solution of the equation y'' - 3y' + 2y = 0?
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A.
y = C1 e^(x) + C2 e^(2x)
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B.
y = C1 e^(2x) + C2 e^(x)
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C.
y = C1 e^(3x) + C2 e^(0)
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D.
y = C1 e^(0) + C2 e^(3x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. Thus, the general solution is y = C1 e^(2x) + C2 e^(x).
Correct Answer: B — y = C1 e^(2x) + C2 e^(x)
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Q. What is the general solution of the equation y'' - 4y' + 4y = 0?
-
A.
y = (C1 + C2x)e^(2x)
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B.
y = C1 e^(2x) + C2 e^(-2x)
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C.
y = C1 e^(4x) + C2 e^(-4x)
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D.
y = C1 cos(2x) + C2 sin(2x)
Solution
The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the geometry of a molecule with two bonding pairs and no lone pairs on the central atom? (2023)
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A.
Linear
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B.
Trigonal planar
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C.
Tetrahedral
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D.
Bent
Solution
The geometry is linear when there are two bonding pairs and no lone pairs on the central atom.
Correct Answer: A — Linear
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Q. What is the geometry of the complex ion [Ag(NH3)2]+? (2019) 2019
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A.
Tetrahedral
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B.
Square planar
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C.
Octahedral
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D.
Linear
Solution
The complex [Ag(NH3)2]+ has a linear geometry due to the presence of two ligands around the silver ion.
Correct Answer: D — Linear
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Q. What is the gravitational field strength at a distance of 10 m from a mass of 100 kg? (G = 6.67 × 10^-11 N m²/kg²)
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A.
6.67 × 10^-10 N/kg
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B.
6.67 × 10^-9 N/kg
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C.
6.67 × 10^-8 N/kg
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D.
6.67 × 10^-7 N/kg
Solution
g = G * M / r² = (6.67 × 10^-11) * (100) / (10²) = 6.67 × 10^-10 N/kg
Correct Answer: A — 6.67 × 10^-10 N/kg
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Q. What is the gravitational force acting on a 50 kg object on the surface of the Earth?
-
A.
490 N
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B.
50 N
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C.
5 N
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D.
500 N
Solution
Weight W = m * g = 50 kg * 9.8 m/s² = 490 N.
Correct Answer: A — 490 N
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Q. What is the gravitational force between two 1 kg masses placed 1 meter apart? (G = 6.67 × 10^-11 N m²/kg²)
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A.
6.67 × 10^-11 N
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B.
1.67 × 10^-10 N
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C.
6.67 × 10^-10 N
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D.
0 N
Solution
F = G * (m1 * m2) / r² = (6.67 × 10^-11) * (1 * 1) / (1²) = 6.67 × 10^-11 N
Correct Answer: A — 6.67 × 10^-11 N
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Q. What is the gravitational force between two objects of mass 5 kg and 10 kg separated by a distance of 2 m? (G = 6.67 × 10^-11 N m²/kg²)
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A.
1.67 × 10^-10 N
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B.
1.25 × 10^-10 N
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C.
1.00 × 10^-10 N
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D.
2.00 × 10^-10 N
Solution
Using the formula F = G * (m1 * m2) / r², we have F = (6.67 × 10^-11) * (5 * 10) / (2²) = 1.67 × 10^-10 N.
Correct Answer: A — 1.67 × 10^-10 N
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Q. What is the gravitational force between two objects of mass m1 and m2 separated by a distance r? (2021)
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A.
G * (m1 * m2) / r^2
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B.
G * (m1 + m2) / r^2
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C.
G * (m1 - m2) / r^2
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D.
G * (m1 * m2) * r^2
Solution
The gravitational force is given by Newton's law of gravitation: F = G * (m1 * m2) / r^2.
Correct Answer: A — G * (m1 * m2) / r^2
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Q. What is the gravitational potential energy of a 10 kg mass at a height of 10 m?
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A.
100 J
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B.
200 J
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C.
500 J
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D.
1000 J
Solution
Gravitational potential energy (PE) is given by PE = mgh. Here, PE = 10 kg * 9.81 m/s² * 10 m = 981 J.
Correct Answer: D — 1000 J
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Q. What is the gravitational potential energy of a 10 kg object at a height of 5 m? (g = 9.8 m/s²)
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A.
49 J
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B.
98 J
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C.
245 J
-
D.
490 J
Solution
Gravitational potential energy (PE) is given by PE = mgh. Here, m = 10 kg, g = 9.8 m/s², and h = 5 m. PE = 10 * 9.8 * 5 = 490 J.
Correct Answer: B — 98 J
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Q. What is the gravitational potential energy of a 2 kg object at a height of 10 m? (2023)
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A.
20 J
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B.
15 J
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C.
10 J
-
D.
25 J
Solution
Gravitational potential energy (PE) = mgh = 2 kg × 9.8 m/s² × 10 m = 196 J.
Correct Answer: A — 20 J
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Q. What is the group number of Alkali metals in the periodic table? (2023)
Solution
Alkali metals are located in group 1 of the periodic table.
Correct Answer: A — 1
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Q. What is the group number of Halogens in the periodic table? (2021)
Solution
Halogens are located in group 17 of the periodic table.
Correct Answer: B — 17
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Q. What is the heat required to raise the temperature of 2 kg of aluminum from 25°C to 75°C? (Specific heat of aluminum = 0.9 J/g°C) (2022)
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A.
9000 J
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B.
18000 J
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C.
36000 J
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D.
4500 J
Solution
Heat required Q = mcΔT = (2000 g)(0.9 J/g°C)(50°C) = 90000 J.
Correct Answer: B — 18000 J
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Q. What is the hybridization of the central atom in [Ni(CO)4]? (2022)
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A.
sp
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B.
sp2
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C.
sp3
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D.
d2sp3
Solution
In [Ni(CO)4], the nickel atom undergoes sp3 hybridization to accommodate four CO ligands.
Correct Answer: C — sp3
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Q. What is the hybridization of the central metal ion in [Cr(NH3)6]3+? (2020) 2020
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A.
sp
-
B.
sp2
-
C.
d2sp3
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D.
dsp2
Solution
In [Cr(NH3)6]3+, the hybridization is d2sp3 due to the octahedral arrangement of six ligands.
Correct Answer: C — d2sp3
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Q. What is the impedance of a circuit with a resistance of 4 ohms and a reactance of 3 ohms? (2021)
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A.
5 ohms
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B.
7 ohms
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C.
1 ohm
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D.
12 ohms
Solution
Impedance Z = √(R² + X²) = √(4² + 3²) = √(16 + 9) = √25 = 5 ohms.
Correct Answer: A — 5 ohms
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Q. What is the impedance of a circuit with a resistor of 10Ω and an inductor of 0.1 H at a frequency of 50 Hz? (2022)
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A.
10Ω
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B.
15.71Ω
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C.
20Ω
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D.
5Ω
Solution
Impedance Z = √(R² + (XL)²), where XL = 2πfL = 31.42Ω. Thus, Z = √(10² + 31.42²) = 15.71Ω.
Correct Answer: B — 15.71Ω
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Q. What is the impedance of a circuit with a resistor of 4Ω and an inductor of 3Ω at resonance? (2022)
Solution
At resonance, the impedance is equal to the resistance, which is 4Ω.
Correct Answer: C — 4Ω
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Q. What is the impedance of a series circuit containing a resistor (R) and an inductor (L) at angular frequency ω? (2020)
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A.
R
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B.
√(R² + (ωL)²)
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C.
R + jωL
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D.
R + L
Solution
The impedance Z in a series R-L circuit is given by Z = √(R² + (ωL)²).
Correct Answer: B — √(R² + (ωL)²)
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Q. What is the impedance of a series circuit containing a resistor (R) and an inductor (L) at a frequency (f)? (2020)
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A.
R
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B.
√(R² + (2πfL)²)
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C.
R + 2πfL
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D.
R - 2πfL
Solution
The impedance Z in a series RL circuit is given by Z = √(R² + (2πfL)²).
Correct Answer: B — √(R² + (2πfL)²)
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Q. What is the integral of 1/x? (2022)
-
A.
ln
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B.
x
-
C.
+ C
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D.
1/x + C
-
.
x + C
-
.
x^2 + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer: A — ln
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Q. What is the integral of cos(2x) dx? (2021)
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A.
(1/2)sin(2x) + C
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B.
sin(2x) + C
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C.
(1/2)cos(2x) + C
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D.
2sin(2x) + C
Solution
The integral of cos(2x) is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. What is the integral of e^x with respect to x? (2023)
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A.
e^x + C
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B.
xe^x + C
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C.
e^x/x + C
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D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of e^x? (2023)
-
A.
e^x + C
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B.
xe^x + C
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C.
e^x/x + C
-
D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of sin(x) dx? (2021)
-
A.
-cos(x) + C
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B.
cos(x) + C
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C.
sin(x) + C
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D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. What is the integral of tan(x) dx? (2023)
-
A.
-ln
-
B.
cos(x)
-
C.
+ C
-
D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of tan(x)? (2021)
-
A.
-ln
-
B.
cos(x)
-
C.
+ C
-
D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of x^2 with respect to x? (2021)
-
A.
(1/3)x^3 + C
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B.
(1/2)x^2 + C
-
C.
(1/4)x^4 + C
-
D.
(1/5)x^5 + C
Solution
The integral of x^n is (1/(n+1))x^(n+1) + C. Here, n=2, so the integral is (1/3)x^3 + C.
Correct Answer: A — (1/3)x^3 + C
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