JEE Main
Q. Solve the differential equation dy/dx = 3x^2.
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
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D.
y = 3x + C
Solution
Integrating both sides gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. Solve the differential equation dy/dx = x^2 + y^2.
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A.
y = x^3/3 + C
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B.
y = x^2 + C
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C.
y = x^2 + x + C
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D.
y = Cx^2 + C
Solution
This is a non-linear differential equation. The solution can be found using substitution methods.
Correct Answer: A — y = x^3/3 + C
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Q. Solve the differential equation y' = 3y + 6.
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A.
y = Ce^(3x) - 2
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B.
y = Ce^(3x) + 2
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C.
y = 2e^(3x)
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D.
y = 3e^(3x) + 2
Solution
Using the integrating factor method, we find y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
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Q. Solve the differential equation y'' + 4y = 0.
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A.
y = C1 cos(2x) + C2 sin(2x)
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B.
y = C1 e^(2x) + C2 e^(-2x)
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C.
y = C1 cos(x) + C2 sin(x)
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D.
y = C1 e^(x) + C2 e^(-x)
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. Solve the differential equation y'' - 5y' + 6y = 0.
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A.
y = C1 e^(2x) + C2 e^(3x)
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B.
y = C1 e^(3x) + C2 e^(2x)
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C.
y = C1 e^(x) + C2 e^(2x)
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D.
y = C1 e^(2x) + C2 e^(x)
Solution
The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).
Correct Answer: B — y = C1 e^(3x) + C2 e^(2x)
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Q. Solve the equation 2sin(x) + √3 = 0 for x in the interval [0, 2π].
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A.
5π/3
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B.
π/3
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C.
2π/3
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D.
4π/3
Solution
Rearranging gives sin(x) = -√3/2, so x = 4π/3 and x = 5π/3.
Correct Answer: A — 5π/3
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Q. Solve the equation 2sin(x) - 1 = 0 for x in the interval [0, 2π].
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A.
π/6
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B.
5π/6
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C.
π/2
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D.
7π/6
Solution
The solution is x = π/2.
Correct Answer: C — π/2
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Q. Solve the equation 3cos^2(x) - 1 = 0.
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A.
x = π/3, 2π/3
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B.
x = π/4, 3π/4
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C.
x = 0, π
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D.
x = π/6, 5π/6
Solution
Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.
Correct Answer: A — x = π/3, 2π/3
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Q. Solve the equation 3sin(x) - 4 = 0 for x in the interval [0, 2π].
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A.
π/6
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B.
π/3
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C.
2π/3
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D.
5π/6
Solution
The solution is x = π/3.
Correct Answer: B — π/3
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Q. Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].
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A.
π/4
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B.
π/2
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C.
3π/4
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D.
0
Solution
The only solution is x = π/2.
Correct Answer: B — π/2
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Q. Solve the equation cos(x) = -1/2 for x in the interval [0, 2π].
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A.
2π/3, 4π/3
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B.
π/3, 5π/3
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C.
π/2, 3π/2
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D.
0, π
Solution
The solutions are x = 2π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — 2π/3, 4π/3
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Q. Solve the equation dy/dx = y^2 - x.
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A.
y = sqrt(x + C)
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B.
y = x + C
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C.
y = 1/(C - x)
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D.
y = x - C
Solution
This is a separable equation. Separating variables and integrating gives y = 1/(C - x).
Correct Answer: C — y = 1/(C - x)
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Q. Solve the equation sin(2x) = 1 for x in the interval [0, 2π].
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A.
π/4
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B.
3π/4
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C.
π/2
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D.
5π/4
Solution
The equation sin(2x) = 1 gives 2x = π/2 + 2nπ, hence x = π/4 + nπ/2. In [0, 2π], the solutions are π/4 and 5π/4.
Correct Answer: C — π/2
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Q. Solve the equation sin(2x) = √3/2 for x in the interval [0, 2π].
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A.
π/12
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B.
5π/12
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C.
7π/12
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D.
11π/12
Solution
The solutions are x = π/12, 5π/12, 7π/12, and 11π/12.
Correct Answer: A — π/12
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Q. Solve the equation sin(3x) = 0 for x in the interval [0, 2π].
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A.
0, π, 2π
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B.
0, π/3, 2π/3
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C.
0, π/2, π
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D.
0, π/4, π/2
Solution
The solutions are x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
Correct Answer: A — 0, π, 2π
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Q. Solve the equation sin(x) = 0.5 for x in the interval [0, 2π].
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A.
π/6
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B.
5π/6
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C.
7π/6
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D.
11π/6
Solution
The solutions are x = π/6 and x = 5π/6 in the interval [0, 2π].
Correct Answer: A — π/6
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Q. Solve the equation tan(x) = √3 for x in the interval [0, 2π].
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A.
π/3
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B.
2π/3
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C.
4π/3
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D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3 for x in the interval [0, 2π].
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A.
π/3
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B.
2π/3
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C.
4π/3
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D.
5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — π/3
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Q. Solve the equation tan^2(x) = 3.
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A.
x = π/3
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B.
x = 2π/3
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C.
x = 4π/3
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D.
x = 5π/3
Solution
The solutions are x = π/3 and x = 4π/3.
Correct Answer: A — x = π/3
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Q. Solve the equation y' = y(1 - y).
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A.
y = 1/(C - x)
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B.
y = 1/(C + x)
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C.
y = C/(1 + x)
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D.
y = C/(1 - x)
Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. Solve the first-order linear differential equation dy/dx + y/x = x.
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A.
y = x^2 + C/x
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B.
y = Cx^2 + x
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C.
y = C/x + x^2
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D.
y = x^2 + C
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we can solve the equation.
Correct Answer: A — y = x^2 + C/x
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Q. Solve the inequality -2x + 4 > 0. What is the solution?
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A.
x < 2
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B.
x > 2
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C.
x < -2
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D.
x > -2
Solution
-2x + 4 > 0 => -2x > -4 => x < 2
Correct Answer: B — x > 2
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Q. Solve the inequality -2x + 4 < 0.
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A.
x > 2
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B.
x < 2
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C.
x > 4
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D.
x < 4
Solution
-2x + 4 < 0 => -2x < -4 => x > 2.
Correct Answer: A — x > 2
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Q. Solve the inequality -2x + 5 > 1.
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A.
x < 2
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B.
x > 2
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C.
x < 3
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D.
x > 3
Solution
-2x + 5 > 1 => -2x > -4 => x < 2.
Correct Answer: B — x > 2
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Q. Solve the inequality -3x + 2 ≤ 5. What is the solution?
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A.
x ≤ -1
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B.
x ≥ -1
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C.
x < -1
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D.
x > -1
Solution
-3x + 2 ≤ 5 => -3x ≤ 3 => x ≥ -1.
Correct Answer: B — x ≥ -1
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Q. Solve the inequality -4x + 1 > 9.
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A.
x < -2
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B.
x > -2
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C.
x < 2
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D.
x > 2
Solution
Subtract 1 from both sides: -4x > 8. Then divide by -4 (reverse the inequality): x < -2.
Correct Answer: B — x > -2
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Q. Solve the inequality -4x + 8 > 0.
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A.
x < 2
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B.
x > 2
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C.
x < 4
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D.
x > 4
Solution
-4x + 8 > 0 => -4x > -8 => x < 2.
Correct Answer: B — x > 2
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Q. Solve the inequality -4x + 8 ≤ 0. What is the solution?
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A.
x < 2
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B.
x > 2
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C.
x ≤ 2
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D.
x ≥ 2
Solution
-4x + 8 ≤ 0 => -4x ≤ -8 => x ≥ 2.
Correct Answer: C — x ≤ 2
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Q. Solve the inequality -5x + 3 < 2. What is the solution?
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A.
x > 1
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B.
x < 1
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C.
x ≤ 1
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D.
x ≥ 1
Solution
-5x + 3 < 2 => -5x < -1 => x > 1.
Correct Answer: B — x < 1
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Q. Solve the inequality -x/2 + 4 > 0. What is the solution set?
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A.
x < 8
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B.
x > 8
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C.
x ≤ 8
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D.
x ≥ 8
Solution
-x/2 + 4 > 0 => -x/2 > -4 => x < 8.
Correct Answer: B — x > 8
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