CBSE Class 12 Maths - MCQ and Online Tests - Unit 9 - Differential Equations

CBSE Class 12 Maths – MCQ and Online Tests – Unit 9 – Differential Equations

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

 

CBSE Class 12 Maths – MCQ and Online Tests – Unit 9 – Differential Equations

Question 1.
The order and degree of the differential equation
\(\frac{d^2y}{dx^2}\) + (\(\frac{dy}{dx}\))^{\(\frac{1}{4}\)} + x^{\(\frac{1}{3}\)} = 0 respectvely, are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3

Answer

Answer: (a) 2 and not defined

---

Question 2.
If y = e^-x (A cos x + B sin x), then y is a solution of
(a) \(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) = 0
(b) \(\frac{d^2y}{dx^2}\) – 2\(\frac{dy}{dx}\) + 2y = 0
(c) \(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) + 2y = 0
(d) \(\frac{d^2y}{dx^2}\) + 2y = 0

Answer

Answer: (c) \(\frac{d^2y}{dx^2}\) + 2\(\frac{dy}{dx}\) + 2y = 0

---

Question 3.
The differential equation for y = A cos αx + B sin αx where A and B are arbitary constants is
(a) \(\frac{d^2y}{dx^2}\) – α²y = 0
(b) \(\frac{d^2y}{dx^2}\) + α²y = 0
(c) \(\frac{d^2y}{dx^2}\) + αy = 0
(d) \(\frac{d^2y}{dx^2}\) – αy = 0

Answer

Answer: (b) \(\frac{d^2y}{dx^2}\) + α²y = 0

---

Question 4.
Integration factor of differential equation \(\frac{dy}{dx}\) + py = Q, where P and IQ are functions of x is
(a) ∫e^pdx
(b) \(_{e}\)∫pdx
(c) \(_{e}\)-∫pdx
(d) None of these

Answer

Answer: (d) None of these

---

Question 5.
Solution of differential equation xdy – ydx = Q represents
(a) a rectangular hyperbola
(b) parabola whose vertex is at origin
(c) straight line passing through origin
(d) a circle whose centre is at origin

Answer

Answer: (c) straight line passing through origin

---

Question 6.
Integrating factor of the differential equation cos x \(\frac{dy}{dx}\) + y sin x = 1 is
(a) cos x
(b) tan x
(c) sec x
(d) sin x

Answer

Answer: (c) sec x

---

Question 7.
Solution of the differential equation tan y sec² x dx + tan x sec² y dy + 0 is .
(a) tan x + tan y = k
(b) tan x – tan y = k
(c) \(\frac{tan x}{tan y}\) = k
(d) tan x.tan y = k

Answer

Answer: (d) tan x.tan y = k

---

Question 8.
Family r = Ax + A³ of curves is represented by the differential equation of degree
(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: (b) 2

---

Question 9.
The radius of a circle is increasing at the rate of 0.4 cm/ s. The rate of increasing of its circumference is
(a) 0.4 π cm/s
(b) 0.8 π cm/s
(c) 0.8 cm/s
(d) None of these

Answer

Answer: (b) 0.8 π cm/s

---

Question 10.
The degree of differential equation
[1 + (\(\frac{dy}{dx}\))²]^{\(\frac{3}{2}\)} = \(\frac{d^2y}{dx^2}\) is
(a) 4
(b) \(\frac{3}{2}\)
(c) 2
(d) not defined

Answer

Answer: (c) 2

---

Question 11.
Integrating factor of \(\frac{xdy}{dx}\) – y = x⁴ – 3x is
(a) x
(b) log x
(c) \(\frac{1}{2}\)
(d) -x

Answer

Answer: (c) \(\frac{1}{2}\)

---

Question 12.
Solution of \(\frac{dy}{dx}\) – y = 1 y(0) = 1 is given by
(a) xy = -e^x
(b) xy = -e^-x
(c) xy = -1
(d) y = 2e^x – 1

Answer

Answer: (d) y = 2e^x – 1

---

Question 13.
The number of solutions of \(\frac{dy}{dx}\) = \(\frac{y+1}{x-1}\) when y(1) = 2 is
(a) none
(b) one
(c) two
(d) infinite

Answer

Answer: (b) one

---

Question 14.
Which of the following is a second order differential equation?
(a) (y’)² + x = y²
(b) y’y” + y = sin x
(c) y” + (y”)² + y = 0
(d) y’ = y²

Answer

Answer: (b) y’y” + y = sin x

---

Question 15.
Integrating factor of the differential equation
(1 – x²) \(\frac{dy}{dx}\) – xy = 1 is
(a) -x
(b) \(\frac{x}{1+x^2}\)
(c) \(\sqrt{1-x^2}\)
(d) \(\frac{1}{2}\) log(1 – x²)

Answer

Answer: (c) \(\sqrt{1-x^2}\)

---

Question 16.
The differential equation y \(\frac{dy}{dx}\) + x = c represents
(a) Family of hyperbolas
(b) Family of parabolas
(c) Family of ellipses
(d) Family of circles

Answer

Answer: (d) Family of circles

---

Question 17.
The general solution of e^x cos y dx – e^x sin y dy = 0 is
(a) e^x cos y = k
(b) e^x sin y = k
(c) e^x = k cos y
(d) e^x = k sin y

Answer

Answer: (a) e^x cos y = k

---

Question 18.
The solution of \(\frac{dy}{dx}\) = 1 + x + y + xy is
(a) x – y = k(1 + xy)
(b) log (1 + y) = x + \(\frac{x^2}{2}\) + k
(c) log (1 + x) + y + \(\frac{y^2}{2}\) = k
(d) None of these

Answer

Answer: (b) log (1 + y) = x + \(\frac{x^2}{2}\) + k

---

Question 19.
The degree of the differential equation
\(\frac{d^2y}{dx^2}\) + (\(\frac{dy}{dx}\))³ + 6y⁵ = 0 is
(a) 1
(b) 2
(c) 3
(d) 5

Answer

Answer: (a) 1

---

Question 20.
The degree of the differential equation
(\(\frac{d^2y}{dx}\))² + (\(\frac{dy}{dx}\))² = x sin \(\frac{dy}{dx}\) is
(a) 1
(b) 2
(c) 3
(d) not defined

Answer

Answer: (d) not defined

---

Question 21.
Family y = Ax + A³ of curves will correspond to a differential equation of order
(a) 3
(b) 2
(c) 1
(d) not finite

Answer

Answer: (b) 2

---

Question 22.
The solution of \(\frac{dy}{dx}\) + y = e^-x, y (0) = 0 is
(a) y = e^x(x – 1)
(b) y = xe^-x
(c) y = xe^-x + 1
(d) y = (x + 1 )e^-x

Answer

Answer: (b) y = xe^-x

---

Question 23.
The solution of the differential equation \(\frac{dy}{dx}\) = \(\frac{1+y^2}{1+x^2}\)
(a) y = tan^-1 x
(b) y – x = k(1 + xy)
(c) x = tan^-1 y
(d) tan (xy) = k

Answer

Answer: (b) y – x = k(1 + xy)

---

Question 24.
The integrating factor of the differential equation \(\frac{dy}{dx}\) + y = \(\frac{1+y}{x}\) is
(a) \(\frac{x}{e^x}\)
(b) \(\frac{e^x}{x}\)
(c) xe^x
(d) e^x

Answer

Answer: (b) \(\frac{e^x}{x}\)

---

Question 25.
y = ae^mx + be^-mx satisfies which of the following differential equation?
(a) \(\frac{dy}{dx}\) + my = 0
(b) \(\frac{dy}{dx}\) – my = 0
(c) \(\frac{d^2y}{dx^2}\) – m²y = 0
(d) \(\frac{d^2y}{dx^2}\) +m²y = 0

Answer

Answer: (c) \(\frac{d^2y}{dx^2}\) – m²y = 0

---

Question 26.
The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is
(a) \(\frac{sin x}{sin y}\) = c
(b) sin x sin y = c
(c) sin x + sin y = z
(d) cos x cos y = c

Answer

Answer: (b) sin x sin y = c

---

Question 27.
The solution of x \(\frac{dy}{dx}\) + y = e^x is
(a) y = \(\frac{e^x}{x}\) + \(\frac{k}{x}\)
(b) y = xe^x + cx
(c) y = xe^x + k
(d) x = \(\frac{e^vy}{y}\) + \(\frac{k}{y}\)

Answer

Answer: (a) y = \(\frac{e^x}{x}\) + \(\frac{k}{x}\)

---

Question 28.
Integrating factor of the differential equation \(\frac{dy}{dx}\) + y tan x – sec x = 0 is
(a) cos x
(b) sec x
(c) e^{cos x}
(d) e^{sec x}

Answer

Answer: (b) sec x

---

Question 29.
The differential equation of the family of cuves x² + y² – 2ay = 0, where a is arbitrary constant is
(a) (x² – y²)\(\frac{dy}{dx}\) = 2xy
(b) 2 (x² + y²)\(\frac{dy}{dx}\) = xy
(c) 2(x² – y²)\(\frac{dy}{dx}\) = xy
(d) (x² + y²) \(\frac{dy}{dx}\) = 2xy

Answer

Answer: (a) (x² – y²)\(\frac{dy}{dx}\) = 2xy

---

Question 30.
tan^-1 x + tan^-1 y = c is the general solution of the differential equation
(a) \(\frac{dy}{dx}\) = \(\frac{1+y^2}{1+x^2}\)
(b) \(\frac{dy}{dx}\) = \(\frac{1+x^2}{1+y^2}\)
(c) (1 + x²)dy + (1 + y²)dx = 0
(d) (1 +x²2)dx+(1 + y²)dy = 0

Answer

Answer: (c) (1 + x²)dy + (1 + y²)dx = 0

---