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.
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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) ∫epdx
(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 = -ex
(b) xy = -e-x
(c) xy = -1
(d) y = 2ex – 1
Answer
Answer: (d) y = 2ex – 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 ex cos y dx – ex sin y dy = 0 is
(a) ex cos y = k
(b) ex sin y = k
(c) ex = k cos y
(d) ex = k sin y
Answer
Answer: (a) ex 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 = ex(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) xex
(d) ex
Answer
Answer: (b) \(\frac{e^x}{x}\)
Question 25.
y = aemx + 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 = ex is
(a) y = \(\frac{e^x}{x}\) + \(\frac{k}{x}\)
(b) y = xex + cx
(c) y = xex + 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) ecos x
(d) esec 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
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