CBSE Class 12 Mathematics Vector Algebra Important Questions Set 01

Very Short Answer Questions:

Question. What is the degree of the following differential equation: \(5x\left(\frac{dy}{dx}\right)^2 - \frac{d^2y}{dx^2} - 6y = \log x\) ? 
Answer: 1

Question. Write the degree of the following differential equation: \(x^3 \left(\frac{d^2y}{dx^2}\right)^2 + x\left(\frac{dy}{dx}\right)^4 = 0\) 
Answer: 2

Question. Write the sum of the order and degree of the following differential equation: \(\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^4 = 0\) 
Answer: 4

Question. Find the product of the order and degree \(x\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 + y^2 = 0\).
Answer: 4

Question. Write the differential equation formed from the equation \(y = mx + c\), where \(m\) and \(c\) are arbitrary constants. 
Answer: \(\frac{d^2y}{dx^2} = 0\)

Question. Write the integrating factor of \((x \log x)\frac{dy}{dx} + y = 2\log x\) . 
Answer: \(\log x\)

Question. Solve : \(e^{dy/dx} = x^2\)
Answer: \(y = 2(x \log x - x) + C\)

Question. State whether \(y = e^{-x}(x + a)\) is the solution of differential equation: \(\frac{dy}{dx} + y = e^{-x}\)
Answer: Yes

Question. Solve : \(\frac{dy}{dx} - \frac{y(x + 1)}{x} = 0\)
Answer: \(y = x e^{x+C}\)

Short Answer Questions–I:

Question. Write the general solution of the differential equation \(\frac{dy}{dx} = \frac{y}{x}\) .
Answer: \(y = C x\)

Question. Write the integrating factor of \(\frac{dy}{dx} + y = \frac{1 + y}{x}\) .
Answer: \(\frac{e^x}{x}\)

Question. Given that \(\frac{dy}{dx} = e^{-2y}\) and \(y = 0\) when \(x = 5\). Find the value of \(x\) when \(y = 3\).
Answer: \(\frac{e^6+9}{2}\)

Question. Find the general solution of the differential equation \(\frac{dy}{dx} = 2^{y-x}\) .
Answer: \(2^{-x} - 2^{-y} = C\)

Question. Find the differential equation of the family of curves \(y = A e^{2x} + B e^{-2x}\).
Answer: \(\frac{d^2y}{dx^2} - 4y = 0\)

Short Answer Questions–II:

Question. Solve the following differential equation: \(\cos^2 x \frac{dy}{dx} + y = \tan x\) 
Answer: \(y = \tan x - 1 + C e^{-\tan x}\)

Question. Solve the differential equation: \((x^2 + 1)\frac{dy}{dx} + 2xy = \sqrt{x^2 + 4}\) 
Answer: \((x^2 + 1)y = \frac{x}{2}\sqrt{x^2 + 4} + 2 \log|x + \sqrt{x^2 + 4}| + C\)

Question. Solve : \((1 + x^2)\frac{dy}{dx} + y = \tan^{-1} x\) 
Answer: \(y e^{\tan^{-1} x} = (\tan^{-1} x - 1) e^{\tan^{-1} x} + C\)

Question. Solve : \((x^2 - 1)\frac{dy}{dx} + 2xy = \frac{2}{x^2 - 1}\) 
Answer: \(y(x^2 - 1) = \log \left| \frac{x - 1}{x + 1} \right| + C\)

Question. Solve the differential equation: \(y + x \frac{dy}{dx} = x - y \frac{dy}{dx}\) 
Answer: \(y^2 + 2xy - x^2 = C^2\)

Question. Solve the differential equation: \((x^2 + 3xy + y^2)dx - x^2 dy = 0\) given that \(y = 0\), when \(x = 1\)
Answer: \(y = \frac{x \log|x|}{1 - \log|x|}\)

Question. Solve the differential equation : \((x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^3\) 
Answer: \(\frac{y}{x + 1} = \frac{e^{3x}}{3} \frac{(x + 1)^2}{9} + C\)

Question. Solve the following differential equation : \(y^2 dx + (x^2 - xy + y^2)dy = 0\) 
Answer: \(y = C e^{\tan^{-1} \frac{y}{x}}\)

Question. Find the particular solution of the differential equation: \((1 + e^{2x})dy + (1 + y^2)e^x dx = 0\), given that \(y = 1\), when \(x = 0\) 
Answer: \(\tan^{-1} y + \tan^{-1} e^x = \frac{\pi}{2}\)

Question. Find the particular solution of this differential equation \(x^2 \frac{dy}{dx} - xy = 1 + \cos\left(\frac{y}{x}\right) , x \neq 0\). Find the particular solution of this differential equation, given that when \(x = 1, y = \frac{\pi}{2}\) . 
Answer: \(\tan\left(\frac{y}{2x}\right) = -\frac{1}{2x^2} + \frac{3}{2}\)

Question. Find the particular solution of the differential equation \(\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\) given that \(y = 1\), when \(x = 0\). 
Answer: \(-\frac{x^2}{2y^2} + \log|y| = 0\)

Question. Solve the following differential equation: \((x^2 - 1)\frac{dy}{dx} + 2xy = \frac{2}{x^2 - 1}\) 
Answer: \(y = \frac{1}{x^2 - 1} \log \left| \frac{x - 1}{x + 1} \right| + \frac{C}{x^2 - 1}\)

Question. Solve the differential equation \(\frac{dy}{dx} + y \cot x = 2 \cos x\), given that when \(x = \frac{\pi}{2}, y = 0\). 
Answer: \(2y \sin x = -(1 + \cos 2x)\)

Question. Solve the following differential equation: \((1 + x^2)\frac{dy}{dx} + y = \tan^{-1} x\) 
Answer: \(y = (\tan^{-1} x - 1) + C e^{-\tan^{-1} x}\)

Question. Solve the differential equation \((x^2 - yx^2) dy + (y^2 + x^2y^2) dx = 0\), given that \(y = 1\), when \(x = 1\). 
Answer: \(\log|y| + \frac{1}{y} = -\frac{1}{x} + x + 1\)

Question. Solve the differential equation: \(\frac{dy}{dx} = \frac{x + y}{x - y}\) 
Answer: \(\tan^{-1}\left(\frac{y}{x}\right) = \frac{1}{2}\log(x^2 + y^2) + C\)

Question. Solve the differential equation: \((1 + x^2)dy + 2xy dx = \cot x dx\) 
Answer: \(y = \frac{1}{1 + x^2} \log|\sin x| + \frac{C}{1 + x^2}\)

Question. Find the general solution of the differential equation \(x^2 y dx - (x^3 + y^3) dy = 0\). 
Answer: \(\log|y| = \frac{x^3}{3y^3} + C\)

Long Answer Questions:

Question. Solve the following differential equation, given that \(y = 0\), when \(x = \frac{\pi}{4}\) : \(\sin 2x \frac{dy}{dx} - y = \tan x\) 
Answer: \(y = \tan x - \sqrt{\tan x}\)

Question. Find the differential equation for all the straight lines, which are at a unit distance from the origin. 
Answer: \(y = x y' \pm \sqrt{(y')^2 + 1}\)

Question. Solve the following differential equation: \(\left[y - x \cos\left(\frac{y}{x}\right)\right]dy + \left[y \cos\left(\frac{y}{x}\right) - 2x \sin\left(\frac{y}{x}\right)\right]dx = 0\) 
Answer: \(y^2 - 2x^2 \cos\left(\frac{y}{x}\right) = C\)

Question. Find the particular solution of the differential equation \((1 + x^2)\frac{dy}{dx} = (e^{m \tan^{-1} x} - y)\) given that \(y = 1\), when \(x = 0\). 
Answer: \(y e^{\tan^{-1} x} = (\tan^{-1} x - 1)e^{\tan^{-1} x} + C\)

Question. Find the particular solution of the following differential equation: \(xy \frac{dy}{dx} = (x + 2)(y + 2)\); \(y = -1\) when \(x = 1\) 
Answer: \(\frac{y}{x+1} = \frac{e^{3x}}{3} - \frac{e^{3x}}{9} + C\)

Question. Find the particular solution of the differential equation \(\frac{dx}{dy} + x \cot y = 2y + y^2 \cot y\), \((y \neq 0)\) given that \(x = 0\) when \(y = \frac{\pi}{2}\) . 
Answer: \(x \sin y = y^2 \sin y - \frac{\pi^2}{4}\)

Question. Find the particular solution of the differential equation \(x(1 + y^2)dx - y(1 + x^2)dy = 0\) given that \(y = 1\) when \(x = 0\). 
Answer: \(\frac{x^2 \log|x|}{1 - \log|x|}\)

Question. Find the particular solution of the differential equation satisfying the given conditions \(x^2 dy + (xy + y^2) dx = 0\); \(y = 1\) when \(x = 1\). 
Answer: \(y = \frac{x \log|x|}{1 - \log|x|}\)

Question. \((x^2 + y^2) dy = xy dx\). If \(y(1) = 1\) and \(y(x_0) = e\), then find the value of \(x_0\). 
Answer: \(x_0 = \sqrt{3}e\)

Question. Find the particular solution of the differential equation \((y - \sin x)dx + (\tan x)dy = 0\) satisfying the condition that \(y = 0\) when \(x = 0\).
Answer: \(y^2 - 2x^2 \cos\left(\frac{y}{x}\right) = C\)