Prove that \(x^2-y^2=c(x^2+y^2)\)is the general solution of differential equation(\(x^3-3xy^2)dx=(y^3-3x^2y)dy\),where \(c\) is parameter.
Show that the points \(A(1,2,7),B(2,6,3)\),and \(C(3,10,-1)\) are collinear.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. What is the value of E(X)?
For each of the exercises given below, verify that the given function (implicit or explicit)is a solution of the corresponding differential equation.
i) \(y=ae^x+be^{-x}+x^2: x\frac{d^2y}{dx^2}+2\frac{dy}{dx}-xy+x^2-2=0\)
ii) \(y=e^x(a \cos x+ b \sin x):\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\)
iii) \(y= x \sin 3x:\frac{d^2y}{dx^2}+9y-6\cos3x=0\)
iv) \(x^2=2y^2\log y:(x^2+y^2)\frac{dy}{dx}-xy=0\)
Find the vector equation of the line passing through(1, 2, 3)and parallel to the planes \(\vec r.=(\hat i-\hat j+2\hat k)=5\) and \(\vec r.(3\hat i+\hat j+\hat k)=6\).
Let X denotes the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Solve the system of the following equations\(\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4\)\(\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1\)\(\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2\)
Determine P(E|F): A dice is thrown three times. E: 4 appears on the third toss,F: 6 and 5 appears respectively on first two tosses.
Find the equation of the plane which contain the line of intersection of the planes \(\hat r.(\hat i+2\hat j+3\hat k)-4=0\), \(\vec r.(2\hat i+\hat j-\hat k)+5=0\) and which is perpendicular to the plane \(\vec r.(5\hat i+3\hat j-6\hat k)+8=0.\)
A class has 15 students whose ages are 14,17,15,14,21,17,19,20,16,18,20,17,16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Using properties of determinants, prove that:\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ \beta&\beta^2 &\gamma+\alpha \\ \gamma&\gamma^2 &\alpha+\beta \end{vmatrix}\)=(\(\beta-\gamma\))( \(\gamma-\alpha\))(\(\alpha-\beta\))(\(\alpha+\beta+\gamma\))
Evaluate\(\begin{vmatrix} x &y &x+y \\ y&x+y &x \\ x+y&x &y \end{vmatrix}\)
Find the particular solution of the differential equation \((1+e^{2x})dy+(1+y^2)e^xdx=0\), given that \(y=1 \) when \(x=0\)