VITEEE 2018 :: Mathematics :: General questions

• Mathematics - General questions

The mean and variance of a random variable X having binomial distribution are 4 and 2 respectively the P(x=1) is

Answer:

Explanation:

Given, $P(\overline{A\cup B}) =\frac{1}{6}$

$\Rightarrow P({A\cup B}) =1-\frac{1}{6}=\frac{5}{6}$

$P(\overline{A}) =\frac{1}{4}$ $\Rightarrow P(A) =1-\frac{1}{4}=\frac{3}{4}$

We know,

$P(A\cup B) = P(A) + P(B) - P(A \cap B)$

$\Rightarrow \frac{5}{6}=\frac{3}{4}+P(B)-\frac{1}{4}$

$\Rightarrow P(B)=\frac{1}{3}$

 P(A)≠P(B) so they are not equally likely

Also P(A) x P(B)=3/4 × 1/3= 1/4 = $P(A \cap B)$

So A & B are independent

If $\vec{a}=(\hat{i}+\hat{j}+\hat{k})$, $\vec{a}.\vec{b} = 1$ and $\vec{a}\times\vec{b} = \hat{j}-\hat{k}$,

then $\vec{b}$ is

Answer:

Explanation:

$\because(\vec{a}\times\vec{b})\times\vec{a}=(\vec{a}.\vec{a})\vec{b}-(\vec{a}.\vec{b})\vec{a}$

$\therefore(\hat{j}-\hat{k})\times(\hat{i}+\hat{j}+k)=(\sqrt{3})^{2}(\vec{b})-(\hat{i}+\hat{j}+k)$

$\Rightarrow 3\hat{b}=3\hat{i}\Rightarrow \hat{b}=\hat{i}$

 A unit vectorperpendicular to the plane formed by the points (1, 0, 1), (0,2,2) and (3, 3, 0) is

Answer:

Explanation:

Let A(1, 0, 1), B(0, 2,2) andc (3, 3, 0) be the given points, then

$\vec{AB}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{BC}=3\hat{i}+\hat{j}-2\hat{k}$

$\vec{AB}\times\vec{BC}=\begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\-1 & 2 & 1 \\ 3 & 1 & -2\end{vmatrix}= -5\hat{i}+\hat{j}-7\hat{k}$

.'. unit vector $\perp$ to the Plane: ABC

= $\pm\frac{1}{5\sqrt{3}}(-5\hat{i}+\hat{j}-7\hat{k})$

The solution of  $\frac{dv}{dt}+\frac{k}{m}v$ = -g is

Answer:

Explanation:

$\frac{dv}{dt}+\frac{k}{m}v$ = -g → $\frac{dv}{dt}=-\frac{k}{m}\left(v+\frac{mg}{k}\right)$

$\frac{dv}{v+mg/k}=-\frac{k}{m}dt$

$\log_{}{}\left(v+\frac{mg}{k}\right)$ = $-\frac{k}{m}t$+log c

$\left(v+\frac{mg}{k}\right) = Ce^{-\frac{kt}{m}}$

= $v = C\bar{e}^{\frac{k}{m}t}-\frac{mg}{k}$

The order and degree of the differential equation whose solution is y = cx + c² - 3c^3/2 + 2, therefore c is a parameter, is

Answer:

Explanation:

y = cx + c² - 3c^3/2 + 2 .....(1)

Differentiating above with respect to x, we get $\frac{dy}{dx}$ = c

Putting this value of c in (1), we get

$y = x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^{2}-3\left(\frac{dy}{dx}\right)^{3/2}+2$

Clearly its order is ONE and after removing the fractional power we get the degree FOUR.

• Mathematics - General questions