VITEEE 2017 :: Mathematics :: General questions

• Mathematics - General questions

The area bounded by f(x) = x² , 0≤x≤1, g(x) = -x+2, 1≤x≤2 and x- axis is 

Answer:

Explanation:

Required area = Area of OAB + Area of ABC

Now, Area of OAB = $\int_{0}^{1}f(x)dx+\int_{1}^{2}g(x)dx$

= $\int_{0}^{1}x^{2}dx+\int_{1}^{2}(-x+2)dx$

= $\frac{x^{3}}{3}\mid_0^1$ + $\left[\frac{-x^{2}}{2}+2x\right]_1^2$

= $\frac{1}{3}+\left[\left(\frac{-4}{2}+4\right) - \left(\frac{-1}{2}+2\right)\right]$

= $\frac{1}{3}+\left[\left(-2+4\right) - \left(\frac{3}{2}\right)\right]$

= $\frac{1}{3}+\frac{1}{2}$ = $\frac{5}{6}$ Sq.Unit

If $u_{n}\int_{0}^{\pi/4}\tan^{n}\theta d\theta$ , then u_n + u_n-2 is

Answer:

Explanation:

Given :  $u_{n}\int_{0}^{\pi/4}\tan^{n}\theta d\theta$

= $\int_{0}^{\pi/4}\tan^{2}\theta \tan^{n-2}\theta d\theta$

= $\int_{0}^{\pi/4}(\sec^{2}\theta-1) \tan^{n-2}\theta d\theta$

= $\int_{0}^{\pi/4}\sec^{2}\theta \tan^{n-2}\theta d\theta$ - $\int_{0}^{\pi/4}\tan^{n-2}\theta d\theta$

= $\int_{0}^{\pi/4}(\sec^{2}\theta) \tan^{n-2}\theta d\theta$ - u_n-2

u_n + u_n-2 = $\int_{0}^{\pi/4}\sec^{2}\theta\tan^{n-2}\theta d\theta$

= $\frac{\tan^{n-1}\theta}{n-1}\mid_0^\frac{\pi}{4}$ = $\frac{1}{n-1}$

The line which passes through the origin and intersect the two lines

$\frac{x -1}{2}=\frac{y+3}{4}=\frac{z-5}{3}$, $\frac{x -4}{2}=\frac{y+3}{3}=\frac{z-14}{4}$ is

Answer:

Explanation:

Let the line be $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$ ...(1)

If line (1) intersects with the line $\frac{x-1}{2}=\frac{y+3}{4}=\frac{z-5}{3}$, then

$\begin{vmatrix}a & b & c\\2& 4 & 3\\4&-3&14 \end{vmatrix}=0$

9a - 7b - 10c = 0   ...(2)

from (1) and (2) , we have,

$\frac{a}{1}=\frac{b}{-3}=\frac{c}{5}$

.'. The line  is $\frac{x}{1}=\frac{y}{-3}=\frac{z}{5}$

A and B are events such that P(A υ B) = 3/4, $P(A\cap B) =1/4, P(\bar{A}) = 2/3then, P(\bar{A}\cap B)$ is

Answer:

Explanation:

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

3/4  $=1- P(\bar{A}) + P(B) - 1/4$

1 = 1-2/3 + P(B) → P(B) = 2/3

Now ,$P(\bar{A}\cap B)$ = P(B) - $P(A\cap B)$ = 2/3 - 1/4 = 5/12

A and B are two independent witnesses (i.e. there is no collision between them) in a case' The probability that A will speak the truth is x and the probability that B will speak the truth is y. A and B agree on a certain statement. The probability that the statement is true is

Answer:

Explanation:

A and B will agree in a certain statement if both speak truth or both tell a lie. We define following events

E₁ = A and B both speak truth → P(E₁) = xy

E₂ = A and B both tell a lie → P(E₂) = (1-x)(1-y)

E = A and B agree in a certain statement

Clearly, P(E/E1) = 1 and P(E/E₂) = 1 The required probability is P(E₁/E )

Using Baye's theorem P(E₁/E )

 = $\frac{P(E_{1})P(E/E_{1})}{P(E_{1})P(E/E_{1})+P(E_{2})P(E/E_{2})}$

= $\frac{xy.1}{xy.1+(1-x)(1-y).1}=\frac{xy}{1-x-y+2xy}$

• Mathematics - General questions