VITEEE 2019 :: Mathematics :: General questions

• Mathematics - General questions

Let a, b, c, be in A.P. with a common difference d.Then $e^{1/c}, e^{b/ac},e^{1/a}$ are in :

Answer:

Explanation:

a,b,c are in A.P. 2b = a+c Now,

$e^{1/c}\times e^{1/a} = e^{(a+c)/ac}$ = $e^{2b/ac}$ = $\left(e^{b/ac}\right)^{2}$

.'. $e^{1/c}, e^{b/ac},e^{1/a}$ in G.P with common ratio

$\frac{e^{b/ac}}{e^{1/c}}$ = $e^{b-a/ac}$ = $e^{d/(b-d)(b+d)}$

= $e^{d/(b^{2}-d^{2})}$

'.' a, b, c are in A.P. with common difference d .'. b- a = c-b = d

The value of the integral $\int_{-1}^{3}(|x|+|x-1|)dx $ is

Answer:

Explanation:

$|x|+|x-1| = \begin{cases}-x-(x-1)=-2x+1, if & x \leq 0\\ x-(x-1) = 1, if & 0\leq x \leq 1\\ x+(x-1) = 2x-1,if&x\geq1\end{cases}$

.'.$\int_{-1}^{3}(|x|+|x-1|)dx $ =

$\int_{-1}^{0}(-2x+1)dx +\int_{0}^{1}1dx+\int_{1}^{3}(2x-1)dx$

= $\left[-x^{2}+x\right]_{-1}^{0}+\left[x\right]_0^1+\left[x^{2}-x\right]_1^3$ =9

The value of x in the interval [4,9] at which the function f(x) = √x satisfies the mean value theorem is

Answer:

Explanation:

(i) f(x) = √x is continuous in[4,9]

(ii) $f'(x) = \frac{1}{2\sqrt{x}}$

Thus f(x) is differentiable in (4, 9) (iii) f(4) ≠ f(9)

All the three conditions of LMV theorem satisfied then there exist at least one c Ε (4,9) such that.

f'(c) = $\frac{f(b)-f(a)}{b-a}$ = $\frac{1}{2\sqrt{c}}$ = 1/5

c = 25/4

Let f: R→ R, g : R→ R be two functions such that f(x) = 2x - 3, g(x) = x³ + 5. The function (fog)^-1(x) is equal to 

Answer:

Explanation:

We have, f(x) = 2x-3, g(x) = x³ + 5

(fog)x =f (x³ + 5) = 2(x³ + 5) -3 = 2x³+7

Let y= (fog)x = 2x³ + 7

$x = \left(\frac{y-7}{2}\right)^{1/3}$

(fog)^-1$x = \left(\frac{x-7}{2}\right)^{1/3}$

It is given that the events A and B are such that P(A) = 1/4 , P(A/B) = 1/2 and P(B/A) = 2/3 then P(B) is?

Answer:

Explanation:

P(A) = 1/4 , P(A/B) = 1/2 and P(B/A) = 2/3 then P(B) 

By conditional probability

P(A∩B) = P(A)P(B/A) = P(B)P(A/B)

$\frac{1}{4}\times\frac{2}{3} = P(B)\times\frac{1}{2}$

P(B) = $\frac{1}{3}$

• Mathematics - General questions