MHT CET 2017 :: Mathematics :: General questions

• Mathematics - General questions

For the following distribution function F(x)  of a rv.X

$P(3<x\leq5)=$

Answer:

Explanation:

$P(3<x\leq5)=$   p(x=4)+p(x=5)

           =0.14+0.23= 0.37

$\triangle$ ABC  has vertices at A=(2,3,5), B=(-1,3,2) and C= ($\lambda$,5, $\mu$). If the median through A is equally inclined to the axes, then the values of $\lambda$  and $\mu$  respectively are

Answer:

Explanation:

Give , A(2,3,5), B(-1,3,2) and C ($\lambda$,5,$\mu$) be the vertices of $\triangle$ ABC.

Let D be the median through A to BC

=$\left(\frac{\lambda-1}{2},\frac{5+3}{2},\frac{\mu+2}{2}\right)$

=$\left(\frac{\lambda-1}{2},\frac{8}{2},\frac{\mu+2}{2}\right)$

$z=\left(\frac{\lambda-1}{2},4,\frac{\mu+2}{2}\right)$

Now, direction ratio of 

$AD=\left(\frac{\lambda-1}{2}-2,4-3,\frac{\mu+2}{2}-5\right)$

 i.e,  $\left(\frac{\lambda-1-4}{2},1,\frac{\mu+2-10}{2}\right)$

i.e,  $\left(\frac{\lambda-5}{2},1,\frac{\mu-8}{2}\right)$

 Since , the line AD  is equally inclined to the coordinates axes

$\therefore$    $\frac{\lambda-5}{2}=1=\frac{\mu-8}{2}$

 On solving first two, we get

$\frac{\lambda-5}{2}=1$

 $\Rightarrow$      $\lambda$  =7

On solving last two, we get 

     $1=\frac{\mu-8}{2}$

$\Rightarrow$      $\mu$  =10

The value of 

$\cos ^{-1}\left(\cot\left(\frac{\pi}{2}\right)\right)+\cos ^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right)$  is

Answer:

Explanation:

 We have, 

  $\cos ^{-1}\left(\cot\left(\frac{\pi}{2}\right)\right)+\cos ^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right)$ 

=$\cos ^{-1}(0)+\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$

    $\left[\because \cot \frac{\pi}{2}=0  and  \sin \frac{2 \pi}{3}=\frac{\sqrt{3}}{2}\right]$

 =$cos ^{-1}\left( \cos \frac{\pi}{2}\right)+cos ^{-1}\left( \cos \frac{\pi}{6}\right)$

                  $\left[\because \cos  \frac{\pi}{2}=0   and  \cos \frac{\pi}{6}=\frac{\sqrt{3}}{2}\right]$

 = $\frac{\pi}{2}+\frac{\pi}{6}$

$=\frac{3\pi+\pi}{6}=\frac{4\pi}{6}=\frac{2 \pi}{3}$

If the  distance of points $2\hat{i}+3\hat{j}+\lambda \hat{k}$ from the plane $r.(3\hat{i}+2\hat{j}+6\hat{k})=13$ is  5 units. then $\lambda$=

Answer:

Explanation:

Given point be (2,3,$\lambda$) and equation of the plane be

 $r.(3\hat{i}+2\hat{j}+6\hat{k})=13$

$\therefore$    $ (x\hat{i}+y\hat{j}+z\hat{k}).(3\hat{i}+2\hat{j}+6\hat{k})=13$

                                                                    $ [\because r.(x\hat{i}+y\hat{j}+z\hat{k})]$

$\Rightarrow    $          $ 3x+2y+6z=13$

or   3x+2y+6z-13=0

 Now, distance of the plane from the point (2,3, $\lambda$)

$d=|\frac{ax_{1}+by_{1}+cz_{1}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}|$

$5=|\frac{3\times2+2\times3+6 \times\lambda-13}{\sqrt{3^{2}+2^{2}+6^{2}}}|$

                                                                          [given , d=5]

 $\pm5=\frac{6+6+6 \times\lambda-13}{\sqrt{9+4+36}}$

$\Rightarrow $     $\pm5=\frac{6\lambda-1}{\sqrt{49}}$

$\therefore$    $5=|\frac{6\lambda-1}{7}|$

 $\Rightarrow$    $\pm35=6\lambda-1$

 $\Rightarrow$      $35=6\lambda-1$

or                     $-35=6\lambda-1$

 $\Rightarrow$    $\lambda=6,\lambda=-\frac{17}{3}$

Probability that a person will develop immunity after  vaccinations is 0.8 .If 8 people are given the vaccine, then probability  that all develop  immunity is =

Answer:

Explanation:

 Given probability  that a person will develop immunity after vaccination=0.8

 $\therefore$ probability of 8 persons that will develop immunity 

                                            =     $(0.8)^{8}$

• Mathematics - General questions