SNAP 2014

Considering two vowels as one unit ,total number of units = 4

Number of way of arranging these 4 units = 4! = 24 ways

2 vowels can be internally arranged in 2! ways 2 ways

Therefore total number of ways = 24*2 =48 ways.

AC= $$\sqrt{\ AB^2\ +BC^2}=\ \sqrt{\ 6^2+8^2\ }=10$$

In-radius of right angled triangle = $$\frac{\left(a+b-h\right)}{2}$$

Inradius = $$\frac{\left(6+8-10\right)}{2}=2$$

$$m^2-n^2=\ \left(m+n\right)\left(m-n\right)=\left(2\tan\theta\ \right)\left(2\sin\theta\ \right)=4\tan\theta\ \sin\theta\ $$

Going by the options you have only mn or m/n

Case 1 : mn

mn= $$\left(\tan\theta\ +\sin\theta\right)\ \left(\tan\theta\ -\sin\theta\ \right)$$ = $$\tan^2\theta\ -\sin^2\theta$$ = $$\frac{\sin^2\theta}{\cos^2\theta\ }\ -\sin^2\theta\ =\sin^2\theta\ \left(\frac{1}{\cos^2\theta\ }-1\right)=\frac{\sin^2\theta}{\cos^2\theta\ }\ \left(1-\cos^2\theta\ \right)=\tan^{^2}\theta\ \cdot\sin^2\theta\ $$

So $$4\sqrt{\ mn}\ =\ 4\tan\theta\ \cdot\sin\theta\ $$

Since P(B) and P(A) are mutually exclusive and exhaustive , thus, P(B)+P(A)=1

But we know that P(B)=3 P(A) , thus substituting in the above equation we get P(B)=3/4 and P(A)=1/4

Now they have asked P(B_) i.e. 1-P(B)=1-(3/4)=1/4 

The profit will be shared based on amount and time the money was invested by person

For A : $$\left(16000\times\ 3\right)+\left(11000\times\ 9\right)$$ = 147,000

For B : $$\left(12000\times\ 3\right)+\left(17000\times\ 9\right)$$ = 189,000

For C : $$\left(21000\times\ 6\right)$$ = 126,000

Share of B will be $$\frac{189}{147+189+126}\times\ 26400\ =\ 10,800$$

Share of C will be $$\frac{126}{147+189+126}\times\ 26400\ =\ 7,200$$

Difference is 3,600

$$time\ =\ \frac{dis\tan ce}{speed}$$

$$t1=\ \frac{1500}{S}$$

$$t2=\ \frac{1500}{S+250}$$

Given t1 - t2 = 1/2 hours

So, $$\frac{1500}{S}\ -\frac{1500}{S+250}=\frac{1}{2}$$

On solving S=750kmph

For divisibility by 6, number should be divide by 3 and even.

Sum of number = 4 + 4 + 5 + 6 = 19

For divisible by 3, 2 number should be added so,

Number = 4456 + 2 = 4458 

TPO is a right angled triangle .

So $$\sin\left(30\right)=\frac{OP}{TO}=\frac{3}{TO}\ =>TO=6$$

$$PT^2=TO^2-OP^2=36-9=27$$

PT= 3$$\sqrt{\ 3}$$

$$\log_{b^n}\left(a^m\right)\ =\frac{m}{n}\log_ba\ =\frac{m}{n}\cdot\frac{\log\left(a\right)}{\log\left(b\right)}$$

So given equation becomes $$\frac{4}{2}\cdot\frac{4}{2}\cdot\frac{\log\left(3\right)}{\log\left(2\right)}\cdot\frac{\log\left(2\right)}{\log\left(3\right)}$$  = 4

$$\frac{1}{100}\cdot\frac{1}{100}\cdot\frac{25}{100}\cdot1000\ =0.025$$