SRCC GBO 2021

To solve such questions, it is always advisable to convert the number first into decimal (base 10) and then analyse the result. 

As per the question, we have 624 in base 7. To convert it into decimal, we do: 6$$\times\ $$ $$7^2$$ + 2$$\times\ $$ 7 + 4$$\times\ $$ 1, which is 312. 

This is exactly what we have in the question, which means that the number is 312 in base 10. 

Hence, $$10^2$$ = 100

To find $$\log_2(fog(64))$$, we shall first find the value of fog(64). 

The value of g(64): $$\sqrt[3]{f(64)}$$, where f(64) is $$8^9$$

The cube root of the following expression would be $$8^3$$, which is the value of g(64).

Now, fog(64) is f($$8^3$$), which is $$8^{13}$$.

For the final result, $$\log_2\left(8^{13}\right)$$, which is $$\log_2\left(2^{39}\right)$$ , or 39. 

Observing the series shows that the difference between consecutive terms is in and AP. For instance: 3-1=2, 7-3=4, 13-7=6 and so on.

Hence, the general term (when the difference of terms is in an AP) is represented by a$$n^2$$+bn+c, where n is the number of that specific term in the series.

Putting n=1, we get a+b+c=1

Putting n=2, we get 4a+2b+c=3

Putting n=3, we get 9a+3b+c=7

On solving the three equations, a=1, b=(-1), and c=1

To find the $$12^{th}$$ term, we shall put n=12 and substituting  the values of a,b, and c in the equation a$$n^2$$+bn+c, we get it as 144-12+1= 133

Let us assume the total work as 10 units.

Now, Akbar alone takes 10 days to complete the work, making his efficiency as 1 unit/day.

Babar takes 5 days to complete the same work, which means his efficiency is 2 units/day.

It is also given that Akbar, Babar, and Changez together take 2 days to complete the work, which means their combined efficiency is 5 units/day.

Therefore, Changez's efficiency would be 2 units/day (excluding Akbar and Babar from the combined efficiency).

Now, the amount shall be distributed to the three in the ratio of their efficiencies, which is 1:2:2 (Akbar, Babar, and Changez).]

This makes Changez's earning as 40 dinars out of the total 100 dinars. 

Assume that the person traveled for 't' hours. 

As per the question (extra distance because of increased speed), 12t-10t = 20, which implies that t = 10 hours. 

Hence, the distance traveled by the person in 10 hours at the speed of 10 kmph shall be 100 km.  

Let us assume that Shyaam does 3x units of work per day, while Ram does x units per day. 

In 45 days of working together, they must have done work in the ratio of 3:1 as far as the units of work done are considered. Hence, the amount earned shall also be distributed in a similar ratio. 

So out of Rs.13,500, Shyaam would have received $$\left(\frac{\ 3}{4}\right)\times\ 13500$$ for 45 days of work, which is Rs.10,125 for 45 days or Rs.225 per day.

In 10 days, Shyaam would have earned Rs.2250.

In a 5 digit binary number set, there are five places _ _ _ _ _, which shall be filled with 2 numbers (0 or 1).

Hence, for every place, we have two choices. For 5 places, we will have a total of $$2^5$$ choices, making the total count as 32. 

This means n= 32. Therefore, 2n= 64.

$$8\left(1 + \frac{1}{2} + \frac{1}{4}.....\infty\right)$$ is an infinite GP (the expression under the brackets) with the first term being 1, and the common ratio being $$\frac{\ 1}{2}$$or 0.5.

For infinite GP with common ratio less than 1, the sum is $$\ \frac{\ a}{(1-r)}$$

Applying the same in this, we get the sum as $$\ \frac{\ 1}{(1-0.5)}$$, which is 2.

Hence, the value of the entire expression is $$8\times2$$, which is 16.

As per the question (assuming the proportionality quotient be 'k'): x = k($$y^2$$+4)

Substituting the given values, 39 = k(9+4), which makes k as 3. 

Now, the value of $$\sqrt{\ y}$$ when x is 60: 60 = 3($$y^2$$+4)

From the above equation, $$y^2$$ = 16, which makes y as 4 and the square root of y as 2.