Sarvesh K Verma Solutions for Chapter: Number System, Exercise 18: CAT - Test Questions Helping You Bell the CAT

If $p>0, q<0$, then which of the following is correct?

If the integers $m$ and $n$ are chosen at random from the set $\{1, 2, 3,\dots ,100\}$, then, maximum how many distinct numbers of the form $\left(7^m+7^n\right)$ would be divisible by $5$?

A corrupt country receives some educational grant from UNESCO in the form of digital notebooks. The total number of notebooks that this country receives is $N=\frac{P!}{Q!}$, where $0\le P\le 99$ and $0\le Q\le 99$, such that $P-Q=10$. The country is divided into various levels, ${\mathrm{L}}_1, {\mathrm{L}}_2, {\mathrm{L}}_3,\dots$, for administrative conveniences. Initially, all the notebooks are with the head of ${\mathrm{L}}_1$, who keeps half of the notebooks with himself and rest half of the notebooks, he doles out equally to each head of ${\mathrm{L}}_2$. Similarly, each head of ${\mathrm{L}}_2$ keeps half of the notebooks with oneself and rest half of the notebooks, he doles out to each head of ${\mathrm{L}}_3$. This pattern occurs till the level ${\mathrm{L}}_x$, where it becomes impossible for each head of ${\mathrm{L}}_x$ to divide the number of notebooks in two equal parts, so one does not dole out any notebook to any of the heads of ${\mathrm{L}}_{x+1}$ and thus to the expected beneficiary. That is, the public, which is supposed to receive the notebooks directly from ${\mathrm{L}}_{x+1}$, never receives any notebook. Also, at any level, the number of heads is never equal to $2$. What is the maximum possible value of $x$?

In a far land of Lootokhao, an empowered group of ministers was supposed to allocate some coal blocks and $2\mathrm{G}$ spectrums to the aspiring private companies. However, these ministers colluded for their personal gains by allocating the blocks and spectrums to a lobbying corporate coterie and that too after charging the bribe. The bribe fetched by each minister for allocating each block was $₹300$ million and for each spectrum it was $₹500$ million. Each minister allocated the same number of coal blocks and the same number of $2\mathrm{G}$ spectrums. The total bribe they received in this process was $₹13.3$ billion. How many coal blocks did each minister allocate to this corporate coterie by charging the bribe?

Initially, a person had $₹20$, when he sat down to play a game. If he wins, he would get $₹3$, but if he loses, he would have to give away $₹1$ to his opponent. If he played at least $1$ game and at most $20$ games, how many distinct amounts he would have at the end of the game?

During a counter insurgency operation, three paramilitary troops, A, B and C, fought bravely against the rebels and thwarted the attempt of a coup d'état. The fight was so intense and lethal, that exactly half of the paramilitary forces lost their lives. Only $36\%$ of troop A, $60\%$ of troop B and $80\%$ of the troop C could survive the fight. Which of the following is not the possible number of total soldiers in all the three troops together engaged in the fight?

For every natural number, $a, b, c$ and $x$, if $\left(x^a+1\right)\left(x^b+1\right)\left(x^c+1\right)$ is the factor of $S=1+x+x^2+x^3+\dots +x^{111}$, then, what is the HCF of $a, b$ and $c$?

Find the value of $k$, if $15!=1k07674368000$.