Question

Let $\mathrm{S}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{multiple}\mathrm{of}3\mathrm{less}\mathrm{than}100\right\}$, $\mathrm{P}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{number}\mathrm{less}\mathrm{than}20\right\}$. Then $\mathrm{n}\left(\mathrm{S}\right)+\mathrm{n}\left(\mathrm{P}\right)$ is

• A. $34$
• B. $41$
• C. $33$
• D. $30$

Answer 1

The correct option is B

$41$

Explanation of the correct answer.

Step $1$: Compute the number of elements in the set $\mathrm{S}$.

Given: $\mathrm{S}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{multiple}\mathrm{of}3\mathrm{less}\mathrm{than}100\right\}$

So, $S=\left\{3,6,9,12..........,99\right\}$

We can see that it's is an A.P. with $a=3,a_n=99,d=3$.

$N^{th}$ term of an A.P. is given by:

$$\begin{gathered} a_n=a+(n-1)d \\ \Rightarrow 99=3+(n-1)3 \\ \Rightarrow \frac{96}{3}=n-1 \\ \Rightarrow n=33 \end{gathered}$$

Therefore, $n\left(S\right)=33$

Step $2$: Compute the number of elements in the set $\mathrm{P}$.

Given: $\mathrm{P}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{number}\mathrm{less}\mathrm{than}20\right\}$

So, $P=\left\{2,3,5,7,11,13,17,19\right\}$

$n\left(P\right)=8$

Step $3$: Compute total number of elements

Now, $n\left(S\right)+n\left(P\right)=33+8$

$\Rightarrow$ $n\left(S\right)+n\left(P\right)=41$

Hence option $B$ is the correct answer.