A pizza chain offers cheese pizzas with a choice of these toppings: pepperoni, onion, sausage, mushrooms, and anchovies. Explain why the pizza chain can claim to offer 32 different pizzas.

Let $X$ be a non-empty set and let $\mathcal{P}(X)$ be its power set. Define two operations ‘+’ and ‘$\ast$’ on $\mathcal{P}(X)$ as follows. For $A,B\in \mathcal{P}(X)$, let

$$A+B=A\cup B,A\ast B=A\cap B.$$

If $G_1=(\mathcal{P}(X),+)$ and $G_2=(\mathcal{P}(X),\ast )$, then,

The number of people in a small country town increases by 2% per year. If the population at the start of 1982 was 12,500 what is the predicted population at the start of the year 2022?

The set of positive real numbers $x$ for which the series

$$\sum _{n=1}^{\infty }\frac{1}{x^n+x^{-n}}$$

diverges

Evaluate $\int \frac{dx}{\sin x+\sin 2x}$.

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

If two distinct integers $a$ and $b$ are picked from $\{1,2,3,4\dots 100\}$ and multiplied, what is the probability that the resulting number has EXACTLY 3 factors?

The carbonitrided bushes need to meet the specifications on 3 output characteristics namely surface finish, case depth and dimensional variation. Otherwise, the bush is considered as a defective bush. Suppose a batch of 240 bushes have been carbonitrided and 30 of them not met the specification on surface finish, 15 on case depth, 9 on dimensional variation, 6 on both surface finish & case depth, 4 on surface finish & dimensional variation, 4 on case depth & dimensional variation and there is no bush not meeting specification on all the three characteristics. What is the chance that a bush randomly selected from the batch will be a defective bush?

Let $C$ be the circle in the $xy$-plane which passes through the origin and touches the line $y=-2$. If the centre of $C$ lies on the straight line $x+y=1$, then the radius of $C$ is

What is the sum of the maximum and the minimum values of

$$\frac{1}{1+(2\cos x-4\sin x{)}^2}$$

over $x\in \mathbb{R}$?

Let $f(x,y)=a{x}^2+b{y}^2-c$, for $(x,y)\in {\mathbb{R}}^2$, where $a,b,c\in \mathbb{R}$ with $bc>0$ and $a\ne 0$. Let

$$S=\{(x,y)\in {\mathbb{R}}^2:f(x,y)=0\}.$$

Consider the map $p:{\mathbb{R}}^2\to \mathbb{R}$ defined by $p(x,y)=x$. If

$$p(S)=\{p(x,y):(x,y)\in S\},$$

then which of the following can never be true?

The system of equations

$$\begin{array}{rl}x+2y+3z& =c\\ 2x-y+z& =2\\ ax+y+4z& =b\end{array}$$

has no solution if

If $A$ is a $4\times 4$ real matrix with at least 2 non-zero entries, then which of the following necessarily holds?

Let

$$S=\left\{(x_1,x_2,x_3,x_4)\in {\mathbb{R}}^4:\sum _{i=1}^4{x}_i^2=1\right\},$$

and

$$m=\underset{(x_1,\dots ,x_4)\in S}{\min }\sum _{i=1}^4|x_i|,M=\underset{(x_1,\dots ,x_4)\in S}{\max }\sum _{i=1}^4|x_i|.$$

Then, $m/M$ equals