Q. Simplify and obtain a numerical value ${\left(32\right)}^{-\frac{4}{5}}$.

Q. Define: 'Objective function'.

Q. If $A$ and $B$ are any two sets, show that $A^{\prime }-B^{\prime }=B-A$.

Q. If the sum of the first $n$ natural numbers is $S_1$ and that of their squares is $S_2$ and cubes is $S_3$, then show that $9S_2^2=S_3(1+8S_1)$.

Q. Solve the absolute value of inequation $\left|\frac{2x-1}{3}\right|\le 5$.

Q. A shopkeeper sells not more than $30$ shirts of each colour. Atleast twice as many white ones are sold as green ones. If the profit on each of the white be Rs. $20$ and that of green be Rs. $25$; then find out how many of each kind be sold to give him a maximum profit. (Graph need not be drawn)

Q. If a set contains $m$ elements and $B$ contains $n$ elements, then find the number of elements in $A\times B$.

Q. If $lmn=1$, show that
$\frac{1}{1+l+m^{-1}}+\frac{1}{1+m+n^{-1}}+\frac{1}{1+n+l^{-1}}=1$

Q. If $a^x=b,b^y=c,c^z=a$, show that $xyz=1$.

Q. Using graph $y=x^2$, solve the equation $x^2-x-2=0$.

Q. Prove that $A-(B\cup C)=(A-B)\cap (A-C)$ for any three sets $A,B,C$.

Q. Find the sum to $n$ terms $0.7+0.77+0.777+....$

Q. Find the sum to infinity of the G.P.: $5,\frac{20}{7},\frac{80}{49},...$

Q. Find the value of $m$ in order that $x^4-2x^3+3x^2-mx+5$ may be exactly divisible by $x-3$.

Q. Maximize $f=4x-y$, subject to the constraints
$7x+4y\le 28$, $2y\le 7$, $x\ge 0$, $y\ge 0$.

Q. Find the middle term in the expansion ${(\frac{x}{a}+\frac{y}{b})}^6$.

Q. Which term of the A.P. $10,8,6,...$ is $-28$?

Q. Factorise $3x^4-10x^3+5x^2+10x-8$

Q. If $f:R-\left\{3\right\}\to R$ is defined by $f\left(x\right)=\frac{x+3}{x-3}$, show that $f\left[\frac{3x+3}{x-1}\right]=x$ for $x\ne 1$.

Q. Prove: $\sim (p\Rightarrow q)\equiv p\wedge (\sim q)$.

Q. Write the converse and contrapositive of the following conditional. If in $△ABC$, $AB>AC$, then $\angle C>\angle B$.

Q. Let $f,g,h$ are functions defined by $f\left(x\right)=x-1,g\left(x\right)=x^2-2$ and $h\left(x\right)=x^3-3$, show that $(f\circ g)\circ h=f\circ (g\circ h)$.

Q. If a function $f:R\to R$ is defined by $f\left(x\right)=3x+4$ show that $f^{-1}$ (the inverse function of $f$) exists and find it.