No Author Solutions for Exercise 1: Mathematics

If $A_1, A_2$ are the two $\mathrm{A}.\mathrm{M}.\text{'}\mathrm{s}$ between two numbers $a$ and $b$ and $G_1, G_2$ be two $\mathrm{G}.\mathrm{M}.\text{'}\mathrm{s}$ between same two numbers, then $\frac{A_1+A_2}{G_1·G_2}=$

If $p, q$ and $r$ are simple propositions with truth values $T, F$ and $T$, then the truth value of $\left((~p\vee q)\wedge ~p\right)\Rightarrow r$ is

If $n\left(U\right)=700, n\left(A\right)=200, n\left(B\right)=300$ and $n(A\cap B)=100,$ then $n\left(A^{\text{'}}\cap B^{\text{'}}\right)$ is

Statement I The critical points of the function $x\cos x$ will occur between $\frac{\pi }{4}$ and $\frac{\pi }{3}$
Statement II The function $x$ tan$x$ increases on $\left(0,\frac{\pi }{4}\right)$ and decreases on $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$

Statement $\mathrm{I}$:If $I={\int }_0^1\frac{\cos x}{1+x^2}dx$ then $I<\frac{\pi }{4}\cos 1$
Statement $\mathrm{II}$: If $f$ and $\varphi$ are continuous function, on $[a, b],$ then for some $c\in (a,b)$, ${\int }_a^{b}f\left(x\right)\varphi \left(x\right)dx=f\left(c\right){\int }_a^b\varphi \left(x\right)dx$

Statement I Curve satisfying the differential equation $y^{\text{'}}=\frac{y}{2x}$ passing through $(2,1)$ is a parabola with focus $\left(\frac{1}{4},0\right)$
Statement II The differential equation $y^{\text{'}}=\frac{y}{2x}$ is of variable separable.

Statement I Equation of a circle through the origin and belonging to the coaxial system, of which the limiting points are $(1,1)$ and $(3,3)$ is $2x^2+2y^2-3x-3y=0$.
Statement II Equation of a circle passing through the points $(1,1)$ and $(3,3)$ is $x^2+y^2-2x-6y+6=0$.

Statement I If $A\cup B=A\cup C$ and $A\cap B=A\cap C$
then $B=C$.
Statement II $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.