Embibe Experts Solutions for Exercise 7: Assignment

If $a_1, a_2, a_3, a_4, \dots \dots , a_{n-1}, a_n$ are distinct non-zero real numbers such that $\left(a_1^2+a_2^2+a_3^2+\dots \dots +a_{n-1}^2\right)x^2+2\left(a_1{a}_2+a_2{a}_3+a_3{a}_4+\dots \dots +a_{n-1}{a}_n\right)x$$$ $+\left(a_2^2+a_3^2+a_4^2+\dots \dots +a_n^2\right)\le 0$ then $a_1, a_2$ $a_3, \dots \dots , a_{n-1}, a_n$ are in

The equation ${\left({\log }_{x}10\right)}^3-{\left({\log }_{x}10\right)}^2-6{\log }_x 10=0$ is satisfied by a value of $x$ given by

If $1^5+2^5+3^5+............+n^5$ is divided by $1^3+2^3+3^3+.............+n^3$, the remainder is $\left(\mathrm{when} n=3m+1\right)$

The numbers $x_1, x_2, x_3......$ form an infinite decreasing $\mathrm{G}.\mathrm{P}.$ If $x_1=1$, then the common ratio of the progression for which the expression $6x_5-16x_4-3x_3+12x_2$ is maximum, is

If $w, x, y, z\in R^{+}$ and $256xywz\ge {\left(w+x+y+z\right)}^4$ and $3w+5x+7y+9z=24$, then the roots of the equation $\left(w+x\right)t^2+yzt+\sqrt{x+y+z+w}=0$ are

Solve the following equation for $x$ and $y$:

${\log }_{10}x+\frac{1}{2}{\log }_{10}x+\frac{1}{4}{\log }_{10}x+........=y$, $\frac{1+3+5+......\left(2y-1\right)}{4+7+10.......\left(3y+1\right)}=\frac{20}{7{\log }_{10}x},$ then $y$ is

If $n$ is a root of $x^2\left(1-ac\right)-x\left(a^2+c^2\right)-\left(1+ac\right)=0$ and if $n$ harmonic means are inserted between $a$ and $c$, then the difference between first and last mean is

Let $2048$ arithmetic means be inserted between $2^{22}-1$ and $2^{22}+1$. Suppose the sum of these arithmetic means is $S$. Then $\frac{S}{2^{33}}$ is ....