If x^2+xy+y^2=39, y^2+yz+z^2=49, z^2+zx+x^2=19, find all soltion sets for of x, y , z

\lim\limits_{x\to 0} \dfrac{(1+x)^{\tan x}-1}{x^2}=

Show that \sin^2\dfrac{\pi}{14}+\sin^2\dfrac{3\pi}{14}+\sin^2\dfrac{5\pi}{14}=\dfrac{4}{5}

Construct a new equation of which the roots are twice of those of the equation x^3-px^2+qx-r=0

If x_1, x_2 are two real roots of the quadratic equation x^2-mx+m-2=0, then what statement is correct about m to hold the equation \dfrac{1}{x_1}+\dfrac{1}{x_2}=0 true?

If a,b,c are real numbers such that a+b+c=2, \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{2}, show that one of numbers among a,b,c must be equal to 2

If a,b,c are real numbers such than a=2b+\sqrt{2} and ab+\dfrac{\sqrt{3}}{ 2}c^2+\dfrac{1}{4}=0 , find the value of \dfrac{bc}{a}

If \alpha ,\beta are two roots of the equation x^2+2x-4=0, then find the value of α^3-3β^2+2β.

If a,b,c are real numbers such taht a+b+c=0, abc=2. Find the minimum value of |a|+|b|+|c|

Find exact value of \sin\dfrac{\pi}{7} \sin\dfrac{2\pi}{7} \sin\dfrac{3\pi}{7} or sin(pi/7)sin(2pi/7)sin(3pi/7)

Let p(x) be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1) + p(−1) − 2(1 + p(0))

The product of two of the four roots of the quartic equation x^4 − 18x^3 + kx^2 + 200x − 1984 = 0 is −32. Determine the value of k.

If the quartic x^4 + 3x^3 + 11x^2 + 9x + A has roots k, l, m, and n such that kl = mn, find A.

If x_1 and x_2 are the two roots of the quadratic equation x^2-px+q=0, (x_1>x_2). Find the values of the following expressions.

Let m and n be the two roots of the equation x^2 − 15x + 28 = 0. Find the value of (m + 1)(n + 1)