Problem
Solve for x, y, and z from the following simultaneous equations.
 

$z^x \, z^y = 100\,000$   ←   equation (1)

$(z^x)^y = 100\,000$   ←   equation (2)

$\dfrac{z^x}{z^y} = 10$   ←   equation (3)
 

Problem
Find the sum and product of roots of the quadratic equation x² - 2x + 5 = 0.
 

Problem
In a quadratic equation problem, one student made a mistake in copying the coefficient of x and got roots of 3 and -2. Another student made a mistake in copying the constant term and got the roots of 3 and 2. What are the correct roots?
 

Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.
 

Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.
 

Area,

Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2
 

Derivation for area

Square

 

 

Area, $A = a^2$

Perimeter, $P = 4a$

Diagonal, $d = a\sqrt{2}$

Problem
From the right triangle ABC shown below, AB = 40 cm and BC = 30 cm. Points E and F are projections of point D from hypotenuse AC to the perpendicular legs AB and BC, respectively. How far is D from AB so that length EF is minimal?