What is the common ingredient in antacid, mild abrasive in tooth paste, a constituent of chewing gum, and a filler in cosmetics ?

The sum of squares of all the roots of the equation:
​$$\left(x + \frac{1}{x}\right)^2 - 4 = \frac{3}{2} \left(x - \frac{1}{x}\right), x \neq 0$$ is:

One of the factors of 6x² – 24xy + 17x + 24y² – 34y + 5 is:

The cost of 2 pens and 5 pencils is ₹ 57. If the cost of a pen is increased by ₹ 1.50 and that of a pencil is decreased by ₹ 1, then the cost of 5 pens and 3 pencils is ₹ 109. What is the original cost of 3 pens and 4 pencils?

What is the remainder when the polynomial x¹⁰⁰ – 2x⁵¹ + 1 is divided by (x² – 1) ?

If x + $$\frac 1x$$​= –1, then compute: $$x^4 + \frac{1}{x^4} + 2x^2 + \frac{2}{x^2}$$​​

Let x be the greatest number which, when dividing 31, 61 and 91, leaves the same remainder. The LCM of x and 48 is:

The value of $$\frac{x}{x-y}+\frac{y}{x+y}+\frac{2xy}{y^{2}-x^{2}}$$​ is