Solving Quadratic Equations — CAT Previous-Year Questions

The sum of all distinct real values of x that satisfy the equation 10^x + $\frac{4}{{10}^x}$ = $\frac{81}{2}$, is

The number of integral solutions of equation 2|x|(x² + 1) = 5x² is?

The number of integral solutions of the equation ${({\mathrm{x}}^2-10)}^{({\mathrm{x}}^2-3\mathrm{x}-10)}$ = 1 is

A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

How many distinct positive integer-valued solutions exist to the equation ${\left(x^2-7x+11\right)}^{(x^2-13x+42)}$ = 1?

The number of distinct real roots of the equation ${\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)}^2-3\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)+2$ = 0 equals

In how many ways can a pair of integers (x , a) be chosen such that x² − 2|x| + |a - 2| = 0?

The number of integers that satisfy the equality (x² - 5x + 7)^x+1 = 1 is

The number of solutions to the equation |x|(6$x^2$ + 1) = 5$x^2$ is

The product of the distinct roots of ∣x² − x − 6∣ = x + 2 is

The real root of the equation 2^6x + 2^3x+2 - 21 = 0 is

If u² + (u−2v−1)² = −4v(u + v), then what is the value of u + 3v?

What values of x satisfy $x^{\frac{2}{3}}+x^{\frac{1}{3}}-2\le 0$?

Two positive integers differ by 4 and sum of their reciprocals is $\frac{10}{21}$. Then one of the numbers is