algebra

If a positive real x satisfies the following equation log₂ x + log√₂ x = 48, then the value of x is

P and Q are two positive integers such that P² = Q² + 13. The product of the numbers P and Q is _________

It is given that x and y are integers in the following equation: $(x + y - 7)^2 + (y + 3x - 13)^2 = 0$ The value of $(x^3 + y^3)$ is ________ (in integer).

A circle with center at $(x, y)=(0.5,0)$ and radius $=0.5$ intersects with another circle with center at $(x, \mathrm{y})=(1,1)$ and radius $=1$ at two points. One of the points of...

For positive integers p and q, with $\frac{p}{q} \neq 1$, $(\frac{p}{q})^{\frac{p}{q}} = p^{(\frac{p}{q}-1)}$. Then,

The ratio of the number of girls to boys in class VIII is the same as the ratio of the number of boys to girls in class IX. The total number of students (boys and girls) in classes...

The smallest positive root of the equation x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 45 = 0 lies in the range

For positive integers $p$ and $q$, with $\frac{p}{q} \neq 1$, $\left(\frac{p}{q}\right)^\frac{p}{q}= p^{\left(\frac{p}{q}-1\right)}$. Then,

The smallest positive root of the equation $$x^5 - 5 x^4 - 10 x^3 + 50 x^2 + 9 x - 45 = 0$$ lies in the range

A student was supposed to multiply a positive real number $p$ with another positive real number $q$. Instead, the student divided $p$ by $q$. If the percentage error in the student...

The ratio of the number of girls to boys in class VIII is the same as the ratio of the number of boys to girls in class IX. The total number of students (boys and girls) in classes...

If x satisfies the equation $4^{8x} = 256$, then x is equal to ________.

If x satisfies the equation $\rm 4^{8^x}=256$ , then x is equal to ______.

x : y : z = $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$ What is the value of $\frac{x+z-y}{y}$?

Both the numerator and the denominator of $\frac{3}{4}$ are increased by a positive integer, x, and those of $\frac{15}{17}$ are decreased by the same integer. This operation resul...

$$x:y:z = {1 \over 2}:{1 \over 3}:{1 \over 4}$$ What is the value of $${{x + z - y} \over y}$$ = ?

Both the numerator and the denominator of $${3 \over 4}$$ are increased by a positive $${15 \over 17}$$ integer, x, and those of _____ are decreased by the same integer. This opera...

⊕ and ⊙ are two operators on numbers p and q such that p ⊕ q = p^2 + q^2 / pq and p ⊙ q = p^2 / q; If x ⊕ y = 2 ⊙ 2, then x =

The sum of two positive numbers is 100. After subtracting 5 from each number, the product of the resulting numbers is 0. One of the original numbers is ______.

Consider the system of equations $\begin{bmatrix} 1 & 3 & 2 \\ 2 & 2 & -3 \\ 4 & 4 & -6 \\ 2 & 5 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}...

A square has sides 5 cm smaller than the sides of a second square. The area of the larger square is four times the area of the smaller square. The side of the larger square is ____...

$a + a + a + \dots + a = a^2b$ (n times) and $b + b + b + \dots + b = ab^2$ (m times), where a, b, n and m are natural numbers. What is the value of $(m + m + m + \dots + m)$ (n ti...

Hema's age is 5 years more than twice Hari's age. Suresh's age is 13 years less than 10 times Hari's age. If Suresh is 3 times as old as Hema, how old is Hema?

For non-negative integers, a, b, c, what would be the value of a + b + c if log a + log b + log c = 0?

The temperature T in a room varies as a function of the outside temperature T₀ and the number of persons in the room p, according to the relation T = K (Θp + T₀), where Θ and K are...

Given that $\frac{\log P}{y-z} = \frac{\log Q}{z-x} = \frac{\log R}{x-y} = 10$ for $x \neq y \neq z$, what is the value of the product PQR?

In manufacturing industries, loss is usually taken to be proportional to the square of the deviation from a target. If the loss is Rs. 4900 for a deviation of 7 units, what would b...

Hema's age is 5 years more than twice Hari's age. Suresh's age is 13 years less than 10 times Hari's age. If Suresh is 3 times as old as Hema, how old is Hema?

Given that $${{\log \,P} \over {y - z}} = {{\log \,Q} \over {z - x}} = {{\log \,R} \over {x - y}} = 10$$ for $$x \ne y \ne z$$, what is the value of the product PQR?

Given that one root of the equation $$\,{x^3} - 10{x^2} + 31x - 30 = 0\,\,$$ is $$5$$ then other roots are