Math Formulas

We write a cone on the sphere. The center of the sphere divides the height of the cone into two sections such that the longer section equals the geometric mean of the shorter section and the total height. Determine the volume ratio of the sphere to the cone.

MR-537

Solve the fallowing equation:

$\log_{2} x = - 3$

MR-575

Solve the following equation:

$10 x + 20 = 40$

MR-1

The two hypotenuses of a right triangle are 8 cm and 6 cm. Calculate the perimeter and area of the triangle.

MR-97

Simplify the following expression:

$\sqrt{4 x y} \cdot \sqrt[6]{x^{2} y^{3}} \cdot \sqrt[6]{x^{9} y^{8}}$

MR-368 / 12.

Determine the true value of the following expressions:

$p \equiv \frac{3 + 2 x}{3} - \frac{2 - 3 x}{4} = \frac{29}{24}; x = 0, 5$

$q \equiv 3 (5 - y) - 2 (y - 1) = 1 + 3 y; y = 2$ Then determine the true value of the following two terms:

a) $(p \lor q) \land p$

b) $(p \land q) \lor p$

MR-527 / 12.

The class organizes a lottery with three numbers from numbers: 1, 2, 3, 4, 5 . Tom markes 2, 3 and 5 number on the coupon.
Calculate the probability that Tom will have a total win!

Set S is given undefined. Determine the elements of A i B sets.

$A = \{x | x \in S \land \frac{2 x}{12 - x} \in S\}$ undefined

Determine the elements of the fallowing sets undefined

MR-125 / 125.

Simplify the following expression:

${(\frac{a^{- 1} + b^{- 1}}{b a^{- 1} + a b^{- 1}})}^{- 1} + {(\frac{a^{- 1} + b^{- 1}}{2})}^{- 1} - \frac{a^{- 1} - b^{- 1}}{a^{- 1} \cdot b^{- 1}}$

MR-126 / 126.

Simplify the following expression:

$(3 - 4 i) (3 + 4 i)$

MR-134 / 134.

Simplify the following expression:

${(\frac{2^{x} + 2^{- x}}{2})}^{2} - {(\frac{2^{x} - 2^{- x}}{2})}^{2} + 1$

MR-135 / 135.

Determine the first derivative of the following function:

$y = 10 e^{x} - 6 e^{2 x}$

MR-136 / 136.

Determine the first derivative of the following function:

$y = \frac{e^{2 x} - e^{- 3 x}}{e^{5 x} - e^{7 x}}$

MR-138 / 138.

Determine the first derivative of the following function:

$y = 2 a^{x} + a^{3 x} + \frac{1}{4} a^{- x}$

MR-139 / 139.

Determine the first derivative of the following function:

$y = 8^{\frac{x^{2}}{\cos x}}$

MR-140 / 140.

Find the first derivative of the function below:

$y = 9^{6 x + \sqrt{x}}$

MR-143 / 143.

Determine the integral of the function:

$\int x^{7} d x$

MR-147 / 147.

Determine the value of the expression below:

$i^{225} + {(- i)}^{60} + i^{83}$

MR-152 / 152.

Determine the value of the expression below:

${(2 i)}^{2} + (- 2 i)^{4}$

MR-153 / 153.

Determine the value of the expression below:

${(\frac{1 + i}{\sqrt{2}})}^{1000} + {(\frac{1 - i}{\sqrt{2}})}^{1000}$

MR-162 / 156.

MR-163 / 163.

$\frac{x + 2}{x - 2} + \frac{x - 2}{x + 2} = 0$

MR-192 / 192.

Calculate the following expression for α=30°.

$\frac{4 - s i n α}{1 - s i n α} - \frac{25}{4 c o s^{2} α} + \frac{2}{1 + s i n α}$

MR-199 / 199.

Solve the following equation:

$2 x^{5} + x^{4} - 19 x^{3} + 19 x^{2} - x - 2 = 0$

MR-206 / 206.13

Solve the following equation:

$\sqrt{2 x - 3} = \sqrt{x + 10} - 1$

MR-221 / 221.

MR-247 / 247.

Determine the first derivative of the function:

$y = \ln (3 x + 4)$

MR-248 / 248.

Determine the first derivative of the function>:

$y = \ln (3 x^{2} + 2 x - 6)$

MR-249 / 249.

Determine the first derivative of the function below:

$y = \ln \sin^{2} x$

MR-255 / 255.

Solve the following equation::

$\sqrt{x + \sqrt{x^{2} - 1}} + \sqrt{x - \sqrt{x^{2} - 1}} = \sqrt{2 (x + 1)}$

MR-273 / 273.

MR-302 / 302.779

Solve the following system of equations:

$\begin{matrix} x^{2} + y^{2} = 29 \\ x + y = 7 \end{matrix}$

MR-334 / 334.

Solve the following equation:

$\log_{10} (x^{2} + 19) - \log_{10} (x - 8) = 2$

MR-343 / 343.

Calculate the area of the parallelogram if its altitudes are 3 and $2 \sqrt{3}$ cm, and the angle $between tham is$ 60°!

MR-349 / 349.

The area of the rhombus is 100 cm², while its sharp angle is 30°. Calculate the diagonals of the rhombus!

MR-366 / 366.

Solve the following inequality on the set of real numbers!

$x^{2} - x - 2 < 0$

MR-441 / 441.

Solve the following equation::

$x^{2} - 2 x - 3 = 0$

MR-560 / 560.

Simplify the following expression:

$\frac{3 x - 3 y}{2 x + 2 y} \cdot \frac{x^{2} - y^{2}}{x^{2} - 2 \cdot x \cdot y + y^{2}}$

MR-569 / 569.

Solve the equation below:

$x^{2} - 2 x - 35 = 0$

MR-574 / 574.

Solve the following equation:

$2 x + 4 = 6$

MR-367 / 588.

Vektori $\vec{a}$ i $\vec{b}$ obrazuju ugao $ψ = \frac{π}{6}$ . Ako je $|\vec{a}| = \sqrt{3}$ i $|\vec{b}| = 1$ izračunati ugao između vektora $\vec{p} = \vec{a} + \vec{b}$ i $\vec{q} = \vec{a} - \vec{b}$ .

MR-680 / 680.

Inscribed circle of trapezoid ABCD (AB || CD), the side AB touches at point E. If AE=15, BE=10 and CD=8, calculate the area of the trapezoid.

MR-828 / 828.

Prove that 0.117117117... is a rational number.

MR-831 / 831.

$\underset{n \to \infty}{l i m} \frac{n + 3}{n^{2} + 3 n + 2}$

MR-242 / 1175.

Simplify the following expression:

$\cos^{2} x - \cos^{4} x + \sin^{4} x$

MR-243 / 1176.

Simplify the following expression:

$t g^{2} x - s i n^{2} x = t g^{2} x \cdot s i n^{2} x$