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Math

Given the function `f(x)=\sqrt{1-x^2}` . We have to find the inverse of this given function i.e. `f^{-1}(x)` . An inverse function (also known as anti function) is defined as a function which can...

Latest answer posted December 8, 2022, 12:05 pm (UTC)

1 educator answer

Math

The prime numbers less than 10 are 2, 3, 5, and 7. It is obvious that `2^607 - 1 ` is not divisible by 2. To prove the same for other primes, use the facts that `( a + b ) mod c = ( a mod c ) + ( b...

Latest answer posted December 4, 2022, 4:59 pm (UTC)

1 educator answer

Math

Denote the length `A G ` as `x , ` then by the conditions `G E = A E = C E = x . ` By the median property `G D = 1 / 2 x ` and `G B = 2 x .` By the cosine theorem, `E D^2 = x^2 + 1 / 4 x^2 + 1 / 2...

Latest answer posted December 4, 2022, 8:11 am (UTC)

1 educator answer

Math

Denote `AD = 2 a ` and `BC = 2 b , ` then the question becomes what is `b / a . ` Also, denote the height of the trapezoid as `h .` Introduce a coordinate system such that `AD ` lies on the x-axis...

Latest answer posted December 1, 2022, 10:17 pm (UTC)

1 educator answer

Math

It is given that `P ( x ) = q ( x ) ( x^2 + x + 1 ) + ( x + 2 ) = q ( x ) ( x^2 + 1 ) + ( 2x + 1 ) , ` where q and r are polynomials. p and q cannot have degree less than 0, so P cannot have degree...

Latest answer posted December 5, 2022, 2:23 pm (UTC)

1 educator answer

Math

We have two equations for `x ` and `y: ` `x ^ y = 2 ^ 64 ` and `( log _ 2 x ) ^ ( log _ 2 y ) = 2 ^ 27 . ` Let's apply `log _ 2 ` to both sides of them: `y * log _ 2 x = 6 4 , ` `log _ 2 y * log...

Latest answer posted December 5, 2022, 11:03 am (UTC)

1 educator answer

Math

It is simple to find the coordinates of the points A, B, C: `A ( -5 , 0 ) , ` `B ( 0 , -15 ) , ` `C ( 3 , 0 ) . ` Because of this, it is simple to find the lengths of the sides AB, BC, AC: `AB =...

Latest answer posted December 5, 2022, 10:37 am (UTC)

1 educator answer

Math

There are two somewhat different types of tangent circles: they may lie on different sides of their common tangent, or on one side. Because the first and the third circles lie inside the second one...

Latest answer posted December 5, 2022, 8:48 am (UTC)

1 educator answer

Math

The set of all complex numbers of magnitude of at most 1 is he unit circle. Its area is `pi * 1 ^ 2 = pi approx 3 . 1 4 .` Addition of complex numbers is component-wise, so when we add all points...

Latest answer posted November 28, 2022, 12:21 am (UTC)

1 educator answer

Math

Well, there might be some linguistic difficulty because there is actually one word for "circle as a curve" and "circle as a curve and everything inside it." For the curve, we can also say...

Latest answer posted November 29, 2022, 7:53 pm (UTC)

1 educator answer

Math

When analyzing the equality `1 / a + 1 / b + 1 / c = 1 / ( a + b + c ) , ` we noticed that it is equivalent to `( a + b ) ( b + c ) ( a + c ) = 0 ` (provided of course that `a ! = 0 , ` `b ! = 0 ,...

Latest answer posted November 6, 2022, 9:19 am (UTC)

1 educator answer

Math

No problem. I used your image to show where the section resides. Also, I have drawn an image with the section itself. The section of the cube `C ` is a rectangle `Y _ 1 Z _ 1 Z _ 3 Y _ 3 . ` Its...

Latest answer posted November 6, 2022, 8:33 am (UTC)

1 educator answer

Math

Suppose that the parabola has the equation of the form `y = a x ^ 2 + b x + c .` For each given point with the coordinates `( x _ k , y _ k ) ` we obtain an equation for `a , ` `b ` and `c ` in the...

Latest answer posted November 6, 2022, 10:59 am (UTC)

1 educator answer

Math

Actually, the condition `c = b + 1 ` is sufficient to conclude that `g c d ( b , c ) = 1 , ` the more `g c d ( a , b , c ) = 1 .` Indeed, let some integer `n ` divides both `b ` and `c , ` i.e. `b...

Latest answer posted November 2, 2022, 6:47 pm (UTC)

1 educator answer

Math

The equations may be reduced by n! , (n + 4)!, and (n + 7)! , respectively, and the augmented matrix becomes `( ( 1 , n + 1 , ( n + 1 ) ( n + 2 ) , ( n + 1 ) ( n + 2 ) ( n + 3) ), ( 1 , n + 5 , ( n...

Latest answer posted November 1, 2022, 5:17 pm (UTC)

1 educator answer

Math

Determine what the condition `1 / a + 1 / b +1 / c = 1 / ( a + b + c ) ` actually means. Namely, multiply the equation by `a , ` `b , ` `c ` and `a + b + c :` `( b c + a c + a b ) ( a + b + c ) =...

Latest answer posted October 31, 2022, 4:50 pm (UTC)

1 educator answer

Math

Consider a section of the pyramid by a vertical plane that goes through a diagonal of its base. It forms an equilateral triangle of the height `h ` and base, say, `2 a , ` in which a rectangle of...

Latest answer posted October 31, 2022, 2:45 pm (UTC)

1 educator answer

Math

Given `f(x)=(tanx)^{\frac{1}{x}}` . We have to find `f'(x)` using logarithmic differentiation. Taking natural log on both sides of the equation we get: `lnf(x) = ln((tanx)^\frac{1}{x})` Applying...

Latest answer posted October 28, 2022, 10:19 am (UTC)

1 educator answer

Math

We have to calculate the area under the curve `y=x^k.k` (where `k` is a constant) from `x=0` to `x=1` (upper and lower limit). We can use the integration process to find the area under the curve....

Latest answer posted October 25, 2022, 11:53 am (UTC)

1 educator answer

Math

The multiple-choice questions aren't connected to one another, meaning the student could get either, both, or neither right. In a situation where you've got two independent events, the probability...

Latest answer posted October 18, 2022, 4:12 am (UTC)

1 educator answer

Math

The total area covered by the three circles is 942.48. The formula for calculating this is A = π r² (314.16) multiplied by three. However, the wording of the question is ambiguous. Since the...

Latest answer posted October 12, 2022, 3:10 pm (UTC)

1 educator answer

Math

`A D ` and `B E ` are both heights of the triangle `A B C , ` so their point of intersection `H ` is the orthocenter. The third height goes through the same point `H , ` i.e., the line `C H ` is...

Latest answer posted October 2, 2022, 5:53 pm (UTC)

1 educator answer

Math

The answer is: it is impossible with these numbers. Indeed, let the right angle be B. Denote the leg length as a and place the triangle inside a coordinate system such that B(0, 0), A(a, 0) and...

Latest answer posted October 4, 2022, 8:34 am (UTC)

1 educator answer

Math

Let's use the coordinate method. The origin will be `D , ` the x-axis will be `D C ` and the y-axis will be `D A .` Let `D C = a , ` `D A = b , ` then `A ( 0 , b ) , ` `B ( -a , 0 ) , ` `C ( a , 0...

Latest answer posted September 29, 2022, 8:09 pm (UTC)

1 educator answer

Math

The equation of the line tangent to `f(x)=-x^3+x` when x = 2 is y = -11x + 16. To find the equation of a line, we need the slope of the line and a point. Since the tangent line intersects the curve...

Latest answer posted September 22, 2022, 4:52 am (UTC)

1 educator answer

Math

Given the system of equations: `x+xy+xyz=33` ------>(1) `y+yz+yzx=60` ------>(2) `z+zx+zxy=42` ------>(3) We have to find whether `x=33/7` is one element of all triples of real numbers x,...

Latest answer posted August 24, 2022, 6:18 pm (UTC)

1 educator answer

Math

Given the system of equations: `x+xy``+xyz=33`------>(1) `y+yz+yzx=60` ------>(2) `z+zx+zxy = 42` ------>(3) We have to find whether `x=5.5` is one element of all triples of real numbers...

Latest answer posted August 23, 2022, 4:16 pm (UTC)

1 educator answer

Math

"Within distance d from the circle" probably means not "the distance from the circle is d" but "the distance from the circle is less than d." Denote this distance as `a. ` Then the solution...

Latest answer posted August 20, 2022, 4:31 pm (UTC)

1 educator answer

Math

The numbers that form all sums in question are 1, 2, 3, 4, 5, 6, 7, 8, and their squares are 1, 4, 9, 16, 25, 36, 49, 64. Determine the sums of all numbers for `4 lt= n lt= 8 : ` 10, 15, 21, 28,...

Latest answer posted August 20, 2022, 8:35 am (UTC)

1 educator answer

Math

Denote the center of the circle as `O , ` then denote `O R = O Q = r , ` then `O P = 3 r .` Also denote `P Q = Q R = x` and `/_ R Q O ` as `alpha .` Then apply the Cosine law to the triangles `O Q...

Latest answer posted August 19, 2022, 3:11 pm (UTC)

1 educator answer

Math

I suppose the third equation should be `z + zx + zxy = 42 ` (zx , not yz). Subtract the first equation from the second, then subtract the third equation from the second and express `z ` from the...

Latest answer posted August 19, 2022, 9:16 am (UTC)

1 educator answer

Math

Let the circle be a unit circle centered at the origin: this does not affect the probability. Moreover, the position of the first point does not change the probability, so we may even suppose that...

Latest answer posted August 19, 2022, 8:28 am (UTC)

1 educator answer

Math

We are given right triangle ABC with right angle at A. We assume M to be a point on BC (the hypotenuse) with BM = MC (thus, M is the midpoint of the hypotenuse). Locate points P on AB and Q on AC...

Latest answer posted August 16, 2022, 12:49 am (UTC)

1 educator answer

Math

Denote the side BC as `x, ` then `AB = 2x ` and `AC = x sqrt(3) . ` The area in question is `A = sqrt(3) / 2 x^2 .` Also denote `CE = y, ` then `BE = x - y .` By the Pythagorean theorem `3x^2 +...

Latest answer posted August 9, 2022, 5:31 pm (UTC)

1 educator answer

Math

First, we need to find the area of the square. Denote the side length as `a , ` the distance from P to AB as `x ` , and the distance from P to AD as `y . ` Then we have three equations for three...

Latest answer posted August 7, 2022, 6:04 pm (UTC)

1 educator answer

Math

It is evident that `1 lt= n lt= 52 . ` There are two subsets of `n , ` namely such that `n^2 lt 53 ` and `n^2 gt 53 , ` which are the same as `1 lt= n lt= 7 ` and `8 lt= n lt= 52 .` For `n ` from...

Latest answer posted June 6, 2022, 12:53 pm (UTC)

1 educator answer

Math

Consider two adjacent vertices of the polygon, `A_1 ` and `A_2 , ` and its center `O .` Clearly `O A_1 = O A_2 ` and the median `O C ` is also the height and the bisector of the triangle `A_1 O...

Latest answer posted June 6, 2022, 8:31 am (UTC)

1 educator answer

Math

We know that `( x - r ) ( x - s ) ( x - t ) = x^3 - 8x + 1 ` for any `x . ` This means that `rst = -1 , ` `r + s + t = 0 , ` `rs + st + tr = -8 . ` Differentiate the first equality w.r.t. `x:`...

Latest answer posted May 26, 2022, 8:31 am (UTC)

1 educator answer

Math

First, we should represent the function under the integral as a sum of partial fractions. Then it will integrate easily using arctangent and/or logarithm. The decomposition should exist in the...

Latest answer posted May 4, 2022, 6:34 am (UTC)

1 educator answer

Math

To begin, let's move the second root to the right side and raise the equation to the third power: `20x + 22 = 8 + 12 ( 20x - 22 )^( 1 / 3 ) + 6 ( 20x - 22 )^( 2 / 3 ) ,` or `3 ( 20x - 22 )^( 2 / 3...

Latest answer posted June 6, 2022, 7:39 pm (UTC)

1 educator answer

Math

Let the two rightmost digits will be `a ` and `b , ` then `0 lt= a , b lt= 9 . ` The two-digit number composed of the two leftmost digits is equal to `( bar ( a b ) - 1 ) ,` and it must be less...

Latest answer posted July 11, 2022, 2:35 pm (UTC)

1 educator answer

Math

We are finding F(x) if x=0, so substitute 0 for x in the equation and start within the parentheses (PEMDAS). F(0)=67sin(12(0.223))+70 Finish the parenthetical, so multiply 12*0.223...

Latest answer posted June 28, 2022, 3:41 pm (UTC)

1 educator answer

Math

To find the hundredth digit of a number, it is sufficient to compute this number modulo 1000. Because 11 and 1000 are coprime, Euler's totient theorem may be applied, so`11^( phi ( 1000 ) ) = 1...

Latest answer posted July 2, 2022, 8:29 pm (UTC)

1 educator answer

Math

The result of the operation "111+11" is 122. You can find the solution in a number of ways, and the correct method depends on what makes the most sense to you. Some people find it easier to break...

Latest answer posted June 29, 2022, 5:47 pm (UTC)

1 educator answer

Math

We'll use some area considerations. Let's drop the perpendiculars `C C_2 ` and `K C_3 ` to the side `A B ` (look at the attached picture). Then the triangles `C C_2 C_1 ` and `K C_3 C_1 ` are...

Latest answer posted June 25, 2022, 9:04 am (UTC)

1 educator answer

Math

Algebra is a branch of mathematics that deals with variables (like x, y, and z) that represent unknown quantities. It builds on the fundamental arithmetic skills of addition, subtraction,...

Latest answer posted May 25, 2022, 2:03 am (UTC)

1 educator answer

Math

You will be able to solve for one variable in terms of another, but without a second equation to create a system of equations, you won't be able to calculate the actual numerical value of any...

Latest answer posted May 23, 2022, 8:29 pm (UTC)

1 educator answer

Math

In this question, we have seven positive integers (p, q, r, s, t, u, and v). It's said that the sum of each group of three consecutive letters equals 35. Also, q+u=15. So, by inference, we can...

Latest answer posted May 23, 2022, 3:24 pm (UTC)

1 educator answer

Math

For the number 385, the middle digit equals the sum of the other two digits (8=3+5). To find numbers between 100 and 999 with this same property, we perform the following calculations. Note that...

Latest answer posted May 23, 2022, 2:45 pm (UTC)

1 educator answer

Math

Given that `a_1=2021` for `n>=1` and let `a_{n+1}=\sqrt{4+a_n}` ----->(1) then, `a_5` can be written as `a_5=\sqrt{\frac{m}{2}+\frac{\sqrt{n}}{2}}+\sqrt{\frac{m}{2}-\frac{\sqrt{n}}{2}}`...

Latest answer posted April 8, 2022, 4:49 pm (UTC)

1 educator answer

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