[tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
What are Hyperbolic Functions?
Hyperbolic functions are a set of mathematical functions that are analogs of the trigonometric functions. While trigonometric functions are defined based on the unit circle, hyperbolic functions are defined using the hyperbola.
To show that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$:[/tex]
[tex]\textbf{1. Derivative of $\cosh(x)$:}[/tex]
Starting with the definition of [tex]\cosh(x)$:[/tex]
[tex]\[\cosh(x) = \frac{1}{2}(e^x + e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]e^x$:[/tex]
[tex]\cosh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) + \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x - e^{-x}) \\\\&= \sinh(x)[/tex]
Therefore, [tex]\cosh'(x) = \sinh(x)$.[/tex]
[tex]\textbf{2. Derivative of $\sinh(x)$:}[/tex]
Starting with the definition of [tex]\sinh(x)$:[/tex]
[tex]\[\sinh(x) = \frac{1}{2}(e^x - e^{-x})\][/tex]
Taking the derivative with respect to x using the chain rule and the derivative of [tex]$e^x$[/tex]:
[tex]\sinh'(x) &= \frac{1}{2}\left(\frac{d}{dx}(e^x) - \frac{d}{dx}(e^{-x})\right) \\\\&= \frac{1}{2}(e^x + e^{-x}) \\\\&= \cosh(x)[/tex]
Therefore, [tex]\sinh'(x) = \cosh(x)$.[/tex]
Hence, we have shown that [tex]\cosh'(x) = \sinh(x)$ and $\sinh'(x) = \cosh(x)$[/tex] using the rules for differentiating [tex]e^x$.[/tex]
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Cost of a cell phone: $249.50
Markup: 30%
Answer:
$324.35
Step-by-step explanation:
249.50 x 1.30 = $324.35
please solve for x!!!
Answer:
x = 20
Step-by-step explanation:
The two angles are verticals angles and vertical angles are equal
7x -99 = 2x+1
Subtract 2x from each side
7x-99-2x =2x+1-2x
5x-99 =1
Add 99 to each side
5x-99+99 = 1+99
5x =100
Divide each side by 5
5x/5 =100/5
x = 20
Answer:
x=20
set them equal to eac hother
7x-99=2x+1
-2x+99-2x+99
---------------------
5x=100
---- ------
5 5
x=20
Choose all of the points that are reflections of each other across both axes. (1.5,−2) ( 1 . 5 , - 2 ) and (−112,2) ( - 1 1 2 , 2 ) (4.5,−2) ( 4 . 5 , - 2 ) and (5.4,−2) ( 5 . 4 , - 2 ) (−112,3) ( - 1 1 2 , 3 ) and (−3,112) ( - 3 , 1 1 2 ) (1.75,−4) ( 1 . 75 , - 4 ) and (−134,4) ( - 1 3 4 , 4 )
Answer:
The points which are reflections across both axis are;
1) (1.5, -2) and [tex]\left (-1\dfrac{1}{2} , \ 2\right )[/tex]
2) (1.75, -4) and [tex]\left (-1\dfrac{3}{4} , \ 4\right )[/tex]
Step-by-step explanation:
The coordinate of the image of a point after a reflection across the 'x' and 'y' axis are given as follows;
[tex]\begin{array}{ccc}& Preimage&Image\\Reflection \ about \ the \ x-axis&(x, \ y)&(x, \, -y)\\Reflection \ about \ the \ y-axis&(x, \ y)&(-x, \, y)\end{array}[/tex]
Therefore, a reflection across both axis changes (only) the 'x' and 'y' value signs
The given points which are reflections across both axis are;
(1.5, -2) and [tex]\left (-1\dfrac{1}{2} , \ 2\right )[/tex]
We note that [tex]\left (-1\dfrac{1}{2} , \ 2\right )[/tex] = (-1.5, 2)
The reflection of (1.5, -2) across the x-axis gives the image (1.5, 2)
The reflection of the image (1.5, 2) across the y-axis gives the image (-1.5, 2)
Similarly, we have;
(1.75, -4) and [tex]\left (-1\dfrac{3}{4} , \ 4\right )[/tex]
We note that [tex]\left (-1\dfrac{3}{4} , \ 4\right )[/tex] = (-1.75, 4)
The reflection of (1.75, -4) across the x-axis gives the image (1.75, 4)
The reflection of the image (1.75, 4) across the y-axis gives the image (-1.75, 4).
The other points have changes in the values of the 'x' and 'y' between the given pair and are therefore not reflections across both axis
Two very long, parallel wires are separated by d = 0.065 m. The first wire carries a current of I1 = 0.65 A. The second wire carries a current of I2 = 0.35 A.
1) Express the magnitude of the force between the wires per unit length, f, in terms of I1, I2, and d.
2)Calculate the numerical value of f in N/m.
3)Is the force repulsive or attractive?
4) Express the minimal work per unit length needed to separate the two wires from d to 2d.
5)Calculate the numerical value of w in J/m.
1. The work magnitude of the force between the wires per unit length, f, can be expressed using Ampere's Law:
f = μ₀ * I₁ * I₂ / (2πd)
2. The numerical value of f is 2 × 10⁻⁶ N/m.
3. Since the currents I₁ and I₂ are both positive, the force between the wires will be attractive.
4. The minimal work per unit length needed to separate the two wires from d to 2d can be calculated using the equation:
W = f * (2d - d) = f * d.
5. The numerical value of the minimal work per unit length needed to separate the two wires from d to 2d is 1.3 × 10⁻⁷ J/m.
What is Ampère's law?Ampère's law, one of the fundamental correlations between electricity and magnetism, quantifies the relationship between an electric field's changing magnetic field and the electric current that creates it.
1. The work magnitude of the force between the wires per unit length, f, can be expressed using Ampere's Law:
f = μ₀ * I₁ * I₂ / (2πd),
where μ₀ is the permeability of free space, I₁ and I₂ are the currents in the wires, and d is the separation between the wires.
2. To calculate the numerical value of f in N/m, we need to substitute the given values into the formula:
μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
f = (4π × 10⁻⁷ T·m/A) * (0.65 A) * (0.35 A) / (2π * 0.065 m)
Simplifying:
f = 2 * 10⁻⁶ N/m
Therefore, the numerical value of f is 2 × 10⁻⁶ N/m.
3. The force between the wires is attractive when the currents flow in the same direction and repulsive when the currents flow in opposite directions. In this case, since the currents I₁ and I₂ are both positive, the force between the wires will be attractive.
4. The minimal work per unit length needed to separate the two wires from d to 2d can be calculated using the equation:
W = f * (2d - d) = f * d.
5. Substituting the value of f (2 × 10⁻⁶ N/m) and d (0.065 m) into the equation, we get:
W = (2 × 10⁻⁶ N/m) * (0.065 m)
Simplifying:
Work = 1.3 × 10⁻⁷ J/m
Therefore, the numerical value of the minimal work per unit length needed to separate the two wires from d to 2d is 1.3 × 10⁻⁷ J/m.
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Please helppp What's the slope and y intercept of 3x - y = 7
Answer:
slope: 3
y intercept:(0,-7)
Find the slope of the line perpendicular to AB. A(3, 5) and B(3, -8).
Answer:
m = 0
Step-by-step explanation:
AB has a slope of y2 - y1 / x2 - x1 so -8-5/3-3 so -13/0 = undefined so it is a vertical line, so any line that is horizontal will be perpendicular to it, so horizontal lines have a zero slope
True or false is the graph of y = 2/3 x + 4
Answer:
False. 2/3 is positive so therefore, it would be increasing, not decreasing.
Step-by-step explanation:
i need help. what’s the answer to this .
Answer:
12
Step-by-step explanation:
I know bc i did this before
Answer:
72 unit cubes (72 units³)
Step-by-step explanation:
[tex]6\times3\times4=72[/tex]
Which of the following transformations shifts all points graphed in the standard (wy)
coordinate plane down 5 coordinate units?
A. (X, y-5)
B. (X, y+5)
C. (X-5y)
D. (K-5, y)
E. (X+5,y)
Answer:
B is your answer
Step-by-step explanation:
PLSS HELPP D: WILL MARK BRAINLIEST :D
Answer:
√49
Step-by-step explanation:
It's a perfect square root.
Please brainliest ;)
Does anyone know number 1 and 2 if so pls help!
Answer:
364×180=65,520 baskets
Step-by-step explanation:
since its a leap year(4days as shown in the document)
2nd question:
365×180=65,700 baskets
since its not a leap year
if u dont know leap years end with an even number which is 364 and non leap years end with odd number which is 365.
Does anyone know a vb knowledge matters admin login?
I'm trying to complete the Sports & Entertainment Mogul
Answer:
Sadly no but also brainly isn't for this
Step-by-step explanation:
(Ima get so much hate rn from y'all)
Determine Px H=34, o=12, n=35 ox (Round to three decimal places F and a from the given parameters of the population and sample size. 11 Determine and o from the given parameters of the population and sample size. u-34, a 12, n=35 K- (Round to three decimal places as needed).
The value of μₓ is 77 and σₓ is 3 when [tex]\mu[/tex] = 77, [tex]\sigma[/tex] = 21, and n = 49.
We have
[tex]\mu[/tex] = 77,
[tex]\sigma[/tex] = 21,
and Sample size, n = 49.
The formula to determine the sample mean (μₓ) is
=μx
= μ
=77
Similarly, σₓ= σ/(√(n))
Therefore, we substitute the given values,
σx = 21/(√(49))
σx= 3
The standard deviation is a statistical metric used to quantify the extent of variation or dispersion within a dataset. It gauges the degree to which values deviate from the average (mean) value.
A larger standard deviation suggests a greater degree of variability, while a smaller standard deviation suggests less variability.
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A box of macaroni & cheese says that it makes 25% more than a regular box. If a regular box makes 3 cups of macaroni & cheese, how many cups will this box make?
Thanks :)
Answer:
3.75
Step-by-step explanation:
25% = 0.25
3(0.25) = 0.75
3 + 0.75 = 3.75
Historical data indicated that the time students took to complete their CML quizzes could be modelled as a normal distribution with a variance of 123.7 minutes squared. A random sample of 37 students revealed a mean time of 439 minutes. Determine a 94% confidence interval for the average time students take to complete their CML quiz. State the lower bound of this interval (in minutes) to 2 decimal places
The 94% confidence interval for the average time students take to complete their CML quiz can be determined using the sample mean, sample size, and the given variance.
To calculate the confidence interval, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
First, we need to find the standard deviation, which is the square root of the variance:
Standard Deviation = sqrt(123.7) = 11.11
Next, we find the critical value corresponding to a 94% confidence level. Since the sample size is large (n > 30), we can use the Z-distribution. For a 94% confidence level, the critical value is approximately 1.88.
Substituting the values into the formula:
Confidence Interval = 439 ± (1.88) * (11.11 / sqrt(37))
Calculating the confidence interval, we get:
Confidence Interval ≈ 439 ± 3.32
Therefore, the lower bound of the confidence interval is:
Lower bound ≈ 439 - 3.32 = 435.68 minutes (rounded to 2 decimal places).
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PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!
Given that x=6.2•10^12 and y=2•10^-8, find the value of x•2y
Answer:
Step-by-step explanation:
Let M = {a E Ra > 1). Then M is a vector space under standard addition and scalar multiplication of real numbers. False True * Let W = {a + 2x + bx² € Pz: a, b E R} with the standard operations in P2. Which of the following statements is true? 1+xEW W is a subspace of P2. The above is true W is not a subspace of P.
The statement "W is a subspace of P2" is true because the set W, defined as W = {a + 2x + bx² ∈ P2: a, b ∈ R}, is a subspace of P2.
To determine if the set W = {a + 2x + bx² ∈ P2: a, b ∈ R} is a subspace of P2, we need to check if it satisfies three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector.
Closure under addition: For any two polynomials p(x) = a + 2x + bx² and q(x) = c + 2x + dx² in W, their sum p(x) + q(x) = (a + c) + 4x + (b + d)x² is also a polynomial in W. This shows that W is closed under addition.
Closure under scalar multiplication: For any polynomial p(x) = a + 2x + bx² in W and any scalar c, the scalar multiple c * p(x) = ca + 2cx + cbx² is also a polynomial in W. Therefore, W is closed under scalar multiplication.
Contains the zero vector: The zero vector in P2 is the polynomial 0x² + 0x + 0, which can be expressed as a + 2x + bx² with a = 0 and b = 0. Since this polynomial satisfies the conditions of W, W contains the zero vector.
Since W satisfies all three conditions, it is a subspace of P2.
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Tan(x)-sqrt (3)=0, I know that one of the solutions should be pi/3, but I don’t know how to get to that, does anyone know how to solve this?
If V is a finite-dimensional inner product space and V = WW2 is the direct sum of two subspaces, which of the following must be true? • If Ty : W1 + W1 and T2 : W2 + W2 are linear transformations, then there is a unique linear transformation T:V + V such that T(W1) =T1(wi) for all w1 € W1, and T (W2) = T2(W2) for all W2 E W2. • If {V1, ..., Un} is a basis for V such that {V1, ..., Uk} is a basis for W1, then {Uk+1,..., Un} is a basis for W2. • If projw, is the orthogonal projection map onto W1, then for all v EV, we have v-projw, (v) € W2.
The correct statement is:
• If projw₁ is the orthogonal projection map onto W₁, then for all v in V, we have v - projw₁(v) ∈ W₂.
What is finite-dimensional inner product space?
In a finite-dimensional inner product space V, if V = W₁ ⊕ W₂ is the direct sum of two subspaces W₁ and W₂, the orthogonal projection map projw1 onto W₁ is a linear transformation that projects any vector v onto the subspace W₁. The projection of v onto W₁ is the closest vector in W₁ to v.
The statement v - projw₁(v) ∈ W₂ means that the difference between v and its projection onto W₁ lies in the subspace W₂. This is true because V is the direct sum of W₁ and W₂, which means any vector in V can be uniquely decomposed as the sum of a vector in W₁ and a vector in W₂. Therefore, the difference v - projw₁(v) will be in W₂.
This property holds for any vector v in V, so the statement is true.
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Suppose sales of shoe companies follow normal distribution with unknown population mean and a known population standard deviation of 3. It is suspected that on an average the revenue of the shoe companies is $8 million. A random sample of 400 companies was taken and the sample average was found to be $8.30 million. We want to determine whether the average revenue is significantly different than $8 million. The critical value (upper) is_____________ therefore we can __________the Null at the 1% level of significance
2.33, reject
2.57, not reject
1.96, not reject
2.57, reject
The critical value (upper) for a 1% level of significance is 2.33. Therefore, we can reject the null hypothesis.
To determine whether the average revenue is significantly different from $8 million, we perform a hypothesis test using the sample data. The null hypothesis (H0) states that the average revenue is equal to $8 million, while the alternative hypothesis (Ha) states that the average revenue is significantly different.
Given a sample of 400 companies, the sample average revenue is found to be $8.30 million, and the population standard deviation is known to be 3. We can use a z-test since the population standard deviation is known and the sample size is large.
Next, we calculate the test statistic (z-score) using the formula: z = (sample mean - hypothesized mean) / (population standard deviation / √sample size).
Plugging in the values, we get z = (8.30 - 8) / (3 / √400) = 1.73.
To determine the critical value (upper) at a 1% level of significance, we look up the z-value from the standard normal distribution table, which is approximately 2.33.
Since the calculated z-score of 1.73 is less than the critical value of 2.33, we do not have enough evidence to reject the null hypothesis. Therefore, we cannot conclude that the average revenue is significantly different from $8 million at a 1% level of significance.
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Which expression is equivalent to
Answer:
Step-by-step explanation:
the first expression
Use the diagram to identify 4 collinear points
Answer:
the answer is the (L,N,K,O)
a 90% confidence interval for the proportion of americans with cancer was found to be (0.185, 0.210). the margin of error for this confidence interval is:
The margin of error for the 90% confidence interval is 0.012.
Given, a 90% confidence interval for the proportion of Americans with cancer was found to be (0.185, 0.210)
To calculate the margin of error, we can use the following formula:
margin of error = (upper limit of the confidence interval - lower limit of the confidence interval) / 2
Substitute the given values,
margin of error = (0.210 - 0.185) / 2 = 0.0125 ≈ 0.012
Therefore, the margin of error for the confidence interval (0.185, 0.210) is 0.012.
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what is the volume of this sphere? Use a 3.14 and round your answer to the nearest hundredth. cubic feet Submit Work it out Not feeling ready yet? This Area of circles 47)
Answer:
4.19 cubic feet
Step-by-step explanation:
If the coefficient of determination is 0.422, what percentage of the variation in the data
about the regression line is unexplained?
2 pts
42.2%
17.8%
82.2%
57.8%
The percentage of the variation in the data about the regression line that is unexplained is 100% - 42.2% = 57.8%. So, the correct option is D, 57.8%.
If the coefficient of determination is 0.422, the percentage of the variation in the data about the regression line that is unexplained is 57.8%. Coefficient of determination, denoted by R² is a statistical tool that measures how well the regression line approximates the real data points. It is also called the square of the correlation coefficient between the dependent and independent variables.
The coefficient of determination varies from 0 to 1, and it represents the proportion of the total variation in the dependent variable that is explained by the variation in the independent variable. A coefficient of determination of 0.422 indicates that the regression line explains only 42.2% of the total variation in the dependent variable. Hence, the percentage of the variation in the data about the regression line that is unexplained is 100% - 42.2% = 57.8%.
Therefore, the correct option is D, 57.8%.
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find a rational number between 3/5 and 4/5 ( see I just only want one so please don't give me all five)
give me the process too
Thanks
Answer:
[tex]\dfrac{7}{10}[/tex]
Step-by-step explanation:
We need to find a rational number between 3/5 and 4/5.
We can find a rational number between two fractions as :
[tex]\dfrac{a+b}{2}[/tex]
We have, a = 3/5 and b = 4/5
So,
[tex]\dfrac{\dfrac{3}{5}+\dfrac{4}{5}}{2}\\\\=\dfrac{\dfrac{7}{5}}{2}\\\\=\dfrac{7}{10}[/tex]
So, a rational number between 3/5 and 4/5 is equal to 7/10.
Is 3. [3 marks] Use Gauss Divergence theorem to calculate S (5xi + ay 3 – 23 k).n dA over the sphere S: 1+ y + x2 = 9. splats J = ?? It-t? Lose - 2. [6 marks] Calculate the surface integral || G(r)da, where G = (1212 + 36)/2, the suru loob the parametrization r(u, v) = (3u, 2v, u), and 0 su 1, 0 Sv < 2.
1. Using Gauss Divergence theorem, we have to calculate S. (5xi + ay3 - 23k). ndA over the sphere S: 1+ y + x2 = 9. We have the following information: S.(5xi + ay3 - 23k).ndA over the sphere S: 1+ y + x2 = 9.
Gauss Divergence Theorem states that, The surface integral of a vector field F over a closed surface S equals the volume integral of the divergence of F over the enclosed volume V. To calculate the surface integral S.(5xi + ay3 - 23k).ndA, we need to first calculate the volume integral of the divergence of the vector field over the enclosed volume V which in this case is a sphere.
The divergence of the given vector field can be calculated as, div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z = 5 + 3ay + 0 = 5 + 3ay
Thus, the volume integral of the divergence of F over the sphere S: 1+ y + x2 = 9 is given as ∭V div(F) dV = ∭V (5 + 3ay) dV.
The volume integral can be calculated using spherical coordinates.
We have the equation of the sphere as 1 + y + x2 = 9, substituting x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, and simplifying, we get the limits of integration as 0 ≤ r ≤ 2, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π.
Therefore, the volume integral becomes:∭V div(F) dV = ∭V (5 + 3ay) dV = ∫0^2 ∫0^π ∫0^2π (5 + 3a(r cos θ sin φ)) r2 sin θ dr dθ dφ = 60πa
The surface integral of F over the sphere S can be calculated using the Gauss Divergence Theorem, which states that the surface integral of F over a closed surface S is equal to the volume integral of the divergence of F over the enclosed volume V.
Thus, S.(5xi + ay3 - 23k).ndA = ∭V div(F) dV = 60πa
Answer: S.(5xi + ay3 - 23k).ndA = 60πa2.
We are to calculate the surface integral ∬S G(r) da, where G = (12 + 12 + 36)/2 = 30.
The surface is given by r(u, v) = (3u, 2v, u), 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2.
The surface area element da can be calculated as, da = |r/∂u x r/∂v| dudv = |(6, 0, 3) x (0, 2, 1)| dudv = |(-6, -3, 0)| dudv = 3 dudv
Hence, the surface integral ∬S G(r) da becomes ∬S G(r) da = ∫0^2 ∫0^1 G(r(u, v)) da = ∫0^2 ∫0^1 30 * 3 dudv = 180
Answer: ∬S G(r) da = 180
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State the null hypothesis for a one-way ANOVA test if there are four groups.
In a one-way ANOVA test with four groups, the null hypothesis states that there is no significant difference between the means of all four groups.
This means that any observed differences in the sample means are due to chance or random integral alone and not because of any systematic or real differences between the groups.
The null hypothesis assumes that the population means for each group are equal, which implies that there is no effect or influence of the independent variable on the dependent variable. If the null hypothesis is accepted, it means that any observed differences between the groups are not statistically significant and do not support the alternative hypothesis.
To determine whether to accept or reject the null hypothesis, researchers calculate the F-statistic, which compares the variability between the sample means to the variability within each group. If the calculated F-value is greater than the critical F-value for a given level of significance, the null hypothesis is rejected in favor of the alternative hypothesis, indicating that there is a significant difference between at least two of the group means.
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Let p be a prime and let f(x,y) be a polynomial of degree 2 whose only 0 in Z/pZ is 0 (the trivial) (Has no zero divisors).
i) Explain why f(ka,kb) = k^2f(a,b).
ii) Use i) to prove that if a is not 0 mod p, then f(x,y) is congruent to a mod p always has a solution.
Given a prime number p and a polynomial f(x, y) of degree 2 with the only zero in Z/pZ being 0, we can show that f(ka, kb) = k^2f(a, b) using properties of polynomials.
i) To show that [tex]f(ka, kb) = k^2f(a, b)[/tex], we consider the polynomial f(x, y) and apply the properties of polynomials. Since f(x, y) has a degree of 2, we can write it as [tex]f(x, y) = ax^2 + bxy + cy^2,[/tex] where a, b, and c are coefficients. Now, substituting ka for x and kb for y, we get f(ka, kb) = [tex]a(ka)^2 + b(ka)(kb) + c(kb)^2[/tex]. Simplifying this expression, we obtain f(ka, kb) = [tex]k^2(ax^2 + bxy + cy^2) = k^2f(a, b),[/tex] which demonstrates the desired result.
ii) Using the result from part i), we can prove that if a is not congruent to 0 modulo p, then the equation f(x, y) ≡ a (mod p) always has a solution. Suppose f(x, y) ≡ a (mod p) has no solution for some value of a not congruent to 0 modulo p. Therefore, if a is not congruent to 0 modulo p, we can choose an appropriate value of k such that [tex]k^2f(a, b)[/tex] ≡ b (mod p), leading to a solution for f(x, y) ≡ b (mod p). Thus, if a is not congruent to 0 modulo p, f(x, y) ≡ a (mod p) always has a solution.
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