The general solution to the differential equation y'' + 2y = −18x^2e^(2x) by undetermined coefficients is y = (-9x^2)e^(2x).
The given differential equation is y'' + 2y = −18x^2e^(2x)
To solve this equation by undetermined coefficients, we first find the characteristic equation of the homogeneous equation, which is r^2 + 2 = 0.
The solutions to this equation are r = i and r = -i. Therefore the general solution to the homogeneous equation is yh(x) = c1 cos(x) + c2 sin(x)
Now, we use the method of undetermined coefficients to guess the form of the particular solution. Since the non-homogeneous term is −18x^2e^(2x), we guess that the particular solution is of the form
yp = (Ax^3 + Bx^2)e^(2x) + Cx^2e^(2x).
Substituting this into the differential equation, we get
y'' + 2y = (6Ax + 2B)e^(2x) + 2Cxe^(2x) = −18x^2e^(2x)
Equating the coefficients of the like terms, we get:
6A = 0, 2B = −18, and 2C = 0
Solving for A, B and C, we get A = 0, B = −9 and C = 0
Therefore, the general solution to the differential equation y'' + 2y = −18x^2e^(2x) by undetermined coefficients is y = (Ax^3 + Bx^2)e^(2x) + Cx^2e^(2x) = (-9x^2)e^(2x)
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1.
A 50 foot rope is stretched tight from the roof of a building to a spot 20 feet from the base of the
building. How tall is the building? Round your answer to TWO decimal places.
The required height of the building is approximately 45.82 feet.
Given that, a 50-foot rope is stretched from a building's top to a point 20 feet from the building's base.
We can use the Pythagorean theorem to solve the question.
Let h be the height of the building.
Pythagoras's theorem states that in a right-angled triangle, "the square of one side is equal to the sum of the squares of the other two sides".
Then, we have:
h² + 20² = 50²
Simplifying and solving for h, we get:
h² = 50² - 20²
h² = 2500 - 400
h² = 2100
h = √(2100)
h ≈ 45.82
Therefore, the height of the building is approximately 45.82 feet.
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what is the co efficient of x in(2.x+3)²
A coefficient is any constant or numerical term that is in front of one or more variables and is defined as a fixed number that is multiplied by a variable. For example, in the expression 3x+2y+4, 3 is the coefficient of x, 2 is the coefficient of y but 4 is not a coefficient, as it is not being multiplied by a variable.
How do you find the coefficient of x?
To find the coefficient of x, we can encircle it or underline it. Then, take everything else except for x, i.e. 5y. So, the coefficient of x in the term 5xy is 5y. Similarly, the coefficient of y in the term 5xy is 5x.
Solving it with Binomial Theorem(2.x+3)² = (2x+3)(2x+3)
2x(2x+3)+3(2x+3)=
2x(2x+3)+3(2x+3)=
4x²+6x+3(2x+3)=
4x²+12x+9
so if your using the binomial theorem it would equal 4x²+12x+9
if you just want the coefficient of x that would be 4 ↑
A triangle has side lengths of 13, 18, and 24. Is it a right triangle?
Answer:
No, it is not a right triangle.
[tex] \sqrt{ {13}^{2} + \ {18}^{2} } = \sqrt{169 + 324} = \sqrt{493} [/tex]
√493 is not equal to 24.
Answer:
it doesn't
Step-by-step explanation:
it doesn't follow pythagoras theorem
a² = b² + c²
In a class of 26 students, 15 play an instrument and 5 play a sport. There are 3 students who play an instrument and also play a sport. What is the probability that a student chosen randomly from the class plays a sport and an instrument?
Answer:
3:26
Step-by-step explanation:
.
please answer all 3 questions
1. The equation of the form y = a • bˣ is y = 81 x (¹/₃)ˣ
2. His stamp should be worth approximately $7,851.47 after 6 years.
3. The equation of the form y = a • bˣ is y = (¹/₁₆) x 2²ˣ ⁺ ⁵
How did we get our values?1. One will see that y is decreasing by a factor of 3 as x increases by 1. Therefore, we can write the equation as:
y = 81 x (¹/₃)ˣ
2. The increase in value of the stamp can be calculated using the formula:
V = P(1+r)ᵗ
where V is the future value, P is the present value, r is the annual interest rate as a decimal, and t is the number of years.
Substituting the given values:
V = 4900(1+0.075)⁶
V ≈ $7,851.47
Therefore, the stamp should be worth approximately $7,851.47 after 6 years.
3. One will see that y is increasing by a factor of 2 as x increases by 1. Therefore, we can write the equation as:
y = (1/16) x 2²ˣ ⁺ ⁵
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What is the domain of the function
Answer:
xs7
Step-by-step explanation:
2) Factor by CTS: x² +12
please show work
The factored form of x² + 12 using the difference of squares formula is
(x + 2√3)(x - 2√3).
We have,
To factor x² + 12 using the difference of squares formula, we need to express it as the difference between two squares:
x² + 12 = x² + (2√3)²
Now we can use the difference of squares formula, which states that:
a² - b² = (a + b)(a - b)
In this case, we have a = x and b = 2√3. So we can write:
= x² + 12
= x² + (2√3)²
= (x + 2√3)(x - 2√3)
Therefore,
The factored form of x² + 12 using the difference of squares formula is
(x + 2√3)(x - 2√3).
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i dont know the answer to this
more equation things
For the given linear equation y = (1/4)*x + 5/4.
(1, 1.5) is a solution.(12, 4) is not a solution.The x-intercept is (-16/5, 0).Are these points solutions of the linear equation?Here we have the linear equation:
y = (1/4)*x + 5/4.
to check if (1, 1.5) is a solution we need to evaluate this in x = 1 and see if we get 1.5.
y = (1/4)*1 + 5/4
y = 6/4 = 3/2 = 1.5
Then (1, 1.5) is a solution.
For the second point we evaluate in x = 12.
y = (1/4)*12 + 5/4
y = 3 + 5/4 = 4.25
So (12, 4) is not a solution.
Finally, to find the x-intercept, we evaluate in y = 0.
0 = (1/4)*x + 5/4
-4/5 = (1/4)*x
4*(-4/5) = x
-16/5 = x
The x-intercept is (-16/5, 0).
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Mary has a lemonade stand.Last week she spent $18.00 on supplies,but earned only $12.00.Did Mary make a profit ?Explain and show problem as a mathematical equation.
They need to sell at least 13 cups to gain profits.
Here, we have,
Givens
The initial cost is $1.20.
Each cup of lemonade costs 6 cents to make.
May's children sell 10 cents a cup.
First, we need to represents the cost of their investment.
0.06x = 1.20, because each cup costs 6 cents, and the initial costs of all is $1.20.
That means,
x= 20
They can produced 20 cups.
However, they already sell a cup for 10 cents. So, how many cups they need to sell to gain profits? We need to establish a similar relation,
0.10 x = 1.20
x = 12
This means if they sell 12 cups, they will just cover the investment, that is, zero profits.
Therefore, they need to sell at least 13 cups to gain profits.
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complete question
Mary's children decide to run a lemonade stand to earn some extra money.The cost to start the business is $1.20 and each cup of lemonade costs 6 cents to make. If lemonade sells for 10 cents a cup,how many cups must Mary's children sell to make a profit?----Apply systems of equations
Use the following sets to answer the question.
A={1,2,3,4,5}
B={5,6,7,8}
Which answer shows the union of sets A
and B?
{5}
{1,2,3,4,6,7,8,9}
{1,2,3,4,5,6,7,8}
{2,4,8}
The union of sets A and B is {1, 2, 3, 4, 5, 6, 7, 8}.
The union of two sets A and B is the set of all elements that are in A, or B, or both. In this case, the elements in set A are {1, 2, 3, 4, 5} and the elements in set B are {5, 6, 7, 8}.
To find the union of these sets, we simply combine all the elements from both sets but remove any duplicates.
Therefore, the answer that shows the union of sets A and B is {1, 2, 3, 4, 5, 6, 7, 8}, since it contains all the elements from both sets without any duplicates.
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The community health clinic you volunteer for aims to develop a strategy allowing the most vulnerable members of the community to have first access to the Flu vaccine. The clinic has collected data finding that the total number of people in their service area is 50,000, 27% of the community is over the age of 65 and immunocompromised, and 11% of the community under 65 is immunocompromised. While the clinic wishes to market to all members of the community, immunocompromised people over 65 are their current priority.2. Explain in a one well-written paragraph, how you will analyze the data in this scenario to arrive at the answer you will provide in number 2.
In order to understand and prioritize the vaccination strategy for the community health clinic, first accurately assess the demographics of the community based on the data provided.
How to analyze the data ?Calculating the absolute numbers of people in each group: over 65 and immunocompromised, and under 65 and immunocompromised. For this, we'll apply the given percentage to the total population of 50,000.
Knowing the total number of people in each category will help us understand the size of each target group and prioritize accordingly. The clinic's current priority is immunocompromised people over 65, so having the exact figure will aid in planning for vaccine supply and outreach strategies.
Furthermore, the analysis of this data can also aid in projecting the needs for future vaccine campaigns for the remaining community members. Finally, this detailed analysis will help in crafting an effective communication plan that emphasizes the clinic's priority of serving the most vulnerable first, while still acknowledging the needs of all community members.
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Work out the area of the shaded shape on the millimetre (mm) grid.
State the units with your answer.
The diagram is not drawn to scale.
The requried area of the shaded shape is 5 square millimeters.
From the figure,
The area of the green shaded area is given by:
The area of a single square is given as = 1 * 1 = 1 square millimeters.
Now there is 5 square in the shaded region, So the area is given as:
= 5 * 1 = 5 square millimeters.
Thus, the requried area of the shaded shape is 5 square millimeters.
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Following is a table for the present value of an annuity of $1 at compound interest
On the map, the grocery store is 2 inches away from the library. The actual distance is 1.5 miles. The same map shows that the movie theater is 20 inches from the school.
What is the actual distance from the movie theater to the school, rounded to the nearest mile?
A: 15
B:27
C:30
D:60
The actual distance from the movie theater to the school is given as follows:
A. 15 miles.
How to calculate the actual distance?The actual distance from the movie theater to the school is obtained applying the proportions in the context of the problem.
On the map, the grocery store is 2 inches away from the library. The actual distance is 1.5 miles, hence the scale factor is of:
2 inches = 1.5 miles
1 inch = 0.75 miles.
The same map shows that the movie theater is 20 inches from the school, hence the actual distance is given as follows:
20 x 0.75 = 15 miles.
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An Inverse Variation Includes The Points(3, 3)and(1, n).Find
n .
The inverse variation is y = 9/x, using that equation we can see that n = 9.
How to find the value of n?An inverse variation between two variables x and y can be written as:
y = k/x
Where k is a constant.
We know that this inverse variation contains the point (3, 3), replacing these values we have:
3 = k/3
3*3 = k
9 = k
Then the inverse variation is:
y = 9/x
Now we want to find n such that (1, n) is on the relation above, then we will get:
n = 9/1
n = 9
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Can someone help me with this problem please
Suppose M is a matrix of size 9x10, c is a scalar, and the matrix computation cM is defined. What is the size of matrix cM?
----------------
If the size of matrix "M" is 9×10, and a scalar "c" is multiplied by matrix, then the size of "cM" will be 9×10.
We know that when a scalar is multiplied to a matrix, each element of the matrix gets multiplied by that scalar.
In this case, the scalar "c" is multiplied with the Matrix "M";
So if a scalar "c" is multiplied by a matrix "M" of size 9×10, then the resulting matrix "cM" will also have the same number of rows and columns as the original matrix "M".
Therefore, the size of "cM" will also be 9×10.
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What is the instantaneous rate of change at x=2 for the function
f(x)= 2x - 5
The instantaneous rate of change at x = 2 is equal to the derivative, which is 2.
How to solve for the rate of changeThe derivative of f(x) = 2x - 5 with respect to x can be found by applying the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1).
Taking the derivative of f(x) = 2x - 5:
f'(x) = 2 * (d/dx)(x) - (d/dx)(5)
= 2 * 1 - 0
= 2.
The derivative of f(x) with respect to x is a constant, 2, indicating that the function has a constant slope.
Therefore, the instantaneous rate of change at x = 2 is equal to the derivative, which is 2.
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By using the trapezoidal rule with 5 ordinates, approximate [sin(x²+1) dx to 4 decimal places.
Using the trapezoidal rule with 5 ordinates, we approximate the integral [sin(x²+1) dx] over the interval [0,1] to be 0.5047 to 4 decimal places.
To approximate the integral [sin(x²+1) dx] using the trapezoidal rule with 5 ordinates, we can use the following formula:
∫[a,b]f(x)dx ≈ [(b-a)/2n][f(a) + 2f(a+h) + 2f(a+2h) + 2f(a+3h) + 2f(a+4h) + f(b)]
where n is the number of ordinates (in this case, n = 5), h = (b-a)/n is the interval width, and f(x) = sin(x²+1).
First, we need to find the interval [a,b] over which we want to integrate. Since no interval is given in the problem statement, we'll assume that we want to integrate over the interval [0,1].
Therefore, a = 0 and b = 1.
Next, we need to find h:
h = (b-a)/n = (1-0)/5 = 0.2
Now, we can apply the trapezoidal rule formula:
∫[0,1]sin(x²+1)dx ≈ [(1-0)/(2*5)][sin(0²+1) + 2sin(0.2²+1) + 2sin(0.4²+1) + 2sin(0.6²+1) + 2sin(0.8²+1) + sin(1²+1)]
≈ (1/10)[sin(1) + 2sin(0.05²+1) + 2sin(0.15²+1) + 2sin(0.35²+1) + 2sin(0.65²+1) + sin(2)]
≈ (1/10)[0.8415 + 2sin(1.0025) + 2sin(1.0225) + 2sin(1.1225) + 2sin(1.4225) + 1.5794]
≈ 0.5047
Therefore, using the trapezoidal rule with 5 ordinates, we approximate the integral [sin(x²+1) dx] over the interval [0,1] to be 0.5047 to 4 decimal places.
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The nth term of an arithmetic sequence is given by un=15-3n.
a. [1 mark] State the value of the first term, u1.
b. [2 marks] Given that the nth term of this sequence is -33, find the value of n.
c. [2 marks] Find the common difference, d.
a. The first term of the arithmetic sequence is 12.
b. The value of n for which the nth term is -33 is 16.
c. The common difference of the arithmetic sequence is -3.
a. The first term, u1, can be found by substituting n=1 into the given formula for the nth term:
u1 = 15 - 3(1) = 12
b. To find the value of n for which the nth term is -33, we set the formula for the nth term equal to -33 and solve for n:
un = 15 - 3n = -33
Adding 3n to both sides, we get:
15 = -33 + 3n
Adding 33 to both sides, we get:
48 = 3n
Dividing both sides by 3, we get:
n = 16
c. The common difference, d, is the difference between any two consecutive terms of the sequence. To find d, we can subtract any two consecutive terms, such as u2 and u1:
u2 = 15 - 3(2) = 9
u1 = 15 - 3(1) = 12
d = u2 - u1 = 9 - 12 = -3
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The cosine of θ is −0.95. What is sin(θ)?
The value of sin(θ) is 0.31 and it lies in the third and fourth quadrants. if the value of the cosine of θ is −0.95.
cos(θ) = -0.95
To calculate the value of sin(θ), we can use the Pythagorean theorem:
[tex]sin^{2}[/tex] (θ) +[tex]cos^{2}[/tex] (θ) = 1
We can rearrange the Pythagorean identity to solve for sin(θ):
[tex]sin^{2}[/tex] (θ)= 1 - [tex]cos^{2}[/tex]
sin(θ) = ±[tex]\sqrt{1-cos^{2}}[/tex] *(θ)--------- (equation 1)
Substitute the value of the cosine of θ = −0.95 in Equation 1
sin(θ) = ±[tex]\sqrt{1-cos^{2}}[/tex] *(θ)
sin(θ) = ±[tex]\sqrt{(1 - 0.9025)}[/tex]
sin(θ) = ±[tex]\sqrt{0.0975}[/tex]
The positive value determines that the value is in the first and second quadrants, Negative shows the third and fourth quadrants.
sin(θ) = [tex]\sqrt{ 0.0975}[/tex]
sin(θ) = 0.312
Therefore, we can conclude that the value of sin(θ) is 0.312.
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Is this negative and odd or even and positive or negative and even and positive and odd?
Answer:
Hold on, our servers are swamped. Wait for your answer to fully loadHold on, our servers are swamped. Wait for your answer to fully loadHold on, our servers are swamped. Wait for your answer to fully loadHold on, our servers are swamped. Wait for your answer to fully load
Step-by-step explanation:
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Olivia, Muhammad, and Cameron had a challenge to see who could bike the farthest in one day. Olivia biked 8 miles, Muhammad biked 2 times as many miles as Cameron and Cameron biked 4 times as many miles as Olivia. How many miles did Muhammad bike?
Answer:
Muhammad biked 64 miles
Step-by-step explanation:
Olivia biked 8 miles.
Cameron biked 4 times as many miles as Olivia, and Muhammad biked 2 times as many miles as Cameron.
Cameron: 4 x 8 = 32
Cameron biked 32 m.
Muhammad: 32 x 2 = 64
Muhammad biked 64 miles.
You are considering a 5/1 ARM. What does the 1 represent?
A. The total number of years in the loan
B. The number of years that a fixed interest rate will be applied to the
loan
OC. The number of years between adjustments in the interest rate
D. The interest rate of the initial, fixed-rate loan period
SUBMIT
As far as a 5/1 ARM is concerned, note that the "1" refers to how often the rate can be adjusted after the initial fixed-rate period ends.
What is 5/1 ARM?A 5/1 ARM is an adjustable rate mortgage loan (ARM) that has a fixed interest rate for the first five years. Following that, the 5/1 ARM transitions to an adjustable interest rate for the remainder of its term. The terms "variable" and "adjustable" are frequently used synonymously.
If you want a low monthly payment and don't expect to stay in your house for long, a 5/1 adjustable-rate mortgage (ARM) loan may be worth considering. For the first five years, rates on 5/1 ARMs are typically lower than rates on 30-year fixed-rate mortgages.
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Alondra has $350,000 saved for retirement in an account earning 2.9% interest, compounded monthly. How much will she be able to withdraw each month if she wants to take withdrawals for 22 years? Round your answer to the nearest dollar.
Alondra will be able to withdraw approximately $1,427 per month to take withdrawals for 22 years from her retirement account.
To calculate the monthly withdrawal amount, we need to use the present value formula, which is:
PMT = (PV * r) / (1 - (1 + r)⁻ⁿ)
where:
PV = present value = $350,000
r = monthly interest rate = 2.9% / 12 = 0.002417
n = number of months = 22 years * 12 months/year = 264 months
Substituting the values into the formula, we get:
PMT = (350000 * 0.002417) / (1 - (1 + 0.002417)⁻²⁶⁴)
PMT ≈ $1,427.07
This calculation assumes that the interest rate remains constant throughout the 22-year period, which may not be the case in reality.
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Find the range of the number of points scored.
Range: – =
answer is 56, 41, 15
A small toy rocket is launched from a 48-foot pad. The height (h, in feet) of the rocket t seconds after
taking off is given by the formula h = - 3t2 +0t + 48. How long will it take the rocket to hit the
ground?
t =
Okay, here are the steps to solve this problem:
1) The height (h) of the rocket t seconds after launch is given as: h = - 3t2 + 0t + 48
2) We want to find the time (t) when the rocket hits the ground (h = 0)
3) Set the formula equal to 0: - 3t2 + 0t + 48 = 0
4) Factor the left side: - 3(t2 - 0t) + 48 = 0
5) Solve for t2 - 0t: t2 - 0t = 16
6) Add 0t to both sides: t2 = 16 + 0t
7) Take the square root of both sides: t = 4
Therefore, the time for the rocket to hit the ground is 4 seconds.
So in this case, t = 4
Let me know if you have any other questions!
Answer:
4 seconds.
Step-by-step explanation:
When the rocket hits the ground, its height will be 0. Therefore, since we are given an expression for the height of the rocket dependent on the time, we can simply set it equal to 0 and solve for the time and find how long the rocket will take to hit the ground. I'm assuming the equation is[tex]h = -3t^{2} + 48[/tex]
Now set this equal to 0
[tex]0 = -3t^2+48[/tex]. Solve for t by isolating it.
[tex]-48 = -3t^2[/tex]
[tex]16 = t^2[/tex]
From here, by taking the square root, we see that t is either equal to 4 or -4 in seconds. Since we can't have negative time, we can clearly see that the answer is 4 seconds.
Hope this helps
Find the greatest common factor of 56xy and 16y³.
Answer:
8y
Step-by-step explanation:
Lets put these two parts into addition to make it easier.
(y)56x + 16y^2
(8y)7x + 2y^2
8y is the GCF(grates common factor)
Select the correct answer. Consider this function. Which graph represents the inverse of function f? f(x)= x+4
The inverse of the function f(x) = x + 4 is given as f⁻¹(x) = x - 4
What is inverse of a function?An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f-1 or F-1.
In this problem, the function given is f(x) = x + 4;
We can find the inverse of the function as;
y = x + 4;
Let's switch the variables by replacing x as y and y as x;
x = y + 4
Solving for y;
y = x - 4
f⁻¹(x) = x - 4
The graph of the function is attached below
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a) The monthly basic salary of the married Chief of Army Staffs (COAS) General is Rs 79,200 with Rs 2,000 dearness allowance. He gets Dashain allowance which is equivalent to his basic salary of one month. He contributes 10% of his basic salary in Employee's Provident Fund (EPF) and he pays Rs 50,000 as the premium of his life insurance. Given that 1% social security tax is levied upon the income of Rs 6,00,000, 10% and 20% taxes are levied on the next incomes of Rs 2,00,000 and up to Rs 3,00,000 respectively. Answer the following questions. (i) What is his monthly basic salary? (ii) Find his taxable income.
(i). His monthly basic salary is Rs 79,200.
(ii) His taxable income is Rs 1,02,480.
(i) The monthly basic salary of the married COAS General is given as Rs 79,200, with an additional Rs 2,000 as a dearness allowance.
Total salary = Basic salary + Dearness allowance
Total salary = Rs 79,200 + Rs 2,000
Total salary = Rs 81,200
So his monthly basic salary is Rs 79,200.
(ii)
Total income = Basic salary + Dearness allowance + Dashain allowance
Total income = Rs 79,200 + Rs 2,000 + Rs 79,200
Total income = Rs 1,60,400
From this, we subtract his EPF contribution and life insurance premium:
Taxable income = Total income - EPF contribution - Life insurance premium
Taxable income = Rs 1,60,400 - 10% of Rs 79,200 - Rs 50,000
Taxable income = Rs 1,60,400 - Rs 7,920 - Rs 50,000
Taxable income = Rs 1,02,480
So his taxable income is Rs 1,02,480.
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