a particle in the infinite square well has the initial wave function Ψ (x,0) = {Ax, 0 < x < a/2
{A(a-x), a/2 < x < a
(a) Sketch Ψ(x, 0), and determine the constant A. (b) Find Ψ (x, t). (c) What is the probability that a measurement of the energy would yield the value E1? (d) Find the expectation value of the energy, using Equation 2.21.2
Answers
\((a)A =\sqrt{\frac{12}{a^3}}}\) and i cannot provide the sketch of \(\psi(x,t)\).
(b)\(\psi(x, t) = \psi(x, 0) * e^{\frac{-iEt}{\hbar}}\)
(c)The probability is given by the square of the coefficient corresponding to the energy eigenstate \(E_{1}\).
(d)\(< E > = \int\limits\psi'(x, t)}{\hat{H}}\psi(x,t)dx\)
What is the wave function?
The wave function, denoted as \(\psi(x, t)\), describes the state of a quantum system as a function of position (x) and time (t). It provides information about the probability amplitude of finding a particle at a particular position and time.
(a) To sketch \(\psi(x, 0)\) and determine the constant A, we need to plot the wave function\(\psi(x, 0)\) for the given conditions.
The wave function Ψ(x, 0) is given as:
\(\psi(x, 0)\) = {Ax, 0 < x < \(\frac{a}{2}\)
{A(a-x), \(\frac{a}{2}\) < x < a
Since we have a particle in the infinite square well, the wave function must be normalized. To determine the constant A, we normalize the wave function by integrating its absolute value squared over the entire range of x and setting it equal to 1.
Normalization condition:
\(\int\limits|\psi(x, 0)|^2 dx = 1\)
For 0 < x <\(\frac{a}{2}\):
\(\int\limits |Ax|^2dx = |A|^2 \int\limits^\frac{a}{2}_0 x^2 dx \\ = |A|^2 *\frac{1}{3} * (\frac{a}{2})^3 \\= |A|^2 * \frac{a^3}{24}\)
For \(\frac{a}{2}\) < x < a:
\(\int\limits |A(a-x)|^2 dx = |A|^2 \int\limits^a_\frac{a}{2} (a-x)^2 dx\\ = |A|^2 * \frac{1}{3} * (\frac{a}{2})^3 \\= |A|^2 * \frac{a^3}{24}\)
Now, to normalize the wave function:\(|A|^2 * \frac{a^3}{24}+ |A|^2 * \frac{a^3}{24} = 1\)
Since the integral of \(|\psi(x, 0)|^2\) over the entire range should be equal to 1, we can equate the above expression to 1:
\(2|A|^2 * \frac{a^3}{24} = 1\)
Simplifying, we have:
\(|A|^2 * \frac{a^3}{12} = 1\)
Therefore, the constant A can be determined as:
\(A =\sqrt{\frac{12}{a^3}}}\)
(b) To find \(\psi(x, t)\), we need to apply the time evolution of the wave function. In the infinite square well, the time evolution of the wave function can be described by the time-dependent Schrödinger equation:
\(\psi(x, t) = \psi(x, 0) * e^{\frac{-iEt}{\hbar}}\)
Here, E is the energy eigenvalue, and ħ is the reduced Planck's constant.
(c) To find the probability that a measurement of the energy would yield the value \(E_{1}\), we need to find the expansion coefficients of the initial wave function \(\psi(x, 0)\) in terms of the energy eigenstates. The probability is then given by the square of the coefficient corresponding to the energy eigenstate \(E_{1}\).
(d) The expectation value of the energy can be found using Equation 2.21.2:
\(< E > = \int\limits\psi'(x, t)}{\hat{H}}\psi(x,t)dx\)
Here, \(\psi'(x,t)\) represents the complex conjugate of Ψ(x, t), and Ĥ is the Hamiltonian operator.
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Related Questions
a forester measured 40 of the trees in a large woods that is up for sale. he found that their mean diameter was 187 inches and their standard deviation 20.4 inches. suppose that these trees provide an accurate description of the whole forest and that the diameter of the tree follows a normal distribution. answer the following: a) what percentage of trees would be below 191 inches in diameter? b) what percentage of trees would be between 176 and 196 inches? c) what size diameter represents the bottom 35% of the trees? d) what size diameter represents the top 35% of the trees?
Answers
Approximately 57.83% of trees would be below 191 inches in diameter.
Approximately 37.48% of trees would be between 176 and 196 inches in diameter.
A diameter of approximately 194.83 inches represents the top 35% of the trees.
To find the percentage of trees below 191 inches in diameter, we need to calculate the cumulative probability below that value.
Using the mean and standard deviation given:
Mean (μ) = 187 inches
Standard Deviation (σ) = 20.4 inches
We can standardize the value 191 inches using the z-score formula:
z = (x - μ) / σ
z = (191 - 187) / 20.4 ≈ 0.1961
Next, we find the cumulative probability using the z-score:
P(Z < 0.1961)
Using a standard normal distribution table or calculator, we find that P(Z < 0.1961) ≈ 0.5783.
b) To find the percentage of trees between 176 and 196 inches in diameter, we need to find the cumulative probability between these values.
First, we calculate the z-scores for both values:
z1 = (176 - 187) / 20.4 ≈ -0.5392
z2 = (196 - 187) / 20.4 ≈ 0.4412
Next, we find the cumulative probabilities for each z-score:
P(Z < -0.5392) ≈ 0.2946
P(Z < 0.4412) ≈ 0.6694
To find the probability between these two values, we subtract the smaller probability from the larger probability:
P(-0.5392 < Z < 0.4412) = 0.6694 - 0.2946 ≈ 0.3748
c) To find the diameter that represents the bottom 35% of the trees, we need to find the z-score corresponding to that percentile.
Using a standard normal distribution table or calculator, we find the z-score that corresponds to the bottom 35% is approximately -0.3853.
Next, we can solve for the diameter using the z-score formula:
-0.3853 = (x - 187) / 20.4
Solving for x, we get:
x ≈ 179.17 inches
So, a diameter of approximately 179.17 inches represents the bottom 35% of the trees.
d) To find the diameter that represents the top 35% of the trees, we can use the same approach as in part c) but with the z-score corresponding to the top 35% (which is the same as the bottom 65%).
Using a standard normal distribution table or calculator, we find the z-score that corresponds to the top 35% (bottom 65%) is approximately 0.3853.
Using the z-score formula:
0.3853 = (x - 187) / 20.4
Solving for x, we get:
x ≈ 194.83 inches
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11 Finding a difference quotient for a linear or quadratic function V Find the difference quotient f(x)=-3x²-2x+5 Simplify your answer as much as possible. f(x +h)-f(x) h f(x+h)-f(x) h = ( where h#0,
Answers
The difference quotient for the given function is 9 -2/h.
The difference quotient for the given function can be calculated as:
[f(x+h) - f(x)]/h
= [(3(x+h)² - 2(x+h) + 5) - (3x² - 2x + 5)]/h
= (3x² + 6xh + 3h² - 2x - 2h + 5 - 3x² + 2x - 5)/h
= (6xh + 3h² - 2h)/h
= (6x + 3h -2)/h
Simplifying the expression further, we get:
(6x + 3h -2)/h = 6 + 3h/h -2/h
= 6 + 3 -2/h
= 9-2/h
Therefore, the difference quotient for the given function is 9 -2/h.
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"Your question is incomplete, probably the complete question/missing part is:"
Find the difference quotient [f(x+h)-f(x)]/h, where h≠0, for the function below.
f(x)=3x² -2x+5. Simplify your answer as much as possible.
Question 1 P(x) is a polynomial and r is a number. Which of the following is NOT equivalent to the others? a.(x-r) is a factor of P(x) b.r is a zero of P(x) c.P(0) = r
d. P(r) = 0
Answers
The statement that is NOT equivalent to the others is option c. "P(0) = r."
In the context of polynomials, a factor of a polynomial is a term or expression that divides evenly into the polynomial. So, if (x - r) is a factor of P(x), it means that when P(x) is divided by (x - r), the remainder is zero. This is equivalent to saying that r is a zero or root of the polynomial P(x). In other words, if r is a zero of P(x), then P(r) = 0.
Option c, on the other hand, states that P(0) = r. This means that when x is equal to zero, the value of the polynomial P(x) is equal to r. This statement does not provide any information about whether (x - r) is a factor of P(x) or if r is a zero of P(x). It simply relates the value of the polynomial at x = 0 to the constant value r.
To summarize, options a, b, and d are equivalent because they all refer to the fact that r is a zero of the polynomial P(x), while option c does not provide the same information and is therefore not equivalent to the others.
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13. The emf result at the junction of a thermocouple is given by the equation e=0.4T−e T−100. The thermocouple is then calibrated using a standard thermometer. When the standard thermometer reads 50∘C, what is the reading of the thermocouple?
O a. 50.09
O b. 50.11
O c. 50.13
O d. 50.15
Answers
The standard thermometer reads 50°C, the reading of the thermocouple is 0.3922.
To find the reading of the thermocouple when the standard thermometer reads 50°C substitute T = 50 into the equation e = 0.4T - e(T - 100). Let's calculate it:
e = 0.4(50) - e(50 - 100)
e = 20 - e(-50)
e = 20 + 50e
to solve this equation for e. Let's rearrange it:
50e + e = 20
51e = 20
e = 20/51 ≈ 0.3922
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28. Given M₁ = 35, M₂ = 45, and SM1-M2= 6.00, what is the value of t? -2.92 -1.67 O-3.81 2.75
Answers
The t-distribution value is -1.67 for the given mean samples of 35 and 45. Thus, option B is correct.
M₁ = 35
M₂ = 45
SM1-M2 = 6.00
The t-value or t-distribution formula is calculated from the sample mean which consists of real numbers. To calculate the t-value, the formula we need to use here is:
t = (M₁ - M₂) / SM1-M2
Substituting the given values into the formula:
t = (35 - 45) / 6.00
t = -10 / 6.00
t = -1.67
Therefore, we can conclude that the value of t is -1.67 for the samples given.
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The t-distribution value is -1.67 for the given mean samples of 35 and 45. Thus, option B is correct.
Given, M₁ = 35
M₂ = 45
SM1-M2 = 6.00
The t-value or t-distribution formula is calculated from the sample mean which consists of real numbers.
To calculate the t-value,
the formula we need to use here is:
t = (M₁ - M₂) / SM1-M2
Substituting the given values into the formula:
t = (35 - 45) / 6.00
t = -10 / 6.00
t = -1.67
Therefore, we can conclude that the value of t is -1.67 for the samples given.
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Please help me solve this problem... I will mark you brainliest
Answers
Answer:
(8,1)
Step-by-step explanation:
PLEASE HELP DUE IN 10 MINS!!
Which of the following statements describes the solution set of -5x < 10?
All real numbers that are negative two or less.
All real numbers that are greater than negative two.
All real numbers that are negative two or greater.
All real numbers that are less than negative two.
Answers
Answer:
Its B
Step-by-step explanation:
The correct statement is "All real numbers that are greater than negative two".
What is inequality?A relationship between two expressions or values that are not equal to each other is called inequality.
Given an inequality, -5x<10
On solving, we get,
x > -2
Which is saying that, solution is all real number that is greater than -2.
Hence, All real numbers that are greater than negative two.
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This list shows the numbers of times a gym member visited their gym in a month.
8, 10, 2, 5, 1, 22, 16, 15, 20
What is the range of the number of gym visits?
A. 12
B. 20
C. 21
D. 22
Answers
A w=-8
B w=-5
C w=5
D w=8
Answers
Answer:
D
Step-by-step explanation:
Plug in every scenario, in see if it equal to each other.
D is the only one that is equal.
if your height is 5 feet 3 inches and your weight is 120 pounds, what is your bmi? round the answer to the nearest whole number.
Answers
If your height is 5 feet 3 inches and your weight is 120 pounds, then the BMI is 21.23
To calculate your Body Mass Index (BMI), use the following formula:
BMI = weight in kilograms / (height in meters)²
First, convert your height to meters :
5 feet 3 inches = 63 inches
63 inches = 160.02 cm
160.02 cm = 1.6002 meters
Next, convert your weight to kilograms :
120 pounds = 54.43 kg
Now, plug in these values to calculate your BMI:
BMI = 54.43 kg / (1.6002 meters)²
Divide the terms
BMI = 21.23
Therefore, your BMI is approximately 21.23
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Write 6-12 as an addition expression.
Answers
Answer:
-12 + 6
Step-by-step explanation:
Answer:
-12+6
Step-by-step explanation:
The two terms are
-12
and
+6
Put them together in the reverse order and you have -12+6
find the steady state solution of the heat conduction equation
Answers
The steady-state solution of the heat conduction equation refers to the temperature distribution that remains constant over time. This occurs when the heat flow into a system is balanced by the heat flow out of the system.
To find the steady-state solution of the heat conduction equation, follow these steps:
1. Set up the heat conduction equation: The heat conduction equation describes how heat flows through a medium and is typically given by the formula:
q = -k * A * dT/dx,
where q represents the heat flow, k is the thermal conductivity of the material, A is the cross-sectional area through which heat flows, and dT/dx is the temperature gradient in the direction of heat flow.
2. Assume steady-state conditions: In the steady-state, the temperature does not change with time, which means dT/dt = 0.
3. Simplify the heat conduction equation: Since dT/dt = 0, the equation becomes:
q = -k * A * dT/dx = 0.
4. Apply boundary conditions: Boundary conditions specify the temperature at certain points or surfaces. These conditions are essential to solve the equation. For example, you might be given the temperature at two ends of a rod or the temperature at the surface of an object.
5. Solve for the steady-state temperature distribution: Depending on the specific problem, you may need to solve the heat conduction equation analytically or numerically. Analytical solutions involve techniques like separation of variables or Fourier series expansion. Numerical methods, such as finite difference or finite element methods, can be used to approximate the solution.
It's important to note that the exact method for solving the heat conduction equation depends on the specific problem and the boundary conditions given. However, the general approach is to set up the heat conduction equation, assume steady-state conditions, simplify the equation, apply the boundary conditions, and solve for the steady-state temperature distribution.
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A rectangle is drawn so that the width is 3 feet shorter than the length. The area of the rectangle is 10 square feet. Find the length of the rectangle.
Answers
width= 2ft
10ft=l x w
w= l -3
10ft= l(l-3)
0= l^2 -3l-10
0= (l-5)(l+2)
l=5, l≠-2 (because a measurement can’t be negative)
w= 5-3
w=2
HELP ME GUYSSSSSS PLZZZZ I NEED HELP RIGHT NOW ANWSERS ANYONE??????
Answers
Answer:
45 x (1/3) = 15
Step-by-step explanation:
Note that when you divide with a fraction, you will change the division into multiplication, and flip the second term. In this case:
15/(1/3) = 15 * 3 = 45
In the answer's case, when you multiply a fraction, you are really multiplying the term with the numerator, and then dividing the denominator:
(45 x 1)/3 = 45/3 = 15
~
PLEASE HELP ME
I need the answer for CD and EC.
Answers
The length of EC is 8 and the length of CD is 16
How to solve the question?
Pythagoras theorem is a fundamental theorem in mathematics named after the ancient Greek mathematician Pythagoras. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).
The theorem can be expressed mathematically as:
c^2 = a^2 + b^2
where c is the length of the hypotenuse and a and b are the lengths of the other two sides. This means that if we know the lengths of any two sides of a right-angled triangle, we can use Pythagoras theorem to find the length of the third side.
The theorem has many practical applications, including in construction, engineering, and physics. It is also a key concept in trigonometry and is used extensively in various fields of science and mathematics.
We can use the Pythagorean theorem to solve this problem. According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In this case, we know that the length of the hypotenuse (c) is 10 and the length of one leg (a) (the base) is 6. We can use this information to find the length of the other leg (b) (the perpendicular) as follows:
c² = a²+ b²
10²= 6² + b²
100 = 36 + b²
b²= 64
b = 8
Therefore, the length of the perpendicular (EC) is 8 units.
for CD it is given in question that EC=ED there fore
ED=8
CD=EC+ED
CD=8+8
CD=16
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I just this one please someone help
Answers
Answer:
100
Step-by-step explanation:
because 4, 3, and 2 all make 180 and 4 and 2 are both 40 so you add that which makes 80 and then you take 180 and subtract 80 which gives you 100
Answer:
100
Step-by-step explanation:
This is because angle 1 and angle 2 are congruent, so both of them are 40 degrees. To find angle 3, you do 40+40 which is 80. You then subtract 180-80=100. This is because angles 1, 2, and 3 are all on 1 straight line.
in the right triangle ABC below, segment BD is parallel to segment AE, and segment BD is perpendicular to segment EC at D. The length of segment AC is 20 feet, the length of segment BD is 3 feet, and the length of segment CD is 4 feet. What is the length, In feet, of segment AE?
а10
b12
с15
d16
e17
Answers
Answer:
It is a number 10 thank u very much ok ok
What is the x-intercept of 4x+6y= 24
Answers
Answer:
i think its between 2-5
Step-by-step explanation:
Check all equations that are equivalent.
EXPLAIN please
Answers
Step-by-step explanation:
A = ½ h (b₁ + b₂)
Multiply both sides by 2:
2A = h (b₁ + b₂)
Distribute the h:
2A = h b₁ + h b₂
So the second option is not correct.
Instead of distributing, divide by h:
2A/h = b₁ + b₂
Subtract b₂ from both sides:
b₁ = 2A/h − b₂
Third option is correct.
Next, use common denominator to combine.
b₁ = 2A/h − h b₂/h
b₁ = (2A − h b₂) / h
Factor out 2.
b₁ = 2 (A − ½ h b₂) / h
Fourth option is correct.
Use cylindrical coordinates to evaluate ∭E√x2+y2dV, where E is the region that lies inside the cylinder x2+y2=16 and between the planes z=−5 and z=4.
Answers
The value of the triple integral is 72π.
To evaluate this triple integral in cylindrical coordinates, we first need to express the region E using cylindrical coordinates.
The cylinder x² + y² = 16 can be expressed in cylindrical coordinates as r² = 16, or r = 4. The planes z = -5 and z = 4 define a region of height 9.
So, the region E can be expressed in cylindrical coordinates as:
4 ≤ r ≤ 4
-5 ≤ z ≤ 4
0 ≤ θ ≤ 2π
The integrand √(x² + y²) can be expressed in cylindrical coordinates as r, so the integral becomes:
∭E√x²+y²dV = ∫0²π ∫4⁴ ∫-5⁴ r dz dr dθ
Note that the limits of integration for r are from 0 to 4, which means we are only integrating over the positive x-axis. Since the integrand is an even function of x and y, we can multiply the result by 2 to get the total volume.
The integral with respect to z is easy to evaluate:
∫₋₅⁴ r dz = r(4 - (-5))
= 9r
So the triple integral becomes:
∭E√x²+y²dV = 2 ∫0^2π ∫4⁴ 9r dr dθ
= 2(9) ∫0^²π 4 dθ
= 72π
Therefore, the value of the triple integral is 72π.
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Classify/name the shape based on the number of sides and markings.
Hint: Two Words
Answers
Consider function f(x) = 6√x + 10 on interval [2, 8].
Find the mean slope:
Answers
The mean slope of the function f(x) = 6√x + 10 on the interval [2, 8]. Here are the steps:
1. Determine the function values at the endpoints of the interval:
f(2) = 6√2 + 10
f(8) = 6√8 + 10
2. Calculate the difference in function values (Δy) and the difference in input values (Δx):
Δy = f(8) - f(2)
Δx = 8 - 2
3. Compute the mean slope: Mean slope = Δy / Δx
Now, let's perform the calculations:
1. f(2) = 6√2 + 10 ≈ 18.49
f(8) = 6√8 + 10 ≈ 26.97
2. Δy = 26.97 - 18.49 ≈ 8.48
Δx = 8 - 2 = 6
3. Mean slope = 8.48 / 6 ≈ 1.41
So, the mean slope of the function f(x) = 6√x + 10 on the interval [2, 8] is approximately 1.41.
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I need help again with these
Please and thank you
17points
Answers
Is the correct mixed number
Find the value of x.
2х
120
70°
ОА. 35
ОВ. 55
С. 25
OD. 10
Answers
According to the figure,
→ 2x° + 70° = 120°
→ 2x° = 120° - 70°
→ 2x° = 50°
→ x = 50°/2
→ x= 35Option A!I need help with questions in the photo this is from Khan Academy precalculus trigonometry section
Answers
Answer:
Step-by-step explanation:
okay first stop trying to make me do your test, and second get smarter its so easy.
{(x, y)} : x3 }{ ( x, y ) : y-3} CAN ANYONE GRAPH THIS
Answers
The graph of {(x, y)} : x3 } and { ( x, y ) : y-3} is shown in image.
We have to given that;
Expression is,
⇒ {(x, y)} : x3 } and { ( x, y ) : y-3}
Now, We can find that;
Point (1.672, 4.672) is a solution of the give expression.
Hence, We get;
The graph of {(x, y)} : x3 } and { ( x, y ) : y-3} is shown in image.
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The time spent by a teacher on schoolwork during the weekend is normally distributed with a mean of 6 hours and a standard deviation of 1.2 hours.
a) What amount of time do the bottom 20% of teachers work?
b) What amount of time do the top 20% of teachers work?
c) What amount of time do the middle 40% of teachers work?
Answers
Length (cm) of fish in tank:
7, 2, 1, 4, 5, 2, 3
Round to the nearest tenth.
Mean: [?] cm
A certain substance in an experiment was being stored at −1.7°F. It was then placed on a table where the temperature was raised by 3.1F
What was the new temperature of the substance?
the answers are
-4.8F
-1.4F
1.4F
4.8F
Answers
1.4F. When you add 3.1 to -1.7 you get 14F
Answer:
The answer is 1.4.
Step-by-step explanation:
suppose that a single chip is drawn at random from the bag. find the probability that the chip is red and the probability that the chip is blue
Answers
To find the probability that a chip drawn at random from a bag is red or blue, we need to consider the number of red and blue chips in the bag and the total number of chips.
Let's assume that the bag contains a certain number of red and blue chips. To find the probability that the chip drawn is red, we need to determine the number of red chips in the bag and divide it by the total number of chips.
Similarly, to find the probability that the chip drawn is blue, we need to determine the number of blue chips in the bag and divide it by the total number of chips.
The probabilities can be expressed as:
Probability of drawing a red chip = Number of red chips / Total number of chips
Probability of drawing a blue chip = Number of blue chips / Total number of chips
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Find the value of k, if x + 3 is a factor of 3x² + kx + 6. *
Answers
Given:
Polynomial = P(x) = 3x² + kx + 6
Factor of the ablove polynomial = (x+3)
To Find:
Value of K, for which (x+3) become the factor of P(x) = 3x² + kx + 6
Solution :
Now,
x + 3 = 0
⇒x = (-3)
So,
As (x+3) is a factor so x = (-3) is one root of the polynomial.
Therefore,
P(-3) = 0
→ P(-3) = 3(-3)² + k(-3) + 6 = 0
→ 3(9) - 3k + 6 = 0
→ 27 - 3k + 6 = 0
→ 27 + 6 - 3k = 0
→ 33 - 3k = 0
→ - 3k = -33
→ k = -33 ÷ -3
→ k = 11
Hence,For the value of k = 11, (x+3) is a factor of 3x²+ kx + 6V E R I F I C A T I O N :3x²+ kx + 6, by putting the value of k = 11 and taking -3 as root the remainder should be zero
→ 3x²+ 11x + 6
→ 3(-3)² + 11(-3) + 6
→ 3(9) - 33 + 6
→ 27 - 33 + 6
→ 27 + 6 - 33
→ 33 - 33
→ 0
Hence verified !