A number from 1-54 is chosen at random. Find each probability as a fraction (in simplest form), decimal, and percent.

P(odd or at most 14)

a: 7/27, 0.2593, 25.93%

b: 17/27, 0.62, 62%

c: 7/27, 0.25, 25%

d: 17/27, 0.6296, 62.96%
-------------------------------------
A card is chosen at random from a standard deck. Find each probability as a fraction in simplest form.

P(king)

a:1/52

b: 1/13

c: 4/52

d: 13/52
-----------------------
A card is chosen at random from a standard deck. Find each probability as a fraction in simplest form.

P(spade or heart)

Group of answer choices

a: 1/26

b: 1/2

c: 26/52

d: 2/52
------------------------
i dont need any explanations, so just the answers please ^^

If a student is equally likely to select pizza, nachos, or chicken for lunch, the probability that the student does not select chicken is 2/3.

Since the student has three equally likely options to choose from, the probability of selecting each option is 1/3. Therefore, the probability of the student selecting chicken is 1/3.

To calculate the probability that the student does not select chicken, we need to subtract the probability of selecting chicken from 1. This is because the sum of all possible outcomes must equal 1, and if we know the probability of one outcome, we can find the probability of its complement (not selecting chicken) by subtracting it from 1.

Therefore, the probability that the student does not select chicken is:

1 - 1/3 = 2/3

So, the correct answer is 2/3.

In summary, when the probability of each possible outcome is equal, we can find the probability of an event (in this case, not selecting chicken) by subtracting the probability of that outcome from 1.

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Answer:

The answer is C.

Step-by-step explanation:

The midway point is (19/2, -6)

19/2 = 9.5

Hope this helps!

ps. you can type this in a calculator and ask it to find the midpoint, if that helps :)

The given geometric progression's fourth, seventh, and ninth term is 24, 192, and 768 respectively.

In mathematics, a sequence known as a geometric progression (GP) is one in which each following term is generated by multiplying each preceding term by a constant integer, known as a common ratio. This progression is sometimes referred to as a pattern-following geometric sequence of numbers. Learn development in mathematics here as well. Here, each phrase is multiplied by the common ratio to generate the subsequent term, which is a non-zero value.

The given geometric progression: 3,6,12,24.......

The first term of GP is a = 3 and the Common ratio r = 2,

We know that the nth term of GP is,

[tex]T_n=ar^{n-1}[/tex]

We have to Find the 4, 7, and 9th terms of  above mentioned GP,

[tex]T_4\\\\T_4=ar^{4-1}\\\\T_4=3*2^3\\\\T_4=24\\\\\\T_7\\\\T_7=3*2^{7-1}\\\\T_7=192\\\\\\T_9\\\\T_9=3*2^{9-1}\\\\T_9=768[/tex]

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Another representation of the given algebraic expression 52s + 30b as required is; 2 (26s + 15b).

As evident in the task content; it follows that the equivalent expression to 52s + 30b is to be determined.

Given; 52s + 30b

Since 2 is a common factor to both terms of the equation; we have that;

2 (26s + 15b)

Ultimately, the equivalent expression is; 2 (26s + 15b).

Complete question:

What is 52s+30b in a factorized form.

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We can write the following system of equations:

x + y = 28 (equation 1)

50x + 80y = 1850 (equation 2)

An equation is a statement that two expressions are equal. It consists of two expressions, separated by an equals sign. The expression on the left-hand side of the equals sign is equal to the expression on the right-hand side. Equations can be used to represent relationships between variables, to solve problems, and to describe mathematical models. They are an important tool in algebra, calculus, and other areas of mathematics.

A linear equation is an algebraic equation of the first degree, which means that the variables in the equation have an exponent of one and are not multiplied or divided by any other variables. Linear equations can have one or more variables and are typically represented graphically by a straight line.

In the given question,

Let x be the number of small boxes shipped and y be the number of large boxes shipped.

From the problem, we know that each small box of paper weighs 50 pounds and each large box of paper weighs 80 pounds. We also know that a total of 28 boxes of paper were shipped weighing 1850 pounds altogether.

Using this information, we can write the following system of equations:

x + y = 28 (equation 1)

50x + 80y = 1850 (equation 2)

In equation 1, x + y = 28 represents the total number of boxes shipped, which is the sum of the number of small boxes shipped (x) and the number of large boxes shipped (y).

In equation 2, 50x + 80y = 1850 represents the total weight of the boxes shipped, which is the sum of the weight of each small box (50x) and the weight of each large box (80y).

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Answer:yo

Step-by-step explanation:

15 miles in 6 hours was covered, hence with the average speed formula we find that the average speed is 2.5 miles per hour.

To find the average speed, we can use the formula:

Average Speed = Total Distance / Total Time

In this case, the total distance covered is 15 miles, and the total time taken is 6 hours. Substituting these values into the formula, we get:

Average Speed = 15 miles / 6 hours

Average Speed = 2.5 miles per hour

Therefore, the average speed for the 15 miles covered in 6 hours is 2.5 miles per hour. Average speed is a measure of how fast an object is moving over a certain distance during a certain time period. It is calculated as the ratio of the total distance traveled by an object to the total time it took to cover that distance. The unit of measurement for average speed is distance divided by time, such as miles per hour (mph), kilometers per hour (km/h), or meters per second (m/s). Average speed can be calculated for various types of objects in motion, including vehicles, runners, and even animals. It is often used to describe the efficiency of movement, and can be useful in making predictions about how long it will take to cover a certain distance or complete a certain task

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We estimated the margin of error for the population mean with 95% confidence using the given data and formula. The margin of error is approximately 3.32, providing a reasonable range for the true population mean.

To find the margin of error for estimating the population mean with 95% confidence, we can use the formula:

Margin of error = t(α/2) × (s/√n)

where t(α/2) is the critical value of the t-distribution with (n - 1) degrees of freedom and a significance level of α/2 (0.025 for 95% confidence), s is the sample standard deviation, and n is the sample size.

From the given data, we can calculate the sample mean and standard deviation as:

x = (10 + 8 + 12 + 15 + 13 + 11 + 6 + 5) / 8 = 9.75

s = √[((10-9.75)² + (8-9.75)² + (12-9.75)² + (15-9.75)² + (13-9.75)² + (11-9.75)² + (6-9.75)² + (5-9.75)²) / 7] = 3.36

Using a t-table or calculator, we can find the critical value t(0.025) with 7 degrees of freedom to be approximately 2.365.

Substituting these values into the formula, we get:

Margin of error = 2.365 × (3.36 / √(8)) ≈ 3.32

Therefore, with 95% confidence, the margin of error for estimating the population mean is approximately 3.32.

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The length of the ribbon used by Lilly Mae is 15.7 inches.

A circle is a two-dimensional shape made up of points that are equidistant from a center point.

The perimeter of a circle is also known as its circumference and the formula for perimeter of circle is given by  

                                  C = 2πr

Where 'r' is the radius of the circle.

Here we have

Lilly Mae measures the length of the radius of her bedroom clock

From the figure, the Radius of the clock is 2.5 inch

Here the length of the ribbon around clock will equal the perimeter of the circular clock

From the formula, the Perimeter of the circle = 2πr  

Length of the ribbon used = 2(3.14)(2.5) = 15.7 inches

Therefore,

The length of the ribbon used by Lilly Mae is 15.7 inches.

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Angle 1 = Angle 2 = Angle 3 = Angle 4, we can infer.

A right angle is an angle with a measure that is exactly equal to 90 degrees (or /2).

We can observe that a transversal divides two parallel lines in the given diagram. Let's give the angles names:

The angle created by the junction of line m and line p is Angle 1, which is located in the top left corner.

The angle created by the intersection of line n and line p is Angle 2, which is located in the top right corner.

Line n and line q intersect at angle 3, which is located in the lower right corner.

The angle created by the junction of line m and line q is angle 4, which is located in the lower left corner.

As they are on the same side of the transversal and in corresponding locations with respect to the parallel lines, we can see that angles 1 and 2 are corresponding angles. When lines are cut by a transversal, corresponding angles are congruent, hence we know that:

1 + 2 = an angle.

Due to their placement on opposing sides of the transversal and in-between the parallel lines, angles 2 and 3 can also be seen to represent alternate internal angles. When lines are cut by a transversal, alternate inner angles are congruent, hence we know that:

2 + 3 = a right angle

As they are on the same side of the transversal and in corresponding locations with respect to the parallel lines, we can see that angles 3 and 4 are corresponding angles. When lines are cut by a transversal, corresponding angles are congruent, hence we know that:

Angle 3 = angle 4 =...

So, Angle 1 = Angle 2 = Angle 3 = Angle 4 follows.

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Answer:

21,265.65 in³

Step-by-step explanation:

Formula: Volume (V) = πr²h

V = πr²h

Slotting in values,

V = 3.14 x 1.5² x 3.01

V = 3.14 x 2.25 x 3.01

V = 21.26565 in³

But that is for 1 cupcake

To fill a thousand of them, you will need to multiply your answer by the number of cupcakes in total.

Therefore,

Answer = V x 1000

Answer = 21.26565 x 1000

Answer = 21,265.65 in³

The answer is already rounded to the nearest hundredth,

So they would need 21,265.65 cubic inches (in³) to fill 1000 cupcakes.

Please mark brainliest.

Thanks.

To answer the question, let's start by preparing a scatterplot graphing the volume of oil changes (x-axis) against the company's monthly operating expenses (y-axis). From the data provided, the scatterplot appears as follows:

It appears that there is a fairly strong correlation between the number of oil changes performed and the company's operating expenses each month. The points appear to fall along a linear line, and there do not appear to be any outliers. This indicates that the company's operating costs are fixed.

Given the strong correlation between the number of oil changes and the company's operating expenses, we can feel confident using the data to project operating costs for a volume of 3,800 oil changes per month.

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Answer:

$1920

Step-by-step explanation:

If Tammy's water bill is deducting $160 every other month for 2 years, (or $160 a month for 1 year since we skip a month every two months), then you would multiply $160 by 12 months to get $1920.

Thus, if the weather is warm, you can make a total of 20 combinations (5 shirts x 4 pants), and if the weather is cold, you can make a total of 40 combinations (5 shirts x 4 pants x 2 sweaters).

The total combination of clothing items you can wear is 5 shirts, 4 pants, and either 2 neckties or 2 sweaters. If the weather is warm, you can opt to omit the sweater and keep the two neckties. If the weather is cold, then you can omit the neckties and wear the two sweaters. To calculate the total number of clothing combinations you can make, you need to multiply the items you are allowed to wear. For example, if you are wearing a shirt and a pair of pants, you can make 5 x 4 = 20 combinations. If you are wearing a shirt, pants, and either a necktie or a sweater, you can make 5 x 4 x 2 = 40 combinations.

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Answer: 42$

Step-by-step explanation:

egiryofhvjg

The solutions can be written as x = 8 + 10i and x = 8 - 10i

The graph of a quadratic equation is a parabola, which is a symmetrical U-shaped curve. The direction and shape of the parabola depend on the value of a. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward.

Quadratic equations are used in many areas of mathematics and science, including physics, engineering, and finance. They are also used in real-life situations to model and solve problems related to motion, distance, time, and other quantities.

To solve this quadratic equation, we can rearrange it to be in standard form:

x² - 16x + 164 = 0

We can then use the quadratic formula to find the solutions:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 1, b = -16, and c = 164. Substituting these values into the formula, we get:

x = (16 ± √((-16)² - 4(1)(164))) / 2(1)

Simplifying, we get:

x = (16 ± √(256 - 656)) / 2

x = (16 ± √(-400)) / 2

We have a negative number inside the square root, which means that the solutions are complex. To express them in the form x = a + bi and x = a - bi, we can simplify the square root by factoring out -1:

x = (16 ± √(400) i) / 2

x = (16 ± 20i) / 2

x = 8 ± 10i

Therefore, the solutions to the quadratic equation x² - 16x + 164 = 0 are:

x = 8 + 10i and x = 8 - 10i

So, a = 8 and b = 10. Therefore, the solutions can be written as:

x = 8 + 10i and x = 8 - 10i

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As a result, the answer to the following question, We may use the SAS criteria to argue that the two triangles are congruent if we know that KM is congruent to SU.

A triangle is a polygon because it contains four or more parts. It features a simple rectangular shape. A triangle ABC is a rectangle with the edges A, B, and C. When the sides are not collinear, Euclidean geometry produces a single plane and cube. If a triangle contains three components and three angles, it is a polygon. The corners are the points where the three edges of a triangle meet. The sides of a triangle sum up to 180 degrees.  

Option b: KM is congruent to SU is the additional information required by SAS to demonstrate Triangle KLM is congruent to Triangle UTS.

To demonstrate that two triangles are congruent using the SAS (Side-Angle-Side) congruence criterion, we must demonstrate:

Two comparable side pairs are congruent.

The angle included (the angle formed by these two sets of sides) is congruent.

In this situation, we are told that LM is congruent to ST, but we must still demonstrate that another pair of sides is congruent. We may use the SAS criteria to argue that the two triangles are congruent if we know that KM is congruent to SU.

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16 weeks is 277

3800 insects in about 27.4 weeks

The answer to the question is given below.An entomologist has an insect colony for an experiment. The population of insects is increasing at a continuous rate of 5.6 percent per week. The initial population of the insect colony was 100.The formula that represents the insect colony population after tt weeks is P(t) = 100 (1 + 0.056)^tUsing the formula from section (a), we can calculate that the insect population after 16 weeks is 277. The nearest whole number is 277.After how many weeks would the population of insects reach 3800?Let’s apply the formula P(t) = 100 (1 + 0.056)^t3800 = 100 (1 + 0.056)^t(1 + 0.056)^t = 38(1.056)^t = 38log 1.056^t = log 38t log 1.056 = log 38t = log 38 / log 1.056t ≈ 27.4 weeksThe insect population will reach 3800 insects in about 27.4 weeks. We rounded the answer to one decimal place.

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Therefore , the solution of the given problem of expressions comes out to be  students can develop their factoring abilities and avoid mistakes with practise and instruction.

Estimates that combine joining, disabling, and rather than randomly divide should be produced when variables are shifting. If they got together, they could solve a mental puzzle, provide some data, and instead software. A declaration of truth contains formulas, components, and mathematical processes like combination, subtraction, omission, and grouping. Both phrases and words can be assessed and analysed.

Here,

are some pointers to assist students B and C prevent incorrect factoring in the future:

Regularly practise factoring: Since factoring is a talent that must be honed, students should make sure to factor problems frequently. They will be better able to understand the procedure and spot trends in the expressions they are factoring as a result of this.

Verify the indications again because they are prone to error when factoring. Students should double-check their distribution of the signs and make sure no negative indicators are being left out.

Students should double-check their work by multiplying the factors back together after factoring an equation. They'll be able to correct any errors they may have made thanks to this.

If students need assistance with factoring, they should ask an instructor or tutor for assistance. Although factoring can be a challenging ability to master, students can develop their factoring abilities and avoid mistakes with practise and instruction.

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Recursive fοrmula οf a1 = 3, d = -8 is -29 = -21 + (-8)

Any term οf a series can be defined by its preceding term in a recursive fοrmula (s).

The recursive fοrmula fοr an arithmetic sequence with first term a1=3 and cοmmοn difference d=-8 is:

an = an-1 + d, where n >= 2

Expanding the fοrmula fοr the first few terms, we get:

a2 = a1 + d = 3 - 8 = -5

a3 = a2 + d = -5 - 8 = -13

a4 = a3 + d = -13 - 8 = -21

a5 = a4 + d = -21 - 8 = -29

Nοw,

Recursive fοrmula οf a1 = 3, d = -8 is -29 = -21 + (-8)

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7. The speed of the car is approximately 73.3 miles per hour. 10.  a. 7.07 ft/sec b. 9.87 feet from the center. c. the person on the outside is approximately 30 ft.

It is a fundamental concept in physics and is used to describe the motion of objects in both everyday life and scientific contexts. In addition to speed, other related concepts include velocity (which includes direction) and acceleration (the rate at which an object's speed or velocity changes).

7. The circumference of the wheel is given by C = πd, where d is the diameter of the wheel. Therefore, the circumference of each wheel is:

C = πd = π(22 in) = 22π in

Since the wheels are rotating at 600 rpm (revolutions per minute), the distance covered by each wheel in one minute is:

distance = circumference × revolutions = 22π in/rev × 600 rev/min = 13,200π in/min

To convert this to miles per hour, we need to convert inches to miles and minutes to hours. There are 12 inches in 1 foot and 5280 feet in 1 mile, so there are 63360 inches in 1 mile. There are 60 minutes in 1 hour, so:

distance = 13,200π in/min = (13,200π in/min) × (1 ft/12 in) × (1 mi/5280 ft) × (60 min/1 hr) ≈ 73.3 mi/hr

Therefore, the speed of the car is approximately 73.3 miles per hour.

10. a) The linear speed of a point on the carousel is given by v = rω, where r is the distance from the center of the carousel and ω is the angular velocity in radians per second. Since the carousel makes 3 revolutions per minute, the angular velocity is:

ω = (3 rev/min) × (2π rad/rev) × (1 min/60 sec) = π/10 rad/sec

The linear speed of a person riding a horse that is 22.5 ft from the center is:

v = rω = (22.5 ft) × (π/10 rad/sec) ≈ 7.07 ft/sec

b) Let d be the distance of the person on the inside from the center. Since the carousel makes 3 revolutions per minute, the angular velocity is still ω = π/10 rad/sec. The linear speed of the person on the inside is given as 3.1 ft/sec, so:

v = rω

3.1 ft/sec = d × (π/10 rad/sec)

d = (3.1 ft/sec) × (10/π rad/sec) ≈ 9.87 ft

Therefore, the person on the inside is approximately 9.87 feet from the center.

c) The linear speed of a point on the carousel is proportional to its distance from the center. Therefore, the ratio of the linear speeds of the person on the outside and the inside is equal to the ratio of their distances from the center:

v_outside/v_inside = r_outside/r_inside

We know that the linear speed of the person on the inside is 3.1 ft/sec and the distance of the person on the inside is approximately 9.87 feet. Therefore:

v_outside/3.1 ft/sec = r_outside/9.87 ft

v_outside = (r_outside/9.87 ft) × (3.1 ft/sec)

We can solve for r_outside by using the fact that the carousel makes 3 revolutions per minute:

v_outside = r_outsideω

v_outside = r_outside × (3 rev/min) × (2π rad/rev) × (1 min/60 sec)

v_outside = r_outsideπ/10 rad/sec

Combining these equations, we get:

r_outsideπ/10 rad/sec = (r_outside/9.87 ft) × (3.1 ft/sec)

r_outside ≈ 30.9 ft

Therefore, the person on the outside is approximately 30.

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The event C is: C = { HHT, HTH, THH, HTT, THT, TTH }
P(C) = 1/2

(a) P(notB) = 1/2
(b) P((notA) or C) = 5/8
(c) P(A or B or C) = 7/8

Solution:
A fair coin is tossed three times, and the events A, B, and C are defined as follows: A: { At least one head is observed } B: { At least two heads are observed } C: { The number of heads observed is odd }

The sample space of the given event is as follows:
S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
Probability of the occurrence of A:
If at least one head is observed, then it can be possible in the following ways:
(i) HHT, HTH, THH, or
(ii) HHT, HTH, THH, or HTT, THT, TTH, or
(iii) HHT, HTH, THH, or HTT, THT, TTH, or HHH
Therefore, the event A is: A = { HHT, HTH, THH, HTT, THT, TTH, HHH }
P(A) = 7/8
Probability of the occurrence of B:
If at least two heads are observed, then it can be possible in the following ways:
(i) HHT, HTH, THH, or
(ii) HTT, THT, TTH, or
(iii) HHH
Therefore, the event B is: B = { HHT, HTH, THH, HTT, THT, TTH, HHH }
P(B) = 7/8
Probability of the occurrence of C:
If the number of heads observed is odd, then it can be possible in the following ways:
(i) HHT, HTH, THH, or
(ii) HTT, THT, TTH
Therefore, the event C is: C = { HHT, HTH, THH, HTT, THT, TTH }
P(C) = 1/2

Probability of the occurrence of notB:
Not B means "at least one head" or "no head." So, the event notB is not the event B, which means the event (notB) is the event in which we do not get at least two heads.
The (notB) event is: (notB) = { TTH, THT, HTT, TTT }
P(notB) = 4/8 = 1/2

Probability of the occurrence of (notA) or C:
The event (notA) means that we don't get at least one head, which means we get all tails. The (notA) event is: (notA) = { TTT }
P(notA) = 1/8
Now, we will find P((notA) or C).
For this, we can take the union of (notA) and C.
(notA) ∪ C = { HHT, HTH, THH, HTT, THT, TTH, TTT }
P((notA) or C) = 7/8

Probability of the occurrence of A or B or C:
The probability of the occurrence of A or B or C can be found as follows:
P(A or B or C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)
Here, A ∩ B = { HHT, HTH, THH, HHH }, A ∩ C = { HHT, HTH, THH }, B ∩ C = { HHT, HTH, THH }, and A ∩ B ∩ C = { HHT, HTH, THH }.
Substituting these values, we get:
P(A or B or C) = 7/8 + 7/8 + 1/2 - 3/8 - 3/8 - 1/8 + 1/8
P(A or B or C) = 7/8

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Let's label the regions in the Venn diagram as follows:

F: students who do only field events

T: students who do only track events

S: students who do only swimming events

FT: students who do both field and track events

ST: students who do both swimming and track events

FST: students who do all three events

From the information given in the problem, we can fill in some of the values in the Venn diagram:

F + FT + 15 = 33 (33 students do field events, and 15 do field events only)

T + FT + 14 = 40 (40 students do track events, and 14 do both field and track events)

S + ST + 10 = 23 (23 students do swimming, and 10 do both swimming and track events)

F + T + S + 2FT + ST + FST = 68 (there are 68 students in total)

We can simplify these equations to:

F + FT = 18

T + FT = 26

S + ST = 13

F + T + S + 2FT + ST + FST = 68

To solve for the remaining unknowns, we need to use some algebra. We can start by solving for FT:

F + FT = 18

T + FT = 26

Adding the two equations, we get:

F + T + 2FT = 44

Rearranging, we get:

FT = (44 - F - T) / 2

Now we can substitute this expression for FT into the equation for the total number of students:

F + T + S + 2FT + ST + FST = 68

F + T + S + 2((44 - F - T) / 2) + ST + FST = 68

F + T + S + 44 - F - T + ST + FST = 68

Simplifying, we get:

S + ST + FST = 26

Now we have two equations involving S, ST, and FST:

S + ST = 13

S + ST + FST = 26

We can solve for S and ST by subtracting the first equation from the second:

S + ST + FST = 26

(S + ST) + FST = 26

Substituting S + ST = 13:

13 + FST = 26

FST = 13

So there are 13 students who do all three events. Now we can use the equation S + ST = 13 to solve for S:

S + ST = 13

ST = 10 (given in the problem)

S = 13 - 10 = 3

So there are 3 students who do swimming only. Similarly, we can use the equation T + FT = 26 to solve for T:

T + FT = 26

FT = (44 - F - T) / 2

Substituting F + FT = 18:

T + (18 - F) / 2 = 26

Multiplying both sides by 2:

2T + 18 - F = 52

In the number 7,326.64, the digit 2 appears in the thousands place, and its value is 200. Therefore, the answer is (Choice C) 7,326.64.

A number is a mathematical concept used to represent a quantity or value. Numbers can be used to count, measure, or calculate, and they are fundamental to many fields of mathematics, science, and everyday life. Numbers can be represented and manipulated using mathematical symbols, such as digits, operations, and functions. They are used in many applications, such as in finance, engineering, physics, and computer science.

Here,

To solve this problem, we need to find a number that has a digit 2 with a value that is 1/10 times as great as the digit 2 in the number 6,257.11.

In the number 6,257.11, the digit 2 appears in the thousands place, and its value is 2,000.

To find a number in which the digit 2 has a value that is 1/10 times as great, we need to divide 2,000 by 10, which gives us 200.

So, we need to look for a number in which the digit 2 appears in the thousands place and has a value of 200. The only option that satisfies this condition is (Choice C) 7,326.64.

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Answer:72

Step-by-step explanation:

71+1=72

The solution for the given equation isθ = π/2, 5π/6, 13π/6.

The given equation is cos(θ) - sin(θ) = √2sin(θ/2)cos(θ/2).Use double-angle formula for sin to write sin(θ) in terms of cosθ as follows:sin(θ) = 2sin(θ/2)cos(θ/2).Substitute sin(θ) with the above equation. We have,cos(θ) - 2sin(θ/2)cos(θ/2) = √2sin(θ/2)cos(θ/2)cos(θ) - 3sin(θ/2)cos(θ/2) = 0(cosθ - 3sin(θ/2) = 0 or cosθ/3 = sin(θ/2)Squaring both sides of the equation gives us:cos2θ/9 = 1 - cos2(θ/2)cos2(θ/2) = (1 - cos2θ/9) / 2cos2(θ/2) = (9 - cos2θ) / 18sin2(θ/2) = 1 - cos2(θ/2)sin2(θ/2) = 1 - ((9 - cos2θ) / 18)sin2(θ/2) = (9 + cos2θ) / 18Therefore, the solution for the given equation isθ = π/2, 5π/6, 13π/6.

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Option D is the graph that represent the variation of the gravitational energy (E) with the distance (r) from the centre of a spherical shell of mass M radius R

When R is less than r, the graph we would have  is a straight line. When R = r, it has constant value and finally if R is more than r it is a hyperbola.

The intensity that is inside the spherical shell would be given as 0. Then we would have this represented as

I = 0

upto r = R

I ∝ I/r²

r > R

Hence D

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