## Question 77 In A Parliament Election A Political Party Hired A Public Relations Firm To Promote Its Candidates In Three Ways Telephone House Calls And Letters The Cost Per Contact Is Given In Matrix A As

**The number of contacts of each type made in two cities X and Y is given in the matrix B as**

**Telephone House calls Letters**

**Find the total amount spent by the party in the two cities. What should one consider before casting his/her vote partys promotional activity of their social activities?**

**Solution:**

## = Questions Together With Extra Space For Answer = Questions Only With No Extra Space For Answers = Solutions

**Topic 3. Geometry and trigonometry**

Care has been taken for the questions to be in order of difficulty. For each topic there are 3 types of exercises

**A. Practice Questions:**My questions on basic understanding of theory**B. Past Paper Questions :**SECTION A questions**C. Past Paper Questions**

** **

**For an amazing, remarkably elegant questionbank on Math AA, made by Kevin Bertman, Osaka International School, please visit**

## Rational Exponents And Radical Functions Chapter Test

Question 1.Solve the inequality \ â 2 â¤ 13 and the equation 5\ â 2 = 13. Describe the similarities and differences in solving radical equations and radical inequalities.

**Describe the transformation of f represented by g. Then write a rule for g.**

Question 2.

Question 8.\

Question 9.Write two functions whose graphs are translations of the graph of y = \. The first function should have a domain of x â¥ 4. The second function should have a range of y â¥ â2.

Question 10.In bowling, a handicap is a change in score to adjust for differences in the abilities of players. You belong to a bowling league in which your handicap h is determined using the formula h = 0.9, where a is your average score. Find the inverse of the model. Then find the average for a bowler whose handicap is 36.

Question 11.The basal metabolic rate of an animal is a measure of the amount of calories burned at rest for basic functioning. Kleiberâs law states that an animalâs basal metabolic rate R can be modeled by R = 73.3w3/4, where w is the mass of the animal. Find the basal metabolic rates of each animal in the table.

Question 12.Let f = 6×3/5 and g = âx3/5. Find and and state the domain of each. Then evaluate and .

Question 13.Let f = \x3/4 and g = 8x. Find and ) and state the domain of each. Then evaluate and ).

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## Properties Of Rational Exponents And Radicals 52 Exercises

**Vocabulary and Core Concept Check**

Question 1.**WRITING **How do you know when a radical expression is in simplest form?Answer:

Question 2.**WHICH ONE DOESNT BELONG?** Which radical expression does not belong with the other three? Explain your reasoning.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 312, use the properties of rational exponents to simplify the expression.**

Question 3.\^\)Answer:

\^\)Answer:

\Answer:

\Answer:

\^\)Answer:

\^\)Answer:

\)-1Answer:

\)-1/4Answer:

\Answer:

\Answer:

**In Exercises 1320, use the properties of radicals to simplify the expression.**

Question 13.

\Answer:

\Answer:

\Answer:

**In Exercises 2936, write the expression in simplest form.**

Question 29.\Answer:

\Answer:

\Answer:

\Answer:

\Answer:

\Answer:

\Answer:

\Answer:

**In Exercises 3746, simplify the expression.**

Question 37.\ + \Answer:

\ \Answer:

\\) + \\)Answer:

\\) \\)Answer:

\ \Answer:

\ + \Answer:

\ + \Answer:

\ \Answer:

\\) 4\))Answer:

\ + 6Answer:

**ERROR ANALYSIS **Describe and correct the error in simplifying the expression.Answer:

**MULTIPLE REPRESENTATIONS** Which radical expressions are like radicals?A. \left^\)B. \^}\)C. \D. \ \E. \ + \F. \ \Answer:

**In Exercises 4954, simplify the expression.**

Question 49.\Answer:

\Answer:

\Answer:

\Answer:

\Answer:

**ERROR ANALYSIS** Describe and correct the error in simplifying the expression.Answer:

**In Exercises 5764, write the expression in simplest form. Assume all variables are positive.**

\Answer:

\Answer:

\Answer:

## Linear Algebra And Its Applications Exercise 252

from exercise 2.5.1 and any vector in the column space of show that . Prove the same result based on the rows of . What is the implication for the potential differences around a loop?

Answer: From exercise 2.5.1 we have the incidence matrix

If is in the column space of then we have

for some set of scalar coefficients ,

in the column space of .

Turning to the rows of if

which corresponds to the system of equations

We then have

so that

The 3 by 3 incidence matrix represents a graph with three nodes and three edges and hence one loop. Each node of the graph is represented by a column of and each edge by a row of . The first row represents edge 1 from node 2 to node 1 . The second row represents edge 2 from node 3 to node 2. The third row represents edge 3 from node 3 to node 1.

If the vector represents potentials at the nodes ( at node 1,

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## Lesson 56 Inverse Of A Function

**Essential Question**How can you sketch the graph of the inverse of a function?

**EXPLORATION 1Graphing Functions and Their InversesWork with a partner. **Each pair of functions are inverses of each other. Use a graphing calculator to graph f and g in the same viewing window. What do you notice about the graphs?a. f = 4x + 3c. f = \ = x2 + 3, x â¥ 0d. f = \frac\)g = \

**EXPLORATION 2Sketching Graphs of Inverse FunctionsWork with a partner.** Use the graph of f to sketch the graph of g, the inverse function of f, on the same set of coordinate axes. Explain your reasoning.

**Communicate Your Answer**

How can you sketch the graph of the inverse of a function?

Question 4.In Exploration 1, what do you notice about the relationship between the equations of f and g? Use your answer to find g, the inverse function off = 2x â 3.Use a graph to check your answer.

**5.6 Lesson**

**Solve y = f for x. Then find the input when the output is 2.**

Question 1.

**Determine whether the functions are inverse functions.**

Question 10.f = x + 5, g = x â 5

Question 11.f = 8×3, g = \

Question 12.The distance d that a dropped object falls in t seconds on Earth is represented by d = 4.9t2. Find the inverse of the function. How long does it take an object to fall 50 meters?

## Graphing Radical Functions 53 Exercises

**Vocabulary and Core Concept Check**

Question 1.**COMPLETE THE SENTENCE** Square root functions and cube root functions are examples of __________ functions.Answer:

Question 2.**COMPLETE THE SENTENCE** When graphing y = a\ + k, translate the graph of y = a\h units __________ and k units __________.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3â8, match the function with its graph.**

Question 3.

h = \Answer:

**In Exercises 19â26, describe the transformation of f represented by g. Then graph each function.**

Question 19.f = \, g = \ + 8Answer:

f = \, g = 2\Answer:

f = \, g = â\ â 1Answer:

f = \, g = \ â 5Answer:

f = \, g = \^\)Answer:

f = \, g = \ + 6Answer:

f = \, g = \ â 4Answer:

f = \, g = \ + 3Answer:

**ERROR ANALYSIS** Describe and correct the error in graphing f = \ â 2.Answer:

Question 28.**ERROR ANALYSIS** Describe and correct the error in describing the transformation of the parent square root function represented by g = \ + 3.Answer:

**USING TOOLS In Exercises 29â34, use a graphing calculator to graph the function. Then identify the domain and range of the function.**

Question 29.g = \Answer:

h = \Answer:

f = \Answer:

f = \Answer:

f = \Answer:

h = \Answer:

**ABSTRACT REASONING In Exercises 35â38, complete the statement with sometimes, always, or never.**

Question 35.The domain of the function y = a\ is ______ x â¥ 0.Answer:

The range of the function y = a\ is ______ y â¥ 0.Answer:

The domain of the function y = a\ + k is ________ x â¥ 0.Answer:

Answer:

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## Access Answers To Maths Rd Sharma Solutions For Class 12 Chapter 5 Algebra Of Matrices Exercise 53

**1. Compute the indicated products:**

**Solution:**

**2. Show that AB BA in each of the following cases:**

**Solution:**

From equation and , it is clear that

AB BA

From equation and , it is clear that

AB BA

From equation and , it is clear that

AB BA

**3. Compute the products AB and BA whichever exists in each of the following cases:**

**Solution:**

Because the number of columns in B is greater than the rows in A

Consider,

AB =

AB = 11

**4. Show that AB BA in each of the following cases:**

**Solution:**

From equation and , it is clear that

AB BA

From equation and it is clear that,

AB BA

First we have to add first two matrix,

On simplifying, we get

First we have to multiply first two given matrix,

= 82

First we have subtract the matrix which is inside the bracket,

**Solution:**

Now, consider,

We have,

Now, from equation , , and , it is clear that A2 = B2= C2= I2

**Solution:**

Now we have to find,

**Solution:**

From equation and AB = BA = 03×3

**Solution:**

From equation and AB = BA = 03×3

**Solution:**

Again consider, BA we get,

Hence BA = B

Now again consider, B2

Now by subtracting equation from equation we get,

**16. For the following matrices verify the associativity of matrix multiplication i.e. C = A **

**Solution:**

From equation and , it is clear that C = A

Given,

Now consider RHS,

From equation and , it is clear that C = A

**17. For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A = AB + AC.**

**Solution:**

From equation and , it is clear that A = AB + AC

Given,

= = 0

## Performing Function Operations 55 Exercises

**Vocabulary and Core Concept Check**

Question 1.**WRITING **Let f and g be any two functions. Describe how you can use f, g, and the four basic operations to create new functions.Answer:

**WRITING **What x-values are not included in the domain of the quotient of two functions?Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3â6, find and and state the domain of each. Then evaluate f + g and f â g for the given value of x.**

Question 3.f = \, g = 19\ x = 16Answer:

f = \, g = â11\ x = â4Answer:

f = 6x â 4×2â 7×3, g = 9×2â 5x x = â1Answer:

f = 11x + 2×2, g = â7x â 3×2 + 4 x = 2Answer:

**In Exercises 7â12, find and ) and state the domain of each. Then evaluate fg and \ for the given value of x.**

Question 7.f = 2×3, g = \ x = â27Answer:

f = x4, g = \ x = 4Answer:

f = 4x, g = 9×1/2 x = 9Answer:

f = 11×3, g = 7×7/3 x = â8Answer:

f = 7×3/2, g =â14×1/3 x = 64Answer:

f = 4×5/4, g = 2×1/2 x = 16Answer:

**USING TOOLS In Exercises 13â16, use a graphing calculator to evaluate , , , and ) when x = 5. Round your answers to two decimal places.**

Question 13.

Answer:

Question 19.**MODELING WITH MATHEMATICS** From 1990 to 2010, the numbers of female F and male M employees from the ages of 16 to 19 in the United States can be modeled by F =â0.007t2 + 0.10t + 3.7 and M = 0.0001t3 â 0.009t2 + 0.11t + 3.7, where t is the number of years since 1990.a. Find .b. Explain what represents.Answer:

**Solve the literal equation for n. **

Question 28.

\ = 7bAnswer:

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## Rd Sharma Solutions For Class 12 Maths Exercise 53 Chapter 5 Algebra Of Matrices

**RD Sharma Solutions for Class 12 Maths Exercise 5.3 Chapter 5 Algebra of Matrices** are available at BYJUS website in PDF format with solutions prepared by experienced faculty in a precise manner. It can mainly be used by the students as a study material to clear the Class 12 exams with a good score according to the **CBSE** syllabus.

Exercise 5.3 of the fifth chapter contains problems, which are solved based on the **transpose of a matrix**. The solutions prepared are in an explanatory manner to bring about conceptual clarity among the students. RD Sharma Solutions for Class 12 Maths Chapter 5 Algebra of Matrices Exercise 5.3 are provided here.

## Identifying Zeros And Their Multiplicities

Graphs behave differently at various *x*-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and “bounce” off.

Suppose, for example, we graph the function shown.

Notice in Figure 7 that the behavior of the function at each of the *x*-intercepts is different.

*x*

The *x*-intercept is the solution of equation , so the behavior near the intercept is like that of a lineâit passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.

The *x*-intercept is the repeated solution of equation , so the behavior near the intercept is like that of a quadraticâit bounces off of the horizontal axis at the intercept.

The factor is repeated, that is, the factor appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor, x has multiplicity 2 because the factor (

is the repeated solution of factor , so the behavior near the intercept is like that of a cubicâwith the same S-shape near the intercept as the toolkit function f . We call this a triple zero, or a zero with multiplicity 3.

For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the *x*-axis.

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## Rational Exponents And Radical Functions Chapter Review

**Evaluate the expression without using a calculator.**

Question 1.

Question 8.\

Question 9.\

Question 10.\ + \

Question 11.\ â \

Question 12.)1/2

**Simplify the expression. Assume all variables are positive.**

Question 13.\

Question 14.\

Question 15.\

**Describe the transformation of f represented by g. Then graph each function.**

Question 16.f = \, g = â2\

Question 17.f = \, g = \ â 6

Question 18.Let the graph of g be a reflection in the y-axis, followed by a translation 7 units to the right of the graph of f = \. Write a rule for g.

Question 19.Use a graphing calculator to graph 2y2 = x â 8. Identify the vertex and the direction that the parabola opens.

Question 20.Use a graphing calculator to graph x2 + y2 = 81. Identify the radius and the intercepts.

**Solve the equation. Check your solution.**

Question 21.\ = 20

Question 22.\ = \ â 1

Question 23.

7\ â¥ 21

Question 27.In a tsunami, the wave speeds can be modeled by s = \, where d is the depth of the water. Estimate the depth of the water when the wave speed is 200 meters per second.

Question 28.Let f = 2\ and g = 4\. Find and ) and state the domain of each. Then evaluate and ).

Question 29.Let f = 3×2 + 1 and g = x + 4. Find and and state the domain of each. Then evaluate and .

Question 30.f = â\x + 10

Question 31.f = x2 + 8, x â¥ 0

Question 32.

f = 3\ + 5

**Determine whether the functions are inverse functions.**

Question 34.f = 42, g = \2

Question 35.f = â2x + 6, g = â\x + 3

## Recognizing Characteristics Of Graphs Of Polynomial Functions

Polynomial functions of degree 2 or more have graphs that do not have sharp corners recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. Figure 1 shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.

Which of the graphs in Figure 2 represents a polynomial function?

are graphs of polynomial functions. They are smooth and continuous.

The graphs of are graphs of functions that are not polynomials. The graph of function g has a sharp corner. The graph of function k

**Do all polynomial functions have as their domain all real numbers?**

*Yes. Any real number is a valid input for a polynomial function.*

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## Lesson 54 Solving Radical Equations And Inequalities

**Essential Question**How can you solve a radical equation?

**EXPLORATION 1Solving Radical EquationsWork with a partner. **Match each radical equation with the graph of its related radical function. Explain your reasoning. Then use the graph to solve the equation, if possible. Check your solutions.a. \ â 1 = 0b. \ â \ = 0c. \ = 0d. \ â x = 0e. \ â x = 0f. \ = 0

**EXPLORATION 2Solving Radical EquationsWork with a partner.** Look back at the radical equations in Exploration 1. Suppose that you did not know how to solve the equations using a graphical approach.a. Show how you could use a numerical approach to solve one of the equations. For instance, you might use a spreadsheet to create a table of values.b. Show how you could use an analytical approach to solve one of the equations. For instance, look at the similarities between the equations in Exploration 1. What first step may be necessary so you could square each side to eliminate the radical? How would you proceed to find the solution?

**Communicate Your Answer**

How can you solve a radical equation?

Question 4.Would you prefer to use a graphical, numerical, or analytical approach to solve the given equation? Explain your reasoning. Then solve the equation.\ â \ = 1

**Monitoring Progress**

**Solve the equation. Check your solution.**

Question 1.\ â 9 = -6

Question 2.

\ â 10 = â7Answer:

â2\ + 13 = â1Answer:

8\ â 15 = 17Answer:

\ = 8Answer:

\ = 0Answer:

\ + 7 = 15Answer:

\ â 13 = â15Answer:

Question 15.xâ 6 = \Answer:

## Linear Algebra And Its Applications Exercise 253

from exercise 2.5.1 and any vector in the row space of show that . Prove the same result based on the linear system . What is the implication if , are currents into each node?

Answer: From exercise 2.5.1 we have the incidence matrix

If is in the row space of then we have

for some set of scalar coefficients ,

in the row space of .

which corresponds to the system of equations

We then have

The 3 by 3 incidence matrix represents a graph with three nodes and three edges. The first row represents edge 1 from node 2 to node 1 . The second row represents edge 2 from node 3 to node 2. The third row represents edge 3 from node 3 to node 1.

Each node of the graph is represented by a column of and thus by a row of . If the vector represents current sources at each node then the fact that means that the net current into each node is zero .

NOTE: This continues a series of posts containing worked out exercises from the book

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