In Exercises (PageIndex{1}) - (PageIndex{8}), find the prime factorization of the given natural number.

Exercise (PageIndex{1})

80

**Answer**(80=2 cdot 2 cdot 2 cdot 2 cdot 5)

Exercise (PageIndex{2})

108

Exercise (PageIndex{3})

180

**Answer**(180=2 cdot 2 cdot 3 cdot 3 cdot 5)

Exercise (PageIndex{4})

160

Exercise (PageIndex{5})

128

**Answer**(128=2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2)

Exercise (PageIndex{6})

192

Exercise (PageIndex{7})

32

**Answer**(32=2 cdot 2 cdot 2 cdot 2 cdot 2)

Exercise (PageIndex{8})

72

In Exercises (PageIndex{9})-(PageIndex{16}), convert the given decimal to a fraction.

Exercise (PageIndex{9})

0.648

**Answer**There are three decimal places, so (0.648=frac{648}{1000}=frac{81}{125})

Exercise (PageIndex{10})

0.62

Exercise (PageIndex{11})

0.240

**Answer**There are three decimal places, so (0.240=frac{240}{1000}=frac{6}{25})

Exercise (PageIndex{12})

0.90

Exercise (PageIndex{13})

0.14

**Answer**There are two decimal places, so (0.14=frac{14}{100}=frac{7}{50})

Exercise (PageIndex{14})

0.760

Exercise (PageIndex{15})

0.888

**Answer**There are three decimal places, so (0.888=frac{888}{1000}=frac{111}{125})

Exercise (PageIndex{16})

0.104

In Exercises (PageIndex{17})-(PageIndex{24}), convert the given repeating decimal to a fraction.

Exercise (PageIndex{17})

(0 . overline{27})

**Answer**Let (x=0 . overline{27} .) Then (100 x=27 . overline{27} .) Subtracting on both sides of these equations.

[egin{aligned} 100 x &=27 . overline{27} x &=0 . overline{27} end{aligned}]

yields (99 x=27 .) Finally, solve for (x) by dividing by (99 : x=frac{27}{99}=frac{3}{11}).

Exercise (PageIndex{18})

(0 . overline{171})

Exercise (PageIndex{19})

(0 . overline{24})

**Answer**Let (x=0 . overline{24} .) Then (100 x=24 . overline{24} .) Subtracting on both sides of these equations [egin{aligned} 100 x &=24 . overline{24} x &=0 . overline{24} end{aligned}]

yields (99 x=24 .) Finally, solve for (x) by dividing by (99 : x=frac{24}{99}=frac{8}{33})

Exercise (PageIndex{20})

(0 . overline{882})

Exercise (PageIndex{21})

(0 . overline{84})

**Answer**Let (x=0 . overline{84} .) Then (100 x=84 . overline{84} .) Subtracting on both sides of these equations

[egin{aligned} 100 x &=84 . overline{.84} x &=0 . overline{84} end{aligned}]

yields (99 x=84 .) Finally, solve for (x) by dividing by (99 : x=frac{84}{99}=frac{28}{33})

Exercise (PageIndex{22})

(0 . overline{384})

Exercise (PageIndex{23})

(0 . overline{63})

**Answer**Let (x=0 . overline{63} .) Then (100 x=63 . overline{63} .) Subtracting on both sides of these equations

[egin{aligned} 100 x &=63 . overline{63} x &=0 . overline{63} end{aligned}]

yields (99 x=63 .) Finally, solve for (x) by dividing by (99 : x=frac{63}{99}=frac{7}{11})

Exercise (PageIndex{24})

(0 . overline{60})

Exercise (PageIndex{25})

Prove that (sqrt{3}) is irrational.

**Answer**Suppose that (sqrt{3}) is rational. Then it can be expressed as the ratio of two integers p and q as follows:

[sqrt{3}=frac{p}{q}]

Square both sides, [3=frac{p^{2}}{q^{2}}]

then clear the equation of fractions by multiplying both sides by (q^{2}):

[p^{2}=3 q^{2}]

Now p and q each have their own unique prime factorizations. Both (p^{2}) and (q^{2}) have an even number of factors in their prime factorizations. But this contradicts equation (1), because the left side would have an even number of factors in its prime factorization, while the right side would have an odd number of factors in its prime factorization (there’s one extra 3 on the right side).

Therefore, our assumption that (sqrt{3}) was rational is false. Thus, (sqrt{3}) is irrational.

Exercise (PageIndex{26})

Prove that (sqrt{5}) is irrational.

In Exercises (PageIndex{27})-(PageIndex{30}), copy the given table onto your homework paper. In each row, place a check mark in each column that is appropriate. That is, if the number at the start of the row is rational, place a check mark in the rational column. Note: Most (but not all) rows will have more than one check mark.

Exercise (PageIndex{27})

(mathbb{N}) | (mathbb{W}) | (mathbb{Z}) | (mathbb{Q}) | (mathbb{R}) |

0 | ||||

-2 | ||||

-2/3 | ||||

0.15 | ||||

(0 . overline{2}) | ||||

(sqrt{5}) |

**Answer**(mathbb{N}) (mathbb{W}) (mathbb{Z}) (mathbb{Q}) (mathbb{R}) 0 x x x x -2 x x x -2/3 x x 0.15 x x (0 . overline{2}) x x (sqrt{5}) x

Exercise (PageIndex{28})

(mathbb{N}) | (mathbb{W}) | (mathbb{Z}) | (mathbb{Q}) | (mathbb{R}) |

10/2 | ||||

(pi) | ||||

-6 | ||||

(0 . overline{9}) | ||||

(sqrt{2}) | ||||

0.37 |

Exercise (PageIndex{29})

(mathbb{N}) | (mathbb{W}) | (mathbb{Z}) | (mathbb{Q}) | (mathbb{R}) |

-4/3 | ||||

12 | ||||

0 | ||||

(sqrt{11}) | ||||

(1. overline{3}) | ||||

6/2 |

**Answer**(mathbb{N}) (mathbb{W}) (mathbb{Z}) (mathbb{Q}) (mathbb{R}) -4/3 x x 12 x x x x x 0 x x x x (sqrt{11}) x (1. overline{3}) x x 6/2 x x x x x

Exercise (PageIndex{30})

(mathbb{N}) | (mathbb{W}) | (mathbb{Z}) | (mathbb{Q}) | (mathbb{R}) |

-3/5 | ||||

(sqrt{10}) | ||||

1.625 | ||||

10/2 | ||||

0/5 | ||||

11 |

In Exercises (PageIndex{31})-(PageIndex{42}), consider the given statement and determine whether it is true or false. Write a sentence explaining your answer. In particular, if the statement is false, try to give an example that contradicts the statement.

Exercise (PageIndex{31})

All natural numbers are whole numbers.

**Answer**True. The only difference between the two sets is that the set of whole numbers contains the number 0.

Exercise (PageIndex{32})

All whole numbers are rational numbers.

Exercise (PageIndex{33})

All rational numbers are integers.

**Answer**False. For example, (frac{1}{2}) is not an integer.

Exercise (PageIndex{34})

All rational numbers are whole numbers.

Exercise (PageIndex{35})

Some natural numbers are irrational.

**Answer**False. All natural numbers are rational, and therefore not irrational.

Exercise (PageIndex{36})

Some whole numbers are irrational.

Exercise (PageIndex{37})

Some real numbers are irrational.

**Answer**True. For example, π and √2 are real numbers which are irrational.

Exercise (PageIndex{38})

All integers are real numbers.

Exercise (PageIndex{39})

All integers are rational numbers.

**Answer**True. Every integer b can be written as a fraction b/1.

Exercise (PageIndex{40})

No rational numbers are natural numbers.

Exercise (PageIndex{41})

No real numbers are integers.

**Answer**False. For example, 2 is a real number that is also an integer.

Exercise (PageIndex{42})

All whole numbers are natural numbers.

In Exercises (PageIndex{43})-(PageIndex{54}), solve each of the given equations for x.

Exercise (PageIndex{43})

45x + 12 = 0

**Answer**[egin{aligned} & 45 x+12=0 Longrightarrow quad & 45 x=-12 Longrightarrow quad& x=-frac{12}{45}=-frac{4}{15} end{aligned}]

Exercise (PageIndex{44})

76x − 55 = 0

Exercise (PageIndex{45})

x − 7 = −6x + 4

**Answer**[egin{aligned} & x-7=-6 x+4 Longrightarrow quad & x+6 x=4+7 Longrightarrow quad & 7 x=11 Longrightarrow quad & x=frac{11}{7} end{aligned}]

Exercise (PageIndex{46})

−26x + 84 = 48

Exercise (PageIndex{47})

37x + 39 = 0

**Answer**[egin{aligned} & 37 x+39=0 Longrightarrowquad & 37 x=-39 Longrightarrowquad & x=-frac{39}{37} end{aligned}]

Exercise (PageIndex{48})

−48x + 95 = 0

Exercise (PageIndex{49})

74x − 6 = 91

**Answer**[egin{aligned} & 74 x-6=91 Longrightarrowquad & 74 x=97 Longrightarrowquad & x=frac{97}{74} end{aligned}]

Exercise (PageIndex{50})

−7x + 4 = −6

Exercise (PageIndex{51})

−88x + 13 = −21

**Answer**[egin{aligned} &-88 x+13=-21 Longrightarrowquad &-88 x=-34 Longrightarrow quad & x=frac{-34}{-88}=frac{17}{44} end{aligned}]

Exercise (PageIndex{52})

−14x − 81 = 0

Exercise (PageIndex{53})

19x + 35 = 10

**Answer**[egin{aligned} & 19 x+35=10 Longrightarrowquad & 19 x=-25 Longrightarrowquad & x=-frac{25}{19} end{aligned}]

Exercise (PageIndex{54})

−2x + 3 = −5x − 2

In Exercises (PageIndex{55})-(PageIndex{66}), solve each of the given equations for x.

Exercise (PageIndex{55})

6 − 3(x + 1) = −4(x + 6) + 2

**Answer**[egin{aligned} & 6-3(x+1)=-4(x+6)+2 Longrightarrowquad & 6-3 x-3=-4 x-24+2 Longrightarrowquad &-3 x+3=-4 x-22 Longrightarrowquad &-3 x+4 x=-22-3 Longrightarrowquad & x=-25 end{aligned}]

Exercise (PageIndex{56})

(8x + 3) − (2x + 6) = −5x + 8

Exercise (PageIndex{57})

−7 − (5x − 3) = 4(7x + 2)

[egin{aligned} &-7-(5 x-3)=4(7 x+2) Longrightarrowquad &-7-5 x+3=28 x+8 Longrightarrowquad &-5 x-4=28 x+8 Longrightarrowquad &-5 x-28 x=8+4 Longrightarrowquad &-33 x=12 Longrightarrowquad & x=-frac{12}{33}=-frac{4}{11} end{aligned}]

Exercise (PageIndex{58})

−3 − 4(x + 1) = 2(x + 4) + 8

Exercise (PageIndex{59})

9 − (6x − 8) = −8(6x − 8)

**Answer**[egin{aligned} & 9-(6 x-8)=-8(6 x-8) Longrightarrow quad & 9-6 x+8=-48 x+64 Longrightarrowquad &-6 x+17=-48 x+64 Longrightarrowquad &-6 x+48 x=64-17 Longrightarrowquad & 42 x=47 Longrightarrowquad & x=frac{47}{42} end{aligned}]

Exercise (PageIndex{60})

−9 − (7x − 9) = −2(−3x + 1)

Exercise (PageIndex{61})

(3x − 1) − (7x − 9) = −2x − 6

**Answer**[egin{aligned} &(3 x-1)-(7 x-9)=-2 x-6 Longrightarrowquad & 3 x-1-7 x+9=-2 x-6 Longrightarrowquad &-4 x+8=-2 x-6 Longrightarrowquad &-4 x+2 x=-6-8 Longrightarrowquad &-2 x=-14 Longrightarrowquad & x=7 end{aligned}]

Exercise (PageIndex{62})

−8 − 8(x − 3) = 5(x + 9) + 7

Exercise (PageIndex{63})

(7x − 9) − (9x + 4) = −3x + 2

**Answer**[egin{aligned} &(7 x-9)-(9 x+4)=-3 x+2 Longrightarrowquad & 7 x-9-9 x-4=-3 x+2 Longrightarrowquad &-2 x-13=-3 x+2 Longrightarrowquad &-2 x+3 x=2+13 Longrightarrowquad & x=15 end{aligned}]

Exercise (PageIndex{64})

(−4x − 6) + (−9x + 5) = 0

Exercise (PageIndex{65})

−5 − (9x + 4) = 8(−7x − 7)

**Answer**[egin{array}{ll}{} & {-5-(9 x+4)=8(-7 x-7)} {Longrightarrow} & {-5-9 x-4=-56 x-56} {Longrightarrow} & {-9 x-9=-56 x-56} {Longrightarrow} & {-9 x+56 x=-56+9} {Longrightarrow} & {47 x=-47} {Longrightarrow} & {x=-1}end{array}]

Exercise (PageIndex{66})

(8x − 3) + (−3x + 9) = −4x − 7

In Exercises (PageIndex{67})-(PageIndex{78}), solve each of the given equations for x. Check your solutions using your calculator.

Exercise (PageIndex{67})

−3.7x − 1 = 8.2x − 5

**Answer**First clear decimals by multiplying by 10.

[egin{aligned} &-3.7 x-1=8.2 x-5 Longrightarrowquad &-37 x-10=82 x-50 Longrightarrowquad &-37 x-82 x=-50+10 Longrightarrowquad &-119 x=-40 Longrightarrowquad & x=frac{40}{119} end{aligned}]

Here is a check of the solutions on the graphing calculator. The left-hand side of the equation is evaluated at the solution in (a), the right-hand side of the equation is evaluated at the solution in (b). Note that they match.

Exercise (PageIndex{68})

8.48x − 2.6 = −7.17x − 7.1

Exercise (PageIndex{69})

(-frac{2}{3} x+8=frac{4}{5} x+4)

**Answer**First clear fractions by multiplying by 15.

[egin{aligned} &-frac{2}{3} x+8=frac{4}{5} x+4 Longrightarrowquad &-10 x+120=12 x+60 Longrightarrowquad &-10 x-12 x=60-120 Longrightarrowquad &-22 x=-60 Longrightarrowquad & x=frac{-60}{-22}=frac{30}{11} end{aligned}]

Here is a check of the solutions on the graphing calculator. Note that they match.

Exercise (PageIndex{70})

−8.4x = −4.8x + 2

Exercise (PageIndex{71})

(-frac{3}{2} x+9=frac{1}{4} x+7)

**Answer**First clear fractions by multiplying by 4.

[egin{aligned} &-frac{3}{2} x+9=frac{1}{4} x+7 Longrightarrowquad &-6 x+36=x+28 Longrightarrowquad &-6 x-x=28-36 Longrightarrowquad &-7 x=-8 Longrightarrowquad & x=frac{8}{7} end{aligned}]

Here is a check of the solutions on the graphing calculator. Note that they match.

Exercise (PageIndex{72})

2.9x − 4 = 0.3x − 8

Exercise (PageIndex{73})

5.45x + 4.4 = 1.12x + 1.6

**Answer**First clear decimals by multiplying by 100.

[egin{aligned} & 5.45 x+4.4=1.12 x+1.6 Longrightarrowquad & 545 x+440=112 x+160 Longrightarrowquad & 545 x-112 x=160-440 Longrightarrowquad & 433 x=-280 Longrightarrowquad & x=-frac{280}{433} end{aligned}]

Here is a check of the solutions on the graphing calculator. Note that they match.

Exercise (PageIndex{74})

(-frac{1}{4} x+5=-frac{4}{5} x-4)

Exercise (PageIndex{75})

(-frac{3}{2} x-8=frac{2}{5} x-2)

**Answer**First clear fractions by multiplying by 10. [egin{aligned} &-frac{3}{2} x-8=frac{2}{5} x-2 Longrightarrowquad &-15 x-80=4 x-20 Longrightarrowquad &-15 x-4 x=-20+80 Longrightarrowquad &-19 x=60 Longrightarrowquad & x=-frac{60}{19} end{aligned}]

Here is a check of the solutions on the graphing calculator. Note that they match.

Exercise (PageIndex{76})

(-frac{4}{3} x-8=-frac{1}{4} x+5)

Exercise (PageIndex{77})

−4.34x − 5.3 = 5.45x − 8.1

**Answer**First clear decimals by multiplying by 100.

[egin{aligned} &-4.34 x-5.3=5.45 x-8.1 Longrightarrowquad &-434 x-530=545 x-810 Longrightarrowquad &-434 x-545 x=-810+530 Longrightarrowquad &-979 x=-280 Longrightarrowquad & x=frac{280}{979} end{aligned}]

Here is a check of the solutions on the graphing calculator. Note that they match.

Exercise (PageIndex{78})

(frac{2}{3} x-3=-frac{1}{4} x-1)

In Exercises (PageIndex{79})-50, solve each of the given equations for the indicated variable.

Exercise (PageIndex{79})

P = IRT for R

**Answer**[egin{aligned} & P=I R T Longrightarrowquad & P=(I T) R Longrightarrowquad & frac{P}{I T}=frac{(I T) R}{I T} Longrightarrowquad & frac{P}{I T}=R end{aligned}]

Exercise (PageIndex{80})

d = vt for t

Exercise (PageIndex{81})

(v=v_{0}+a t) for (a)

**Answer**[egin{aligned} & v=v_{0}+a t Longrightarrowquad & v-v_{0}=a t Longrightarrowquad & frac{v-v_{0}}{t}=a end{aligned}]

Exercise (PageIndex{82})

(x=v_{0}+v t) for (v)

Exercise (PageIndex{83})

Ax + By = C for y

**Answer**[egin{aligned} & A x+B y=C Longrightarrowquad & B y=C-A x Longrightarrowquad & y=frac{C-A x}{B} end{aligned}]

Exercise (PageIndex{84})

y = mx + b for x

Exercise (PageIndex{85})

(A=pi r^{2}) for (pi)

**Answer**[egin{aligned} A &=pi r^{2} Longrightarrow quad frac{A}{r^{2}} &=pi end{aligned}]

Exercise (PageIndex{86})

(S=2 pi r^{2}+2 pi r h) for (h)

Exercise (PageIndex{87})

(F=frac{k q q_{0}}{r^{2}}) for (k)

**Answer**[egin{aligned} & F=frac{k q q_{0}}{r^{2}} Longrightarrowquad & F r^{2}=k q q_{0} Longrightarrowquad & frac{F r^{2}}{q q_{0}}=k end{aligned}]

Exercise (PageIndex{88})

(C=frac{Q}{m T}) for (T)

Exercise (PageIndex{89})

(frac{V}{t}=k) for (t)

**Answer**[egin{aligned} & frac{V}{t}=k Longrightarrowquad & V=k t Longrightarrowquad & frac{V}{k}=t end{aligned}]

Exercise (PageIndex{90})

(lambda=frac{h}{m v}) for (v)

Exercise (PageIndex{91})

(frac{P_{1} V_{1}}{n_{1} T_{1}}=frac{P_{2} V_{2}}{n_{2} T_{2}}) for (V_{2})

**Answer**Cross multiply, then divide by the coefficient of (V_{2}).

[egin{aligned} & frac{P_{1} V_{1}}{n_{1} T_{1}}=frac{P_{2} V_{2}}{n_{2} T_{2}} Longrightarrowquad & n_{2} P_{1} V_{1} T_{2}=n_{1} P_{2} V_{2} T_{1} Longrightarrowquad & frac{n_{2} P_{1} V_{1} T_{2}}{n_{1} P_{2} T_{1}}=V_{2} end{aligned}]

Exercise (PageIndex{92})

(pi=frac{n R T}{V} i) for (n)

Exercise (PageIndex{93})

Tie a ball to a string and whirl it around in a circle with constant speed. It is known that the acceleration of the ball is directly toward the center of the circle and given by the formula [a=frac{v^{2}}{r}] where a is acceleration, v is the speed of the ball, and r is the radius of the circle of motion.

i. Solve formula (1) for r.

ii. Given that the acceleration of the ball is 12 m/s2 and the speed is 8 m/s, find the radius of the circle of motion.

**Answer**Cross multiply, then divide by the coefficient of r.

[egin{aligned} a &=frac{v^{2}}{r} a r &=v^{2} r &=frac{v^{2}}{a} end{aligned}]

To find the radius, substitute the acceleration (a=12 mathrm{m} / mathrm{s}^{2}) and speed v = 8 m/s.

[r=frac{v^{2}}{a}=frac{(8)^{2}}{12}=frac{64}{12}=frac{16}{3}]

Hence, the radius is (r=16 / 3 mathrm{m},) or 5(frac{1}{3}) meters.

Exercise (PageIndex{94})

A particle moves along a line with constant acceleration. It is known the velocity of the particle, as a function of the amount of time that has passed, is given by the equation

[v=v_{0}+a t] where v is the velocity at time t, v0 is the initial velocity of the particle (at time t = 0), and a is the acceleration of the particle.

i. Solve formula (2) for t.

ii. You know that the current velocity of the particle is 120 m/s. You also know that the initial velocity was 40 m/s and the acceleration has been a constant (a=2 mathrm{m} / mathrm{s}^{2}). How long did it take the particle to reach its current velocity?

Exercise (PageIndex{95})

Like Newton’s Universal Law of Gravitation, the force of attraction (repulsion) between two unlike (like) charged particles is proportional to the product of the charges and inversely proportional to the distance between them. [F=k_{C} frac{q_{1} q_{2}}{r^{2}}] In this formula, (k_{C} approx 8.988 imes 10^{9} mathrm{Nm}^{2} / mathrm{C}^{2}) and is called the electrostatic constant. The variables q1 and q2 represent the charges (in Coulombs) on the particles (which could either be positive or negative numbers) and r represents the distance (in meters) between the charges. Finally, F represents the force of the charge, measured in Newtons.

i. Solve formula (3) for r.

ii. Given a force (F=2.0 imes 10^{12} mathrm{N}), two equal charges (q_{1}=q_{2}=1 mathrm{C}), find the approximate distance between the two charged particles.

**Answer**Cross multiply, then divide by the coefficient of r.

[egin{aligned} F &=k_{C} frac{q_{1} q_{2}}{r^{2}} F r^{2} &=k_{C} q_{1} q_{2} r^{2} &=frac{k_{C} q_{1} q_{2}}{F} end{aligned}]

Finally, to find r, take the square root.

[r=sqrt{frac{k_{C} q_{1} q_{2}}{F}}]

To find the distance between the charged particles, substitute (k_{C}=8.988 imes 10^{9} mathrm{Nm}^{2} / mathrm{C}^{2}),

(q_{1}=q_{2}=1 mathrm{C},) and (F=2.0 imes 10^{12} mathrm{N}).[r=sqrt{frac{left(8.988 imes 10^{9} ight)(1)(1)}{2.0 imes 10^{12}}}]

A calculator produces an approximation, (r approx 0.067) meters.

Perform each of the following tasks in Exercises (PageIndex{96})-(PageIndex{99}).

i. Write out in words the meaning of the symbols which are written in set-builder notation.

ii. Write some of the elements of this set.

iii. Draw a real line and plot some of the points that are in this set.

Exercise (PageIndex{96})

(A={x in mathbb{N} : x>10})

**Answer**i. A is the set of all (x) in the natural numbers such that (x) is greater than (10.)

ii. (A={11,12,13,14, ldots})

iii.

Exercise (PageIndex{97})

(B={x in mathbb{N} : x geq 10})

Exercise (PageIndex{98})

(C={x in mathbb{Z} : x leq 2})

**Answer**i. C is the set of all (x) in the integers such that (x) is less than or equal to (2.)

ii. (C={ldots,-4,-3,-2,-1,0,1,2})

iii.

Exercise (PageIndex{99})

(D={x in mathbb{Z} : x>-3})

In Exercises (PageIndex{100})-(PageIndex{103}), use the sets A, B, C, and D that were defined in Exercises (PageIndex{96})-(PageIndex{99}). Describe the following sets using set notation, and draw the corresponding Venn Diagram.

Exercise (PageIndex{100})

(A cap B)

**Answer**(A cap B={x in mathbb{N} : x>10}={11,12,13, ldots})

Exercise (PageIndex{101})

(A cup B)

Exercise (PageIndex{102})

(A cup C)

**Answer**(A cup C={x in mathbb{Z} : x leq 2 ext { or } x>10}={ldots,-3,-2-1,0,1,2,11,12,13 dots})

Exercise (PageIndex{103})

(C cap D)

In Exercises (PageIndex{104})-(PageIndex{111}), use both interval and set notation to describe the interval shown on the graph.

Exercise (PageIndex{104})

**Answer**The filled circle at the endpoint 3 indicates this point is included in the set. Thus, the set in interval notation is ([3, infty)), and in set notation ({x : x geq 3}).

Exercise (PageIndex{105})

Exercise (PageIndex{106})

**Answer**The empty circle at the endpoint −7 indicates this point is not included in the set. Thus, the set in interval notation is ((-infty,-7)), and in set notation is ({x : x<-7}).

Exercise (PageIndex{107})

Exercise (PageIndex{108})

**Answer**The empty circle at the endpoint 0 indicates this point is not included in the set. Thus, the set in interval notation is ((0, infty)), and in set notation is ({x : x>0}).

Exercise (PageIndex{109})

Exercise (PageIndex{110})

**Answer**The empty circle at the endpoint −8 indicates this point is not included in the set. Thus, the set in interval notation is ((-8, infty)), and in set notation is ({x : x>-8}).

Exercise (PageIndex{111})

In Exercises (PageIndex{112})-(PageIndex{119}), sketch the graph of the given interval.

Exercise (PageIndex{112})

([2,5))

**Answer**

Exercise (PageIndex{113})

((-3,1])

Exercise (PageIndex{114})

([1, infty))

**Answer**

Exercise (PageIndex{115})

((-infty, 2))

Exercise (PageIndex{116})

({x :-4

**Answer**

Exercise (PageIndex{117})

({x : 1 leq x leq 5})

Exercise (PageIndex{118})

({x : x<-2})

**Answer**

Exercise (PageIndex{119})

({x : x geq-1})

In Exercises (PageIndex{120})-(PageIndex{127}), use both interval and set notation to describe the intersection of the two intervals shown on the graph. Also, sketch the graph of the intersection on the real number line.

Exercise (PageIndex{120})

**Answer**The intersection is the set of points that are in both intervals (shaded on both graphs). Graph of the intersection:

([1, infty)={x : x geq 1})

Exercise (PageIndex{121})

Exercise (PageIndex{122})

**Answer**There are no points that are in both intervals (shaded in both), so there is no intersection. Graph of the intersection:

no intersection

Exercise (PageIndex{123})

Exercise (PageIndex{124})

**Answer**The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

([-6,2]={x :-6 leq x leq 2})

Exercise (PageIndex{125})

Exercise (PageIndex{126})

**Answer**The intersection is the set of points that are in both intervals (shaded in both). Graph of the intersection:

([9, infty)={x : x geq 9})

Exercise (PageIndex{127})

In Exercises (PageIndex{128})-(PageIndex{135}), use both interval and set notation to describe the union of the two intervals shown on the graph. Also, sketch the graph of the union on the real number line.

Exercise (PageIndex{128})

**Answer**The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:

((-infty,-8]={x : x leq-8})

Exercise (PageIndex{129})

Exercise (PageIndex{130})

**Answer**The union is the set of all points that are in one interval or the other (shaded in either graph). Graph of the union:

((-infty, 9] cup(15, infty))

(={x : x leq 9 ext { or } x>15})

Exercise (PageIndex{131})

Exercise (PageIndex{132})

**Answer**The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

((-infty, 3)={x : x<3})

Exercise (PageIndex{133})

Exercise (PageIndex{134})

**Answer**The union is the set of all points that are in one interval or the other (shaded in either). Graph of the union:

([9, infty)={x : x geq 9})

Exercise (PageIndex{135})

In Exercises (PageIndex{136})-56, use interval notation to describe the given set. Also, sketch the graph of the set on the real number line.

Exercise (PageIndex{136})

({x : x geq-6 ext { and } x>-5})

**Answer**This set is the same as ({x : x>-5}), which is ((-5, infty)) in interval notation. Graph of the set:

Exercise (PageIndex{137})

({x : x leq 6 ext { and } x geq 4})

Exercise (PageIndex{138})

({x : x geq-1 ext { or } x<3})

**Answer**Every real number is in one or the other of the two intervals. Therefore, the set is the set of all real numbers ((-infty, infty)). Graph of the set:

Exercise (PageIndex{139})

({x : x>-7 ext { and } x>-4})

Exercise (PageIndex{140})

({x : x geq -1 ext { or } x>6})

**Answer**This set is the same as ({x : x geq-1}), which is ([-1, infty)) in interval notation. Graph of the set:

Exercise (PageIndex{141})

({x : x geq 7 ext { or } x<-2})

Exercise (PageIndex{142})

({x : x geq 6 ext { or } x>-3})

**Answer**This set is the same as ({x : x>-3}), which is ((-3, infty)) in interval notation. Graph of the set:

Exercise (PageIndex{143})

({x : x leq 1 ext { or } x>0})

Exercise (PageIndex{144})

({x : x<2 ext { and } x<-7})

**Answer**This set is the same as ({x : x<-7}), which is ((-infty,-7)) in interval notation. Graph of the set:

Exercise (PageIndex{145})

({x : x leq-3 ext { and } x<-5})

Exercise (PageIndex{146})

({x : x leq-3 ext { or } x geq 4})

**Answer**This set is the union of two intervals, ((-infty,-3] cup[4, infty)). Graph of the set:

Exercise (PageIndex{147})

({x : x<11 ext { or } x leq 8})

Exercise (PageIndex{148})

({x : x geq 5 ext { and } x leq 1})

**Answer**There are no numbers that satisfy both inequalities. Thus, there is no intersection. Graph of the set:

Exercise (PageIndex{149})

({x : x<5 ext { or } x<10})

Exercise (PageIndex{150})

({x : x leq 5 ext { and } x geq-1})

**Answer**This set is the same as ({x :-1 leq x leq 5}), which is [−1, 5] in interval notation. Graph of the set

Exercise (PageIndex{151})

({x : x>-3 ext { and } x<-6})

In Exercises (PageIndex{152})-(PageIndex{163}), solve the inequality. Express your answer in both interval and set notations, and shade the solution on a number line.

Exercise (PageIndex{152})

(-8 x-3 leq-16 x-1)

**Answer**[egin{aligned} & -8 x-3 leq-16 x-1 Longrightarrow quad & − 8x + 16x leq −1 + 3 Longrightarrow quad& 8x leq 2 Longrightarrow quad & x leq frac{1}{4}end{aligned}]

Thus, the solution interval is ((−infty, frac{1}{4}]) = ({x|x leq frac{1}{4}}).

Exercise (PageIndex{153})

(6 x-6>3 x+3)

Exercise (PageIndex{154})

(-12 x+5 leq-3 x-4)

**Answer**[egin{aligned} & -12 x+5 leq-3 x-4 Longrightarrow quad & -12x + 3x leq −4 − 5 Longrightarrow quad& -9x leq -9 Longrightarrow quad & x geq 1end{aligned}]

Thus, the solution interval is ([1,infty) = {x|x geq 1}).

Exercise (PageIndex{155})

(7 x+3 leq-2 x-8)

Exercise (PageIndex{156})

(-11 x-9<-3 x+1)

**Answer**[egin{aligned} & − 11x − 9 < −3x + 1 Longrightarrow quad & − 11x + 3x < 1 + 9 Longrightarrow quad& − 8x < 10 Longrightarrow quad & x > -frac{5}{4}end{aligned}]

Thus, the solution interval is ((−frac{5}{4} ,infty) = {x|x >−frac{5}{4} }).

Exercise (PageIndex{157})

(4 x-8 geq-4 x-5)

Exercise (PageIndex{158})

(4 x-5>5 x-7)

**Answer**[egin{aligned} & 4x − 5 > 5x − 7 Longrightarrow quad & 4x − 5x > −7 + 5 Longrightarrow quad& − x > −2 Longrightarrow quad &x < 2end{aligned}]

Thus, the solution interval is ((−infty, 2) = {x|x < 2}).

Exercise (PageIndex{159})

(-14 x+4>-6 x+8)

Exercise (PageIndex{160})

(2 x-1>7 x+2)

**Answer**[egin{aligned} & 2x − 1 > 7x + 2 Longrightarrow quad & 2x − 7x > 2 + 1 Longrightarrow quad& − 5x > 3 Longrightarrow quad &x < −frac{3}{5}end{aligned}]

Thus, the solution interval is ((−infty, −frac{3}{5}) = {x|x < −frac{3}{5}}).

Exercise (PageIndex{161})

(-3 x-2>-4 x-9)

Exercise (PageIndex{162})

(-3 x+3<-11 x-3)

**Answer**[egin{aligned} & − 3x + 3 < −11x − 3 Longrightarrow quad & − 3x + 11x < −3 − 3 Longrightarrow quad& 8x < −6 Longrightarrow quad &x < -frac{3}{4}end{aligned}]

Thus, the solution interval is ((−infty, −frac{3}{4}) = {x|x < −frac{3}{4}}).

Exercise (PageIndex{163})

(6 x+3<8 x+8)

In Exercises 13-50, solve the compound inequality. Express your answer in both interval and set notations, and shade the solution on a number line.

Exercise (PageIndex{164})

(2 x-1<4) or (7 x+1 geq-4)

**Answer**[egin{aligned} & 2x − 1 < 4 ext{ or } 7x + 1 geq −4 Longrightarrow quad & 2x < 5quad ext{or}quad 7x geq −5 Longrightarrow quad&x

For the union, shade anything shaded in either graph. The solution is the set of all real numbers ((−infty,infty)).

Exercise (PageIndex{165})

(-8 x+9<-3) and (-7 x+1>3)

Exercise (PageIndex{166})

(-6 x-4<-4) and (-3 x+7 geq-5)

**Answer**[egin{aligned} & − 6x − 4 < −4 ext{ and } − 3x + 7 geq −5 Longrightarrow quad & -6x < 0quad ext{and}quad -3x geq −12 Longrightarrow quad&x>0quad ext{and}quad xleq4 Longrightarrow quad & 0< x leq 4 end{aligned}]

The intersection is all points shaded in both graphs, so the solution is ((0, 4] = {x|0 < x leq 4}).

Exercise (PageIndex{167})

(-3 x+3 leq 8) and (-3 x-6>-6)

Exercise (PageIndex{168})

(8 x+5 leq-1) and (4 x-2>-1)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{169})

(-x-1<7) and (-6 x-9 geq 8)

Exercise (PageIndex{170})

(-3 x+8 leq-5) or (-2 x-4 geq-3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{171})

(-6 x-7<-3) and (-8 x geq 3)

Exercise (PageIndex{172})

(9 x-9 leq 9) and (5 x>-1)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-7 x+3<-3) or (-8 x geq 2)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(3 x-5<4) and (-x+9>3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-8 x-6<5) or (4 x-1 geq 3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(9 x+3 leq-5) or (-2 x-4 geq 9)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-7 x+6<-4) or (-7 x-5>7)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(4 x-2 leq 2) or (3 x-9 geq 3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-5 x+5<-4) or (-5 x-5 geq-5)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(5 x+1<-6) and (3 x+9>-4)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(7 x+2<-5) or (6 x-9 geq-7)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-7 x-7<-2) and (3 x geq 3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(4 x+1<0) or (8 x+6>9)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(7 x+8<-3) and (8 x+3 geq-9)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(3 x<2) and (-7 x-8 geq 3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-5 x+2 leq-2) and (-6 x+2 geq 3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(4 x-1 leq 8) or (3 x-9>0)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(2 x-5 leq 1) and (4 x+7>7)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(3 x+1<0) or (5 x+5>-8)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-8 x+7 leq 9) or (-5 x+6>-2)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(x-6 leq-5) and (6 x-2>-3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-4 x-8<4) or (-4 x+2>3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(9 x-5<2) or (-8 x-5 geq-6)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-9 x-5 leq-3) or (x+1>3)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-5 x-3 leq 6) and (2 x-1 geq 6)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-1 leq-7 x-3 leq 2)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(0<5 x-5<9)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(5<9 x-3 leq 6)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-6<7 x+3 leq 2)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-2<-7 x+6<6)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-9<-2 x+5 leq 1)

**Answer**Add texts here. Do not delete this text first.

In Exercises 51-62, solve the given inequality for x. Graph the solution set on a number line, then use interval and set-builder notation to describe the solution set.

Exercise (PageIndex{1})

(-frac{1}{3}

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-frac{1}{5}

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-frac{1}{2}

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-frac{2}{3} leq frac{1}{2}-frac{x}{5} leq frac{2}{3})

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-1

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-2

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-2

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-3

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(x<4-x<5)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-x<2 x+3 leq 7)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(-x

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

(−2x < 3 − x leq 8)

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

Aeron has arranged for a demonstration of “How to make a Comet” by Professor O’Commel. The wise professor has asked Aeron to make sure the auditorium stays between 15 and 20 degrees Celsius (C). Aeron knows the thermostat is in Fahrenheit (F) and he also knows that the conversion formula between the two temperature scales is C = (5/9)(F − 32).

a) Setting up the compound inequality for the requested temperature range in Celsius, we get (15 leq C leq 20). Using the conversion formula above, set up the corresponding compound inequality in Fahrenheit.

b) Solve the compound inequality in part (a) for F. Write your answer in set notation.

c) What are the possible temperatures (integers only) that Aeron can set the thermostat to in Fahrenheit?

**Answer**Add texts here. Do not delete this text first.

Exercise (PageIndex{1})

Add exercises text here.

**Answer**Add texts here. Do not delete this text first.

## Chapter 13

The level surfaces are spheres, centered at the origin, with radius c .

The level surfaces are paraboloids of the form z = x 2 c + y 2 c the larger c , the “wider” the paraboloid.

The level curves for each surface are similar for z = x 2 + 4 y 2 the level curves are ellipses of the form x 2 c 2 + y 2 c 2 / 4 = 1 , i.e., a = c and b = c / 2 whereas for z = x 2 + 4 y 2 the level curves are ellipses of the form x 2 c + y 2 c / 4 = 1 , i.e., a = c and b = c / 2 . The first set of ellipses are spaced evenly apart, meaning the function grows at a constant rate the second set of ellipses are more closely spaced together as c grows, meaning the function grows faster and faster as c increases.

The function z = x 2 + 4 y 2 can be rewritten as z 2 = x 2 + 4 y 2 , an elliptic cone the function z = x 2 + 4 y 2 is a paraboloid, each matching the description above.

## ML Aggarwal Class 8 Solutions for ICSE Maths

**APC Understanding ICSE Mathematics Class 8 ML Aggarwal Solutions 2019 Edition for 2020 Examinations**

ML Aggarwal Class 8 Maths Chapter 1 Rational Numbers

ML Aggarwal Class 8 Maths Chapter 2 Exponents and Powers

ML Aggarwal Class 8 Maths Chapter 3 Squares and Square Roots

ML Aggarwal Class 8 Maths Chapter 4 Cubes and Cube Roots

ML Aggarwal Class 8 Maths Chapter 5 Playing with Numbers

ML Aggarwal Class 8 Maths Chapter 6 Operation on sets Venn Diagrams

ML Aggarwal Class 8 Maths Chapter 7 Percentage

ML Aggarwal Class 8 Maths Chapter 8 Simple and Compound Interest

ML Aggarwal Class 8 Maths Chapter 9 Direct and Inverse Variation

ML Aggarwal Class 8 Maths Chapter 10 Algebraic Expressions and Identities

ML Aggarwal Class 8 Maths Chapter 11 Factorisation

ML Aggarwal Class 8 Maths Chapter 12 Linear Equations and Inequalities in One Variable

## RD Sharma Class 8 Solutions (2020-201 Edition)

Get Latest Edition of RD Sharma Class 8 Solutions Pdf Download on LearnInsta.com. It provides step by step solutions RD Sharma Class 8 Solutions Pdf Download. You can download the RD Sharma Class 8 Solutions with Free PDF download option, which contains chapter wise solutions. In RD Sharma Solutions for Class 8 all questions are solved and explained by expert Mathematic teachers as per CBSE board guidelines. By studying these RD Sharma Class 8 Solutions you can easily get good marks in CBSE Class 8 Examinations.

You can also Download NCERT Solutions for Class 8 Maths to help you to revise complete Syllabus and score more marks in your examinations.

RD Sharma Solutions Class 8 is given here to help students of class 8 while solving the exercise problems. Also RD Sharma Solutions Class 8 is useful for various entrance/ competitive examination preparation. Here you will get all RD Sharma Class 8 Solutions PDF for free.

### RD Sharma Class 8 Solutions 2020 Edition for 2021 Examinations

RD Sharma Solutions for Class 8 are given below for all chapter wise. Select chapter number to view solutions exercise wise.

**Ex 5.1 Class 12 Maths Question 1.**

Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5. **Solution:**

(i) At x = 0. lim_{x–>0} f (x) = lim_{x–>0} (5x – 3) = – 3 and

f(0) = – 3

∴f is continuous at x = 0

(ii) At x = – 3, lim_{x–>3} f(x)= lim_{x–>-3} (5x – 3) = – 18

and f( – 3) = – 18

∴ f is continuous at x = – 3

(iii) At x = 5, lim_{x–>5} f(x) = lim_{x–>5} (5x – 3) = 22 and

f(5) = 22

∴ f is continuous at x = 5

**Ex 5.1 Class 12 Maths Question 2.**

Examine the continuity of the function f(x) = 2x² – 1 at x = 3. **Solution:**

lim_{x–>3} f(x) = lim_{x–>3} (2x² – 1) = 17 and f(3)= 17

∴ f is continuous at x = 3

**Ex 5.1 Class 12 Maths Question 3.**

Examine the following functions for continuity.

(a) f(x) = x – 5

(b) f(x) = , x≠5

(c) f(x) = ,x≠5

(d) f(x) = |x – 5| **Solution:**

(a) f(x) = (x-5) => (x-5) is a polynomial

∴it is continuous at each x ∈ R.

**Ex 5.1 Class 12 Maths Question 4.**

Prove that the function f (x) = x n is continuous at x = n, where n is a positive integer. **Solution:**

f (x) = x n is a polynomial which is continuous for all x ∈ R.

Hence f is continuous at x = n, n ∈ N.

**Ex 5.1 Class 12 Maths Question 5.**

Is the function f defined by continuous at x = 0? At x = 1? At x = 2? **Solution:**

(i) At x = 0

lim_{x–>0-} f(x) = lim_{x–>0-} x = 0 and

lim_{x–>0+} f(x) = lim_{x–>0+} x = 0 => f(0) = 0

∴ f is continuous at x = 0

(ii) At x = 1

lim_{x–>1-} f(x) = lim_{x–>1-} (x) = 1 and

lim_{x–>1+} f(x) = lim_{x–>1+}(x) = 5

∴ lim_{x–>1-} f(x) ≠ lim_{x–>1+} f(x)

∴ f is discontinuous at x = 1

(iii) At x = 2

lim_{x–>2} f(x) = 5, f(2) = 5

∴ f is continuous at x = 2

**Find all points of discontinuity off, where f is defined by**

**Ex 5.1 Class 12 Maths Question 6.**

**Solution:**

at x≠2

**Ex 5.1 Class 12 Maths Question 7.**

**Ex 5.1 Class 12 Maths Question 8.**

Test the continuity of the function f (x) at x = 0

**Solution:**

We have

(LHL at x=0)

**Ex 5.1 Class 12 Maths Question 9.**

**Ex 5.1 Class 12 Maths Question 10.**

**Ex 5.1 Class 12 Maths Question 11.**

At x = 2, L.H.L. lim_{x–>2-} (x³ – 3) = 8 – 3 = 5

R.H.L. = lim_{x–>2+} (x² + 1) = 4 + 1 = 5

**Ex 5.1 Class 12 Maths Question 12.**

**Ex 5.1 Class 12 Maths Question 13.**

Is the function defined by a continuous function? **Solution:**

At x = 1,L.H.L.= lim_{x–>1-} f(x) = lim_{x–>1-} (x + 5) = 6,

R.HL. = lim_{x–>1+} f(x) = lim_{x–>1+} (x – 5) = – 4

f(1) = 1 + 5 = 6,

f(1) = L.H.L. ≠ R.H.L.

=> f is not continuous at x = 1

At x = c < 1, lim_{x–>c} (x + 5) = c + 5 = f(c)

At x = c > 1, lim_{x–>c} (x – 5) = c – 5 = f(c)

∴ f is continuous at all points x ∈ R except x = 1.

**Discuss the continuity of the function f, where f is defined by**

**Ex 5.1 Class 12 Maths Question 14.**

In the interval 0 ≤ x ≤ 1,f(x) = 3 f is continuous in this interval.

At x = 1,L.H.L. = lim f(x) = 3,

R.H.L. = lim_{x–>1+} f(x) = 4 => f is discontinuous at

x = 1

At x = 3, L.H.L. = lim_{x–>3-} f(x)=4,

R.H.L. = lim_{x–>3+} f(x) = 5 => f is discontinuous at

x = 3

=> f is not continuous at x = 1 and x = 3.

**Ex 5.1 Class 12 Maths Question 15.**

At x = 0, L.H.L. = lim_{x–>0-} 2x = 0 ,

R.H.L. = lim_{x–>0+} (0)= 0 , f(0) = 0

=> f is continuous at x = 0

At x = 1, L.H.L. = lim_{x–>1-} (0) = 0,

R.H.L. = lim_{x–>1+} 4x = 4

f(1) = 0, f(1) = L.H.L.≠R.H.L.

∴ f is not continuous at x = 1

when x < 0 f (x) = 2x, being a polynomial, it is

continuous at all points x < 0. when x > 1. f (x) = 4x being a polynomial, it is

continuous at all points x > 1.

when 0 ≤ x ≤ 1, f (x) = 0 is a continuous function

the point of discontinuity is x = 1.

**Ex 5.1 Class 12 Maths Question 16.**

At x = – 1,L.H.L. = lim_{x–>1-} f(x) = – 2, f(-1) = – 2,

R.H.L. = lim_{x–>1+} f(x) = – 2

=> f is continuous at x = – 1

At x= 1, L.H.L. = lim_{x–>1-} f(x) = 2,f(1) = 2

∴ f is continuous at x = 1,

R.H.L. = lim_{x–>1+} f(x) = 2

Hence, f is continuous function.

**Ex 5.1 Class 12 Maths Question 17.**

Find the relationship between a and b so that the function f defined by

is continuous at x = 3 **Solution:**

At x = 3, L.H.L. = lim_{x–>3-} (ax+1) = 3a+1 ,

f(3) = 3a + 1, R.H.L. = lim_{x–>3+} (bx+3) = 3b+3

f is continuous ifL.H.L. = R.H.L. = f(3)

3a + 1 = 3b + 3 or 3(a – b) = 2

a – b = or a = b + , for any arbitrary value of b.

Therefore the value of a corresponding to the value of b.

**Ex 5.1 Class 12 Maths Question 18.**

For what value of λ is the function defined by

continuous at x = 0? What about continuity at x = 1? **Solution:**

At x = 0, L.H.L. = lim_{x–>0-} λ (x² – 2x) = 0 ,

R.H.L. = lim_{x–>0+} (4x+ 1) = 1, f(0)=0

f (0) = L.H.L. ≠ R.H.L.

=> f is not continuous at x = 0,

whatever value of λ ∈ R may be

At x = 1, lim_{x–>1} f(x) = lim_{x–>1} (4x + l) = f(1)

=> f is not continuous at x = 0 for any value of λ but f is continuous at x = 1 for all values of λ.

**Ex 5.1 Class 12 Maths Question 19.**

Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. **Solution:**

Let c be an integer, [c – h] = c – 1, [c + h] = c, [c] = c, g(x) = x – [x].

At x = c, lim_{x–>c-} (x – [x]) = lim_{h–>0} [(c – h) – (c – 1)]

= lim_{h–>0} (c – h – (c – 1)) = 1[∵ [c – h] = c – 1]

R.H.L. = lim_{x–>c+} (x – [x])= lim_{h–>0} (c + h – [c + h])

= lim_{h–>0} [c + h – c] = 0

f(c) = c – [c] = 0,

Thus L.H.L. ≠ R.H.L. = f (c) => f is not continuous at integral points.

**Ex 5.1 Class 12 Maths Question 20.**

Is the function defined by f (x) = x² – sin x + 5 continuous at x = π? **Solution:**

Let f(x) = x² – sinx + 5,

**Ex 5.1 Class 12 Maths Question 21.**

Discuss the continuity of the following functions:

(a) f (x) = sin x + cos x

(b) f (x) = sin x – cos x

(c) f (x) = sin x · cos x **Solution:**

(a) f(x) = sinx + cosx

**Ex 5.1 Class 12 Maths Question 22.**

Discuss the continuity of the cosine, cosecant, secant and cotangent functions. **Solution:**

(a) Let f(x) = cosx

**Ex 5.1 Class 12 Maths Question 23.**

Find all points of discontinuity of f, where

**Solution:**

At x = 0

**Ex 5.1 Class 12 Maths Question 24.**

Determine if f defined by is a continuous function? **Solution:**

At x = 0

**Ex 5.1 Class 12 Maths Question 25.**

Examine the continuity of f, where f is defined by **Solution:**

**Find the values of k so that the function is continuous at the indicated point in Questions 26 to 29.**

**Ex 5.1 Class 12 Maths Question 26.**

**Solution:**

At x =

L.H.L =

**Ex 5.1 Class 12 Maths Question 27.**

**Ex 5.1 Class 12 Maths Question 28.**

**Solution:**

**Ex 5.1 Class 12 Maths Question 29.**

**Ex 5.1 Class 12 Maths Question 30.**

Find the values of a and b such that the function defined by

to is a continuous function. **Solution:**

**Ex 5.1 Class 12 Maths Question 31.**

Show that the function defined by f(x)=cos (x²) is a continuous function. **Solution:**

Now, f (x) = cosx², let g (x)=cosx and h (x) x²

∴ goh(x) = g (h (x)) = cos x²

Now g and h both are continuous ∀ x ∈ R.

f (x) = goh (x) = cos x² is also continuous at all x ∈ R.

**Ex 5.1 Class 12 Maths Question 32.**

Show that the function defined by f (x) = |cos x| is a continuous function. **Solution:**

Let g(x) =|x|and h (x) = cos x, f(x) = goh(x) = g (h (x)) = g (cosx) = |cos x |

Now g (x) = |x| and h (x) = cos x both are continuous for all values of x ∈ R.

∴ (goh) (x) is also continuous.

Hence, f (x) = goh (x) = |cos x| is continuous for all values of x ∈ R.

**Ex 5.1 Class 12 Maths Question 33.**

Examine that sin |x| is a continuous function. **Solution:**

Let g (x) = sin x, h (x) = |x|, goh (x) = g (h(x))

= g(|x|) = sin|x| = f(x)

Now g (x) = sin x and h (x) = |x| both are continuous for all x ∈ R.

∴f (x) = goh (x) = sin |x| is continuous at all x ∈ R.

**Ex 5.1 Class 12 Maths Question 34.**

Find all the points of discontinuity of f defined by f(x) = |x|-|x+1|. **Solution:**

f(x) = |x|-|x+1|, when x< – 1,

f(x) = -x-[-(x+1)] = – x + x + 1 = 1

when -1 ≤ x < 0, f(x) = – x – (x + 1) = – 2x – 1,

when x ≥ 0, f(x) = x – (x + 1) = – 1

We hope the NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.1 help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.1, drop a comment below and we will get back to you at the earliest.

### Samacheer Kalvi 10th Maths Guide Pdf Download 2020 English Medium

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## Samacheer Kalvi 10th Maths Guide Pdf Free Download

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## Samacheer Kalvi 7th Maths Guide Pdf Free Download

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## Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board chapter 1 - Two Gentlemen of Verona [Latest edition]

Who did the narrator meet at the outskirts of Verona?

Why did the driver not approve of the narrator buying fruits from the boys?

The boys did not spend much on clothes and food. Why?

Were the boys saving money to go to the States? How do you know?

Why did the author avoid going to Lucia&rsquos room?

What was Lucia suffering from?

What made the boys join the resistance movement against the Germans?

What made the boys work so hard?

Why didn&rsquot the boys disclose their problem to the author?

#### Answer the following questions in three or four sentences each.

Describe the appearance of Nicola and Jacopo.

What were the various jobs undertaken by the little boys?

How did the narrator help the boys on Sunday?

Who took the author to the cubicle?

Describe the girl with whom the boys were talking to in the cubicle.

Recount the untold sufferings undergone by the siblings after they were rendered homeless.

The narrator did not utter a word and preferred to keep the secret to himself. Why? Substantiate the statement with reference to the story

#### Answer the following in a paragraph of 100–150 words each.

What was the driving force that made the boys do various jobs?

How was the family affected by the war?

Write a character sketch of Nicola and Jacopo.

What message is conveyed through the story &lsquoTwo Gentlemen of Verona&rsquo?

Justify the title of the story &lsquoTwo Gentlemen of Verona&rsquo

Adversity brings out the best as well as the worst in people. Elucidate this statement with reference to the story

Which character do you like the most in the story and why?

### Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board Chapter 1 Two Gentlemen of VeronaVocabulary [Pages 6 - 7]

**Read the following words taken from the story. Give two synonyms and one antonym for each of these words. Use a dictionary, if required.**

Word | Synonyms | Antonym | |

cautious | |||

disapprove | |||

brisk | |||

engaging | |||

humble | |||

eager | |||

resistance | |||

persuade | |||

scarce | |||

nobility |

**Given below is a list of common confusable. Distinguish the meaning of each pair of word by framing your own sentence.**

**Given below is a list of common confusable. Distinguish the meaning of each pair of word by framing your own sentence.**

**Given below is a list of common confusable. Distinguish the meaning of each pair of word by framing your own sentence.**

**Fill in the blanks with suitable homophones or confusables.**

wallet | ______ | hoard | ______ |

fairy | ______ | desert | ______ |

medal | ______ | night | ______ |

wait | ______ | sweet | ______ |

yoke | ______ | plain | ______ |

grown | ______ | might | ______ |

earn | ______ | quite | ______ |

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Give the meaning of the following phrasal verb and frame sentence using them**

**Read the list of words formed by adding suffixes.**

frequently | satisfaction | willingness |

comfortable | resemblance | nobility |

**Form two derivatives from each of the following words by adding prefixes and suffixes.**

Word | Prefix | Suffix |

patient | patient | patiently |

honour | ||

respect | ||

manage | ||

fertile | ||

different | ||

friend | ||

obey |

### Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board Chapter 1 Two Gentlemen of VeronaListening [Page 7]

**Now, you are going to listen to the cautionary instructions that are given to the general public living in flood-prone areas. Listen carefully and complete the following sentences.**

Floods are an inevitable natural disaster that can occur in any part of the world. Floods can prove all the more disastrous in localities, where population density is high. Preparation for Disaster Management has become imperative for any city, town, or village during monsoons. The Government Department entrusted with Disaster Management makes periodic announcements about the precautions to be taken whenever floods are anticipated.

Now, you are going to listen to the cautionary instructions that are given to the general public living in flood-prone areas. Listen carefully and complete the following sentences. For the attention of the public, here is an announcement from the Department of Disaster Management. As per the warning issued today by the Meteorological Department, there exists a high probability of a widespread heavy downpour from the early hours of Sunday and consequent flooding of low-lying areas. In order to ensure the safety of life and property, everyone is hereby warned and advised to take certain precautionary measures:

- First of all, prepare a household flood plan and be ready to respond to the situation.
- Find out the locations of the closest flood shelters available and routes to reach them.
- Maintain an emergency kit comprising water bottles, biscuit packets, medical supplies, a torchlight, and a whistle to signal for help.
- Paste or fix a list of emergency telephone numbers on the wall in a visible spot.
- Switch off hazardous items like gas cylinders and disconnect electrical gadgets.
- Secure important personal documents and valuables in a waterproof case and place it in an accessible location.
- Place small pieces of furniture and clothing on tables and cots.
- Shift all the small objects safely to the loft.
- Empty your refrigerators and leave their doors open to avoid damage in case they float.
- Charge your mobile phones as well as your battery banks so as to communicate with friends, relatives, and emergency services.
- Place sandbags in the toilet bowls and bathroom drain holes to prevent sewage inflow.
- Prepare and pack food with a long shelf life.
- Finally, listen to the periodic news updates through your portable communication devices and follow the instructions implicitly.

- The announcement was made by the Department of ______.
- Widespread heavy rains are expected from the early hours of ______.
- The public is asked to find out the locations of ______.
- An emergency kit should contain water bottles, biscuit packets, and a ______.
- A list of ______should be displayed on the wall. f) Important documents can be secured by keeping them in a______ case.
- Damage to refrigerators can be avoided by ______.
- Mobile phones should be charged to enable the marooned to contact their friends, relatives and ______.
- ______should be placed in the toilet bowls to prevent sewage inflow.
- Listen to the ______and follow the instructions implicitly.

### Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board Chapter 1 Two Gentlemen of VeronaSpeaking [Pages 8 - 9]

#### Task 1

On the occasion of World Environment Day, you have been asked to deliver a speech during morning assembly on the importance of tree planting. Write the speech in about 100 &ndash 150 words.

- Introduction
- Suggested value points Pollution control &ndash Medicine &ndash Necessary for wildlife &ndash Cause rainfall
- Conclusion

#### Task 2

Prepare a speech on &ldquoThe importance of a reading habit&rdquo in about 100&ndash150 words using the hints given below together with your own ideas.

- Introduction
- Suggested value points Knowledge enrichment &ndash Skill development &ndash Meaningful usage of time &ndash Overall development
- Conclusion

### Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board Chapter 1 Two Gentlemen of VeronaReading [Page 9]

**Read the passage given below and make notes.**

To match the best cities across the world, the Government of India initiated &lsquosmart cities&rsquo to drive economic growth and improve the quality of life of people. The agenda under smart city promises to resolve urban sustainability problems. Urban forests provide a range of important ecosystem services that are critical for the sustainability of cities. Urban forestry, which is defined more as &lsquoManagement of Trees&rsquo contributes to the physiological, sociological and economic well-being of the society. Mangroves, lakes, grasslands, and forests in and around our cities, act as sponges that absorb the air and noise pollution and they present themselves as our cultural and recreational hotspots. However, these spots are rapidly being reclaimed and replaced in the name of development. Presence of urban green has shown to increase the economic value of the place.

Urban forests contribute to reduce the cost of building storm water drain systems for municipalities and neutralizing urban heat island effect. Plants not only provide shade but also help in regulating the micro-climate. They help regulate energy budgets, improve air quality, and curtail noise pollution. Trees, herbs, shrubs, and grasses arrest sedimentation and prevent other pollutants from entering our water systems. This will give a chance for our urban lakes and rivers to recover and help improve aquatic ecosystems. Biodiversity also gets a boost through the urban forestsand helps create corridors connecting the forest areas. High biodiversity areas can also help to build resilient ecosystems. Availability of forests within our urban areas gives an opportunity for children to connect to the natural environment and learn about native species.

### Samacheer Kalvi solutions for English Class 12th HSC Tamil Nadu State Board Chapter 1 Two Gentlemen of VeronaGrammar- Tenses [Pages 11 - 12]

#### Task 1

**Choose the correct options and complete the dialogue.**

A | : Hello. What do you watch/are you watching? |

B | A programme about the Jallian Wala Bagh Massacre, which I recorded last night. I study/ I&rsquom studying about it this term. |

A | All that I know/ I&rsquove known about it is that hundreds of people died/had died in it. |

B | Yes, it was much, much worse than anyone has expected/had expected. It went on/has gone on for hours. Do you want/Have you wanted to watch the programme with me? |

A | No, thanks. I&rsquove got to do some veena practice. I&rsquove just remembered/I just remembered that we&rsquove got a concert tomorrow, and I don&rsquot have/ haven&rsquot had time to practise my new piece this week. |

B | OK. I&rsquove already done / I already did my practice, so I&rsquove got time to watch TV. See you later. |

#### Task 2

**Complete the sentence with the correct tense form of the verb in brackets.**

(tell) me exactly what (happen) last night!

**Complete the sentence with the correct tense form of the verb in bracket.**

Mrs. Mageswari is my Maths teacher. She (teach) me for four years.

**Complete the sentence with the correct tense form of the verb in bracket.**

I (never) think of a career in medicine before I spoke to my Biology teacher but now I think (seriously) it.

**Complete the sentence with the correct tense form of the verb in bracket.**

Oh no! I (forget) to bring my assignment! What am I going to do? This is the second time I (do) this!

**Complete the sentence with the correct tense form of the verb in bracket.**

I can&rsquot remember what my teacher (say) yesterday about our homework. I (not listen) properly because Hussain (talk) to me at the same time.

**Complete the sentence with the correct tense form of the verb in bracket.**

Last year we (go) on a school trip to Kanyakumari. We (have) a very interesting time.

**Complete the sentence with the correct tense form of the verb in bracket.**

At the moment I (think) about what course to pursue next year but I ( not make) a final decision yet.

**Complete the sentence with the correct tense form of the verb in bracket.**

I (get) up at 7 every morning but this morning I (sleep) for a long time and I (not get) up until 8.

#### Task 3

**Fill in the blank with the correct form of the verb given in the bracket.**

Everyone______ when the earthquake hit the small town. (sleep)

**Fill in the blank with the correct form of the verb given in the bracket.**

Evangelene ______ her job a couple of years ago. (quit)

**Fill in the blank with the correct form of the verb given in the bracket.**

Where ______ your last holidays? (you spend)

**Fill in the blank with the correct form of the verb given in the bracket.**

I think Suresh ______ for Tiruvallur next morning. (leave)

**Fill in the blank with the correct form of the verb given in the bracket.**

I was angry that I ______ such a mistake (make).

**Fill in the blank with the correct form of the verb given in the bracket.**

My mother was tired yesterday because she ______well the night before (not sleep).

**Fill in the blank with the correct form of the verb given in the bracket.**

Her parents ______ in Coimbatore for two weeks from today (be).

**Fill in the blank with the correct form of the verb given in the bracket.**

Nothing much ______when I got to the meeting (happen).

**Fill in the blank with the correct form of the verb given in the bracket.**

Scientists predict that by 2050, man______ on Mars. (land)

**Fill in the blank with the correct form of the verb given in the bracket.**

Sh! Someone______ to our conversation! (listen)

**Fill in the blank with the correct form of the verb given in the bracket.**

The plane ______off in a few minutes. (take)

**Fill in the blank with the correct form of the verb given in the bracket.**

They ______about me when I interrupted their conversation. (talk)

**Fill in the blank with the correct form of the verb given in the bracket.**

Justin and his parents ______ in an apartment right now because they can&rsquot find a cheap house. (live)

**Fill in the blank with the correct form of the verb given in the bracket.**

Rajini Prem&rsquos family ______ in Chengalpet now. (be).

**Fill in the blank with the correct form of the verb given in the bracket.**

Yusuf ______ to the movies once in a while (go)

**Fill in the blank with the correct form of the verb given in the bracket.**

This ______an easy quiz so far (be).

**Fill in the blank with the correct form of the verb given in the bracket.**

Our team ______ any games last year. (not win)

**Fill in the blank with the correct form of the verb given in the bracket.**

We ______ a wonderful film at the cinema last night. (see)

**Fill in the blank with the correct form of the verb given in the bracket.**

## Highly Effective Class 10 Maths Preparation Tips

Here we are providing few preparation tips for Class 10. You can boost your mark in the Mathematics board exam results. You should understand and go through the study material regularly, so your entire syllabus can be covered well in time before the exam.

- Before starting your preparations, you should know the syllabus of Class 10 and prepare a study plan according to the Class 10 Maths Syllabus. Prioritize your toughest section first and cover all syllabus in detail before the exam with RD Sharma Class 10 Solutions.
- You should maintain a separate sheet of important formulas, theorems, and their derivations. It will help you during last-minute preparations. Also, you need to revise them regularly to avoid forgetting concepts during the examination time.
- You should solve as many as possible questions based on the practical application of concepts. For better preparations, refer to RD Sharma Class 10 Solutions.
- You have to practice the previous 5 to 6 years of question papers. By solving those, you will become more familiar with the exam pattern and learn time management skills while solving questions. It will also help you to enhance the speed and accuracy of solving a problem.

We hope this article on RD Sharma Class 10 Solutions If you have any queries, you may let us know through the comments section below. Keep visiting our blog for more updates.