# 2.9: Exercise Supplement - Mathematics

## Exercise Supplement

### Symbols and Notations

For the following problems, simplify the expressions.

Exercise (PageIndex{1})

(12 + 7(4 + 3))

(61)

Exercise (PageIndex{2})

(9(4 - 2) + 6(8 + 2) - 3(1 + 4))

Exercise (PageIndex{3})

(6[1 + 8(7 + 2)])

(438)

Exercise (PageIndex{4})

(26 div 2 - 10)

Exercise (PageIndex{5})

(dfrac{(4+17+1)+4}{14-1})

(2)

Exercise (PageIndex{6})

(51 div 3 div 7)

Exercise (PageIndex{7})

((4 + 5)(4 + 6) - (4 + 7))

(79)

Exercise (PageIndex{8})

(8(2 cdot 12 div 13) + 2 cdot 5 cdot 11 - [1 + 4(1 + 2)])

Exercise (PageIndex{9})

(dfrac{3}{4} + dfrac{1}{12}(dfrac{3}{4} - dfrac{1}{2}))

(dfrac{37}{47})

Exercise (PageIndex{10})

(48 - 3[dfrac{1 + 17}{6}])

Exercise (PageIndex{11})

(dfrac{29 + 11}{6 - 1})

(8)

Exercise (PageIndex{12})

(dfrac{dfrac{88}{11} + dfrac{99}{9} + 1}{dfrac{54}{9} - dfrac{22}{11}})

Exercise (PageIndex{13})

(dfrac{8 cdot 6}{2} + dfrac{9 cdot 9}{3} dfrac{10 cdot 4}{5})

(43)

For the following problems, write the appropriate relation symbol (=,<,>) in place of the ∗.

Exercise (PageIndex{14})

(22 * 6)

Exercise (PageIndex{15})

(9[4 + 3(8)] * 6[1 + 8(5)])

(252 > 246)

Exercise (PageIndex{16)

(3(1.06 + 2.11) * 4(11.01 - 9.06))

Exercise (PageIndex{17})

(2 * 0)

(2 > 0)

For the following problems, state whether the letters or symbols are the same or different.

Exercise (PageIndex{18})

(<) and ( ot ge)

Exercise (PageIndex{19})

(>) and ( ot <)

Different

Exercise (PageIndex{20})

(a = b) and (b = a)

Exercise (PageIndex{21})

Represent the sum of (c) and (d) two different ways.

(c + d) ; (d + c)

For the following problems, use algebraic notataion.

Exercise (PageIndex{22})

(8) plus (9)

Exercise (PageIndex{23})

(62) divided by (f)

(dfrac{62}{f}) or (62 div f)

Exercise (PageIndex{24})

(8) times ((x + 4))

Exercise (PageIndex{25})

(6) times (x), minus (2)

(6x - 2)

Exercise (PageIndex{26})

(x + 1) divided by (x - 3)

Exercise (PageIndex{27})

(y + 11) divided by (y + 10), minus (12)

((y + 11) div (y + 10) - 12) or (dfrac{y + 11}{y + 10} - 12)

Exercise (PageIndex{28})

zero minus (a) times (b)

### The Real Number Line and the Real Numbers

Exercise (PageIndex{29})

Is every natural number a whole number?

Yes

Exercise (PageIndex{30})

Is every rational number a real number?

For the following problems, locate the numbers on a number line by placing a point at their (approximate) position.

Exercise (PageIndex{31})

(2) Exercise (PageIndex{32})

(3.6)

Exercise (PageIndex{33})

(-1dfrac{3}{8}) Exercise (PageIndex{34})

(0)

Exercise (PageIndex{35})

(-4dfrac{1}{2}) Exercise (PageIndex{36})

Draw a number line that extends from 10 to 20. Place a point at all odd integers.

Exercise (PageIndex{37})

Draw a number line that extends from (−10) to (10). Place a point at all negative odd integers and at all even positive integers. Exercise (PageIndex{38})

Draw a number line that extends from (−5) to (10). Place a point at all integers that are greater then or equal to (−2) but strictly less than (5).

Exercise (PageIndex{39})

Draw a number line that extends from (−10) to (10). Place a point at all real numbers that are strictly greater than (−8) but less than or equal to (7). Exercise (PageIndex{40})

Draw a number line that extends from (−10) to (10). Place a point at all real numbers between and including (−6) and (4).

For the following problems, write the appropriate relation symbol (=,<,>).

Exercise (PageIndex{41})

(-3) (0)

(-3 < 0)

Exercise (PageIndex{42})

(-1) (1)

Exercise (PageIndex{43})

(-8) (-5)

(-8 < -5)

Exercise (PageIndex{44})

(-5) (-5dfrac{1}{2})

Exercise (PageIndex{45})

Is there a smallest two digit integer? If so, what is it?

Yes, (-99)

Exercise (PageIndex{46})

Is there a smallest two digit real number? If so, what is it?

For the following problems, what integers can replace x so that the statements are true?

Exercise (PageIndex{47})

(4 le x le 7)

(4, 5, 6) or (7)

Exercise (PageIndex{48})

(-3 le x < 1)

Exercise (PageIndex{49})

(-3) (0)

(-3 < 0)

Exercise (PageIndex{50})

(-3 < x le 2)

(-2, -1, 0, 1), or (2)

Exercise (PageIndex{51})

The temperature today in Los Angeles was eighty-two degrees. Represent this temperature by real number.

Exercise (PageIndex{52})

The temperature today in Marbelhead was six degrees below zero. Represent this temperature by real number.

(-6°)

Exercise (PageIndex{53})

On the number line, how many units between (-3) and (2)?

(-3 < 0)

Exercise (PageIndex{54})

On the number line, how many units between (-4) and (0)?

(4)

## Properties of the Real Numbers

Exercise (PageIndex{55})

(a + b = b + a) is an ilustration of the property of addition.

Exercise (PageIndex{56})

(st = ts) is an illustration of the _________ property of __________.

commutative, multiplication

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems.

Exercise (PageIndex{57})

(y + 12)

Exercise (PageIndex{58})

(a + 4b)

(4b + a)

Exercise (PageIndex{59})

(6x)

Exercise (PageIndex{60})

(2(a-1))

((a-1)2)

Exercise (PageIndex{61})

((-8)(4))

Exercise (PageIndex{62})

((6)(-9)(-2))

((-9)(6)(-2)) or ((-9)(-2)(6)) or ((6)(-2)(-9)) or ((-2)(-9)(6))

Exercise (PageIndex{63})

((x + y)(x - y))

Exercise (PageIndex{64})

(△ cdot ⋄)

( ⋄cdot △)

Simplify the following problems using the commutative property of multiplication. You need not use the distributive property.

Exercise (PageIndex{65})

(8x3y)

Exercise (PageIndex{66})

(16ab2c)

(32abc)

Exercise (PageIndex{67})

(4axyc4d4e)

Exercise (PageIndex{68})

(3(x+2)5(x−1)0(x+6))

(0)

Exercise (PageIndex{69})

(8b(a−6)9a(a−4))

For the following problems, use the distributive property to expand the expressions.

Exercise (PageIndex{70})

(3(a + 4))

(3a + 12)

Exercise (PageIndex{71})

(a(b + 3c))

Exercise (PageIndex{72})

(2g(4h + 2k)

(8gh+4gk)

Exercise (PageIndex{73})

((8m+5n)6p)

Exercise (PageIndex{74})

(3y(2x+4z+5w))

(6xy+12yz+15wy)

Exercise (PageIndex{75})

((a+2)(b+2c))

Exercise (PageIndex{76})

((x+y)(4a+3b))

(4ax+3bx+4ay+3by)

Exercise (PageIndex{77})

(10a_z(b_z + c))

### Exponents

For the following problems, write the expressions using exponential notation.

Exercise (PageIndex{78})

(x) to the fifth.

(x^5)

Exercise (PageIndex{79})

(y + 2) cubed.

Exercise (PageIndex{80})

((a+2b)) squared minus ((a+3b)) to the fourth.

((a + 2b)^2 - (a + 3b)^4)

Exercise (PageIndex{81})

(x) cubed plus (2) times ((y−x)) to the seventh.

Exercise (PageIndex{82})

(aaaaaaa)

(a^7)

Exercise (PageIndex{83})

(2 cdot 2 cdot 2 cdot 2)

Exercise (PageIndex{84})

((−8)(−8)(−8)(−8)xxxyyyyy)

((-8)^4x^3y^5)

Exercise (PageIndex{85})

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

Exercise (PageIndex{86})

(2zzyzyyy + 7zzyz(a - 6)^2(a-6))

(2y^4z^3 + 7yz^3(a-6)^3)

For the following problems, expand the terms so that no exponents appear.

Exercise (PageIndex{87})

(x^3)

Exercise (PageIndex{88})

(3x^3)

(3xxx)

Exercise (PageIndex{89})

(7^3x^2)

Exercise (PageIndex{90})

((4b)^2)

(4b cdot 4b)

Exercise (PageIndex{91})

((6a^2)^3(5c-4)^2)

Exercise (PageIndex{92})

((x^3+7)^2(y^2-3)^3(z+10))

((xxx+7)(xxx+7)(yy−3)(yy−3)(yy−3)(z+10))

Exercise (PageIndex{93})

Choose values for (a) and (b) to show that:

a. (a+b)^2) is not always equal to (a^2 + b^2)

b. ((a+b)^2) may be equal to (a^2 + b^2)

Exercise (PageIndex{94})

Choose value for (x) to show that

a. ((4x)^2) is not always equal to (4x^2).

b. ((4x)^2) may be equal to (4x^2)

(a) any value except zero

(b) only zero

### Rules of Exponents - The Power Rules for Exponents

Simplify the following problems.

Exercise (PageIndex{95})

(4^2 + 8)

Exercise (PageIndex{96})

(6^3 + 5(30))

(366)

Exercise (PageIndex{97})

(1^8 + 0^{10} + 3^2(4^2 + 2^3))

Exercise (PageIndex{98})

(12^2 + 0.3(11)^2)

(180.3)

Exercise (PageIndex{99})

(dfrac{3^4 + 1}{2^2 + 4^2 + 3^2})

Exercise (PageIndex{100})

(dfrac{6^2 + 3^2}{2^2 + 1} + dfrac{(1+4)^2 - 2^3 - 1^4}{2^5-4^2})

(10)

Exercise (PageIndex{101})

(a^4a^3)

Exercise (PageIndex{102})

(2b^52b^3)

(4b^8)

Exercise (PageIndex{103})

(4a^3b^2c^8 cdot 3ab^2c^0)

Exercise (PageIndex{104})

((6x^4y^{10})(xy^3))

(6x^5y^{13})

Exercise (PageIndex{105})

((3xyz^2)(2x^2y^3)(4x^2y^2z^4))

Exercise (PageIndex{106})

((3a)^4)

(81a^4)

Exercise (PageIndex{107})

((10xy)^2)

Exercise (PageIndex{108})

((x^2y^4)^6)

(x^{12}y^{24})

Exercise (PageIndex{109})

((a^4b^7c^7z^{12})^9)

Exercise (PageIndex{110})

((dfrac{3}{4}x^8y^6z^0a^{10}b^{15})^2)

(dfrac{9}{16}x^{16}y^{12}a^{20}b^{30})

Exercise (PageIndex{111})

(dfrac{14a^4b^6c^7}{2ab^3c^2})

(7a^3b^3c^5)

Exercise (PageIndex{112})

(dfrac{11x^4}{11x^4})

Exercise (PageIndex{113})

(x^4 cdot dfrac{x^{10}}{x^3})

(x^{11})

Exercise (PageIndex{114})

(a^3b^7 cdot dfrac{a^9b^6}{a^5b^{10}})

Exercise (PageIndex{115})

(dfrac{(x^4y^6z^{10})^4}{(xy^5z^7)^3})

(x^{13}y^9z^{19})

Exercise (PageIndex{116})

(dfrac{(2x-1)^{13}(2x+5)^5}{(2x-1)^{10}(2x+5)})

Exercise (PageIndex{117})

((dfrac{3x^2}{4y^3})^2)

(dfrac{9x^4}{16y^6})

Exercise (PageIndex{118})

(dfrac{(x+y)^9(x-y)^4}{(x+y)^3})

Exercise (PageIndex{119})

(x^n cdot x^m)

(x^{n+m})

Exercise (PageIndex{120})

(a^{n+2}a^{n+4})

Exercise (PageIndex{121})

(6b^{2n+7} cdot 8b^{5n+2})

(48b^{7n+9})

Exercise (PageIndex{122})

(dfrac{18x^{4n+9}}{2x^{2n+1}})

Exercise (PageIndex{123})

((x^{5t}y^{4r})^7)

(x^{35t}y^{28r})

Exercise (PageIndex{124})

((a^{2n}b^{3m}c^{4p})^{6r})

Exercise (PageIndex{125})

(dfrac{u^w}{u^k})

(u^{w-k})

## 2.9: Exercise Supplement - Mathematics

Math 275 is an introduction to rigorous probability at the graduate level. The Fall quarter will focus on foundations and sequences of independent random variables, including: measure theory background independence laws of large numbers weak convergence and characteristic functions central limit theorems.

While you may have encountered some of these topics in an undergraduate probability course, we will take a much deeper look at them here. This course will be followed by (and required) for Math 275B (Winter 2011) and 275C (Spring 2011) which develop the theory of stochastic processes in discrete and continuous time. It should appeal both to students interested in pure mathematics (esp. analysis) and in applications (esp. physics, engineering, biology, economics).

Prerequisites: Although prior or concurrent coursework on measure theory (typically Math 245A) will be useful, all required measure-theoretic material will be covered in class.

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## Subsection 2.10.1 Exercises

Referring to the function (h) shown in Figure 2.9.2, state the values of (t) where the function is continuous from the right but not the left. Then state the values of (t) where the function is continuous from the left but not the right.

Referring again to the function (h) shown in Figure 2.9.2, state the values of (t) where the function has removable discontinuities.

/>

## The Derivatives.

= sec 2 (cos 5x).(&ndashsin5x).5.1 = &ndash 5sec 2 (cos 5x)sin5x.

Differentiating both sides w.r.t. to x.

= &ndash sin(sin(3x 2 + 2)).cos(3x 2 + 2).3.2x

= &ndash 6x sin(sin3x 2 + 2).cos(3x 2 + 2).

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

= 5tan 4 (sin(px &ndash q)).sec 2 (sin(px &ndash q)).cos(px &ndash q).p.1

= 5p.tan 4 (sin(px &ndash q))sec 2 (sin(px &ndash q)).cos(px &ndash q).

Differentiating both sides w.r.t. to x.

= 3cosec 2 (cot 4x).(&ndashcosec(cot4x)).cot(cot 4x).(&ndashcosec 2 4x).4.1

= 12cosec 3 (cot 4x).cot(cot4x).cosec 2 4x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

= 2sin(cos 6x).cos(cos 6x).(&ndashsin6x).6.1

Soln:
Let y = (x 2 + 3x).sin 5x.

Differentiating both sides w.r.t. to x.

= (x 2 + 3x).cos5x.5 .1 + sin5x.(2x + 3)

= 5(x 2 + 3x).cos5x.5.1 + sin5x.(2x + 3).

= 5(x 2 + 3x).cos5x + (2x + 3)sin5x

Soln:
Let y = x 3 tan(2x 3 + 3x)

Differentiating both sides w.r.t. to x.

= x 3 sec 2 (2x 3 + 3x).(6x 2 + 3) + tan(2x 3 + 3x).3x 2

= 3x 3 (2x 2 + 1)sec 2 (2x 3 + 3x) + 3x 2 .tan(2x 3 + 3x).

Differentiating both sides w.r.t. to x.

Soln:
Let y = (x + sin2x)sec3x 2

Differentiating both sides w.r.t. to x.

= 6x(x + sin2x)sec3x 2 .tan3x 2 + sec3x 2 (1 + cos2x.x)

=6x(x + sin2x)sec3x 2 tan3x 2 + sec3x 2 (1 + 2cos 2 x).

Soln:
Let y = ax 3 .cosec(p &ndash qx)

Differentiating both sides w.r.t. to x.

= aqx 3 cosec(p &ndash qx)cot(p &ndash qx) + 3ax 2 cosec(p &ndash qx)

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

Differentiating both sides w.r.t. to x.

= 2(secx + tanx).secx(tanx + secx) = 2secx(secx + tanx) 2 .

Differentiating both sides w.r.t. to x.

Let y = sin6x.cos4x = $frac<1><2>$(2sin6x.cos4x)

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

= (m + n)sin (2m + 2n)x &ndash (m &ndash n)sin(2m &ndash 2n)x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Put y = sin&theta. Then y = sin &ndash1 (1 &ndash 2sin 2 &theta)

Differentiating both sides w.r.t. x

Put x = cos&theta. Then y = cos &ndash1 (4cos 3 &theta &ndash 3cos&theta)

Differentiating both sides w.r.t. x

Or, y = cos &ndash1 (cos 2&theta) = 2&theta = 2tan &ndash1 x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Or, y = tan &ndash1 (tan 2&theta) = 2&theta = 2tan &ndash1 x.

Differentiating both sides w.r.t. x

Differentiating both sides with w.r.t. &lsquox&rsquo.

Differentiating both sides with w.r.t. &lsquox&rsquo.

Differentiating both sides with w.r.t. &lsquox&rsquo.

Differentiating both sides with w.r.t. &lsquox&rsquo.

Or, x 2 $frac<<< m>>><<< m>>>$ + 2xy = 2xy.sec xy 2 . tanxy 2 $frac<<< m>>><<< m>>>$ + y 2 secxy 2 tanxy 2

Or, 2xy &ndash y 2 sec xy 2 tan xy 2 = 2xy sec xy 2 tan xy 2 $frac<<< m>>><<< m>>>$ &ndash x 2 $frac<<< m>>><<< m>>>$

Or, 2xy &ndash y 2 . sec xy 2 . tan xy 2 = (2xy. secxy 2 tan xy 2 &ndash x 2 ) $frac<<< m>>><<< m>>>$

Differentiating both sides with w.r.t. &lsquox&rsquo.

Differentiating both sides with w.r.t. &lsquox&rsquo.

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Differentiating both sides w.r.t. x

Or, x.$frac<<< m>>><<< m>>>$ + y = 2xsec 2 (x 2 + y 2 ) + 2ysec 2 (x 2 + y 2 ) $frac<<< m>>><<< m>>>$

## Exercise 2.6 (Solutions)

Identify the following statements as true or false. (i) $sqrt<-3>cdotsqrt <-3>= 3$
(ii) $i^<73>=-i$
(iii) $i^ <10>= -1$
(iv) Complex conjugate of $(-6i + i^2) is (-1 + 6i)$
(v) Difference of complex numbers $z = a + ib$ and its conjugate is a real number.
(vi) If $(a-1)-(b+3)i = 5+8i$, then a = 6 & b = -11
(vii) Product of complex number and its conjugate is always a non-negative real number.

### Question 2

Express each complex number in the standrad form $a+ib$, where a and b are real numbers. (i) $(2+3i)+(7-2i)$

### Question 5

Calculate (a) $overline$ (b) $z + overline$ © $z - overline$ (d) $zoverline$ , for each of the following. (i) $z = -i$
(ii) $z = 2 + i$
(iii) $frac<1+i><1-i>$
(iv) $frac<4-3i><2+4i>$

### Question 6

If $z = 2 + 3i , w = 5 - 4i$, show that

### Question 7

Solve the following equations for real x and y

Solution
7(i) $egin (2-3i)(x+yi) = 4+i 2(x+yi)-3i(x+yi) = 4+i 2x+2yi-3xi-3yi^2 = 4+i 2x+2yi-3yi-3y(-1) = 4+i (2x+3y)+(2y-3x)i = 4+i 2x+3y =4 (i) 2y-3x = 1 (ii) 3 imes(i) + 2 imes(ii) 6x+9y =12 (iii) -6x+4y= 2 (iv) 13y = 14 y = 14/13 hbox 2x+3(14/13) =4 hbox 2 imes13x + 42/13 imes13 = 52 26x+42 = 52 26x = 52-42 26x = 10 x = 10/26 x = 5/13 * x = 5/13 , y = 14/13 end$

7(ii) $egin (3-2i)(x+yi)= 2(x-2yi)+2i-1 3(x+yi)-2i(x+yi) = 2x-4yi+2i-1 3x+3yi-2xi-2yi^2 = 2x-1+(2-4y)i 3x+(3y-2x)i-2y(-1) = 2x-1 +(2-4y)i (3x+2y) +(3y-2x)i = (2x-1)+(2-4y)i 3x+2y = 2x-1 3x-2x+2y = -1 x+2y = -1 (i) 3y-2x = 2-4y -2x+3y+4y = 2 -2x+7y = 2 (ii) 2(i) + (ii) 2x+4y = -2 -2x+7y = 2 11y =0 y=0 hbox x+2(0) = -1 x = -1 * x = -1 ,y =0 end$

7(iii) $egin (3+4i)^2-2(x-yi) = x+yi 3^2+24i+16i^2-2x+2yi = x + yi 9+16(-1)-2x+24i+2yi = x+yi 9-16-2x+(24+2y)i = x+yi (-7-2x)+(24+2y)i = x +yi -7-2x = x -2x-x = 7 -3x = 7 x = -7/3 24+2y = y 2y-y = -24 y = -24 * x =-7/3 , y = -24 end$ Note:-This Subject Applied Mathematics-2 PDF Free Download For Polytechnic PDF In Hindi Or English available in both language so students who are convenient in their language they can download it according to their own language.

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UP Polytechnic Applied Mathematics-2 PDF Free Download Contents of this subject (Applied Mathematics – II: For Second Semester Diploma) provide fundamental base for understanding Advance mathematics and their uses in solving engineering mathematics problems.

Contents of engineering mathematics this course will help students to use basic mathematical functions like logarithms, partial fractions, matrices and basic 2D, curves in solving various engineering mathematics problems of all fields. And Bellow You Will Get The Details Information For engineering mathematics for Up Polytechnic Applied Mathematics-2.

Okay So Here Are The Short Details Of The applied Mathematics-2 that you learn & Understand while reading this Applied mathematics-2 Subject.

1. apply Binomial theorem to solve engineering problems
2. apply determinants properties and Crammer’s rule to solve engineering problems
3. apply dot & cross product of vectors to find the solution of engineering problems
4. use complex numbers in various engineering problems
5. apply differential calculus and higher order to solve engineering problems
6. find velocity, acceleration, errors and approximation in engineering problems with application of derivatives.

List Of Chapter in Applied Mathematics-2 with details topics.

1. Integral Calculus-I

Method Of Indefinite Integration:

1.1 Integration by substitution

1.2 Integration by rational function

1.3 Integration by partial function

1.5 Integration of special function

2. Integral Calculus –II

2.1 Meaning and properties of definite integrals, Evaluation of defining integrals.

2.2 Application: Length of simple curves, Finding areas bounded by simple curves, Volume of Solids of revolution, center of mean of plane areas.

2.3 Simpson’s 1/3 rd and Simpson’s 3/8 th and Trapezoidal Rule: Their Application in simple case. Numerical solutions of algebraic equation Bisections method, Regula – Falsi Method, Newton-Raphson’s Method (Without Proof), Numerical Solutions of Simultaneous equation, Gauss elimination method (without proof).

3. Co-ordination Geometry (2-Dimension)

3.1 Circle: Equation Of Circle in standard form, Centre-Radius form of circle, Diameter of circle, intercept form circle (two).

4. Co-ordination Geometry (3-Dimension)

4.1 Straight line and planes in space:

Distance between two points in space, direction cosine and direction ratios, Finding Equation of a straight line (without proof).

खंड – 1 : समाकलन गणित – 1

2. प्रतिस्थापन द्वारा समाकलन

3. खण्डश: समाकलन

4. आंशिक भिन्नों द्वारा समाकलन

5. कुछ विशिषट समाकलन

खंड – 2 : समाकलन गणित – 2

6. निश्चित खंड समाकलन

7. समाकलन के अनुप्रयोग

9. आंकिक समाकलन `

10. बीजीय समीकरणों का हल : अंकीक विधियाँ

खंड – 3 : दिविमीय निर्देशांक जयमिती

खंड – 4 : त्रिविमीय निर्देशांक जयमिती

12. अन्तरिक्ष मे बिन्दु

14. सरल रेखा

Books Recommended For Applied Mathematics-2 Up polytechnic

1. Elementary Engineering Mathematics of BS Grewal, Khanna Publishers, New Delhi
2. Engineering Mathematics, Vol I & II by SS Sastry, Prentice Hall of India Pvt. Ltd.,
3. Applied Mathematics-I by Chauhan and Chauhan, Krishna Publications, Meerut.
4. Applied Mathematics-I (A) by Kailash Sinha and Varun Kumar Aarti Publication, Meerut

Yes Books Are Also Available Online To Buy Book Applied Mathematics-2 Amazon For Up Polytechnic Diploma Engineering Students Of 2 nd Semester Only.

Up Polytechnic 2nd Semester Applied Maths – 2 in Hindi

## Ordering Decimals Worksheets

This webpage encompasses a combination of printable worksheets based on ordering decimals with a view to enhance 4th grade and 5th grade students' knowledge on decimals and their place values. A number of pdf worksheets are stacked with a variety of exercises include ordering decimals in place value boxes, using the number line, and using the greater than and less than symbols. Riddle worksheets require you to order decimals to decode the riddles that are sure to tickle your funny bone! Our free ordering decimal worksheets are perfect launchpads! Observe the digits in the whole number and decimal parts and order each set of decimals in either increasing or decreasing order as directed. Level 1 involves up to hundredths decimal places. Read the number line. Arrange each set of decimals in either increasing or decreasing order as specified. Rule: Decimals to the right of the number line will always be greater than the decimals to the left of it. Decimal numbers are given in random order. Set them in the correct order in accordance with the greater than and less than symbols provided. There are seven problems in each pdf worksheet for grade 4 and grade 5. Levitate your ordering practice with these worksheets featuring decimals with up to thousandths places. Write the decimals in the ascending order in part A and the descending order in part B. Keenly observe each set of decimals and fill them in the correct place value boxes provided. Order the decimals from the least to the greatest and vice versa. Read each decimal number displayed on these vivid theme-based pdf worksheets. Order them in the increasing order and decode the rib-tickling riddles! Grade 4 and grade 5 children are expected to figure out the largest decimal and move on till the smallest one write down the corresponding letters solve the intriguing riddles. Identify the correct sequence of decimals in either increasing or decreasing order with this set of MCQs. This activity forms a perfect tool in evaluating a child's analytical and logical skills. Transcend your peers in ordering decimals with these level 3 worksheets. Swap the positions of the numbers incorporating up to ten thousandths places and arrange them in the indicated order. This assortment of 70+ worksheets consists of captivating exercises and activities on comparing decimals using greater than, lesser than and equal to symbols.