Quadratic equations Questions and Answers

3 z 5 29 10 Combine like terms on the left side of equation Divide by 3 Distribute 3 to the binomial Subtract 14 from each side

Two companies working together can clear a parcel of land in 30 hours Working alone it would take Company A 3 hours longer to clear the land than it would Company B How long would it take Company B to clear the parcel of land alone Round your answer to the nearest tenth

A rectangular dining room table is twice as long as it is wide If the perimeter of the table is 192 inches what is the width 64 inches 32 inches 47 inches 48 inches

the conclusion is true Therefore the argument is valid This form of valid argument is called the law of detachment because the conclusion q is detached from a premise namely p q It is also called the law of direct inference ii Law of contraposition or modus tollens To understand this law consider the following argument If Kali can draw then she will get a job Kali will riot get a job Therefore Kali can t draw Taking p arid q as in i above you can see that the premises arc p q and q The conclusion is p So the argument is P q 9 P If you check you ll find that this is a valid form of argument There are two more rules of inference that most commonly form tlie basis of several proofs The following exercise is about them 3 E4 You will find three arguments below Convert each of them into the language of symbols and check if they are valid i Either the eraser is white or oxygen is a metal The eraser is black Therefore oxygen is a metal i c pq Aq p ii If Madhu is a sarpanch she will head the panchayat If Madhu heads the panchayat she will decide on property disputes Therefore if Madhu is a sarpanch she will decide on property disputes iii Either Munna will cook or Mumi will practise Karate If Munni practises Karate then Munna studies 1 Munna does not study Therefore Munni will practise Karate E5 Write down one example each of modus ponens and modus tollens As you must have discovered the arguments in E4 i and ii arc valid The first one is an example of a disjunctive syllogism The second one is an example of a hypothetical syllogism Thus a disjunctive syllogism is of the form pvq P i e pv q Ap q q And a hypothetical syllogism is of the form 2 P q qr i e p q qr pr Method Modus tollen method of de

Subtract the product of 2x 1 x 3 from the sum of 2x 4x 5 and x 9x 6 This gives x 2 x 2 r2 10x 4

5 g x f X 10 9 8 7 5 4 if x 1 if x 1 tr 4 4 t if 1 x 1 2 domain range

sin x COS X 05 05 if n x 4 if 14 x domain range

f x 1 x 1 x DAR BAR SUN MOD 100 10 9 7 6 5 4 m if x 0 if x 0 2 121 21 m 10 domain range K

f x X 3 x x 3 10 9 8 7 if x 0 if 0 x 3 if x 3 126 19 domain range

Name GOZ 8 100 Prof Meangru The following functions were taken from Essential Calculus by James Stewart 2007 For each function 1 4 f x a Graph each function neatly b Describe the domain and range 1 x if x 1 if x 1 8 7 6 5 4 3 10 domain range

To prove 1 2 2 l 1 z z 1 Zz z z Z Z 1 22 2 2 2 z 2 2 z 1 z 1 1 z 0 1 2 1 0 Which is obvious as z 1 z Pe cose 3 sine shortest distance exists along the common normal Slope of normal at P esece 3 cos ece 2 tan 6 1 so cose and sine 3 Hence PE 2 1 A A I a b c a b c bcabca ca bc a b 0 0 1 SOLUTIONS a b2 c ab bc ca ab bc ca 100 ab bc ca a b c ab bc ca 010 ab bc ca ab bc ca a 6 0 0 0 1 100 010 a b c 1 1 and ab bc ca 0 2 Now a b c a b c a b c ab bc ca 3abc a b c 3 3 Now a b c a b 2 ab bc ca 1 2 0 1 a b c 1 Now from 3 a b c 1 3 4 a b c 4 1 2 1 2 4 C 4C Alternate A A 1 AA A 1 a b c 3abc 1 a b c 3abc 1 since a b c are positive real number a b c 2 3abc from AM2 GM since a b c are real positive number 1 1 2 n k k r k r x y 7 n r n n r r n k k r

Graph one complete cycle of the following Label the axes accurately A A M A 3 7x 4 4 4 0 4 right unit O right unit O left units Identify the horizontal translation for the graph I Oleft unit 4 KIN EIN

Match each graph with its equation 1 x C Vz 1 GIF A 1 y z y z b d EZ Rad 13 2 0 1 pt

The plot below represents the function f x T 5 14 3 2 1 4 3 ba 2 7 hond 1 Evaluate f 2 f 2 1 2 3 4 LA 3 a

atch each function name with its equation y x y x y x y y x y x y x 1 x 881 8 a Absolute Value b Square Root c Cubic d Reciprocal e Cube root f Linear g Reciprocal Squared h Quadratic Question Help Message instructor

Which graph represents f z 22 2 5 80 31 8

What is the remainder when the polynomial 5x 10x Enter your answer in the box 15 is divided by x 5

Cassidy wants to divide 42 2z 3 by x 2 using synthetic division Which answer shows the correct process 2 4 2 3 8 20 4 10 23 2 3 4 2 8 14 4 7 11 2 4 2 8 4 6 24 2 3 12 15 4 2 3 8 20 4 10 17

What is the result of dividing 2x 6x 6x 2 by x 1 2x 4x 10 2x 4x 2 O 2x 6x 2 2x 6x 12

Graph the function y cos x Give the period Choose the correct graph of the function O A Ay 2 0 2 NA 8x Q OB Ay 4 0 B BA O C Ay 4 0 A 8x Q 5 OD M 2 8x Q 51

Find an equation for the line below so 4

K Graph the function y 3 cos Choose the correct graph of the function O A Give the period and the amplitude 5 B 5 BICECHO OC 0 OD of

y cos x Which of the following is the correct graph of the function y cos x OA 2 Give the amplitude 2 O B 2x Q G CO O C Ay O D 2 G

Graph the following function over the interval 2x 2x Give the amplitude y 4sin x Which of the following is the correct graph of the function y 4sin x SOA OB Ay Q Q O C Ay O D

Graph the following function over a two period interval Give the period and the amplitude 1 y cos x What is the period of the function y cos x HOLLA Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the express

Question 1 4 1 1 Complete the sentence below The amplitude of the graphs of the sine and cosine functions is Type exact answers using as needed HW Score 0 0 of 14 points O Points 0 of 1 and the period of each is

Linear Inequalities Sketch the graph of each Bacar inequality 1 A 8 Z D here to search 2 27 C D

2 A A B 3 4 D 5 6 72 A zoom

5 5 3x 2y 6 A here to search 7 6 6 x 2y 8 AJ D

7 5x 3y 0 A 8 8 4r y24 D F ZOO

2 Write the equation of the line that passes through the point 3 1 and is PARALLEL to the line x 3y 9

Write the slope of the line that passes through the given points 4 y 3 x 5 A y x 7 1 B y 47 1747 7 4 y x D y x Write the slope intercept form of the equation of the line through the given points 5 through 5 4 and 2 2 A y 6x 2 C y 2x 6 B y 3x 2 D y 2x 3

1 2 EXTRA CREDIT for SUM 3 01 Determine the slope of the line represented by the equation 1 75 15y 12x Write the slope of the line that passes through the given points 2 through 3 2 and 0 5

From the data given it is clear that 1 Rate is independent of B because on doubling the initial concentration of B alone the initi rate remains unaffected while it is doubled when initial concentration of A is doubled ii Secondly data also suggest us that initial conc of A and initial rate have direct relationship The above findings can be checked as Let the order with respect to A and B are m and n then Rate K A B Now for A 0 05 k 0 11 0 1 0 1 K 0 2 0 1 CHEMISTRY Now comparing 1 and 2 we get Secondly for B 2 2 or m 1 0 05 k 0 1 0 1 0 05 k 0 1 0 2 Now comparing 1 and 2 we get 1 2 or Hence rate equation for the reaction is Rate k A BP Rate k A B 0 1 k 0 2 0 11 k 0 2 0 1 0 2 k hence AU q W For adiabatic process q 0 hence AU W and so Hence Now 0 5 sec n 0 W p AV P V V AU 100 99 100 100 1 AU 100 bar ml AH AU A PV Here AU already calculated above and APV P V PV So AH 100 100x99 1x100 9900 bar ml The shape of XeF is square planar or square pyramidal and structure is octahedral with sp d hybridisation The molecule looks like Structure and shape of OSF is irregular trigonal bipyramidal with less e o

4 8 points Solve for x Express your answer in interval notation 3x 8216

2 8 points Solve for x 2x 13 1 x 3 6 points Solve for x Express your answer using interval notation 2 3 r 4 5r 9

3x 8216 5 6 points Find the domain and range of the function f x defined by the graph to the right Domain Range a Domain b 432A Range ATH 6 4 points Identify the domain and the range of the function defined by f 1 2 2 2 5 0 3 2 11 2 12345 7 4 points Determine whether the following are functions simply answer yes or no 1 2 2 2 3 2 X

What is the domain of the function distance from post in feet 9 5 3 2 Nour answer 20 40 60 80 100 120 140 160 time in seconds

4 4x 2 x 8 Check your answer

5 5 y2 8x 4 A B C D 6 14 22 6 4 6 44 2 y y y 2 4 0 x 2 4 x 6 7 6 y x 3 A B C D 6 6 16 1 A 21 44 2 AV y V 4 x x X

1 ity Solutions Linear Inequalit Determine which point or points are solutions of each linear inequality gfiven 2 3x 5y 10 1 x y 3 6 54 3 2 1 3 y 2 5 615 4 3 2 1 Ay y 6 x X 2 4 6 5 4 3 2 1 4 5x 2y 10 6 4 3 2 1 4 V y 6 x 6 x

Find the difference quotient f x x 3x f x h f x x h 3 x h x 3x h 2xh h 3h h f x h f x h 2xh h 3 3h 7 3x h h 2x h 3 h 2x h 3 h f x x 2x 2

ame PopQuiz 3 Prof Rudy Meangru Use the definition of derivative f x lim f x h f x to find f x h 0 h f x x 2x

1 Given 4x 7 3x 7 28 find 2 Given 5 x 1 2 2x 3 17 find 6x 9 3 Given 2p 2 8p 8 5p find 3p 4 Solve the equation and justify each step 3 6x4 24 5 Write the steps to solve the following equation 2x 7 5

9 Suppose a b means ab a b For example 3 5 3 5 3 5 17 Now if 4 x 36 then what number is x If you can find the answer show how you found it

Evaluate the determinants in Exercises 1 and 2 1 2 5 2 i 3 If A 82 cos 0 sin 0 4 iii 4 If A 0 1 8 If i 4 sin 0 cos e MATHEMATICS 6 IfA 2 5 Evaluate the determinants 3 1 2 i 0 0 1 3 5 0 A 6 2 X 2 18 X of be rep 004 1 0 2 1 0 3 2 3 0 1 1 54 9 7 Find values of x if 2 4 5 1 2 then show that 2A 4 A T 2 3 N 6 2x 4 4 X 6 2 18 6 ii then show that 3 A 27 A find A x x 1 x 1 B 6 11 Rationalised 2023 24 then x is equal to 2 iv 0 3 1 2 3 1 4 TK ber 5 1 2 2 1 3 5 0 4 3 Area of a Triangle In earlier classes we have studied that the area of 2 13 31 12 x x y 1 X 3 1 X Y x 1 x 35 2x 5 olished triangle whose vertices a x 3 x vz V x y x y and x y is given by the expression x3 Now this expression can be written in the form of a determinant as Remarks i Since area is a positive quantity we always take the absolute value of th 1

80 a31 a32 a 1 a 2 a33 a32 a 13 a22 a 1 a23 a 1 a32 a3 a A a 1 a 12 a33 a21 932 913 a23 a31 a12 a11 a22 a33 913 931 922 Expansion along first Column C By expanding along C we get A a 1 1 922 MATHEMATICS a11 912 a13 A a21 922 923 a31 a32 A33 iii Let A 923 a32 a33 t to me jepub 2 2 23 4 0 a12 a21 a33 a12 923 a31 and B 921 1 1 Rationalised 2023 24 1 20 a31 a 13 a 12 a13 931 1 1 922 923 a 1 a 2 a33 a23 a32 a21 a 12 a 33 a 3 a32 a31 a 12 a23 a13 a22 13 a22 a31 a 13 A a a22 a33 a 11 a23 a32 a21 a 12 a33 a31 a 13 a22 a 1 a 22 a33 a 11 a 23 a32 a 12 a21 A33 A 2 A23 A31 a 13 a21 a32 12 A 0 8 8 and B 0 2 2 Observe that A 4 2 2 B or A square matrices A and B 912 a13 a32 a33 1 24 3 0 4 1 0 a23 a11 a32 a13 a31922 3 Clearly values of A in 1 2 and 3 are equal It is left as an exercise to the reader to verify that the values of A by expanding along R C and C are equal to the value of A obtained in 1 2 or 3 Hence expanding a determinant along any row or column gives same value Remarks i For easier calculations we shall expand the determinant along that row or column which contains maximum number of zeros Example 3 Evaluate the determinant A 1 a13 a21 a32 a21 a 13 a32 ii While expanding instead of multiplying by 1 we can multiply 1 or 1 according as i j is even or odd 2 a31 a 12 a23 verify that A 2B Also In general if A kB where A and B are square matrices of order n then A k B where n 1 2 3 where n 2 is the order of

and Now So II 1 i 18 3 7 21 25 21 25 4 3 22 M32 M 33 ON62 12 1 31 10 545 1 0 0 2 1 0 1 0 00 1 30 0 30 0 5 If A 21 14 5 19 a11 912 12 0 12 CERT 8 30 22 0 18 18 ii be requalis a 2 a 2 3 a 3 5 A 12 A2 22 A 18 12 13 31 a 1 A31 a12 A32 a13 A33 2 12 3 22 5 18 24 66 90 0 a13 1 0 4 ii 35 1 0 1 2 EXERCISE 4 3 Write Minors and Cofactors of the elements of following determinants 2 4 0 3 a b d a31 a32 a33 A a A a A a 4 4 1 2 2 19 19 A23 1 3 13 13 Rationalised 2023 24 A 1 12 12 A32 1 3 2 22 22 B A33 1 3 18 18 4 Using Cofactors of elements of third column evaluate 5 3 8 3 Using Cofactors of elements of second row evaluate A 20 1 3 DETERMINANTS ignas yz A a A a A ZX Z xy 87 then value of A is given by

Factor the expression completely 5x 5x 10 5x 5x 10

As x y O 0 oo Question 19 1 point What are the x intercepts of y x x 3 x 3 0 3 3 3 0 3