ALGEBRA OF MATRICES (A)

EXERCISE - A

TYPE-1

A) Construct a 2x 3 matrix A= [aᵢⱼ] whose elements aᵢⱼ are given by:

a) aᵢⱼ= i x j                    1    2    3

                                      2   4     6

b) aᵢⱼ= 2i - j                        1      0       -1

                                           3      2        1

c) aᵢⱼ= i+ j                          2      3      4

                                           3     4       5

d) aᵢⱼ= (i+ j)²/2.              2      9/2     8

                                      9/2     8     25/2

B) Construct a 2 x 2 matrix A= [aᵢⱼ] whose elements are given by:

a) aᵢⱼ = (i+ j)²/2 .               2         9/2

                                          9/2        8

b) aᵢⱼ= (i - j)⅖/2.              0          1/2

                                         1/2         0

c) aᵢⱼ= (i - 2j)²/2.             1/2        9/2

                                           0            2

d) aᵢⱼ = |2j -3j|/2.            1/2          2

                                        1/2          1

C) Construct a 3 x 4 matrix A= [aᵢⱼ] whose elements aᵢⱼ are given by:

a) aᵢⱼ= i + j.            2      3        4     5

                               3      4        5     6

                               4      5        6     7

b) aᵢⱼ= 2i.                 2     2     2       2

                                 4     4     4       4

                                 6     6     6       6

c) aᵢⱼ= (1/2) |-3i + j|        1       1/2      0        1/2

                                       5/2       2      3/2       1

                                        4        7/2     3         5/2

D)1) If A= [aᵢⱼ] is a matrix given by

A= [aᵢⱼ]= 4     -2      1       3

               5      7      9       6

              21    15    18   -25

Write

a) the order of A.               3x 4

b) the elements a₂₄.               6

b) the element a₃₄.              -25

c) verify a₃₂ = a₂₃ + a₂₄.     

2)  If A= [aᵢⱼ]= 2   3   -5 and B= 2      -1

                        1  4    9              -3       4

                                                   1       2 then find

a) a₂₂ + b₂₁.                   1

b) a₁₁b₁₁ + a₂₂b₂₂.        20

Type - 2

1) If 3x+4y      2      x- 2y = 2       2        4

        a+ b      2a - b     -1     5      -5       -1 then find x, y, a, b.     2, -1, 0, 5

2) If x         3x - y  = 3     2

       2x+z   3y- a      4      7 then find x, y, z ,a.       3,7,-2,14

3) if  x- y       z  = -1      4

        2x - y    a       0      5 then find x, y, z, a.           1,2,4,5

4) If x+ 3     z+ 4      2y -7      0         6      3y-2

       4x+6     a-1          0     =  2x       -3     2c+2

       b -3        3b      z+ c       2b+4   -21      0 find a, b, c, x, y and z.       -2, -7, -1, -3, -5, 2

5) If xy       4     = 8        a

       z+ 6   x+y      0        6 then find the value of x, y, z, a.         2.4, -6,4 or 4,2,-6,4

6) If A= 2x +1     2y & B= x +3       y²+2

                 0       y²- 5y         0           -6 and A = B then find the value of x and y.         

7) if x+ 10        y²+ 2y = 3x +4       3

           0               -4            0       y²- 5y then find the value of x, y.        3,1

8) If A= a+ 4       3b  & B= 2a +2      b²+2 

              8            -6                8         b²-10 with the relation A= B, then find x, y.      2,2

EXERCISE - B

Type - 1

1) If A= 3     -2 & B= -2        4

              1      4           1        3

Find A+ B.                    1      2

                                      2      7

2)       2     1     3      1       -2      3

 If A= 0     3     5      2        6       1

         -1     2     5      0       -3       1

Find A+ B.          3     -1    6

                            2.     9    6

                           -1     -1    6

3) Let A= 2     4  B= 1     3  & C= -2      5

                 3     2       -2    5             3     4 then find

a) 2A - 3B.              1       -1

                                12    -11

b) B - 2C.                 9        -17

                               -14      -11

c) 3A - C.                8        7

                                6        2

d) 3A - 2B + 3C.        -2         21

                                   22         8

4) If A= 2      3 B= -1    0     2 & C= -1      2      3

              5      7       3    4      1           2      1      0 find

a) A+ B.           

b) B + C.               -2      2     5

                               5      5     1

c) 2B + 3A.            

d) 3C - 4B.           1        6         1

                            -6       -13      -4

5) Let A= -1    0     2 & B= 0      -2     5 & C= 1   -5   2

                  3    1     4          1     -3      1          6    0  -4

Find 2A - 3B + 4C.       2.      -14        -3

                                      27      11        -11

6) 

                 

ᵢⱼᵢⱼᵢⱼᵢⱼᵢⱼ

SAMPLING

5% level of confidence= 1.96

1%= 2.58

S. E= √(npq)

Test of significance= diff/S. E

S. Eₚ= √{pq/n}

S. E of the difference between the proportions: 

S. E(p₁ - p₂)= √{pq(1/n₁ + 1/n₂) where p= (n₁p₁ + n₂p₂)/(n₁ + n₂)

       e.g., diff/S.E or (p₁ - p₂)/S. E

S. E of mean = s.d/√n

S. E of the difference between sample means: √{σ²(1/(n₁ + 1/n₂)}

1) in 324 throws of a six-faced die odd point appeared 181 times. Would you say that die is "fair"? (5% level ) 2.1

2) 160 heads and 240 tails were obtained in tossing a coin 400 times. Find a 1% confidence interval for the probability of a head. Does this appear to be true coin ? 4

3) In a sample of 500 people from a village in Rajasthan, 280 are found to be rice eaters and the rest wheat eaters. Can we assume that both are food articles are equally popular ? (1% level ) 2.7

4) In a hospital full 480 female and 520 male babies were born in a week. Do these figures confirm the hypothesis that males and females are born in equal numbers ? (5% level) 1.265

5) 500 apples are taken to random from a large basket and 50 are found to be bad. Estimate the proportion of bad apples in the basket. 0.013

6) In a random sample of 1000 person from town A, 400 are found to be consumers of wheat. In a sample of 800 from Town B, 400 are to be found to be consumers of wheat . Do these data reveal a significant difference between Town A and town B, so far as the proportion of wheat consumers is concerned? (1% level). 4.167

7) In a random sample of 500 persons from Maharashtra, 200 are found to be consumers of vegetable oil. In another sample of 400 persons from Gujarat, 200 are found to be consumers of vegetable oil. Discuss whether the data reveal a significant difference between Maharashtra and Gujarat so far as the proportion vegetable oil consumers is concerned. (1% level)

8) Before an increase in excise duty on a tea 400 people out of sample of 500 persons were found to be tea drinkers. After an increase in the duty , 400 persons one known to be tea drinkers in a sample of 600 people. Do you think that there has been significant decrease in the consumption of the consumption of tea after the increasing in the excise duty. (1% level ). 5.2

9) A machine puts out 10 imperfect articles in a sample of 200, After the machine is overhauled it puts out 4 imperfect articles in a batch of 100. Has the machine been improved ? (5% level). 0.388

10) 500 articles from factory are examined and found to be 2% defective. 800 similar articles for a second factory are found to have only 1.5% defectives . Can it reasonably be concluded that the products of the first factory are inferior to these of the second. 0.78

11) There are 1000 students college. Out of 20000 in the whole University, in a study 200 were found smokers in the college and 1000 in the whole University. Is there a significant difference between the proportion of smokers in the college and university ? (1% level). 22.39

12) A sample of 100 iron rods is said to be drawn from a large number of rods whose lengths are normally distributed with mean 3 ft, and standard deviation 0.6 ft. If the sample mean is 3.2 ft., can the sample be regarded as a truly random sample? (1%). 3.33

13) A sample of 100 students is taken from a large population. The mean height of these students is 64 inches and the standard deviation 4 inches. Can it reasonably regarded that in the population mean height is 66 inches. (1% level)

14) 400 labourers were selected at random from a certain district. Their median income was Rs140.5 p.m with standard deviation of Rs25.2. Do you believe that the average income of the labour community in the district is Rs150 ? (1% level). 6.01

15) Random samples drawn from two countries gave the following data relating to height of adult males:

                                          Country A country B 

Mean height(in inches) 67.42 67.25 

Standard deviation 2.58 2.5 

Number of observations 1000 1200 

Is the difference between the mean height significant? (5% level). 1.55

16) A man buys 100 electric bulbs each of two well-known makes, taken at random from stock for testing purposes . He finds that 'make A' has a mean life of 1300 hours with a standard deviation of 82 hours, and 'make B' has a mean life of 1248 hours with a standard deviation of 93 hours. Discuss the significance of these results. (1% level). 4.19

17) 490 men students and 450 female students appeared at an examination in statistics. The mean and standard deviation in marks of male students are 54.3 and 17.5 respectively whereas those of female students are 50.6 and 18.0. Is there a significant difference in marks of male and female students? (1% level). 3.19

18) In a survey of buying, 400 women shoppers are chosen at random in supermarket A located in a certain section of Bombay city. Their average monthly food expenditure is Rs400 with a standard deviation of Rs 12. For 400 women shoppers chosen at random in supermarket B in another section of the city, the average monthly food expenditure is Rs395 with a standard deviation of Rs15. Test at a level of 0.05, whether the average food expenditure of the two populations of shoppers from which the samples were obtained are equal. 5.21

19) A potential buyer of electric bulbs bought 200 bulbs , 100 bulbs each of two brands. Upon testing these he found that brand A had a mean life of 1210 hours with a standard deviation 40 hours whereas brand B had a mean life of 1250 hours with a standard deviation of 60 hours. Can the buyer be quiet certain that the two brands differ significantly in quality ? (1% level). 5.55

t-test

EXERCISE - A

1) A Company has been producing steel tubes of mean inner diameter of 2cm. A sample of 10 tubes gives an inner diameter of 2.01cm and a variance of 0.04cm². Is the difference in the values of means significant ? (Given t₉(0.05)= 2.262).      is not significant

2) A mechanist is making engine parts with axle diameter of 0.7 inch. A random sample of 10 parts shows mean diameter 0.742 inch with a standard division of 0.04 inch. On the basis of this sample, would you say that the work is inferior ? (Given t₉(0.05)= 2.262).          3.15, the null hypothesis is rejected at 5% level of significance or the alternative hypothesis is accepted at 5% level of significance. Hence , sample mean differs significantly from population means e.g, the work is failure

3) A soap manufacturing company was distributing a particular brand of a soap through a large number of retail shops. Before a heavy advertisement campaign, the mean sales per week per shop was 140 dozens . After the campaign a sample of 26 shops was taken and mean sales was found to be 147 dozens with standard deviation 16. Can you considere the advertisement effective ?( given t₂₅(0.05)= 2.06).       2.187, we reject the null hypothesis and accept the alternative hypothesis. Hence, we conclude that advertisement is effective for sales .

4) A random sample of size 16 has 53 as mean. The sum of the squares of the divisions taken from mean is 150. Can this sample be regarded as taken from the population having 56 as mean ? given t₁₅(0.01)= 2.95).       -3.794, we reject the null hypothesis. Consequently, the alternative hypothesis accepted at 0.01 level of significance. Hence , the sample is not taken from the population having 56 as mean .

5) A random sample of 17 values from a normal population has a mean of 105cm and the sum of the squares of the deviations from this mean is 1225cm².  Is the assumption of a mean of 110cm for the normal population reasonable ? Test under 5% and 1% levels of significance. Also, obtain the 95% and 99% confidence limits. (Given t₁₆(0.05)= 2.11) and t₁₆(0.01)= 2.921).       Yes at 1% level

6) Ten students are selected at random from a college and their heights are found to be 100, 104, 108, 110,118, 120, 122, 124, 126 and 128cms. In the light of these data, discuss the suggestion that the mean height of the Student of the college is 110cms (given t₉(0.05)= 2.262).         Yes

7) A random sample of 10 boys had the following I. Q's: 70, 120, 110, 88, 83, 95, 98, 107, 100. Do these data support the assumption of a population mean IQ of 100 ? Find a reasonable range in which most of them IQ. values of sample of 10 boys lie. (Given t₉(0.05)= 2.262)

Miscellaneous -1

1) Ten cartons are taken at random from an automatic filling machine. The mean net  weight of the cartons is 11.8 kg and the standard deviation 0.15 kg. Does the sample mean differ significantly from the intended weight of 12 kg? (Given t₉(0.05)= 2.262).      Yes

2) A machine is designed to produce insulating washers for electrical devices of average thickness of 0.025cm. A random sample of 10 washers was found to have an average thickness of 0.024 cm with a standard deviation of 0.002cm. test the significance of the deviation (Given t₉(0.05)= 2.262).        Not significant

3) A random sample of size 25 from a normal population has the mean 47.5 and standard deviation 8.4. Does this information refute the claim that the mean of the population is 42%. (Given t₂₄(0.05)= 2.06).        Yes

4) A process of marketing certain bearings is under control if the diameter of the bearings have the mean 0.5cm. What can we say about this process if a sample of 10 of these bearings has a mean diameter of 0.506 cm and standard deviation of 0.004 cm ? (Given t₉(0.05)= 2.262).         Process is not under control

5) A machine is supposed to produce washers of mean thickness 0.12cm. A sample of 10 washers was found to have a mean thickness of 0.128 and standard deviation 0.008.  test whether the machine is working in proper order at 5% level of significance. (Given t₉(0.05)= 2.262).     No

6) A random sample of 16 values from a normal population showed a mean of 41.5 and sum of squares of deviations from mean equal to 135. Can it be assumed that the mean of the population is 43.5 ? (Given t₁₅(0.01)= 2.95).     Yes

7) A sample of size 9 from a normal population mean= 15.8 and s²= 10.3. Find 99% confidence interval for the population mean. (Given t₈(0.01)= 3.335).     (11.99,19.61)

8) A random sample of size 16 has 53 as mean. The sum of the squares of deviations taken from mean is 150. Find 95% and 99% confidence interval for population mean. (Givent₁₅(0.01)= 2.95 and t₁₅(0.05)= 2.13).       (51.32,54.68),(50.67,55.33)

9) A random sample of 16 values from a normal population showed a mean of 41.5 inches and the sum of squares of deviations from this mean equal to 135 square inches. Show that the assumption of a mean of 43.5 inches for a population is not reasonable. Obtained 95% and 99% confidence intervals for the same. (Given t₁₅(0.05)=2.131) and t₁₅(0.01)= 2.947).        (39.902,43.098),(39.29,43.71)

10) The annual rainfall at a certain place is normally distributed with mean 45cm. The rainfall during the last 5 years are 48cm, 42 cm, 40cm, 44cm and 43cm. Can we conclude that the average rainfall during the last 5 years is less than the normal rainfall ? (Given t₄0.05)= 2.132).      No

11) The height of 8 males participating in an athletic championship are found to be 175 cm, 168cm, 170cm, 167cm, 160cm, 173cm and 168 cm. Can we conclude that the average height is greater than 165cm ?(given t₇(0.05)= 1.895).       Yes

12) The mean weakly sells of chocolate bar in general stores was 146.3 bars per store. After an advertising the mean weekly sales in 22 stores for typical week increased to 153.7 bars and showed a standard deviation of 17.2. Was the advertising campaign successful ? (Given t₂₁(0.05)= 2.08).         Yes

13) The foreman of ABC mining company has estimated the average quantity of iron ore extracted to be 36.8 tonnes per shift and the sample standard deviation to be 2.8 tonnes per shift, based upon a random selection of 4 shifts. Consider a 90% confidence interval around this estimate. (Given t₃(0.1)= 2.353).       (34.5, 39.10)

14) A random sample of size 20 from a normal population gives a sample mean of 42 and standard division of 6. Test the hypothesis that the population mean is 44. (give t₁₉(0.05)= 2.09).        True 

15) A random sample of 10 boys had the following I. Q:  70, 120, 110, 101, 88, 83, 95, 93, 107, 100. Do thesr data support the assumption of a population mean IQs of 100? (Given t₉(0.05)=2.262). also, find the 95% confidence interval for the population mean.       Yes (87.494, 107.906)

16) The manufacturer of a certain make of electric bulbs claims that his bulbs have a mean life of 25 months with a standard deviation of 5 months. A random sample of 12 such bulbs gave the following values: 

life in months: 24  26  32  28  20  18  23  27  29  34   20   28 

Can you regard the producer's claim to be valid at 1% level of significance ?     Yes

EXERCISE - B

1) For the following data examination if the means of two samples differ significantly:

                  size   means    standard deviation 

sample I:   6         40                8 

sample II:  5         50              10 

(Given  t₉(0.05)=2.262).               is not differ significantly

2) Two batches of the same product are tested for their mean life. Assumption that the lives of the product follows a normal distribution with an unknown variance, test the hypothesis that the mean life is the same for both the branches, given the following information :

Batch  sam-size mean life(hrs)   sd

I              10            750                  12

II              8             820                  14

(Use t₁₆+0.05)=2.2120).                Yes

3) Samples of two types of electric light bulbs were tested for length of life and following data were obtained:

             Type-I.    Type -II

S size:        8            7

S. means : 1234   1036

S. SD          36         40

Is the difference in the mass sufficient to warrant that type I is superior to type II regarding length of life ? (Given t₁₃(0.05)= 2.216).     No

4) Two different types of drugs A and B were tried on certain patients for increasing weight,5 persons were given A 7 persons were given drug B. The increase the weights in pounds is given below:

Drug A: 8    12     13    9     3

Drug B: 10    8     12   15    6    8    11 

Do the two drugs differ significantly with regard to their effect in increasing the weight (given t₁₀(0.05)= 2.23).          No

5) The height in inches of 6 randomly chosen sailors and 10 randomly chosen soldiers are given as under :

Sailors    Soldiers

  63            61

  65            62

  68            65

  69            66

  71            69

  72            69

                   70

                   71

                   72 

                   73 

Does these figures show that the soldiers are on an average shorter than sailors? (Given t₁₄(0.05)= 2.15).                Yes

6) The mean life of a sample of 10 bulbs was found to be 1456 hours with standard deviations of 423 hours. A second sample of 17 bulbs chosen from a different batch showed a mean in life of 1280 hours with standard deviation of 398 hours. Is there a significant difference between the means of the two batches ?     No

7) Strength tests carried out on samples of 2 years spun to the same count gave the following results:

 Size of sample   samplemean  Sam-size  sample vari.

  Yarn A                      4                   50               42

  Yarn B                      9                   42               56

The strengths are expressed in kg. Is the difference in mean strengths significant of real difference in the mean strengths of the sources from which samples are drawn?   Not significant

8) Samples of two types of electric bulbs were tested for length of life and the following data were obtained:

                                                    Type I    Type II 

No. Of bulbs in the sample         8             7

Mean(in hours)                          1134     1024

S. D(in hours):                             35           40

Is the difference in the sample means significant? (Given t₁₃(0.05)= 2.16).     Significant

9) I. Q test on two groups of boys and girls gave the following results:

Girls: mean=78    n= 50    s.d= 10

Boys : mean=73 n= 100   s.d= 15

Is there a significant difference in the mean scores of boys and girls? (Given t₂₃(0.05)= 2.07)

ELLIPSE

1) Find (i) the co-ordinate of the centre (ii) lengths of axes (iii) equation of axes (iv) co-ordinates of vertices (v) eccentricity (vi) coordinates of focus (vii) equations of directrices (viii) length of lactus rectum for each of the following ellipse .

a) 9x²+ 16y²= 144.       (0,0),8,6,y=0, x=0, (±4,0), √7/4, (±√7,0), x=±16/√7, 9/2

b) x²+ 4y²= 1.        (0,0),2,1,y=0, x=0, (±1,0),√3/2, (±√3/2,0), √3x=±2, 1/2

c) 9x²+ 4y²= 36.       (0,0),6,4,x=0, y=0, (0,±3), √5/3, (0,±√5), y=±9/√5, 8/3

d) 5x²+ 4y²= 1.       (0,0),1,2/√5,x=0, y=0, (0,±1/2), 1/√5, (0,± 1/2√5), 2y=±√5, 4/5

2) Find (i) the co-ordinate of centre (ii) lengths of axis (iii) equations of axis (iv) co-ordinates of vertices (v) eccentricity (vi) coordinates of focus (vii) equation of directrices (viii) length of latus rectum for each of the following ellipse :

a) 4x²+ 3y² - 16x + 12y + 16 =0.     (2,-2),4,2√3, x-2=0, y+2=0, (2,0) and (2,-4), 1/2, (2,-1) and (2,-3), y -2=0 and y+ 6=0, 3

b) 4x²+ 9y² - 24x =0.     (3,0),6,4, y=0, x -3=0, (0,0) and (6,0), √4/3, (3-√5,0) and (3+√5,0), x =3 - 9/√5 and x =3+ 9/√5, 8/3

c) 9x²+ 16y² - 54x + 64y + 1 =0.      (3,-2),8,6, y+2=0, x=3, (-1,-2) and (7,-2), √7/4, (3±√7,-2),  x=3±16/√7, 9/2

d) 9x²+ 5y² + 30y =0.       (0,-3),6,2√5, y+3=0, x=0, (0,0) and (0,-6), 2/3, (0,-1) and (0,-5), 2y -3=0 and 2y+ 15 =0, 10/3

3) The ellipse x²/169 + y²/25 = 1 has the same eccentricity as the ellipse x²/a² + y²/b² = 1. Find the ratio a/b.        13/5

4) Taking x-axis as major axis and y-axis as minor axis, find the equation of the ellipse whose

a) length of the minor axis and latus rectum are 4 and 2 units respectively.   x²+ 4y²= 16

b) length of latus rectum is 6 units and length of major axis is 10 units.  3x²+ 5y²=75

c) length of major axis is 4 units and eccentricity 3/4.      7x²+ 16y²= 28

d) distance between directrices is 16/√7 units and eccentricity is √7/4.   9x²+ 16y²=36

e) distance between the foci is 8 units and distance between the directrices is 18 units.       x²/36 + y²/20 = 1

f) distance between the foci is equals to minor axis and length of the latus rectum is 16 units .       x²+ 2y²=256

g) Sum of the squares of the lengths of major and minor axis is 20 and eccentricity is 1/√3.        2x²+ 3y²= 6

h) eccentricity is 1/2 and distance between the foci is 4 units .      3x²+ 4y²= 48

i) distance between the foci is 4√3 units and length of the minor axis is 4 units. Find also the eccentricity .        x²+ 4y²= 16, √3/2

j) distance between the foci is 10 units and length of the latus rectum is 15 units .    3x²+ 4y²= 300

k) latus rectum is 5 and eccentricity is 2/3.      20x²+ 36y²= 405

5) Find the equation of the ellipse whose vertices are at the point (0,±10) and eccentricity is 4/5.       25x²+ 9y²=900

6) Find the equation of the ellipse whose vertices are at the points (±1/4,0) and the equations of directrices are x= ± 5/12.       16x²+ 25y²= 1

7) If the co-ordinates of the focus of an ellipse are (0,±8) and its eccentricity is 4/5, find its equation.        25x²+ 9y²= 900

8) Find the equation of the ellipse whose co-ordinates of foci are (±5,0) and (±4,0) respectively.       9x²+ 25y²= 225

9) The co-ordinate of the foci of an ellipse are (1,0) and (-1,0) and the length of the minor axis 2 units. Find the equation of the ellipse.     x²+ 2y²= 2

10) The co-ordinates of foci of an ellipse are (0,2) and (0,-2). If its latus rectum is 6 units, find its equation.       4x²+ 3y²= 48

11) If the co-ordinates of the foci of an ellipse which passes through (4,1) are (±3,0), find the equation of the ellipse .       x²+ 2y²= 18

12) If the co-ordinates of the foci of an ellipse are (±2,0) and the equations of the directrices are 2x= ±9, find the ellipse.      5x²+ 9y²= 45

13) The major axis of an ellipse is parallel to x-axis, the co-ordinates of its centre are (-2,3) eccentricity is 1/√3 and the length of the lactus rectum is 4 units . Find the equation of the ellipse .      2x²+ 3y² + 8x - 18y +17=0

14) The major axis of an ellipse is along y-axis, centre is at (0,2), length of the major axis is 6 units and eccentricity is 1/2 0; find the equation of the ellipse.   4x²+ 3y² - 12y - 15=0

15) the major axis of an ellipse is along y- 2 =9, minor axis is along x- 3 =0, eccentricity is 1/2 and length of the latus rectum is 3√3/2; find the equation of the ellipse .      3x²+ 4y² -18x - 16y +34=0

16) The ellipse x²/a² + y²/b² =1 passes through the point (-3,2) and its eccentricity is √3/5; find the length of the latus rectum.      4√19/5

17) Find the equation of the ellipse whose axes are co-ordinate axis and which passes through the points (-3,1) and (2,-2). Find also eccentricity of the ellipse.    3x²+ 5y² = 32 or 5x²+ 3y²=32, √2/5

18) The length of the latus rectum of an ellipse is 8 units and that of the major exis, which lies along the the x-axis, is 18 units . Find the equation in the standard form.   

 Determine the co-ordinates of the foci and the equations of it directrices .      4x²+ 9y² = 324, (±3√5,0), √5 x = ±27

19) The co-ordinates of the vertices of an ellipse are (-5,-1) and (-1,-1) and eccentricity is 1/3; find the equation of the ellipse .    8(x+ 3)² + 9(y +1)²= 32

20) The co-ordinates of the foci of an ellipse are (2,-2) and (2,6) and its eccentricity is 1/2; find the equation of the ellipse.         4(x -2)²+ 3(y -2)²= 192

21) If the foci of an ellipse be (2,3) and (-2,3) and semi-minor axis be √5, find the equation of the ellipse.      5x²+ 9(y -3)² =45

22) The co-ordinates of the centre of an ellipse are (-2,1) and the co-ordinates of a focus are (-1,1); if the length of the major axis of the ellipse is 2√3 units , find its equation.           2(x +2)²+ 3(y -1)² =6

23) Show that the point (2,5/3) lies on the ellipse 5x²+ 9y² =45. Show further that the sum of the distance of this point from the foci is equal to the length of the major axis.             √3/2

24) respectivali of the leaves is 12 find the equation the coordinate of the centre of an ellipse 21 and a coordinate of the focus are 11 of the length of the major Axis 23 units find its equation so the point is 253 lies on the lips so for that the sum of the distance of the point from the focus equals to the length of the major excessity of the ellipso that the sum of the distance of any point and the lips on the pokey is equals to 8 units find the equation of Delhi coordinate of the focus has 23 equation of the directories is nsentitive 12 the coordinate of the focus of an ellipse is 03 the equation of the corresponding directors and it is a cities 12 find the equation of an leaf with the centre city of an ellipse 13 focuses at the point 21 and the point of intersection of major X is directorate is 23 then find the coordinate of the centre of Delhi the coordinate of the focus and its corresponding but its upon A list at 31:24 respectively the central city of the ellipse is 23 then find the coordinator 7676 12 find the equation of find the equation of the auxiliary circle a piece of the following ellipse prove that the point lies on the ellipse if you came the variable quantity so that the locus of the point of intersection of the state lines is an ellipse find the co-ordinate of the point of the ellipse whose eccentricity angle is 30 find the centric angle of the point on the ellipse whose the distance from the centre is 3 units find the century candles of the experimentals of the late to select of the lips so that formali the distance between focus is Q and the distance between two focus 2P find the length of the semi Axis so that the double ordinate of the auxiliary circle up passing through the focus is equal to the minor x aadhe centric angle of two points PQ respectively on the ellipse than so that the equation of the chord will be if the eccentric angle of the extremities of a focal cord of the ellipse so that is an eccentricity to e so that the length of the focal drawn through end of the manufacturing series if the segment PQ find the locus of the point of action nearest to p

SIMPLE INTEREST

EXERCISE - 1

1) a) A sum of Rs1000 was paid as the interest on a loan of Rs20000, What was the amount paid back.           2100

b) If the amount is Rs6000 and the principle is Rs5400, what is the interest ? Rs600

c) Find the amount when the principle is ₹5000 & the interest is Rs300.   ₹5300

d) If the amount is Rs4500 and the interest is Rs500, what is the principle ? Rs4000

2)a) Find the simple interest for the following:

a) On ₹2750 for 6 years at the rate of 5% per annum.       ₹825

b) On ₹3625 for 8 years at the rate of 6% per annum.           Rs1740

c) On ₹65000 for 2 years 6 months at the rate of 4% per annum.         Rs6500

d) On Rs10000 for 1 year 4 months at the rate of 15/2% per annum . 1000

e) On Rs4000 for 2 years at the rate of 4% per annum . Also, find the amount.               Rs32, Rs432

3) Find the amount in the following cases:

a) principle =Rs2400, time= 7 years and interest rate= 6.5%.     Rs3492

b) principle= Rs 3500, time = 2years and interest rate= 8%.         Rs4060

c) A man takes a loan of Rs8000 at the rate of 8% per annum. How much does he pay at the end of 3 years settle his debt ?           Rs9920

4) Find the sum of money on which the interest equals 

a) Rs5700 in 3 years at the rate of 4% per annum.      Rs47500

b) Rs 13875 in 5 years at the rate of 6% per annum . Rs46250

c)  What was the sum on which the simple interest for 5 years at the rate of 25/4% per annum is Rs125?          Rs400

d) Find the sum of money that amounts to Rs11200 in 3 years at the rate of 4% per annum.          Rs10000

e) Find the sum of money that amounts to Rs26000 in 5 years at the rate of interest of 6% per annum. Rs20000

5) In what time will 

a) the interest on Rs6000 becomes Rs1200 at the rate of 5% per annum?    4 years

b) Rs8500 amount to Rs15385 at the interest rate of 9% per annum. 9 years

c) In how many years will Rs1250 to Rs1950 at the interest rate of 8% per annum?    7 years

d) After what time will the interest on Rs3000 be Rs600 at the rate of 5% per annum.      4 years.

6) Find the rate of simple interest if

a) the interest on Rs4000 for 3 years is Rs600.     5%

b) the interest on Rs1000 for 5 years amounts to Rs300. 6%

c) At what rate of interest per annum will the simple interest on Rs1200 in 3 years be Rs288?      8%

d) Find the rate of interest at which sum of the money gets double in 12 years.     25/3%

e) What is the rate of simple interest if

a) the interest on Rs5000 for 2 years is Rs600 .       6%

f) Rs4000 amounts to Rs4600 in 3 years ? 5%

7) Find the sum of money that amounts to :

a) Rs12000 in 4 years at the interest rate of 5% per annum.         10000

b) Rs8250 in 5 years at the interest rate 7.5% per annum .       Rs6000

8)a) Find the rate of interest if a sum of money gets double in 16 years.    25/4%

b) Find the rate of interest if sum of money gets doubled in 8 years.     25/2%

c) Find the time in which a sum of money gets doubled at the interest rate 8% per annum.       25/2 years 

d) Find the rate of interest if a sum of money becomes 4 times as much in 20 years. 15%

MISCELLANEOUS -1

1) A sum of Rs7000 amounts to Rs8269 in 3 years at a certain rate of interest. In what time will Rs5000 amount to Rs6500 at the same rate of interest?    5 years

2) Mr. Sharma borrowed some money from a friend at the interest rate of 8% per annum. He repaid his debt by paying Rs16800 at the end of 5 years. How much did he borrow?   Rs12000

3) Lalita borrowed some money from Praveen at the rate of interest 10% per annum. She repaid her debt by paying Rs13500 after 1 year 3 months. What was sum of money borrowed by Lalita?       Rs12000

4) A farmer takes a loan of Rs8400 at the simple interest of 15/2% per annum. After what time will he have to pay R10920 to clear the debt ? 4 years

5) Sameer borrows Rs60000 from Rahim. He clears his debt by returning Rs64000 to Rahim at the end of 12 months. Find the rate of interest charged by Rahim. 20/3%

6) A man took a loan of Rs 6 lakh from his friend at the simple interest rate of 9% per annum. At the end of 3 years, he cleared his debt by paying Rs3 lakh and giving his flat to his friend. What was the price of the flat? Rs462000

7) A sum of Rs35000 amounts to Rs4060 in 4 years at a certain rate of interest. In what time will Rs6000 amount to Rs7200 at the same rate of interest ? 5 years 

LINER PROGRAMMING

EXERCISE - A

1) A factory produces two products A and B. Each of products A required 2 hours for moulding, 3 hours for grinding and 4 hours for polishing, and each of products B requires 4 hours for moulding, 2 hrs for grinding and 2 hours for polishing . The factory has moulding machine available for 20 hours, grinding machine for 24 hours and polishing machine available for 13 hrs. The profit is Rs5 per unit of A and Rs3 per unit of B and the factory can sell all that it produces. Formulate the problem as LPP to maximize the profit.    Max: Z= 5x + 3y;  eq: 2x+ 4y≤ 20, 3x + 2y ≤ 24, 4x +2y ≤ 13 and x ≥0, y ≥ 0.

2) A toy company manufactures 2 types of doll; a basic version doll A and a deluxe version doll B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day if it produces only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. If the company makes profit of Rs3 and Rs5 per doll respectively on doll A and doll B;  how many of each should be produced per day in order to maximize profit ?     Z= 3x+ 5y; x+ 2y≤ 2000, x + y ≤ 1500, y ≤ 600 and x ≥0, y ≥ 0. 

3) A firm can produce three types of clothes say A, B, C. Three kinds of wool are required for it, say red wool , green wool and blue wool . One unit of length A needs 2 metres of red wool, 3 metres of blue wool; one unit of cloth B needs 3 metres of red wool, 2 metres of green wool and 2 metres of blue wool; and one unit of cloth C needs 5 metres of green wool and 4 metres of blue wool. The farm has only a stock of 16 metres of red wool, 20 metres of green wool and 30 metres of blue wool . It is assumed that the income obtained from one unit of length of cloth A is Rs6, of cloth B is Rs10 and cloth C is Rs8. Formulate the problem as a linear programming problem to maximize the incime.     Max: Z= 6x + 10y + 8z;  2x+ 3y + 0z ≤ 16, 0x + 2y +5z ≤ 20, 3x +2y + 4z ≤ 30  and x ≥0, y ≥ 0, z≥ 0

4) A furniture firm manufacturers chairs and tables, each requiring the use of the three machines A, B and C. Production of one chair requires 2 hours on machine A, 1 hour on machine B, and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit realise by selling one chair is Rs30 while for a table the figure is Rs60.  The total time available per week on machine A is 70 hours, in machine B is 40 hours, and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit ? Develop a mathematical formulation.        Max: Z= 30x + 60y;  2x+ y≤ 70, x + y ≤ 40, x +3y ≤ 90 and x ≥0, y ≥ 0.

 

5) A manufacturer of a line of patient medicine is preparing a production plan on medicine A and B. There are sufficient ingredients available to make to 20000 bottles of A and 40000 bottle of B but there are only 45000 bottles into which either of the medicine can be put p. Further more, it takes 3 hours to prepare enough material to fill 1000 bottles of A , it takes one hour to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is Rs8 per bottle for A and Rs7 per bottle for B. Formulate this  problem as a linear programming problem.     Max: Z= 8x + 7y;  3x+ y≤ 66000, x + y ≤ 45000, 4x ≤ 20000 y≤ 40000 and x ≥0, y ≥ 0.

6) A resourceful home decorator manufacturers two types of lamps say A and B. Both lamps go through two technicians, first a cutter, second a finisher. Lamp A requires 2 hours of the cutter's time and one hour of the finisher's time. Lamp B requires 1 hour of cutter's and 2 hours of finisher's time. The cutter has 104 hours and finisher has 76 hours of time available each month. Profit on one lamp A is Rs6 and on one lamp B is Rs11. Assuming that that he can all that he produces, how many of each type of lamps should he manufacture to obtain the best return.      Max: Z= 6x + 11y;  2x+ y≤ 104, x + 2y ≤ 76, and x ≥0, y ≥ 0.

7) A company makes two kinds of leather belts. A and B Belt A is high quality belt, and B is of lower quality. The respective are Rs 40 and Rs30 per belt. Each belt of type A requires twice as much as time as a belt of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires a fancy buckle, and only 400 buckles per day are available. There are only 700 buckles available for belt B. What should be the daily production of each type of belt? Formulate the problem LPP.   Max: Z= 40x + 30y;  2x+ y≤ 1000, x + y ≤ 800, x ≤400; y ≤ 700 and x ≥0, y ≥ 0.

8) A small manufacturing firm produces two types of gadgets A and B , which are first processed in the foundary, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of A and B , and the number of man-hours the firm has available per week are as follows :

Gadget          foundry         machine-shop 

   A                     10                   5 

   B                      6                    4 

Firm's 

capacity 

per weak         1000              600

The profit on the sale of A is 30% per unit as compared with Rs20 per unit of B. The problem is to determine the weekly production of gadgets A and B , so that the total profit is maximized. Formulate this problem as a LPP. Max: Z= 30x + 20y; 10x + 6y≤ 1000; 5x + 4y ≤600, and x ≥0, y ≥ 0

9) A company is making two products A and B . The cost of producing one unit of products Aand B are Rs 60 and Rs80 respectively. As per the agreement, the company has to supply at least 200 units of products B to its regular customers. One unit of products A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labours hours are available. The company wants to minimise the cost of production of production by satisfying the given requirements. Formulate the problem as a LPP. Max: Z= 60x + 80y; x + y≤ 500; x ≤400, y≥200 and x ≥0, y ≥ 0 

 

10) A firm manufacturers 3 products A, B and C. The profits are Rs3, Rs2, Rs4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product.

 Machine                     Product 

                           A          B           C 

P                         4          3           5

Q                         2          2           4 

Machine P and Q have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's. Set up a LPP to maximize the profit. Max: Z= 3x + 2y + 4z; 4x + 3y + 5z ≤ 2000; 2x + 2y + 4z ≤2500, 100 ≤ x ≤ 150 y≥ 200, z≥ 50 and x ≥0, y ≥ 0, z≥ 0

11) A manufacture two types of products A and B and sells them at a profit of Rs2 on type A and Rs3 on type B. Each products is processed on two machines P and Q. type A requires one minute of processing time on P and 2 minutes of Q; type B requires 1 minute on P and 1 minutes on Q. The machine P is available for not more than 6 hours 40 minutes while machine Q is available for 10 hours during any working day. formulate the problem as a LPP. Max: Z= 2x + 3y; x + y≤ 400; 2x + y ≤600, and x ≥0, y ≥ 0

12) A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10000. If the profit is Rs60 per unit for the product A and Rs40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP. Max: Z= 60x + 40y; 5x + 3y≤ 45000; x ≤7000, y≤ 10000 and x ≥0, y ≥ 0 

EXERCISE - B

1) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units per kg of vitamin A and one unit per kg of v

itamin C while food Ii contains 1 unit per kg of vitamin A and two units per kg of vitamin C. It costs Rs5000 per kg to purchase food I and Rs7000 per kg to produce food II. Formulate the above linear programming problem to minimise the cost of such a mixture.        Max: Z= 5x + 3p7y;  2x+ y≥ 8, x + 2y ≥ 10 and x ≥0, y ≥ 0.

 

2) A diet is to contain at least 400 units of carbohydrates, 500 units of fat, and 300 units of protein. Two foods are available : A, which costs Rs 2 per unit, and B, which costs Rs4 per unit. A unit of food A contains 10 units of carbohydrates, 20 units of fat, and 15 units of protein ; a unit of food B contains 25 units of a carbohydrate, 10 units of fat, and 20 units of protein. Find the minimum cost for A diet that consists of A mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as A linear programming problem.       Max: Z= 2x + 4y;  10x+ 25 y ≥ 400, 20x + 19y ≥ 500, 15x +20y ≥ 300 and x ≥0, y ≥ 0.

3) The objective of A diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited of milk, beaf and eggs, and to vitamins A, B and C. The number of milligrams of each of these vitamins contained within A unit of each food is given below:

Vit  litofmilk  kgofmeat dozofeggs mn d r

 A       1                1               10            1 mg

 B     100             10              10         50 mg

 C     10              100             10        10 mg

Cost Rs100      Rs1.10      Rs0.50

 What is the linear programming formulation for this problem ?      Max: Z= x+ 1.10y + 0.5z,  x+ y +10z ≥ 1; 100x + 10y + 10z≥  50, 10x +100y + 10z ≥ 10 and x ≥0, y ≥ 0, z≥ 0

4) A rubber company engaged in producing three types of tyres A, B and C. Ech type requires processing in two plants, Plant I and Plant II . The capacities of the two plants, in number of tyres per day, are as follows:

Plant              A           B           C

 I                    50        100       100 

II                    60        600       200

The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If Plant I a

costs Rs 2500 per day, and plant II costs Rs3500 per day to operate, how many days should each be run per month to minimise cost while meeting the demand ? Formulate the problem as LPP.      Max: Z= 2500x+ 3500y, 50x+ 60y ≥ 2500; 100x + 60y≥ 3000, 100x +200y ≥ 7000 and x ≥0, y≥0

5) To maintain his health a person must fulfill certain minimum daily requirements for several kinds of nutrients. Assuming that there are only 3 kinds of nutrients-- calcium, protein and calories and the person's diet consists of only two food items, I and II , whose price and nutrients contents are shown in the table below:

               Food I         Food II.           Min req.

Calcium  10                 5                     20

Protein.    5                  4                     20

Calories   2                  6                     13

Price(Rs) 60              100 

What combination of two food items will satisfy the daily daily requirement and entail the least cost? Formulate this as a LPP.          Max: Z= 60x + 100y; 10x + 5y≥ 20; 5x + 4y ≥ 20  and x ≥0, y ≥ 0 

6) A manufacturers can produce two products , A and B , during a given time period. Each of these products requires four different manufacturing operation: grinding, turning, assembling and testing. The manufacturing requirements in hours per units of products A and B are given below.

                              A        B

Grinding               1         2

Turning                 3        1

assembling          6        3 

testing                  5        4 

The available capacities of these operations in hours for the given time period are grinding 30, turning 60, assembling 200, testing 200. The contribution to profit is Rs20 for each unit of A and Rs30 for each of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.       

7) Vitamins A and B are found in two different foods P and Q. one unit of food P contains 2 units of vitamin A and 3 units of vitamin B. One unit of food Q contains 4 units of vitamin A and 2 units of vitamin B. One unit of food P and Q cost of Rs50 and Rs25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any this in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food P and Q at the maximum cost which meets the daily minimum requirement of vitamin A and B. formulate this as a LPP.    Max: Z= 50x + 25y; 2x + 4y≥ 40; 3x +2 y ≥50,  and x ≥0, y ≥ 0.

EXERCISE - B

Shade the region/ Solve graphically 

1) 2x+ 5y≤ 0.

2) 3x - 4y> 12.

3) 4x + 3y≤ 12, x≥ 0, y ≥ 1.

4) 2x + 5y≤ 40, x+ y ≤ 11, x≥ 0, y ≥0.

5) 5x + 2y ≤ 20, 3x + 6y ≤ 18, x ≥ 0, y ≥ 0.

6) x + y ≤ 20, 6y -3x ≤ 48, x ≥ 0, y ≥ 0.

7) 3x + 6y ≤ 8, 5x +2y ≤ 10,  x ≥ 0, y ≥ 0.

8) 2x - y ≥ 4, 4x + 3y ≤ 28, x ≥ 0, y ≥ 0.

9) 2x + y ≥ 4, 3x + 5y ≥ 15, x ≥ 0, y ≥ 0.

10) x - 2y ≤0, 2x - y ≤3, x ≥ 0, y ≥ 0.

11) 2x + y ≥ 4, 2x - y ≥ - 2, x ≥ 0, y ≥ 0.

12) 2x + 3y ≥ 6, x + y ≤8, x ≥ 0, y ≥ 1.

13) 3x + 3y ≤ 17, 3y - 2x ≤  6, x ≥ 0, x ≥ 0, y ≥ 1.

14) x + y ≥ 1, x + 2y ≤ 10, x ≤ 4, x ≥ 0, y ≥ 0.

15) 4x + 5y ≤ 40, x ≥ 3,  y ≥ 4.

16) 3x + 5y ≤ 36, x + y ≤ 10, x ≥ 2, y ≥ 3.

17) 2x + 7y ≥ 22, x + y ≥ 6, 5x + y ≥ 10, x ≥ 0, y ≥ 0.

18) 5x + 7y ≥ 10, x + y ≥ 12, x + 4y ≥ 12, x ≥ 0, y ≥ 0.

19) 2x + 7y ≥ 18, x + y ≥ 12, 3x + 2y ≤ 42, x ≥ 0, y ≥ 0.

20) 6x + y ≥ 18, x + 4y ≥ 12, 2x + y ≥ 10, x ≥ 0, y ≥ 0.

21) 2x + y ≥ 18, x + y ≥ 12, 3x + 2y ≤ 34, x ≥ 0, y ≥ 0.

22) x + y ≤30, x - y ≤0, 0≤ x ≤20, 3≤ y ≤ 12.

23) x + 4y ≤ 12, 2x + 5y ≤ 20, y ≥ 0, 1≤ x ≤ 8.

PLAYING WITH THE NUMBERS

EXERCISE - A

) In a 2 digit number, the unit digit is four times the ten's digit and the sum of the digit is 10. Find the number.

) Without performing a actual addition and division, write the quotient 86 + 68 is divided by

a) 11

b) 14

) Consider the number 73 and 37. Find the quotient if their difference is divided by

a)  9 

b) 4

) Without actual division , obtain the quotient when the difference of 863 and 368 is divided by

a) 11

b) 5

) In a two digit number , the digit at the unit place is double the digit at ten's place. The number exceeds the sum of its digit by 18. Find the number.

) In a 3 digit number, the ten's digit is thrice the unit digit and the hundreds digit is four times the unit digit. Find the number if sum of its digit 16.

) Without actual division, obtain the quotient when the difference of the number 569 and 965 is divided by 

a) 11 

b) 4

) Without actual division, find out the quotient when the sum of the numbers 167, 716 and 671 is divided by

a) 37 

b) 111 

) The product of two digit numbers is 2117. If the product of their unit digit is 27 and that of ten's digit is 14, find the numbers.    

EXERCISE - B

Find the value of lb B 32 a68 1 ABB 6a 679 using digits from 029 solve the puzzle send more money find the value of A and B to ab 1 b18 digits from one to 9 find the value of the letter find the value of a b c d e f g and given division aba1b 266 46 b 6369 859 B A 786

Divisibility test divisibility by 2 a number is divisible by 2 if it ends with 02468 divisibility by 3 a number is divisible by 3 sum of its digit is divisible by 3 divisibility by 4 the number divisible by 4 class 2 digit is the divisible by 4 or ends with 00 divisibility by 5 the number is divisible by 5 times zero and 5 divisibility by 6 a number is divisible by 6 if it is divisible by both the result divisibility by 9 December will be resolved by 9 divisibility by 10 a number is divisible by 10 if it ends with zero divisibility by 11 beginning from the left digit place alternatively the result of divisible by 11 to the number is divisible

Miscellaneous

1828 57 divisible by the 9 then what is the list value of a + b what list number should be subtracted from 26543 show that it is divisible by 30057x is a multiple of 2 X is digit what is the least value of X8 y1 is divisible by 9 then what is the list value of x + Y + Z if 2 6 4 3 6 3 is a multiple of 11 is the value of the difference between the smallest number in the largest two digit number divisible by 3 without actual division obtain the question on the difference of the number 3863 and 368 / 996 what are the possible one digit of an