RANK CORRELATION (C)

RANK CORRELATION COEFFICIENT

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1) The coefficient of rank correlation of the marks obtained by 10 students in Mathematics and Statistics was found to be 0.5. It was then detected that the difference in ranks in the two subjects for one particular student was wrongly taken to be 3 in place of 7. What should be the correct rank correlation coefficient.

A) 0.26. B) -0.26. C) 0.62 D) 0.623

2)  R':  1       2      3       4         5

     R":  5       4      3       2         1                find R.

A) 1.    B) 0. C) - 1.   D) none

3) If sum of squares of difference in two ranks is 33 and Number of variables are 10, Find the value of coefficient of rank correlation.

A) 0.8. B) -0.80. C) 0.94. D) -0.94

4) Find rank correlation coefficient

R' :    1     2     3    4     5

R":.   1      2     3    4     5

A) 1.      B) 0. C) -1.  D) none

5) If n=10, and ∑D² =280 Find R.

A) 0.70. B) -0.7. C) 0.645.  D) none

6) If n=10 and ∑D² =30 find R.

A) 0.28. B) -0.82. C) 0.82. D) -0.28

7) If rank correlation coefficient is 0.60 and N=10 find ∑D² where D is the difference in ranks of the

  two series.

A) 61.  B) 64. C) 74.    D) 66

8) The coefficient of rank correlation between the marks in Statistics and Mathematics obtained by a certain group of students is ⅔ and the sum of the squares of the difference in ranks is 55. How many students are there in the group ?

A) 10  B) 9   C) 12  D) more than 15

9) Find rank correlation coefficient

Roll no.     Marks in. Marks in

                   Maths.       Statistics

1                 78                   84

2                 36                   51

3                 97                   91

4                 25                   60

5                 75                   68 

6                 82                   62

7                 90                   86

8                 62                   58

9                 65                   53

10               39                   47

A) 0.28. B) -0.39. C) 0.49. D) 0.82

10) Find R

Roll no.     Marks in   Marks in

                   English.            Maths

1                   43                      36

2                   29                       6

3                   35                      17 

4                   18                      14 

5                   40                       25

6                   11                       10 

7                   49                       32 

8                   10                         0 

9                    5                          3 

10                 22                       20

A) 0.95. B) -0.95. C) 0.82. D) 0.45

11) Find R

Roll no.     Marks in   Marks in

                   English.            Maths

1                   80                      85

2                   38                      50 

3                   95                      92 

4                   30                      58 

5                   74                      70 

6                   84                       65 

7                   91                       88 

8                   60                       56 

9                   66                       52

10                 40                       46

A) 0.87. B) 0.57. C) -0.57. D)0.49

12) find R

X: 80   91    99   71   61   81   70   59

Y:123 135 154 110 105 134 121106

A) -0.95. B) 0.785. C) 0.95 d) none

13) Find R

X: 75    88   95  70    60   80  81   50 

Y:120 134 150 115 110 140 142100

A) 0.93. B) -0.83. C) 0.85. D) 0.63

14) 

15) In a contest, two judges ranked eight candidates in order of their performances, as shown in the table given below. The rank Correlation coefficient is :

Candidates: A  B  C   D   E   F  G   H

JUDGE 1:     5   2   8   1  4   6   3    7

Judge 2:      4   5   7    3  2   8   1   6

A) - 0.678.            B) 0.875

C) 0.67.                D) none

16) Find R

R. N: 1    2    3     4    5    6    7       8 

X:     62  53  51  25  79  43  60    33

Y:.    52  63  45  36  72  65  45    25

A) 0.489 b) -0.64 c) 0.64 d) none

17) Find the Rank correlation

X: 70  65  71  62  58  69  78  64

X: 91  76  65  83  90  64  55  48

A) 0.3125.           B) - 0.3095

C) 0.2955.           D) - 0.2955

18) Find Rank correlation

Roll no.      Marks in.   Marks in

                   Account.        Statistics

1                     30                    15

2                     20                    40 

3                     40                    40 

4                     50                    45 

5                     30                    20 

6                     20                    30 

7                     30                    15 

8                     50                    50 

9                     10                    20 

10                    0                     10

A) 0.63. B) 0.83.  C) 0.36. D) none

19) 

20) The marks secured by a group of 10 students in Written Selection Test (X) and in the Aptitude Test (Y) are given in the following table. Calculate product-moment Correlation coefficient (r) and rank Correlation coefficient (R). The value of absolute difference between "r" and "R" is :

Test (X).        Test (Y)

  44                  24 

  42                  25 

  40                  28 

  52                  29 

  39                  32 

  32                  35

  24                  36

  46                  41

  41                  45

  50                  50

A) 0.063 b) 0.897 c)0.01 d) 0

21) 

22) 

23) Spearman's Rank correlation formula is given by
R= 1 -  6 ∑D²/(n³-n), where D stand for:
A) Sum of the rank of the two variables.
B) Difference between the rank of two variables
C) both A and B
D) either A or B

24) Spearman's Rank correlation formula  1- 6{∑D² + (m³-m))12}/(n³-n), is used when:
A) There are repeated rank in only one series
B) there are repeated ranks in any one of the series
C) There are repeated ranks in any one of the series or both of the series.
D) it is not used for repeated ranks.

25) The coefficient of rank Correlation of the marks obtained by 10 students in Mathematics and Statistics was found to be 0.5. it was then detected that the difference in ranks in the two subjects for one particular students was wrongly taken to be 3 in place of 7. What should be the correct rank Correlation coefficient?
A) +0.26 B) -0.26 C) 0.62 D) 0.623

26) If the sum of squares of difference in two ranks is 33 and number of variables are 10, find the value of Rank correlation coefficient.
A) 0.8 B) -0.8 C) 0.94 D) -0.94

27) If n=10 and ∑D² = 280, then which of the following represents the value of rank Correlation coefficient?
A) 0.7. B) -0.7. C) 0.645. D) none

28) For two series we have, ∑D²= 30 and n= 10, find the value of R
A) 0.28. B) -0.82. C) 0.82. D) -0.28

29) If R= 0.60 and n= 10. Find the value of ∑D². Where D is the difference in ranks of the two series.
A) 61.  B) 64.  C) 74.   D) 66

30) The coefficient of rank Correlation between the marks in Statistics and Mathematics obtained by a certain group of students is 2/3 and the sum of the squares of the differences in ranks is 55. How many students are there in the group?
A) 10.                        B) 9. 
C) 12.             D) more than 15

1) a 2) c 3) a 4) a 5) b 6) c 7) d 8) a 9) d 10) c 11) a 12) c 13) a 14) d 15) c 16) c 17) b 18) a 19) c 20) a 21) b 22) c 23) b 24) c 25) a 26) a 27) b 28) c 29) d 30) a

PROBLEM ON QUADRATIC EQUATIONS

                      EXERCISE--1

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1) Find two natural numbers which differ by 3 and the sum of the square is 117.                       6 and 9

2) The sum of the squares of two consecutive natural numbers is 41, Find the numbers.                        4, 5

3) Find the natural numbers which differ by 5 and the sum of whose squares is 97.                               4, 9

4) The sum of a number and its reciprocal is 4.25. find the number.      4 or 1/4

5) two natural numbers differ by 3. Find the numbers, if the sum of their reciprocal is 7/10.                 2,5

6) Divide 15 into two parts such that the sum of their reciprocal is 3/10.                                            5, 10

7) A can do a piece of work in 'x' days and B can do the same work in (x+16) days. If both working together can do it in 15 days. Find x.                                                   24

8) the square of a number added to one-fifth of it, is equal to 26. find the number.                       5 or -26/5

9) the sum of the squares of two consecutive positive even numbers is 52. find the numbers.             4, 6

10) find two consecutive positive odd numbers, the sum of the square is 74.                        5 and 7

11) two numbers are in the ratio 3: 5. find the numbers; if the difference between their square is 144.                            9, 15 or -9, -15

12) three positive numbers are in the ratio 1/2;1/3;1/4. Find the numbers, if the sum of their squares is 244.                       12,8,6

13) divide 20 into two parts such that three times the square of one part exceeds the other part by 10.         3, 17

14) the sum of two number is 32 and their product is 175. Find the numbers.                                 7, 25

15) three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60.  Assume the middle number to be x and form a quadratic equation satisfying the above statement. Hence; find the three numbers.                   9,10,11

16) The ages of two Sisters are 11 years in 14 years. In how many years time will the product of their ages be 304 ?                            5 yrs

17) out of the three consecutive positive integers, the middle number is is P. If three times the square of the largest is greater than the sum of the squares of the other two numbers by 67;  calculate the value of P.                                          5

18) Five times a certain whole number is equal to three less than twice the square of the number. Find the number.                            3

18) The product of two consecutive integers is 56. Find the integers.    7 or 8 or -8 and -7

20) Find two consecutive numbers whose squares have the sum 85.       6,7 or -6,-7

21) The sum of two numbers is 48 and their product is 432. Find the numbers.      36,12

22) If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.                        -10,9

23) Find the whole number which when decreased by 20 is equal to 69 times the reciprocal of the number.                                          23

24) Find two consecutive natural numbers whose product is 20.    4,5

25) The sum of the squares of two consecutive odd positive integers is 290. Find them.                         11,13

26) The sum of two numbers is 8 and 15 times the sum of their reciprocal is also 8. Find the numbers.                                    3,5

27) The sum of a number and its positive square root is 6/25. Find the numbers.                              1/25

28) The sum of a number and its square is 63/4, find the numbers.     7/2 or -9/2

29) There are three consecutive integers such that the square of the first increased by the product of the other two gives 154. What are the integers.                                   8,9,10

30) The product of two successive integrals multiples of 5 is 300. Determine the multiples.    15,20, or -20,-15

31) The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers.            8,13

32) Find the consecutive even integers whose squares have the sum 340.                                   12,14

33) Find two consecutive even integers, the sum of whose squares is 164.                                       8,10

34) Find two natural numbers which differ by 3 and whose squares have the sum 117.                          6,9

35) The sum of the squares of three consecutive natural numbers is 149. Find the numbers.            6,7,8

36) Divide 57 into two parts whose products is 782.                       23, 34

37) Determine two consecutive multiples of 3 whose product is 270.                                          15, 18

38) A sum of a number and its reciprocal is 17/4. Find the number.                                   4 or 1/4

39) A two digit number is such that the product of its digits is 8. When 18 is substracted from the number, the digits interchange their places. Find the number.                           42

40) A two digit number is such that the product of its digits is 12. When 36 is added to the number, the digits interchange their places. Find the number.                                   26

41) A two digit number is such that the product of its digits is 16. When 54 is substracted from the number, the digits interchanged their places. Find the number.                            82

42) the product of the digits of a two digit number is 24. If its unit's digit exceeds twice its ten's digit by 2; find the number.                       38.

43) Two numbers differ by 3 and their product is 504, find the numbers.                21, 24 or -24,-21

44) two numbers differ by 4 and their product is 192. find the numbers.                12, 16 or -16,-12

45) A two digit number is 4 times the sum of its digits and twice the product of its digits. find the number.                      36

46) the sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers.                   8, 12

47) A two digit number is such that the product of its digits is 18. When 63 substracted from the number, the digits interchange their places. find the number.                         92

48) A two digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number.                                       27

49) divide 16 into two parts such that the square of the larger part exceeds the the square of the smaller part by 64.                   10,6

50) The sum of the square of two positive integers is 208. If the square of the larger number is 18 times is smaller number, find the numbers.                             8 ,12

51) the difference of the squares of two numbers is 45. the square of the smaller number is four times the larger number. find the numbers.                         9,6 or -6,-9

51) If the sum of n successive odd natural numbers starting from 3 is 48, find the value of n.                    6

52) If the sum of the first n even natural numbers is 420, find the value of n.                                   20

53) The denominator of a fraction is one more than twice the numeritor. if the sum of the fraction and its reciprocal is 58/21, find the fraction.                                      3/7

54) A two digit number is 4 times the sum and the three times the product of its digits. find the number.                                         24

                EXERCISE--2

               **************

1) The sides of a right angled triangle containing the right angl are 4x cm and (2x-1) cm. If the area of the triangle is 30 cm²; calculate the length of its sides.        12, 5, 13

2) The hypotenuse of a right-angled triangle is 26cm the sum of other two sides is 34cm. Find the length of its sides.                       10, 24, 26

3) The sides of a right-angled triangle are (x-1) cm, 3x cm and (3x+1) cm. Find:

A) value of x.                                   8

B) the length of its sides.     7,24,25

C) its area.                                     84

4) The hypotenuse of a right-angled triangle exceeds one side by 1cm and other side by 18cm, find the length of the sides of the triangle.        25,24,7

5) The length of a rectangular plot exceeds its breadth by 12cm and the area is 1260m². Find its length and breadth.                           42, 30

6) The perimeter of a rectangular plot is 104m and its breadth is 640m². Find its length and breadth.        32, 20

7) A footpath of uniform width runs around the inside of a rectangular field 32m long and 24m wide. If the path occupies 208 m², find the width of the footpath.                   2m

8) Two squares have sides x cm and (x+4)cm. The sum of their areas is 656cm². express this as an algebraic equation in x and solve the equation to find the sides of the squares.                                   16, 20

9) the length of a rectangular board exceeds its breadth by 8cm. if the length were decreased by 4 cm and the breadth doubled, the area of the board would be increased by 256 cm². find the length of the board. 24

10) An area is paved with square tiles of a certain size and the number required is 600. if the tires had been 1cm smaller each way. 864 would have been needed. Find the size of the larger tiles.         6cm

11)  A farmer has 70m of fencing, with which he encloses three sides of a rectangle sheep pen; the fourth side being a wall. If the area of the pen is 600 m², find the length of its short side.                        15m when longer side is 40m and 20m when longer side is 30.

12) A square lawn is bounded on three sides by a part 4m wide. If the area of the path is 7/8 that of the lawn, find the dimensions of the lawn.                       Each side is 16m

13) the area of a big room is 300 m². If the length were decreased by 5m and the breadth increased by 5m; the area would be unaltered. Find the length of the room.     20m

14) The hypotenuse of a right angled triangle is 13cm and the difference between the other two sides is 7cm.                            5, 12

15) The length of a verandah is 3m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.

A) taking x as the breadth of the verandah, write an equation in x that represents the above statement.

B) Solve the equation obtained in A above and hence find the dimensions of the verandah.        6,3

16) The perimeter of a rectangular field is 82m and its area is 400 m². find the breadth of the rectangle.           16m

17) the length of a hall is 5m more than its breadth. If the area of the floor of the hall is 84 m², what are the length and the breadth of the hall?                                              7, 12

18) the hypotenuse of a right angle triangle is 25cm. the difference between the lengths of the other two sides of the triangle is 5cm. Find the length of these sides.        15,20  

19)  the hypotenuse of a right angle triangle is 3√10 cm. If the smaller leg is tripled the longer leg doubled, new hypotenuse will be 9√5cm.how long are the legs of the triangle ?     3, 9 cm

20) Two squares have sides x cm and (x+4) cm. the sum of their area is 656 cm². find the sides of the squares.                                    16, 20

21) the area of a right angle triangle is 165 m². determine its base and altitude if the latter exceed the formed by 7m.                      15, 22m

23) The hypotenuse of a right angle triangle is 6 m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.            10,26,24

24) The hypotenuse of right angled triangle is 3√5 cm. If smaller side is tried and the largest side is doubled, the new hypotenuse will be 15cm. Find the length of each side.                                           3, 6

25) Vikram wishes to fit three rods together in the shape of a right angle triangle. the hypotenuse is to be 2cm longer than the base and 4 cm longer than the altitude. what should be the length of the rods?      8, 6,10

26) The hypotenuse of a grassy land in the shape of a right triangle is 1 m more than twice the shortest side. If the third side is 7m more than the shortest side, find the sides of the grassy land.      8,17,15

27) if twice the area of a smaller square is subtracted from the area of a larger square, the result is 14cm². however, if twice the area of the larger square is added to three times the area of the smaller square, the result is 203 cm². Determine the sides of the square.          5,8cm

28) A farmer wishes to grow a 100m² rectangular vegetable garden. Since he has with him only 30m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side fence. Find the dimensions of his garden.          5x20 or 10x10.

29) the area of the right angle triangle is 600cm². If the base of the triangle exceeds the altitude by 10 cm, find the dimension of the triangle.              40, altitude 30cm

30) the perimeter of a rectangular field is 82n and its area is 400m². find the breadth of a rectangle.        25 or 16m

31) the length of the sides forming right angle of a right angle triangle are 5xcm and (3x-1)cm, if the area of the triangle is 60cm², find its hypotenuse.                    20.518cm

32) the area of an isosceles triangle is 60 cm² and the length of each one of its equal side is 13cm.find its base.                          24 or 10cm

33) the perimeter of a right angle triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle. 150cm²

34) the length of a rectangle exceeds its width by 8cm and the area of the rectangle is 240 cm². find the dimension of the rectangle.          12, 20

35) the side of a square exceeds the side of the another square by 4cm and the sum of its areas of the two square is 400cm². find the dimensions of the squares.    12cm

36) there is a square field whose side is 44 m. A square flower bed is prepared in its Centre leaving a gravel path all round the flower bed. the total cost of laying the flower bed and gravelling in the path at ₹2.75 and ₹1.50 per square metre, respectively, is ₹4904. find the width of the gravel path.               2m

37) two squares have sides x cm and (x+4) cm. The sum of their areas is 656 cm². Find the sides of the squares.                        13, 16

             EXERCISE --3

              *************

1) A man travels 200 km with a uniform speed of 'x' km/hr. the distance could have been covered in 2 hours less, had the speed been (x+5) km/hr. calculate the value of x.                20km/hr

2) The speed of a boat in still water is 8km/hr. It can go 15 km upstream and 22km downstream in 5 hours. Find the speed of the stream.                                3 km/hr

3) the speed of an ordinary train is X kilometre per hour and that of an Express train is (x+20) km/hr. 

A) find the time taken by train to convert 300 km.                     300/x

B) if the ordinary train takes 2 hours more than the express train. Calculate speed of express train.                       300/(x+25)

4) If the speed of a car is increased by 10 km/hr, it takes 18 minutes less to cover a distance of 36 km. find the speed of the car.   75 km/hr

5) If the speed of an aeroplane is reduced by 40km/hr. It takes 20 minutes more to cover 1200km. find the speed of the aeroplane.                  400km/hr

6) A train covers a distance of 300 km between two stations at a speed x km/hr. Another train cover the same distance at a speed of (x-5) km/hr

A) find the time with each train takes to cover the distance between the stations.                300/x, 300/(x-5)

B) If the second train takes 3 hours more than the first train, find the speed of each train.      25, 20 km/hr

7) A girl goes to a friend's house, which is at a distance of 12km. She covers half of the distance at a speed of x km/hr, and the remaining at a speed of (x-2)km/hr. if she takes 2 hours 30 minutes to cover the whole distance find x.                4

8) A car made a run of 390 km in x hours. if the speed had been 4 km per hour more, it would have taken 2 hours less for the journey find x.       15

9) A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed?                      30 km/hr

10) A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10km/hr less than that of the fast train, find the speed of the two trains.                     50, 40km/hr

11) A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.               25 km/hr

12) The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction?          20, 30km/hr

13) A plane left 40 minutes late due to had weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane.         800 km/hr.

14) A plane takes 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.           300 km/hr.

15) A goods train leaves a station at 6 p.m. followed by an Express train which leaves at 8 p.m. and travel 39 km per hour faster than the goods train. The express train arrives at a station, 180 km away, 36 minutes before the good train. Assuming that the speeds of both the trains remain constant between the two stations; calculate their speeds.                                   36, 75

16) )  a passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km)hr from its usual speed. Find the usual speed of the train.               25km/hr

17) ) A train travels a distance of 300 km at constant speed. if the speed of the train is increased by 5km/hr, the journey would have taken 2 hours less. find the original speed of the train.              25km/hr

18) The speed of a boat in still water is 15 km per hour. it can go 30km upstream and return downstream to the original point in 4 hours 30 minutes. Find the speed of the stream.                        5km/hr

19) A first train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10km/hr less than that of the fast train. Find the speeds of the two trains.                      40, 50

20) A plane left 30 minutes later than the schedule time and in order to reach its destination 1500km away in time it has to increase its speed by 250 km/hr from its usual speed. Find its usual speed. 750km/hr

21) In a flight of 600km, an aircraft slowed down due to bad weather. Its average speed for the trip was reduced by 200km/hr and the time of flight increased by 30 minutes. find the duration of flight.           1hr

22) Swati can row her boat at a speed of 5km/hr in still water. if it takes 1 hour more to row the boat 5.25 km upstream than to return downstream, find the speed of the stream.                                2km/hr

             EXERCISE --4

            -----------------------

1) A piece of cloth costs ₹35. If the piece were 4m longer and each metre costs ₹ one less, the cost would remain unchanged. How long is the piece ?                             10m

2) By selling an article for ₹24, a trader losses as much as percent as the cost price of the article. calculate the cost price.    60 or 40 

3) A trader bought a number of articles for ₹1200. Ten were damaged and he sold each of the rest at ₹2 more than what he paid for it, thus getting a profit of ₹60 on the whole transaction.   

 Taking the number of articles he bought as x, form an equation in x and solve it.                            100

4) Mr Mehra sends his servant to the market to buy oranges worth ₹15. the servant having eaten three oranges on the way, Mr. Mehra pays 25 paise per orange more than the market price. Taking x to be the number of oranges which Mr. Mehra receives, form a quadratic equation in x. find the value of x. 12

5) A man bought an article for ₹x and sold it for ₹16. If his loss was x percent, find the cost price of the article.                               20, or 80

6) A trader bought an article for ₹x and sold it for ₹52, thereby making a profit of (x-10) percent on his outlay. calculate the cost price, for trader, of the article.                   40

7) by selling chair for ₹75, Mohan gained as much as percent as its cost. calculate the cost of a chair.       50

8) An employer finds that if he increases the weekly wages of each worker by ₹3 and employs one worker less, he reduces his weekly wages bill from ₹816 to ₹781. Taking the original weekly wage of each worker as ₹x; obtain an equation in x and then solve it to find the weekly wages of each worker.                            68

9) a dealer sells an article for ₹24 and gains as much as percentage as the cost price of the article. find the cost price of the article.          20

10) If the list price of a toy is reduced by ₹2, a person can buy 2 toys more for ₹360. find the original price of the toy.                          ₹20

11) A piece of cloth costs ₹200. If the piece was 5m longer and each metre of cloth costs ₹2 less the cost of the piece would have remained unchanged. how long is the piece and what is the original rate per metre ?       20m, ₹10 p/m

12) A shopkeeper buys a number of books for ₹80. If he had bought four more books for the same amount, each book would have cost ₹1 less. how many books did buy?           16

13) If the price of a book is reduced by ₹5, a person can buy 5 more books for ₹300. find the original list price of the book.                       ₹20

14) A factory kept increasing its output by the same percentage every year. find the percentage if it is known that the output is doubled in the last two years.       100(-1+√2)

15) A dealer sells a toy for ₹24 and gains as much percent as the cost price of the toy. Find the cost price of the toy.                                 ₹20

           .         EXERCISE --5

                      **************

1) Ashu is x years old while his mother Mrs Veena is x² years old. 5 years hence Mrs. Veena will be three times as old as Ashu. find their present ages.                   5, 25

2)  the sum of the ages of a man and his son is 45 years. five years ago, the product of their ages was four times the man's age at that time. find their present age.     37, 9

3) the product of Shikha's age 5 years ago and her age 8 years later is 30, Her age at both times being given in years. find her present age. 7yrs.

4)  The product of Ramu's age (in years) five years ago and his age (in years) nine years later is 15. Determine Ramu's present age.     6

5) 1 years ago, A man was 8 times as old as his son. Now his age is equal to the square of his son's age. find their present ages.           7, 49

6) the product of Ramu's age(in years) 5 years ago with his age(in years) 9 years later is 15. find Ramu's present age.            6years.

7) the sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages(in years) was 124. Determine their present ages.                            36, 9

               Exercise -6

               *************

1)  A takes 10 days less than the time taken by B to finish a piece of work. if both A and B together can finish the work in 12 days. find the time taken by B to finish the work.        30 days

2) If two pipes function simultaneously, a reservoir will be filled in 12 hours. one pipe fills the Reservoir 10 hours faster than the other. how many hours will the second pipe take to fill the reservoir?                   30 hrs

3) A takes 6 days less than the time taken by B to finish a piece of work. if both A and B together can finish it in 4 days, find the time taken by B to finish the work.                12 days

4) A swimming pool is filled with three pipes with uniform flow. the first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by the third pipe alone. the second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. find the time required by each pipe to fill the pool separately.     15,10,6

5) two pipes running together can fill a cistern in 40/13 minutes. if one pipe 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.   5,8

6) one pipe can fill a cistern in 3 hours less than the other. the two pipes together can fill a cistern in 6 hours 40 minutes. find the time that is each pipe will take to fill the cistern.                                       12,15

        

                 EXERCISE--7

                ***************

Miscellaneous

1) Some students plant picnic. The budget for food was ₹480. But 8 of these fail to go and this the cost of food for each member increased by ₹10. how many students attended the picnic?                               16

2) ₹ 250 is divided equally among a certain number of children; if there were 25 children more, each would would received 50 paise less. find the number of children.            100

3) out of a group of swans 7/2 times the square root of the total number of playing on the shore of a pond. The two remaining ones are swinging in a water. find the total number of swans.                         16

4)  ₹900 were divided equally among a certain number of persons. Had there been 20 more persons each would have got ₹160 less. find the original number of persons.                                           25

5) some students planned a picnic. the budget for food was ₹500. but, 5 of them failed to go and thus the cost of food for each member increased by ₹5. how many students attended the picnic?    20

6) one fourth of a herd of Camels was seen in the forest. twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of the river. find the total number of camels.              36

7) Out of a group of swans 7/2 times the square root of the number are playing on the Shore of a tank. The two remaining ones are playing, with amarous fight, in the water. what is the total number of the swans?              16

8) A chess board contains 64 equal squares and the area of each square is 6.25 cm². A  border round the board is 2cm wide. find the length of the side of the chess board.               24cm

9) A person on tour has 360 for his expenses. if he extends his tour for 4 days, he has to cut down his daily expenses by ₹3. find the original duration of the tour.              20 days

10) ₹ 6500 were divided equally among a certain number of persons.  Had there been 15 more persons, each would have got ₹30 less. find the original number of persons.                             50

11) the Angry Arjun carried some arrows for fighting with the Bheesm. with half the arrows, he cut down the arrows thrown by Bheesm on him and with six other arrows he killed the rath driver of Bheesm. With one arrow each knocked down respectively the rath, flag and the bow of Bheesm. Finally, with one more than 4 times the square root of arrows he laid Bheesm unconscious on an arrow bed. find the total number of arrow Arjun had.                       100

12) one fourth of a herd of Camels was seen in the forest. Twice the square root of the herd had gone mountains and the remaining 15 camels were seen on the bank of a river. find the total number of camels.                   36.

13) A stone is thrown vertically downwards and the formula d= 16t²+ 4t gives the distance, d metres, that it falls in t seconds. How long does it take to fall 420 metres?                                  5 sec

                 EXERCISE-8

                     ********

1) 8x² -22x-21= 0.             7/2, -3/4

2) x² + 2√2 x -6= 0. -3√2, √2

3) x/(x+1) + (x+1)/x = 34/15, x≠ 0, x≠ -1.                3/2, -5/2

4) (x +3)/(x-2)-(1-x)/x= 17/4. 4, -2/9    

5) 4/x - 3= 5/(2x+3).                  -2, 1

6) 2x/(x-3) + 1/(2x+3)+ (3x+9)/{(x-3)(2x+3)} =0.                                 -1

7) x² - 2ax + a² - b²= 0.        a-b, a+b

8) x² - 4ax + 4a² - b²= 0.   2a-b, 2a+b

9) 4x²-4ax +(a² - b²)= 0.   (a-b)/2, (a+b)/2

10) 4x² - 4a²x +(a⁴ - b⁴)= 0. (a²-b²)/2, (a²+b²)/2

11) 4x² - 2(a²+b²)x + a²b²= 0.   a²/2,b²/2 

12) 9x² - 9(a+b)x + (2a²+5ab +2b²)= 0.                     (2a+b)/3, (a+2b)/3

13) x² + {a/(a+b) + (a+b)/a}x +1= 0.     - a/(a+b), -(a+b)/a

14) x² + x -(a+1)(a+2)= 0.       -(a+2), a+1

15) x² + 3x -(a² + a -2)= 0.      -(a+2), a-1

16) 1/(a+b+x) = 1/a + 1/b + 1/x, a+ b≠ 0.           -a, - b

17) x + 1/x = 626/25.          25, 1/25

18) (x-3)(x-4)= 34/(33)².      98/33, 133/33

19) a²x² - 3abx + 2b²= 0.     2b/a, b/a

20) 4x² + 4bx - (a² - b²)= 0.    (a-b)/2, -(a+b)/2

21) ax² + (4a² - 3b)x - 12ab= 0.    3b/a, -4a

22) (x - 1/2)²= 4.                5/2, -3/2

23) (x+3)/(x+2) = (3x -7)/(2x-3).     -1,5

24) 2x/(x-4) + (2x -5)/(x-3)=25/3.    6, 40/13

25) (x+3)/(x-2) - (1-x )/x= 17/4.     4, -2/9

26) (x-3)/(x+3) - (x +3)/(x-3)= 48/7, x≠ 3, x≠ -3.                 -4, 9/4

27) (x+2)/(x-1) - (x -1)/(x +1)= 5/6, x≠ 1, x≠ -1.                5, -1/5

28) (x-1)/(2x+1)+(2x+1)/(x-1) = 5/2, x≠-1/2.                  1,-1

29) 3x² -14x-5= 0.                    5, -1/3

30) mx²/n + n/m = 1 - 2x.                     -n±√(mn))/m

31) (x-a)/(x-b) + (x -b)/(x-a)= a/b + b/a.        0, a+ b   

32) 1/{(x-1)(x-2)} + 1/{(x -2)(x-3)} + 1/{(x-3)(x-4)}= 1/6.           -2,7

33) (x-5)(x-6) =25/(24)².     145/24, 119/24

34) 7x + 3/x = 178/5.            5, 3/35

35) a/(x-a)+ b/(x-b)= 2c/(x-c).        0, (2ab - bx- ac)/(a+b - 2c)

36) x² + 2ab = (2a+ b)x.           2a, b

37) (a + b)²x² - 4abx - (a-b)²= 0.      1, {(a-b)/(a+b)}²

38) a(x²+1)- x(a²+1)= 0.          a, 1/a

39) x² - x - a(a+1)= 0.              -a, a+1

40) x² + (a+ 1/a)x +1= 0.     - a, -1/a  

                 EXERCISE - 9

                     *********

A) Solve:

1) p²x²+ (p²-q²)x -q= 0.      -1, q²/p²

2) 9x²- 9(a+b)x +(2a²+5ab+2b²)= 0.

    (2a+b)/3, (a+3b)/3

3) abx²+(b²-ac)x -bc= 0.     c/b, -b/a

4) (x-1)/(x+2) + (x-3)/(x-4) =10/3.   (1±√297)/4

5) 1/(x+1) + 2/(x+2)= 4/(x+4), x≠-1,-2,-4.             2±2√3

B) Find the Discriminant:

1) x²- 4x +2= 0.                                8

2) 3x² + 2x - 1= 0.                          16

3) x²- 4x + a = 0.                      16 - 4a

4) √3x²- 2√2x -2√3= 0.                  -3

5) x² + px +2q= 0.                    p² - 8q

6) (x-1)(2x-1)= 0.                              1

C) Find the nature of the roots:

1) 2x²+x -1= 0.              Real, distinct

2) x²- 4x +4= 0.               Real, equal

3) x²+x +1= 0.                      Not real

4) 4x²- 4x +1= 0.              Real, equal

5) 2x² +5x +5= 0.                  Not real

6) (x-1)(2x-5)= 0.                  Not real

7) 3x²/5- 2x/3 +1= 0.           Not real

8) x² + 2√3 x -1= 0.      Real, distinct

9) 3x²- 2√6x +2= 0.         Real, equal

10) (x-2a)(x-2b)= 4ab.  Real, distinct

11) 9a²b³x²- 24abcdx +16c²d³= 0, a≠ 0, b ≠0.               Real, equal

12) 2(a²+b²)x²+2(a+b)x +2= 0.   Not real

13) (b+c)x²- (a+b+c)x +a= 0.     Real, distinct

D) Determine whether the given equations have real roots and if so, find the roots:

1) 9x²+7x -2= 0.                        2/9, -1

2) 2x²+ 5√3 x +6= 0.         -√3/2, 2√3

3) 3x²+ 2√5 x -5= 0.              √5/3,-√5

4) x²+5x +5= 0.                   (-5±√5)/2

5) 6x²+x -2= 0.                      1/2, -2/3

6) 25x²+20x +7= 0.               Not real

7) 3a²x²+ 8abx +4b²= 0, a≠ 0.     -2b/a, -2b/3a 

E) Find the values of k for which the given equations has real and equal roots.

1) 2x²- 10x +k= 0.                     25/2

2) 9x²+3kx +4= 0.                          ±4

3) 12x²+4kx +3= 0.                       ±3

4) 2x²+3x +k= 0.                          9/8

5) 2x²- kx +k= 0.                       ±2√2

6) kx²- 5x +k= 0.                        ±5/2

7) x²+k(4x+k-1) +2= 0.            2/3,-1

8)x²-2x(3k+1) +7(3+2k)= 0.   2,-10/9

9) (k+1)x²- 2(k-1)x +1= 0.          0, 3

10) (k-12)x²+ 2(k-12)x +2= 0.  12,14

11) x²- 2(5+2k)x +3(7+10k)= 0.     2, 1/2

12) (3k+1)x²+ 2(k+1)x +k= 0.   1,-1/2

13) kx²+ kx +1= -4x²- x.               5,-3

14) (k+1)x²+2(k+3)x +(k+8)= 0.   1/3

15) x²- 2kx + 7k -12= 0.                 4, 3

16) (k+1)x²- 2(3k+1)x + 8k+2= 0. 0,3

17) 5x²- 4x +2+k(4x²-2x-1)= 0.        1, -6/5

18) (4-k)x²+(2k+4)x +(8k+1)= 0. 0,3

19) (2k+1)x²+2(k+3)x +(k+5)= 0.   (-5±√41)/2

20) 4x²- 2(k+1)x +(k+4)= 0.       -3,5

21) k²x²- 2(2k-1)x +4= 0.             1/4

                   

F)      

1) For what value of k, (4-k)x² +(2k+4)x + (8k+1)= 0, is a perfect square.                        0, 3

2) If the roots of the equation (b- c)x² + (c-a)x + (a-b)= 0 are real, then show that 2b= a+ c.

3) If the roots of the equation (a²+ b²)x² - 2(ac + bd)x + (c²+ d²)= 0 are equal, then show that a/b= c/d

4) If the roots of the equation ax² + 2bx + c= 0 and bx² - 2 √(ac)x + b= 0 are simultaneously real, then show that b²= ac.   

5) If p, q are real and p+q, then show that the roots of the equation (p- q)x² + 5(p +q)x 2 (p-q)= 0 are real and unequal.

6) If the roots of the equation (c² - ab)x² +2 (a²-bc)x + b² - ac= 0 are equal, then show that a= 0 or a³+ b³+ c³= 3abc.

7) If the equation (1+ m²)x² + 2mcx+ (c²-a²)= 0 are real, then show that c²= a²(1+ m²).

BASIC INTEGRATION (INDIFINITE)

FORMULAE USED

* ∫xⁿdx = (xⁿ⁺¹)/(n+1) + c, n≠-1 where c is constant. 

** ∫ dx/x = log x + c

***∫ eˣ dx= eˣ + c

**** ∫ eⁿˣdx = eⁿˣ/n  + c

*****∫ aˣ dx= aˣ/ log a + c

****** ∫ aⁿˣ dx= aⁿˣ/n log a + c

            EXERCISE --1

              ----------------

1) ∫ x⁶ dx.                               x⁷/7+ c

2) ∫ 5x⁵ dx.                                5x⁶/6

3) ∫ x⁴ dx.                                   x⁵/5

4) ∫ x⁹.                                        x¹⁰/10

5) ∫ 8x⁷.                                           x⁸

6) ∫ 1/x⁷.                                    -1/6x⁶

7) ∫ x⁻¹.                                        logx

8) ∫ x⁵⁾³.                                  3/8 x⁸⁾³

9) ∫ x⁻⁵⁾⁴                                    -4x⁻¹⁾⁴

10) ∫ x⁵⁾⁴.                               4/9 ⁴√x⁹

11) ∫ x⁻²⁾³.                                   3x¹⁾³  

12) ∫ 1/x⁵.                                 -1/4x⁴

13) ∫ 2/x².                                     -2/x

14) ∫ 1/x³⁾².                                  -2/√x

15) ∫ 3x √x + 4 √x+5.            6/5 √x⁵ +8/3 √x³+ 5x

16) ∫ (x³+ 5x² -4+ 7/x + 2/√x).       x⁴/4 +5x³/3 - 4x + 7 log x+ 4√x

17) ∫ 6x⁵-2/x⁴ -7x+ 3/x - 5.     

18) ∫ 8 - x + 2x³ - 6x³+ 2x⁻⁵+ 5x⁻¹.    8x -x²/2+ x⁴/2- 3x⁴/2 -1/2x⁴+ 5 logx

19) ∫ x/a+ a/x.            x²/2a + a logx

20) ∫ 5x³ + 2/x⁵-7x +1/√x+ 5/x.    5x⁴/4 -1/2x⁴ - 7x²/2 + 2√x+ 5 log x

21) ∫ (1-x)√x.             2/3 √x³- 2/5 √x⁵

22) ∫ √x(ax²+ bx + c).      2a/7 √x⁷+ 2b/5 √x⁵ + 2c/3 √x³.

23) ∫ (√x +³√x²)²/x.     x+12⁶√x⁷/7 +3 ³√x⁴/4 

24) ∫ (3x⁴ +7x-11)/x³.      3x²/2 - 7/x + 11/2x²

25) ∫ (2-3x)(3+2x)(1-2x).     3x⁴+4x³/3 - 17x²/2 +6x

26) ∫ (√x - 1/√x)³ dx.             2√x⁵/5 - 2√x³ + 6√x + 2/√x

27) ∫ (3x+4)² dx.         3x³+12x²+16x

28) ∫ dx/x².                               -1/x

29) ∫ (x - 1/x)²dx.         x³/3 -1/x -2x

30) ∫ √x - 1/√x.              2√x³/3 -2√x

31) ∫(1+x)³/√x.        2√x+ 2√x³+ 6/5 √x⁵ + 2/7 √x⁷ 

32) ∫√x(x³ - 2/x).       2/9 √x⁹- 4 √x

33) ∫ (√x+ ³√x²)/x         2√x+3 ³√x²/2

34) ∫ (2x³+3x-7)/³√x².         3³√x¹⁰/5 + 9³√x⁴/4 - 21³√x.

35) ∫ 1/√x(1+ 1/x).           2√x - 2/√x

36) ∫ (x⁶+1)/(x²+1).       x⁵/5- x³/3+ x

37) ∫ (4x+3)².         16x³/3 +12x²+ 9x

38) ∫ dx/x³.                                -1/2x²

39) ∫ (x² + 1/x²)³.            x⁷/7 -1/5x⁵+ x³ -3/x

40) ∫ (1+x)²/√x.                   2√x +4/3 √x³ + 2/5 √x⁵

41) ∫ (1/³√x +√x+2)/³√x.      3 ³√x + 6/7 ⁶√x⁷+ 3 ³√x².

42) ∫ √x(3-5x).                2√x³ - 2 √x⁵.

43) ∫ (x⁴ + x² +1) d(x²).             x⁶/3+ x⁴/2 + x³

44) {(x+1)(x-2)}/√x.           2/5 √x⁵ - 2/3 √x³ - 4√x 

45) ∫(x⁵ + 1/x² +2)/x².      x⁴/4 - 1/3x³ - 2/x.

46) ∫(2x⁴+ 7x³+ 6x²)/(x²+ 2x).    2x³/3 +3x²/2

47) ∫(5x⁴ +12x³+ 7x²)/(x²+x).  5x³/3 + 7x²/2

48) ∫ (x⁴+ x²+1)/(x²- x+1).        x³/3+ x²/2 + x.

49) ∫ {(x³+8)(x-1)}/(x²-2x+4).    x³/3 + x²/2 - 2x

50) ∫ (√x+1)³/x.          2/3 √x³+ 3x + 6√x + log x

51) ∫(1-x³)/(1-x).          x+ x²/2+ x³/3

52) ∫ (3x+2)/(x-2).          3x+8log|x-2|

53) ∫ (x²+5x+2)/(x+2).      x²/2+ 3x- 4 log|x+2|     

54)  ∫x³/(x+2).          x³/3- x²+ 4x -8 log|x+2|

55) ∫ (x⁴- 2x³+3x-7).     x⁵/5- x⁴/2 +3x²/2 -7x

56) ∫ (1+2/√x+3/x).    x+ 4√x+ 3 logx

57) ∫ (x+2)(x-3).            x³/3- x²/2- 6x

58) ∫(3x²-2)².               9x⁵/5- 4x³+ 4x

59) ∫ {(x²+5x)(3x-2)}/x³.    3x+10/x +13 logx

60) ∫ x²/(x+1).     x²/2 - x+ log(x+1)

61)∫(1-x⁴)/(1-x).    x+ x²/2+x³/3+x⁴/4

62) ∫ (x-1)²/(x√x).      2x√x/3- 4√x-2/√x

63) ∫ (2x-3)²/ ³√x.                   3³√x⁸/2 - 36³√x⁵/5 + 27³√x²/2

64) ∫ (2x-1)/(x+1).                    2x - 3 log(x+1)

65) ∫(4+5x)/(3-2x).         -5x/2-23/4 log(3-2x)

66) ∫(x³+5x²-3)/(x+2).       x³/3+3x²/2-6x +9 log(x+2)

67) If f'(x)= x - 1/x² and f(1)= 1/2, find f(x).                x²/2 + 1/x - 1

68) If f'(x)= 3x² - 2/x³ and f(1)= 0, find f(x).             x³ +1/x² -2

69) If f'(x)= x + b, f(1)= 5, f(2)= 13, find f(x).             x²/2+13x/2 - 2

70) If f'(x)= 8x³ -2x, and f(2)= 8, find f(x).                                2x⁴ - x² -20

71) ∫ (2ˣ +3ˣ)/5ˣ. (2/5)ˣ/log(2/5) + (3/5)ˣ/log(3/5)

72) ∫(aˣ+ bˣ)²/aˣbˣ. (a/b)ˣ/log(a/b) + (b/a)ˣ/log(b/a) + 2x

73) ∫ eˣ ˡᵒᵍ ᵃ + eᵃ ˡᵒᵍ ˣ + eᵃ ˡᵒᵍ ᵃ. aˣ/log a + xᵃ⁺¹(a+1)+ aᵃ x

74) ∫(e⁵ˡᵒᵍ ˣ - e⁴ˡᵒᵍ ˣ)/(e³ˡᵒᵍ ˣ - e²ˡᵒᵍˣ). x³/3 

75) ∫ (x-2)³/x².           x²/2- 6x+ 12logx + 8/x

76) ∫ (px²+ qx +r)/x√x.      

77) ∫(2e⁴ˣ - 3e²ˣ+4)/e³ˣ.             2eˣ + 3e⁻ˣ+4e⁻³ˣ/3.     

78) ∫ (e⁶ˣ+ e⁴ˣ)/(eˣ + e⁻ˣ).      e⁵ˣ/5

79) ∫ (e²ˡ⁰ᵍˣ - e⁻ˡᵒᵍˣ)/(e²ˡ⁰ᵍˣ - eˡᵒᵍˣ).   x+ log x - 1/x.

80) ∫ (3²ˣ - 2. 3ˣ+6)/3³ˣ     -1/3ˣlog3 + 1/3²ˣlog 3 - 2/3³ˣlog3.

81) ∫(8²⁺ˣ -4²⁻ˣ)/2ˣ⁺³ .     4.2²ˣ/log2 + 2/3.  2⁻³ˣ/log 2

81) ∫ 2³ˣ.                              2³ˣ/3log 2

82) ∫ e⁵ˡ⁰ᵍˣ.                                    x⁶/6

83) ∫ xⁿ dx

84) ∫ 5x² dx

85) ∫ x⁶⁾⁵ dx

86) ∫³√(x⁴) dx

87) ∫ 1/√(x) dx

88) √(x) - 1/√(x) dx

80) ∫ (9x⁵+ 8x⁴ +x -9) dx

90) ∫(x+1/x)³

91) ∫(6x² -3x +8 - 1/√(x) + 1/x + 1/x²)

92) ∫(ax³ + bx² + cx +d)/x     dx

93) ∫(4x⁸+3x⁵+2x⁴+x³+x²+1)/x³   dx

94) ∫(1-x)(2+3x)(5-4x) dx

95) ∫ (3 - 2x - x⁴) dx

96) ∫ (4x³ + 3x² - 2x +5) dx

97) ∫ (x² -1)² dx

98) ∫ (√x - x/2 + 2/√x)  dx

99) ∫(1-3x)(1+x) dx

100) (x⁴ +1)/x²  dx

101) ∫ (3x⁻¹ + 4x² - 3x +8)  dx

102) ∫ (x - 1/x)³  dx

103) ∫(x² -3x + ³√x +7)/√x     dx

104) ∫ (ax + bx⁻²+ cx⁻⁷)/kx⁻²  dx

105) ∫(1+x)³/x  dx

106) ∫ √x(x⁵ + 3/x) dx

107) ∫(x√x -(1/3) √x + 11/√x) dx

108)  ∫ (1-x⁸)/(1-x)  dx

109) ∫ (a¹⁾³ - x²⁾³) dx

110)  ∫(x+2)(x+3)² dx

111) ∫ (1-2x²)²/x ³√(x)   dx

112)  ∫ (x³ - 4x² +5x -2)/(x²-2x+1) dx

113) (x³-6x +9)/(x+3)  dx

114) ∫(x⁴ +x² +1)/{2(x²+1)}  dx

115) ∫  (2e²ˣ + 3e⁴ˣ +4)/e³ˣ  dx

116)  ∫ (e³ˣ + e⁵ˣ)/(eˣ + e⁻ˣ) dx

117) ∫ e⁵ ˡᵒᵍ ˣ - e⁴ ˡᵒᵍˣ)/(e³ˡᵒᵍˣ -e²ˡᵒᵍˣ)

118) ∫ eᵃ ˡᵒᵍˣ + eˣ ˡᵒᵍ ᵃ) dx

119) ∫ (aˣ + a²ˣ +a³ˣ)/a⁴ˣ dx

120) ∫ (8¹⁺ˣ+4¹⁻ˣ)/2ˣ dx 

----------------------------------***-***********

   FORMULAE USED

_____________________

          

* ∫ sinx dx= - cosx + c

** ∫ sin mx dx= - (cos mx)/m +c

* ∫cosx dx = sin x + c

* ∫ cos mx dx= ∫ (sin mx)/m  + c

* ∫ sec²x dx= tan x+ c

* ∫ sec² mx dx = (tan mx)/m + x 

* ∫ secx tanx dx = secx + c

* ∫ sec mx tan mx dx=(sec mx)m+ x

* ∫ cosec²x dx= - cotx + c

* ∫ cosec²mx dx= -(cot mx)/m + c

* ∫ cosecx cotx dx= - cosecx + c

* ∫cosec mx cot mx dx= -(cosec mx)/m + c 

 

-----------()-------------()------------()-------------

        Commonly used Trigonometric formula:

1) sin²x = (1/2) (1- cos2x)

2) cos²x = (1/2) (1+ cos2x)

3) tan²x= sec²x -1

4) cot²x = cosec²x -1

5) sin³x = (1/4) (3 sinx - Sin3x)

6) cos³x = (1/4) (3 cosx + cos3x)

7) sinA cosB=(1/2) [sin(A+ B)+ sin(A- B)]

8) sinA sinB= (1/2)[cos(A- B) -cos(A+ B)]

9)cosAcosB=(1/2) [cos(A+ B)+cos(A- B)]

10) sinx cosx = (1/2) sin2mx

11) sin²mx = (1/2) (1- cos2mx)

12) cos²mx = (1/2) (1+ cos2mx)       

       

                      EXERCISE--2

                       ____________

1) ∫ cos 2x/sin²x cos²x.            -cotx - tanx

2) ∫(a+ b sinx)/cos²x dx.         a tanx + b secx

3) ∫ (3sinx - 4cosx+ 5/cos²x - 6/sin²x + tan²x - cot²x).       -3cosx- 4sinx +5 tanx +6cotx + tanx + cotx

4) ∫ (cos²x - sin²x)/√(1+ cos4x).    x/√2

5) ∫ (2 cos x/5 sin²x + 1/5cos²x) dx. -2/5 cosec x + 1/5 tan x

6) ∫ (sin²x - cos²x)/(sin²x cos²x)dx. 

       tanx + cot x

7) ∫ sin²(x/2) dx.         x/2 - 1/2 sin x

8) ∫ sin²x dx. x/2 - 1/4 sin 2x

9) ∫ cos² mx dx.              x/2 + (sin 2mx)/4m

10) ∫ cos³x dx.                      3/4 sinx + 1/12 sin3x

11) ∫ sin⁴x dx.               3x/8 - 1/4 sin 2x + 1/32 sin 4x

12) ∫sin²x cos²x dx.           x/8 - 1/32 sin 4x

13) ∫sin³x cos³x dx.       -3/64 cos 2x + 1/192 cos 6x

14) ∫ (cos 2x+ 2sin²x)/cos²x dx.      tan x

15) ∫ sin x sec²x dx.                 sec x

16) ∫ sin x cos x dx.        -1/4 cos 2x

17) ∫ sin 2x cos 2x dx.     1/2[cos x - 1/5 cos 5x]

18) ∫ (cos x cos 2x cos 3x) dx.      x/4 + 1/8 sin 2x + 1/16 sin 4x + 1/24 sin 6x

19) ∫ 1/(1+ sin x) dx.     tan x- sec x

20) ∫1/(1+ cos x).           Tan(x/2)

21) ∫ √(1+ cos x) dx.    2√2 sin(x/2)

22) ∫ √(1- sin 2x) dx.         Sinx+ cosx

23) ∫ √(1+ sin x) dx.        2(sin(x/2) - cos(x/2))

24) ∫ cos x √(1+ cos 2x) dx. 1/√2(x + sin x cos x)

25) ∫ sinx/(1+ sinx) dx.        Sec x - tan x+ x

26) ∫ (cos 2x - cos 2a)/(cos x- cos a) dx.                   2(sinx + x cos a)

27) ∫(cosx - cos 2x)/(1- cosx).   2sinx + x

28) ∫ (sin³x - cos³x)/sin²x cos²x.      Secx + cosecx

29) ∫(5cos³x + 6sin³x)/2sin²x cos²x.     -5/2 cosecx + 3 secx

30) ∫{(cos x- sin x)(2+ 2sin 2x)}/(cos x+ sin x) dx.        

31) ∫ (cos 2x + 2 sin²x)/cos²x dx.  tan x

32) ∫sin(x/2) dx.           -2 cos (x/2)

33) ∫ cos 3x.                   1/3. Sin 3x

34) ∫ (4 - 5sin x)/cos²x.         4 tan x - 5 sec x

35) ∫ (2+ 3cos x)/sin²x.          -2cotx - 3cosecx

36) ∫ sin²mx.                    x/2 - (sin 2mx)/4m

37) ∫ sin³x. 1/4(-3cosx + 1/3 .     Cos 3x)

38) ∫sin 4x cos 2x.          1/12 cos 6x -1/4 cos 2x

39) ∫ 3cosec²x + 2sin 3x.       -3cot x -2/3 cos 3x

40) ∫ 1/(1 - cos x).              -cot(x/2)

41) ∫ 1/(1 + cos x).      -cotx+ cosecx

42) ∫ 1/(1- cos 2x).             -1/2 cot x

43) ∫ √(1+cos 2x).                  √2 sinx

44) ∫ 1/(1- sin x).           Tan x+ sec x

45) ∫ 1/(1+ sin x).            Tan x- sec x

46) ∫ 1/(sin²x cos²x).      - cot x+ tan x

47) ∫ cos⁴x sin⁴x.       1/32(3x - sin 4x +1/8 sin 8x)

48) ∫ sin x sin 2x sin 3x.     -1/4(-1/6 cos 6x +1/2 cos 2x + 1/4 cos 4x)

49) ∫ (1- cos 2x)/(1+ cos 2x).   Tan x - x

50) ∫ (7cos³x + 8 sin³x)/(3sin³x cos²x).           8/3 sec x - 7/3 cosec x

51) ∫(sin⁶x + cos⁶x)/(sin²x cos²x).                   Tan x - cot x - 3x

52) ∫ sin⁶x.              1/32(10x -15/3 sin 2x +3/2 sin 4x -1/6 sin 6x)

53) ∫ sinx/cos²x.                      sec x

54) ∫ √(1+ sin 2x).          sin x- cos x

55) ∫ √(1- sin 2x).          - sin x- cos x

56) ∫ sin x √(1- cos 2x).        1/√2(x - sin x cos x)

57) ∫ (cos²x - sin²x)/(√(1+ cos 4x).      x/√2

59) ∫ √(1+ sin(x/2) 4(sinx/2 - cosx/2)

60) ∫ 2 sin x.                        - 2 cosx

61) ∫ 3 sin x - 2cos x + 4sec²x - 5cosec²x.             -3cos x - 2sin x + 4 tanx + 5 cotx.

62) ∫cosec x(cosec x - cot x) dx.      cosec x - cot x

63) ∫ tan²x dx.                        tan x - x

64) ∫ (tan x+ cot x)² dx.           Tan x - cot x

65) ∫ (tan2x + sec 2x)².         tan 2x+ sec 2x - x

66) ∫ sec x/(sec x + tan x).    tan x - x

67) ∫ cot²x dx.                     - cot x - x

68) ∫ sec²x + cosec²x.        Tanx - cotx

68) ∫ sec²x cosec²x dx.     Tan x- cot x

69) ∫cosec x/(cosec x - cot x) dx. Cot x - cosec x

70) ∫ sec² 2ax dx.        1/2a. Tan 2ax

71) ∫ 1/2 sec²x.                   1/2 tan x

72) ∫ cot x/((cosecx - cotx).               - cosecx - cotx

IMPORTANT QUESTIONS FOR COMPOUND ANGLE, SUM & PRODUCT.

Short Answer Type & Objective Questions::

______________________________

1) Evaluate:

A) sin 15.                          (√3-1)/2√2

B) cos 15.                        (√3+1)/2√2

C) tan 15.                                     2-√3

D) sun 75.                         (√3+1)/2√2

E) cos 75.                          (√3-1)/2√2

F) tan 75.                                   2 + √3

2) If tan(A+B)= 1/2, tan(A-B)=1/3, then find

A) tan 2A.                                         1

B) tan 2B.                                       1/7

3) evaluate: Cos 20+ cos100 + cos140.                                           0

4) A positive acute angle is divided into two parts whose tangents are 1/2 and 1/3. Find the angle.       45°

5) If A+B+C= π and cosA= cosB cosC, Prove that, cotB cotC= 1/2.

6) If x,y,z are in AP, show that, cot y= (sinx - sin z)/(cos z - cos x).

7) If tanA= 1/2 and A+B= π/4, find tan B. 1/3

8) If sinA= 3/5 and If tanB= 5/12, given K being obtuse and B acute, find sin(A+B).                           16/65

9) If A+ B= π/4,Prove that, (1+ tanA) (1+ tanB)= 2.

10) Find the maximum and minimum values of 3 sinx+ 4 cosx.

       5, -5

11) Choose the correct option:

a) The value of tan 75 - Cot 75 is

A) 2√3.  B) 2 + √3  C) 2 - √3.  D) n

b) If (1+ tanA)(1+ tan B)= 2, then the value of A+ B is

A) π/4. B) π/2 C) 3π/4 D) none

c) The value of sin 50 - sin 70+ sin 10 is

A) 1 B) 0. C) 1/2 D) 2

d) value of (cos 36- sin 36)/(cos 36 + sin 36) is

A) tan. 9 B) tan 54 C) tan 81 D) tan 36

e) Value of cos 15+√3cos15 is

A) -1 B) √2. C) -√2 D) 1

f) If A= π/13, the value of (sin 11A - sin 3A)/(sin 3A +sin 15A) is

A) -1. B) 0 C) 1 D) none 

g) Value of tan 20+ tan 50+ tan 110 - tan 20 tan 50 tan 110 is

A) 1 B) 0. C) -1 D) undefined

h) If A+ B+ C=π and tanA= 1, tanB= 2, then tan C is

A) -1 B)-2 C) -3 D) 3.

I) If A+ B+ C=π the value of sin(A+B) sinC - cos(A+B) cosC is

A) -1 B) 0 C) 1 D) none

              ESSAY TYPE

           --------------------------

1) Prove :

a) tan 70= 2tan 50 tan 20.

b) tan 70=tan 20+ 2tan 40+ 4tan 10.

c) sin²(45+A) - sin(30- A) - sin 15 cos(15+ 2A) = sin 2A. 

d) tan(30-A) tan 2A + tan 2A tan(60 - A) + tan(60-A) tan(30 -A) = 1 

e) sin 16 + cos 16 = 1/√2 (sin 1+ √3 cos 1)

f) cot 16 cot 44 + cot 44 cot 76 - cot 76 cot 16 = 3.

2) If cos(A+B) = cos C, show that 1- cos²A - cos²B - cos²C + 2 cosA cosB cosC= 0.

3) An angle a is divided into two parts, such that the ratio of their tangents is K and their difference is x, show that sinx={(K-1)/(K+1)}. Sin a.

4) If cotA, cotB, cotC are in AP, show that cot(B-A), cotB, cot(B-C) are also in AP.

5) If a cosx= b cos(x+120)= c cos(x+240), prove that ab+ bc+ ca= 0.

6) Given A+ B-C=π, prove that, sin²A +sin² B - sin² C= 2 sinA sinB cosC.

7) If sinA= K sin(A+B), prove that tan(A+ B)= sinB/(cos B - K).

8) If the angles A, B, C of a ∆ ABC are in AP, show that √3 tanA tanC - tanA - tan C= √3.

9) If m tan(A-30) = n tan(A+ 120), prove that cos 2A = (m+n)/{2(m-n)}.

10) If {tan(A-B)}/tan A + sin² C/sin²A = 1, show that tan A tan B= tan² C.

11) If x/a = cos(x - A) and y/b = cos(x - B), show that x²/a² + y²/b² - 2xy/ab cos(A- B)= sin²(A-B).

12) If A+ B+ C= 180, show that (cotA + cotB)/(tanA + tanB) + (cotA + cotC)/(tanB + tanC) + (cotC + cotA)/(tanC + tanA) = 1

13) If cos(A +B) cos(C+ D) + cos(A -B) cos(C- D)= 0, then show that tanA tanB tan C tanD= -1.

14) 2 cos x= a+ 1/a, 2 cos y= b+1/b, show that, 2 cos(x-y)= a/b + b/a. 

15) If a/b = cosx/cosy, then show that a tan x + b tan y= (a+b) tan{(x+y)/2}.

16) If A+ B+ C=π, Show that, cotA cot B cot C + cosec A cosec B cosec C = cot A + cot B + cot C. 

17) If (a+b) tan(A -B) = (a-b) tan(A+ B) , and a cos 2B + bcos 2A= c, prove that a² - b² + c²= 2ac cos 2B.

18) If A+ B+ C=π, Show that tanA/(tan B tan C) + tan B/(tanC tan A) + tan C/(tanA tan B) = (tanA + tanB + tan C) - 2(cotA + cotB + cot C).

19) If tanA, tanB are the roots of x² + px + q= 0, Show that the value of sin²(A+B) + p sin(A+B) cos(A+B) + q cos²(A+ B) is q.

20) If √2 cos A = cos B= cos³B and √2 sinA = sin B - sin³B, then show that sin(A- B)= ±1/3.

21) If cos(x-y)+ cos(y-z)+ cos(z-x)= -3/2 show that

A) cosx + cos y+ cos z = 0 and sin x + sin y + sin z= 0

B) cos(x-y)= cos(y-z)= cos(z-x)= -1/2 .

22) If the angle C of a triangle ABC be obtuse, then show that, tanA tanB < 1.

23) A, B, C are the angles of an acute angled triangle, show that, cos A cos B cos C ≤ 1/8.

24) Evaluate:

a) sin 10 sin 50 + sin 50 sin 250 + 

sin 250 sin 10.                           -3/4

b) If 9x= π, then cosx cos 2x cos 3x cos 4x.                                        1/16

c) If A, B, C are the angles of a triangle, find the maximum value

A) SinA+ Sin B + Sin C. 3√3/2

B) cosA + cos B + cos C. 3/2

C) SinA. Sin B. Sin C. 3√3/8

D) cosA cos B cos C. 1/8

d) If tan(A -B)= sin 2B/(5- cos 2B), Find tanA/tanB.                        3/2

e) 1/2 sin 10 - 2 sin 70.                   1

25) Eliminate A and B from,

A) tanA + tanB= a, cotA + cotB= b and A+B= K.

B) sinA + sinB= a, cosA + cosB= b and A-B= π/3.

LOGARITHMS (IX)

logarithmic form of
1) 6⁻¹ = ⅙ is

A) log₆(1/6)= 1 B) log₆(1/6)= -1

C) log₆6= 1       D) none 

2) ³√(27) =3 is

A) log 27¹⁾³= 3    B) log₂₇3 = 1/3 

C) log₂₇3= 1/3    D) log₂₇(1/3)= 3

3) 2⁵= 32 is..

A) log₂32= 5      B) log₂5= 32

C) log₅32= 2     D) log₅2= 32 

4) 2⁰= 1 is..

A) log₁2= 0           B) log₂1= 0

C) log₁0 =2           D) log₂0=2 

5) ³√64= 4 is..

A) log₆₄4= 1/3 B) log₄64= 1/3 

C) log₃64= 1/4 D) log₃4= 64 

6) 8⁻²⁾³ = 1/4 is...

A) log₈(1/4)= -2/3

B) log₈(-1/4)= 2/3

C) log₈(2/3)= -1/4

D) log₈(-2/3)= 1/4 

7) 10⁻² = 0.01 is..

A) log₁₀(0.01) = 2 B) log₁₀(0.01) =-2

C) log₂(0.01)=10  D) log₂(0.01) = 0.1

8) 4⁻¹= 1/4 is..

A) log₄(1/4)= 1

B) log₄(1/4)= -1

C) log₁(1/4)= 4

D) log₁4= 1/4 

*** Exponential form if
9) log₅(625) = 4 is..

A) 5⁴= 625              B) 4⁵= 625 

C) 5/4= 625            D) none 

10) log√₃ 27= 6 is...

A) (√3)²⁷= 6         B) (√3)⁶= 27

C) 3²⁷= 2              D) 3²⁷ = √2

11) log₄(4) = 1

A) 1⁴= 4 B) 4⁴= 0 C) 4⁴= 1 D) 4⁰=1 E) 4¹= 4 

12) log√₅ 625= 8 is 

A) √5= 625            B) (√5)⁸= 625

C) 5=√625             D) none 

13) log₂(1/32)= -5 is..

A) 2⁵= 32               B) 2⁻⁵= 1/32

C) 1/2⁵= 1/32       D) none 

** Value of:
14) log₃(81) is 

A) 1  B) 2    C) 3      D) 4 

15) log₁₀³√(100) is 

A) 1/3  B) 2/3  C) 10 D) 1 

16) log₂(1/32) is..

A) 5 B) 1/5 C) 1/2 D) 2 E) - 5

17) log₉(27) is..

A) 1/2 B) 2/3 C) 1/3 D) √3 E) 3/2

18) log₇343 is..

A) 7 B) 3 C) 7/3 D) 3/7

19) Log₂64 is..

A) 2 B) 4 C) 6 D) 8 

20) log₈32 is..

A) 5 B) 5/2 C) 5/3 D) 5/4

21) log ₃(1/9) is...

A) 1 B) -1 C) 2 D) -2 

22) log₀·₅(16)

A) 2 B) -2 C) 4 D) -4 

23)  log₂(0.125) is..

A) -1 B) -2 C) -3 D) -4 

24) log₇7 is.

A) 0 B) 1 C) -1 D) none 

25) log₅√₅125 

A) 2 B) 4 C) 6 D) 8 

** Find the value of x if..

26) log₃(x)= 4 

A) 3 B) 9 C) 17 D) 81 

27) log₂₅(x)=-1/2 

A) 5 B) -5 C) 1/5 D) -1/5 

28) log₁/₂(x)= -3  

A) 8 B) 1/8 C) 4 D) 2 

29) x=log₁₀(0.001)   

A) 3 B) -3 C) 1/3 D) -1/3 

30) log₂₅(x) = -½  

A) 5 B) 1/5 C) -5 D) -1/5 

31) logₓ(243) = 5   

A) 3 B) 5 C) -3 D) -5  

32) log₂x=-2

A) 4 B) 1/4 C) -4 D) -1/4

33) logₓ9 = 1

A) 1 B) 9 C) 3 D) 1/9 

34) log₉243= x 

A) 2 B) 5 C) 2/5 D) 5/2

35) log₃x= 0 

A) 0 B) 1 C) 3  D) 1/3 

36) log√₃(x-1)= 2 

A) 1 B) -2  C) 3 D) 4 

37) log₅ (x²-19)= 3

A) 12 B) -12 C) ± 12 D) none 

38) logₓ64 = 3/2 

A) 4 B) 8 C) 16 D) 2

39) log ₂ (x²- 9)= 4

A) 2 B) 16 C) 5 D) ±5 

40) logₓ(0.008)= -3

A) 3 B) 5 C) ±5 D) -3 

41) The base when 2 is the logarithm of 9 

A) 2 B) 9 C) 1 D) ±3

42) The base when 3 is the logarithm of 216 is.

A) 6 B) √6 C) 2 D) 3 

43) The base when 6 is the logarithm of 49 is.

A) 7 B) √7 C) ³√7 D) 1/7 

44) The base when (-1/6) is the logarithm of √5 is..

A) 0.8 B) 0.08 C) 0.008 D) 0.0008 

44) the base, when 4 is the logarithm of 1296 is..

A) 6 B) -6 C) 1/6 D) 1 

45) The base, when 6 is the logarithm of 343 is .

A) 7 B) -7 C) ±7 D) √7 

46) The logarithms of 243 to the base 3 is..

A) 2 B) 3 C) 4 D) 5 

47) The logarithms of 16 to the base 32 is .

A) 4 B) 5 C) 4/5 D) 5/4 

48) The logarithms of 81 to the base ³√9 is..

A) 3 B) 4 C) 5 C) 6 

49) If log 2= 0.31 then log 8 is.

A) 0.31 B) 0.62 C) 0 D) 0.93 

50) If log 2= 0.642 then log 5 is 

A) 0.1284 B) 0.358 C) 0.853 D) n

** If log₁₀2=0.3010, and log₁₀3 =0.7781, log₁₀7= 0.8451 then

51) log₅(5) is

A) 0.6990 B) 0.6099 C) 9069 D) n 

52) log₁₀(45)

A) 0.6532 B)1.6532 C) -0.6532 D) n

53) log₁₀(2.4)

A) 1.6811 B) 0.6811 C) 0.7781 D) n 

54) log₁₀(6)

A) 2.7781 B) 0.7781 C) 1.7781 D) n 

55) log₁₀(108)

A) 0.333 B) 1.333 C) 2.0333 D) n

56) log₁₀³√(5) 

A) 0.2330 B) 1.2330 C) 2.2330 D)n 

57) log₁₀(70) 

A) 0.8451 B) 1.8451 C) 2.8451 D)n 

58) log 84

A) 0.9242783       B) 1.9242793 

C) 2.9242783      D) none 

59) log 21.6

A) 0.3344539       B) 1.3344539

C) 2.3343539       D) none

60) log(0.00693)

A) 0.840733           B) 1.840733

C) 2.840733           D) 3.840733

61) log 294

A) 0.4683473    B) 1.4683473 

C) 2.4683473    D) 4.4683473

62) log√(4.5)

A) 0.3266063       B) 1.3266063 

C) 2.3266063        D) 2.3266063 

63) If log₁₀2=0.3010, log₁₀3=0.4771, then log₁₂40 is 

A) 0.485                   B) 1.485 

C) 2.485                  D) 3.485 

64) If log₁₀3= 0.4771 then log₂₅125 is 

A) 2  B) 3  C) 3/2  D) 2/3 

65) If log₁₀3= 0.4771 then log₁₀3000 is..

A)0.4771 B)1.4771 C)2.4771 D) 3.4771

** if log 2 =0.3010 and log(3)=.4771 then 

66) log 8 is..

A) 0.3010 B) 0.6020 C) 0.9030 D) n 

67) log 24 is 

A) 0.4771 B) 0.9030 C) 1.3802 D) n 

68) log 108 is..

A) 0.6020 B) 1.4313 C) 1.0333 D) 2.0333 E) n 

69) log 25 is.

A) 2.0000 B)0.6990 C) 1.3980 D) n 

70) log (0.405)¹/²       

A) 1.9084 B) 3.9084 C) 0.3010 D) 3.6074 E) none 

71) value of log₃log₄log₃81 

A) 1     B) 2      C) 0         D) 3 

72) log₃log₂log₂(256) 

A) 0    B) 1   C) 2    D) 3      E) n

73) log₂log√₂log₃(81)

A) 0  B) 1     C) 2      D) 3     E) n 

74) log₂[log₂{log₃(log₃27³}] 

A) 0      B) 1      C) 2     D) 3     E) n 

*** Value of:

60)a) log 5+ log 20+ log 24 + log 25 - log 60 is 

A) 0 B) 1 C) 2 D) 3 E) n 

b) log 6+ 2log 5+ log 4 - log 3 - log 2 is 

A) 0 B) 1 C) 2 D) 3 E) n 

c) 2log 5+ log 8- 1/2 log 4 is 

A) 0 B) 1 C) 2 D) 3 E) n 

d) log 8+ log 25+2 log 3 - log 18 is 

A) 0 B) 1 C) 2 D) 3 E) n 

e) 5log 2+ 3/2log 25+ 1/2 log 49 - log 28 is 

A) 0 B) 1 C) 2 D) 3 E) n 

f)1/2 log 25 - 2log 3 +log 18 is

A) 0 B) 1 C) 2 D) 3 E) n 

g) log 2 + 16 log(16/15) + 12 log(25/24) + 7 log(81/80) is..

A) 0 B) 1 C) 2 D) 3 E) n 

h) log(81/8) - 2log(3/2) + 3log(⅔) + log(3/4) is

A) 0 B) 1 C) 2 D) 3 E) n 

i) 7log(16/15)+5log(25/24) + 3log(81/80) is

A) 0 B) 1 C) 2 D) 3 E) n 

j) 2 log 2+ log 5 - 1/2 log 36 - log (1/30) is 

A) 0 B) 1 C) 2 D) 3 E) n 

k) log(1.2)+ 2 log (0.75) - log(6.75) is 

A) 0 B) 1 C) 2 D) 3 E) n  

l) log 5+ 16 log(625/6) + 12 log(4/375) + 7 log(81/1250) is..

A) 0 B) 1 C) 2 D) 3 E) n 

m) 2log₁₀8 + log₁₀36 - log₁₀(1.5) - 3 log₁₀ 2 is 

A) 0 B) 1 C) log₁₀ 2 D) log₁₀ 32 E) n 

n) 2 log₁₀5+ 2 log₁₀3 - log₁₀2 + 1 is..

A) 0 B) 1 C) log₁₀ 25 D) log₁₀ 1125 

o) 2+ 1/2 log₁₀9 - 2 log₁₀5 is..

A) 0 B) 1 C) log₁₀ 2 D) log₁₀ 12 E) n 

p) 1/2 log₁₀9 + 1/4 log₁₀81 + 2 log₁₀6 - log₁₀12 is .

A) 0 B) 1 C) log₁₀ 2 D) log₁₀ 7 E) log₁₀ 27 

q) 2log₁₀(11/13) + log₁₀(130/77) - log₁₀(55/91) is .

A) 0 B) 1 C) log₁₀ 2 D) log₁₀ 20 E)n 

r) 1 - 1/3 log₁₀64 is..

A) 0 B) 1 C) log₁₀ 2 D) log₁₀ 5 E) log₁₀ (2.5) 

s) log(32/243) - log(16/75) - 2 log(5/9) is..

A) 0 B) 1 C) 2 D) 3 E) log 2

t) 7 log (15/16) + 6 log (8/3) + 5 log(2/5) + log (32/25) 

A) 0 B) 1 C) log 2 D) log 3 E) log 5 

u) {log√(27)+log(8)+log√(1000)}/ log(120)

A) 0 B) 1 C) 2 D) 3 E) 3/2 

v) log₂(10) - log₁₆(625) 

A) 0 B) 1 C) 2 D) 1/2 E) none 

w) log₃log₂log₂(2⁸) 

A) 0 B) 1 C) 2 D) 1/2 E) none 

x) logᵤa . logᵥx . logₐv 

A) 0 B) 1 C) 2 D) 1/2 E) none 

y) {log√(27) + log√(8) + log√(125)} /{log(6) + log(5)} is 

A) 0 B) 1 C) 2 D) 3 E) 3/2

z) log₃5 x log₂₅27 is...

A) 0 B) 1 C) 2 D) 3 E) 3/2 

a') log₄5 x log₅3 is..

A)log₂3 B)2log₂3 C)log3 D)log2 E) n

b') log₄2 x log₂3 is..

A)log₂3 B)2log₂3 C)log3 D)log2 E) n 

c') log₂10 - log₈125 is 

A) 0 B) 1 C) 2 D) 10 E) none 

d') logₐx . logₓc. log꜀a is..

A) 0 B) 1 C) 2 D) 10 E) none  

**"*Evaluate 

61) a) log₈√[8 {√8√(8)...∞}.

A) 0 B) 1 C) 2 D) 10 E) none 

b) log₄√[4{√4√4...∞}]

A) 0 B) 1 C) 2 D) 10 E) none

**SOLVE:

62)a) 1/2 log (11+4√7)= Log(2+x)

A) 7 B) √ 7 C) 0 D) 1 E) none 

b) log(x+2) + log(x-2)= log 5.

A) 1 B) 2 C) 3 D) 0 E) none 

c) log(x+4) - log(x- 4)= log 2

A) 0 B) 1 C) 2 D) 12 E) none 

d) log(x+3) - log(x- 3)= 1.

A) 0 B) 1 C) 3 D) 11 E) 11/3 

e) log(x² - 21) = 2.

A) 0 B) 1 C) ±2 D) ±11 E) 11/3

f) 2 logx + 1 = log 250.

A) 1 B) 2 C) 5 D) ±6 E) none 

g) logx/log 5 = log 9/log(1/3).

A) 1 B) 2 C) 25 D) 1/25 

h) log 7 - log 2 + log 16 - 2log 3 - log (7/45) = 1+ log x.

A) 0 B) 1 C) 3 D) 4.

i) log₁₀x - log(2x -1)= 1.

A) 1 B) 0 10 D) 19 E) 10/19

j) log₅(x²+x) - log₅ (x+1)= 2

A) ±1 B) ±3 C) ± 5 D) ± 6 E) 25

k) log₁₀5+log₁₀ (5x+1)=log₁₀(x+5) +1

A) 0 B) 1 C) 2 D) 3 

l) log₂ log₃ log₂ x= 1.

A) 5 B) 1 C) 2 D) 512 

m) logₓ (8x-3) - logₓ 4 = 2

A) 3/2 B) 1/2 C) 1 D) 4

n) (log₁₀x - 5)/2 + (13 - log₁₀x)/3 = 2

A) 0 B) 1 C) 2 D) 10

o) log₃ (3+x)+ log₃(8-x) - log₃(9x-8) = 2 - log ₃ 9.

A) 0 B) 2 C) 4 D) 6 D) 8 

p) 5ˡᵒᵍ ˣ + 3ˡᵒᵍ ˣ = 3 ˡᵒᵍ ˣ⁺¹ - 5ˡᵒᵍ ˣ ⁻¹ 

A) 0 B) 1 C) 10 D) 100 E) none

q) log₅ (5¹⁾ˣ+ 125)= log₅6 + 1 + 1/2x

A) 1 B) 1/2 C) 1/3 D) 1/4 E) none 

r) 1/(logₓ 10) + 2= 2/(log₀·₅2)

A) 0.25 B) 0.025 C) 0.0025 E) none

s) logₓ 2. Logₓ/₁₆ 2= log ₓ/₆₄2

A) 0 B) 1 C) 2 D) none 

t) log₂x + log₂(x+4)= 5

A) 0 B) 1 C) 2 D) 3 E) 4

u) ₓˡᵒᵍ10ˣ = 100x.

A) 0 B) 1 C) 10 D) 100 ae) none

v) log₂x + log₄ x+ log₁₆ x= 5.25.

A) 1 B) 1/10/ C) 8 D) none

w) logₓ5 log ₓ/₁₂₅5= log ₓ/₆₂₅5.

A) 25 B) 1/25 C) 50 D)1/50 E) n 

x) Log₇ log₅{√(x+5) + √x}= 0 

A) 0 B) 1 C) 2 D) 4 E) none

63) a²ˣ⁻³. b²ˣ = a ⁶⁻ˣ. b ⁵ˣ then x log(a/b) is 

A) 3 B) log a C) 3 log a D) none 

64) If logₐ b= 6 and log₁₄ₐ(8b)= 3, then the value of a is..

A) 5 B) 6 C) 7 D) none 

65) If the logarithm of y² to the base x³ is equal to the logarithm of x⁸ to the base y¹², then the value of each logarithm is.

A) ±1/3 B) 2/3 C) ±1/5 D) none 

66) If log(x²y³)= a and log(x/y)= b, then values of log x and log y is 

A) 1/5(a+3b), 1/5(a-2b)

B) 1/5(a-3b), 1/5(a+2b)

C) 1/5(a+3b), 1/5(a+2b)

D) none 

67) If log (x³y³)= 6, log(x²/y)= -1/2 then the value of x and y is

A) √10, 10√10 B) 10√10, 10√10

C) 10√10, √10 D) none 

68) If log x +1= 0, then x is 

A) 3 B) 4 C) 5 D) 1/10 E) none 

69) if log√₈ b= 10/3 then b is..

A) 32 B) 33 C) 34 D) 35 

70) If logₓ(1/2)= 1/2 then x is 

A) 1/2 B) 1/3 C) 1/4 D) 1/5 

71) What is the base if the logarithm of 144 is 4 ?

A) 2 B) 3 C) √2 D) 2√3 E) none 

72) value of log₆ log√₂ 8 is 

A) 0 B) 1 C) 2 D) none 

73) value of log₂√6+ log₂√(2/3) is

A) 0 B) 1 C) 10 D) 2 E) none 

74) value of log 144 - log 90 + log(0.0625) is.

A) 0 B) 1 C) -1 D) 2 E) -2

75) If p log a= q and q log b = p, then the value of log (aᑫbᵖ).

A) p+q B) p³+ q³ C) (p³+q³)/p D) (p³+q³)/pq E) none 

76) logₐ m+ logₐ n= logₐ (m+n), then value of m in terms of n is

A) n B) n/(n-1) C) 1/(n-1) D) (n-1)/n 

77) If (log p)/m= (log q)/n = (log r)/l = log x express p²/qr as a power of x.

A) x²⁻ᵐ⁻ⁿ⁻ˡ B) x²⁺ᵐ⁻ⁿ C) x1⁺ᵐ⁻ⁿ D) n 

78) If 3 + log x = 2 log y, express x in terms of y 

A) y/1000 B) y²/1000 C) y³/1000 D) n 

79) If log₁₀ a= r, then value of (a)²⁾ʳ is

A) 1 B) 10 C) 100 D) 1000 E) 10000

80) If a= b²= c³= d⁴, then the value of log ₐ abcd is

A) 2 B) 1 C) 25 D) 25/2 

81) Value of ₐlogₐx is 

A) a B) x C) ax D) a/x E) none 

82) If logₐ log₂ Log₂ 256=2 then a is 

A) 0 B) 1 C) 2 D) 3 

83) If log₃ log₂logₐ81=1, then a is

A) √2 B) √3 C) √5 D) √6 E) none

84) value of log ₘn . logₙm is 

A) 0 B) 1 C) 2 D) √3 

85) If logₘA= log ₙA . P then P is 

A) logₘn B) log n C) log m D) none 

15) If x²+y² = 6xy, prove 

2log(x+y) = logx + log y +3 log 2

16) If a² +b² =23ab prove

log(a+b)/5 = 1/2(log a+ log b).

17) If a³⁻ˣ.b⁵ˣ=aˣ⁺⁵.b³ˣ, prove 

       x log(b/a) = log(a)

18) If a²+b²=14ab, prove 

log{(a+b)/2}= 1/2 (log a + log b)

19) If a²+b² = 27ab prove

log{(a - b)/5 }= 1/2 (log a + log b)

20) If log{(a+b)/3}

                  =1/2(log a+ log b)

prove a/b + b/a =7

21) If log{(a-b)/4} =

                  1/2 (lig a+ log b) 

prove a² + b² = 18ab.

22) If a²+b²= 7ab prove log (a+b) =

                             ½(loga +logb).

23) If x² + y² =11xy prove 

     2log(x-y)=2log3 + log(x) +log(y)

24) If a² =b³=c⁵ =x⁶ then prove

        logₓ(abc) = 31/5.

25)If log(a+b)/7=

   1/2 {log(a) +log(b)} then prove

    a/b +b/a = 47

26)solve

a)log₂log₃log₂(x) = 1

b) logₓ(8x - 3) - logₓ(4) = 2

c) ½ log(11+4√(7)) = log(2+x)

d) {log₁₀(x-5)}/2 +{13-log₁₀(x)}/3=2

e) log₃(3+x)+ log₃(8-x)- log₃(9x-8) =

      2 - log₃(9)

f) 5ˡᵒᵍ ˣ + 3ˡᵒᵍ ˣ = 3logˡᵒᵍ ˣ⁺ ¹ -

    5logx -1

g) log₅(5¹/ˣ) + 125 = log₅(6) + 1

    +1/2x

h) 1/(logₓ 10) +2 = 2/(log₀,₅10)

i) logₓ(2).logₓ/₁₆(2) = logₓ/₆₄(2).

j) log₂x+log₄x+log₁₆x=21/4

27)If log x/(y-z)=log y/(z-x)=logz/(x-y) then prove xyz=1

28) (yz)ˡᵒᵍʸ⁻ˡᵒᵍᶻ (zx)ˡᵒᵍᶻ⁻ˡᵒᵍˣ

       (xy)ˡᵒᵍˣ⁻ˡᵒᵍʸ =1

29)If log a/(w-r)=log b/(r-p)=

      logc/(p-q) prove aᵖbʷcʳ.

30) If a=b²=c³=d⁴, prove 

       logₐ(abcd)=25/12

31) If a,b,c are any three consecutive positive integers, prive that log(1+ac) = 2log(b)

32) prove without using log table that, log₁₀2 > 0.3.

33) If x= 1+logₐvx, y=1+logᵥxa,

    z= 1+logₓav then show

    xy+yz+zx=xyz

34) If x= logₐvx, y=logᵥxa,z=logₓav then show x+y+z+2=xyz