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

CUBES AND CUBE ROOTS

* When a number is multiplied 3 times, it is called cube.

* Cube of a number ending with the digits 0, 1, 4, 5, 6 and 9 ends 0, 1, 4, 5, 6, and 9.

* Cube of even number is even and cube of odd number is odd.

* A perfect cube can be written as the product of triplets of prime factorization.

EXERCISE - A

1) Find which of the following is not a perfect cube ?

a) 121 

b) 1728 

c) 2744

d) 256 

e) 3375

f) 4000

g) 512

h) 100

2) Find the smallest number by each should be multiplied to obtain a perfect cube.

a) 121 

b) 625

c) 72

d) 675

e) 100

f) 500

g) 2662

3) Find the smallest number by each should be divided to obtain a perfect cube :

a) 81

b) 135 

c) 625 

d) 3456

e) 192 

f) 500

g) 2662

4) Find the cube of the following :

a) 6 

b) 12 

c) 21

d) 99 

e) 302

f) 15

g) 11

h) 25

5) Write cubes of 5 natural numbers which are of the form of (3n + 1) and verify the following statement that the cube of a natural number of the form (3n + 1) is a natural number of the same form.

6)  If one side of a cube is 9 metres then find its volume.

7) Rajat makes a cuboid of sides 5 cm, 4 cm and 5 cm. How many such cuboid will he need to form a cube?

8) Namita has a cuboid of sides 2 cm x 1 cm x 1 cm. How many such cuboids will she need to form a cube ?

9) Write cubes of 5 natural numbers which are the form of n² + 1 and verify the following statement that the cube of natural number of the form n²+1 is a natural number of the same form .

10) Find the cube of each of the following using (a+ b)³

a) 12

b) 17

c) 69

d)109

e) 125 

e) 14

f) 102

g) 55

11) Find the cube of each of the following using (a - b)³

a) 69

b) 95

c) 999

d) 88

e) 77

f) 18

g) 99

h) 48

EXERCISE-B

1) Find the cube root of the following by Prime Factorization Method.

a) 64 

b) 125 

c) 512

d) 343

e) 1331

f)1728

g) 2744

h) 15625 

i) 27000

j) 46656

k) 91125

l) 592704

m) 13824

2) Estimate the cube root of the following 

a) 46656

b) 54872

c) 103823

d) 157464 

e) 175616

f) 262144 

g)  389017

h) 551368

i) 804357 

j) 884736 

k) 592704

l) 389017

m) 117649

n) 238328

3) The volume of a cube is 262144 cm³, find its side.

4) The volume of a cuboidal box is 74088 cm³, find its side.

5) Find the cube root of the given negative integers.

a) -64

b) -2197

c) -5832

d) -17576

EXERCISE - C

1)State true or false 

a) There is no perfect cube which ends with 8.

b) The cube of a two digit number maybe a 3 digit number.

c) The cube of a single digit number maybe a single digit.

d) 1729 is a perfect cube.

e) No perfect cube can end with exactly two zeros .

f) if a divides b then a³ divides b³.

2) Choose the Correct Option

1) Find the cube of (-3/4).

a) 9/16 b) (-27/64) c) (-9/16) d) (27/16)

2) Cube of negative numbers are always _____

a) positive b) negative c) even d) odd

3) Solve : ³√343 - ³√-216.

a) 15 b) 17 c) 13  d) 12

4) Find the value of ³√(125 x 343 x 64).

a) 140 b) 135 c) 150  d) 145

5) Three cubes of edges 3cm, 4cm, and 5cm are melted to form a new cube. What is the edge of new cube?

a) 5cm b) 6cm c) 8cm d) 4cm

6) How many hundreds are in there in 31 x 2² x 5³ ?

a) 150 b) 125 c) 155 d) 120

7) How many thousands will be there in 29 x 2³ x 5³ ?

a) 29 b) 30 c) 27 d) 23 

8) Simplify : (³√2197 x ³√1728)/³√1331

a) 152/10  b) 150/12 c) 155/11 d) 156/11

Miscellaneous Exercise

1) Find the cube of (3/5)³ - (1/5)³.

2) If a number is written as 3 x 3x 3x 3x 5x 3 x 7x 7 x 5x 7, then find the smallest number by which this is to be multiplied to form a perfect cube.

3) If ³√x =12. Find x.

4) A cubical box has a volume of 1331 cm³, find its surface area.

5) By which least number 125 x 512 x 4 be divided to make it a perfect cube ?

6) Find the value of ³√1.728 x ³√1.331.

7) Simplifly if 3ˣ = 243, find x.

8) Three solid wooden cubes of different colours with edges 30cm each are placed one along another. How much cubic cm of wood is required to make it ?

9) Find the volume of cube having side 7cm.

10) Find ³√-2.197.

11) Find the cube of 75 using (a+ b)³ and check your result using (a - b)³.

12) Find the cube root of 0.001728.

13) Find the smallest number by which 54 must be multiplied so that the product is a perfect cube.

14) The volume of a cube is 343 cm³, find its side.

15) Three numbers are in the ratio 2:3:4 and sum of their cube is 33957, find the numbers.

16) Estimate the cube root of 456533.

17) Simplify {³√64 x ³√512}/{³√216 x ³√27}

SQUARES AND SQUARE ROOTS

NOTE

* Square number never ends with 2, 3, 7 or 8.

* Square number can end with 0, 1, 4, 5, 6 or 9.

* Square of even number is even and those are odd numbers is odd.

* The set of 3 numbers (x,y,z) is called Pythagorean triplet, if x²+ y²= z².

* For any natural number 'm' greater than 1, (2m, m²-1, m²+1) is a Pythagorean triplet.

* For perfect square x and y

    a) √(x . y)= √x . √y

    b) √(x/y = √x/√y    (y≠ 0).

* The square of a proper fraction is smaller than the fraction.

* (a+ b)²= a²+ 2ab + b².

   (a - b)²= a²- 2ab + b²

* For any natural number 

           Sum of n odd natural numbers= n²

EXERCISE - A

1) What will be the unit digit in the square of the following numbers ?

a) 16 

b) 253

c) 431

d) 9284 

e) 52698

f) 78949 

g) 66866 

h) 4876937

2) Which of the following number would have 6 digit at units place ?

a) 19²

b) 84²

c) 324²

d) 4368²

e) 9487²

f) 14278²

g) 84769²

h) 325846²

3) How many square numbers are there between 1 to 100 ?

4) Find the largest 3-digit number which is perfect square.

5) Find the smallest 4-digit number which is a perfect square.

6) By what least number should the following numbers be divided to get a perfect square.

a) 512

b) 648

c) 1275

d) 3380 

e) 4056

f) 8820

7) By what least number should the following numbers be multiplied to get a perfect square.

a) 648

b) 2475

c) 2925

d) 4056 

e) 4410

f) 4851 

g) 7776

h) 9075 

8) Is the following numbers a perfect square:

a) 196

b) 625

c) 7056

d) 9075

9) Show that 648 is not a perfect square?

10) By what least number should 648 be a multiplied to get a perfect square number ?

11) By what least number 3125 be divided to get a perfect square number?

12) Find the square of following:

a) 15

b) 134

c) 3/19

d) 14/5

e) 0.03

f) 1.3

g) 0.29

h) 4.5

EXERCISE - B

1) Without adding, find the sum.

a) 1+3+5+7+9+11+13.

b) 1+3+5+.......+31.

c) 1+3+5+......+71.

2) Express

a) 64 as the sum of 8 odd natural numbers.

b) 225 as the sum of 15 odd natural numbers.

3) Write a Pythagorean triplet where one member is

a) 6

b) 12

c) 16

d) 35

4) What will be the unit digit of the square of the following:

a) 272

b) 1057

c) 5554

d) 12796

e) 28245

c) 35842

5) The square of which of the following is odd.

a) 248

b) 431

c) 2826

d) 143876

6) Using the given pattern, find the missing numbers.

a) 11²= 121

    101²= 10201

    1001²=1002001

    100001²= 1.........2..........1

    1000000001²= 1.........2......1

b) 11²=121

101²= 10201

10101²= 102030201

( )²= 1020304030201

( )²= 10203040504030201

c) 9²= 81

99²= 9801

999²= 998001

9999²= 99980001

99999²= __________

999999²= _______

d) 4²+5²+20²=21²

     5²+6²+30²=31²

     6²+7²+___=___

7) Evaluate:

a) 38²- 37²

b) 42²- 41².

c) 218²- 217²

d) 428²- 427²

e) 767²- 766²

f) 982²- 981².

9) Without adding find the sum:

a) 1+3+5+7+......+23=?

b) 1+3+5+7+.....+29=?

c) 1+3+5+7+.....+39=?

10) Write the following numbers as the difference of squares of consecutive natural numbers.

a) 21

b) 43

c) 59.

d) 65

EXERCISE - C

1) Write the possible ones digit of the square root of each of the following numbers.

a) 6084

b) 5209

c) 6561

d) 9025

e) 17424

f) 217156 

2) Find the square root of the following by prime factorization method.

a) 144

b) 252

c) 324

d) 441

e) 576

f) 676

g) 784

h) 1225

i) 1764

j) 2601

k) 2916

l) 3528

m) 3969

n) 5328

o) 5625

p) 5929

q) 6084

r) 6561

s) 7056

t) 7921

u) 8281

v) 9075

w)11025

x) 21904

y) 24336

3) Find the square root by method of repeated subtraction.

a) 81

b) 144 

c) 169 

4) Find the smallest number by which the following numbers be multiplied to get a perfect square. Also find this square root of the square numbers so obtained .

a) 396

b) 768 

c) 588

d) 720

e) 847

f) 882

g) 1152

h) 2592

i) 2925 

j) 3332

k) 3380

l) 3645

m) 4802

5) Find the smallest number by which the following numbers be divided to get a perfect square. Also find the square root of the square number so obtained.

a) 245

b) 396 

c) 968

d) 1008

e) 1200

f) 1452

g) 1728

h) 4500

i) 5445

j) 6000

k) 7776

l) 8820

m) 28812

 

6) Find the length of the side of the square having area.

a) 441 cm²

b) 676cm²

c) 729 cm²

d) 2304 cm²

e) 3136 cm²

7) The student of class 8 of a school donoted Rs1521 to PM'srelief fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.    

8) Find the least square number divisible by each one of 8, 9 and 10.

9) Find the least perfect square which is divisible by 5, 6 and 8.

EXERCISE- D

1) Without performing any calculation, find the number of digits in square root of the following numbers :

a) 256

b) 4489

c) 6241

d) 7056

e) 15129 

f) 17956

g) 18769

h) 92416

i) 193600 

j) 403225

k) 999999

l) 1046529

2) Find square root of the following using division method

a) 2401

b) 3249

c) 5184

d) 9225

e) 15129

f) 18769

g) 65536

h) 893304

i) 7.3441

j) 24.01

k) 147.1369

3) Find the value of 

a) √117 x √625 x √13.

b) √1.44 + √6.25 + √506.25 - √3.24.

4) Find the square root of the following character 3 places of decimal.

a) √3

b) √7

c) √13

d) √0.8 

e) √0.9

f) √2.25

g) √2.8

h) √7.29

i) √15.21

5) Evaluate

a) √1_49/576

b) √1_56/169

c) √2_1/4

d) √(1/16 + 1/9).

e) 12_69/121.

f) 9_43/49.

6) What least number must be subtracted from 893304 to get a perfect square. Find the square root of this perfect square.

7) Find the greatest number of 4 digits which is a perfect square.

8) Find the smallest number which must be added to 4931 to make it a perfect square.

9) The area of a square field is 12100 m². Find its side.

10) A Gardner has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this. 

11) The area of a square field 60025 m². A man cycles along its boundary at 18 kmph. In how much time will he return to the original point ?

12) An army commander wishing to arrange his soldiers, who were 8289 in number in the form of square, found that there were 8 soldiers left. How many soldiers were there in each row?

13) What is the number which when multiplied by itself gives √152.5225.

14) The area of a square is equals to the area of a rectangle whose length is 7.26m and breadth is 6m. Find each side of a square.

15) Find the smallest number that must be added to 1750 in order to make it a perfect square. Also find the square root of perfect square.  

16) If √105.0625 = 10.25, find the value of √10506.25.

17) Simplify √[{√(59.29) - √(5.29)}/{59.29 + √(5.29)}].

MULTIPLE CHOICE QUESTIONS:

1) How many 2's are there in the prime factorization of 800 ?

a) 7 b) 5 c) 10 d) 15

2) Solve: 49²- 48²

a) 87 b) 97 c) 90 d) 80

3) Simplify √(63 x 28)

a) 60 b) 62 c) 42 d) 24 

4) Find the √0.0081.

a) 0 b) 0.9 c) 0.09 d) 0.009

5) How many perfect square number lie between 16 and 81?

a) 4 b) 5 c) 8 d) 6 

6) How many perfect square numbers lie between 100 and 500?

a) 13 b) 12 c) 10 d) 77

7) What is the sum of first ten odd numbers?

a) 100 b) 80 c) 200 d) 120

8) Find the sum of 5 + 7+ 9 +...+21.

a) 118  b) 117 c) 120 d) 119

9) Find the missing number. 40 x ____=60².

a) 90 b) 50 c) 60 d) 80

Miscellaneous Exercise - 1

1) Find the square of 24.

2) Can we have a square number that ends with 8?

3) How many 2's are there in the prime factorization of 800 ?

4) How many digits will be there in the square root of 12321.

5) What is the value of 48² - 47²?

6) Find the value of (59 + 41)²?

7) What is the missing digit in (37)²= 1∆∆9.

8) Simplify √(63 x 28 x16).

9) How many natural number lie between 63² and 64²?

10) Find the value of √0.0081.

11) What is the sum of the first twenty odd numbers?

12) Find the sum of 5 + 7+ 9 + 11 + 13+ 15+ 17+ 19+ 21.

13) What should be added to 45² to get 46²?

14) What should be substed from 37² to 35²?

15) What is the value of √(441/1369)=?

16) Give a Pythagorean triplet if one number is 12.

17) The length of a rectangular  park is 80m and its breadth is 60m. Find the length of its diagonal.

18) find the length of diagonal of a square if one side is 47.

Continue

PROBABILITY (J- Basic)

1) A coin is tossed once, what is the probability of getting

a) a head.    1/2

b) a tail.       1/2

c) Both head and tail turn up. 0

d) neither head nor tail turns up. 0

2) A coin is tossed twice, find the probability of getting 

a) no head.        1/4

b)  both tails .     1/4

3) When two coins toss simultaneously, what possible outcomes. Find the probability of getting 

a) both heads.        1/4 

b) at least one head.      3/4

c) atleast one tail.       3/4

d) head turns up exactly once. 1/2

e) Tails turns up atleast once. 3/4

f) Tail does not turn up at all. 1/4

g) two heads. 1/4

h) one head. 1/2

i) One tail. 1/2

j) at most one head. 3/4

k) No head. 1/4

4) If 3 identical coins are tossed. What is the probability of getting:

a) all three heads.         1/8

b) one head and two tails.      3/8

c) all three tails.           1/8

d) atleast two head.       1/2

e) all heads. 1/8

f) two heads . 3/8

g) one head. 3/8

h) exactly 2 heads. 3/8

i) atleast 1 head & 1 tail. 6/8

j) atleast 1 head. 7/8

5) A coin is tossed thrice. Find the probability that:

a) Tail turns up atleast twice. 1/2

b) Head does not turn up at all. 1/8

c) Tail comes on the second toss. 1/2

d) Head turns up atleast once. 7/8

e) getting two or more heads. 1/2

f) the second is not a head. 1/2

) A unbiased dice is thrown. What is the probability of getting

a) prime number.         1/2

b) number greater than 5.       1/6

c) even number.      1/2

d) odd number.         1/2

e) even number greater than 6.         0

f) 3.       1/6

g) multiple of 3?        1/3

h) an even number or a multiple of 3.  2/3

i) an even number and a multiple of 3.    1/6

j) a number 3 or 4.          1/3

k) A number less than 5.         2/3

l) A number greater than 3.    1/2

m) A number between 3 and 6.     1/3

n) two or four. 1/3

C) a multiple of 2 or 3. 2/3

M) a prime number. 1/2

N) 2 or 4. 1/3

P) multiple of 2 and 3.

Q) multiple of 2 but not 3.

R) multiple of 3 but not 2.

S) multiple of either 2 or 3.

T) multiple of neither 2 nor 3.

U) greater than 4. 1/3

V) atmost 4. 2/3

W) between 7 and 10. 0

7) A perfect cubic die is thrown. Find the probability that:

A) A prime number comes up. 1/2

B) A perfect square comes up. 1/3

) A dice is thrown twice. What is its total outcome?       36

) Two dice thrown at a time. Find the probability of getting:

a) doublets.              1/6

b) even number.       1/2

c) an odd number.     1/2

d) equal to 10.      1/12   

e) a double digit number.      1/6

f) a single digit number.      5/6

g) Two heads. 1/4

h) one head. 1/2

i) one tail. 1/2

j) at least one head. 3/4

k) at most one head. 3/4

l) No head. 1/4

m) 8 as the sum. 5/36

n) a doublet of prime numbers.      1/12

o) a doublet of odd numbers. 1/12

p) a sum greater than 9. 1/6

q) an even number on first. 1/2

r) an even number on one and a multiple of 3 on the other. 11/36

s) neither 9 nor 11 as the sum of the numbers on the faces. 5/6

t) a sum less than 6. 5/18

u) a sum less than 7. 5/12

v) a sum more than 7. 5/12

A) an even number as the sum. 1/2

B) the sum as a prime number. 5/12

C) a total of atleast 10. 1/6

D) a doublet of even number. 1/12

E) A multiple of two on one dice & a multiple of 3 on the other. 11/36

F) same number on the both dice i.e. a doublet. 1/6

G) a multiple of 3 as the sum. 1/3

M) an even number on first. 1/2

N) an even number on one and a multiple of 3 on the other. 11/36

O) neither 9 nor 11 as the sum of the numbers on the faces. 5/6

S) The sum of the score is 9. 1/9

T) The product of the score is 12. 1/9

U) The score on the second die is greater than the score on the first die. 5/12

V) The sum of the scores is a multiple of 4. 1/4

W) The sum of the scores is a perfect square. 7/36

X) same score on the first die as on the second. 1/6

Y) the sum of the scores is greater than 9 or even number.                

Z) Sum of the scores on their uppermost face is a perfect square or a multiple of 3. 5/12

) Two unbiased dice are thrown. Find the probably that the total of the numbers on the dice is greater than 10.  1/12

) From a well-shuffled pack of 52 cards, one card is drawn at random. what is the probability that the card is

a) a diamond.          1/4

b) an ace.             1/13

c) 5 of club.         1/52

d) queen of red colours.   1/26

e) a black card.          1/2

f) a club.           1/4

g) 10 of hearts.        1/52

h) a face card.         3/13

i) 2 of red.       1/26

j) 2 of spade.       1/12

k) a red card.       1/2

l) a black Queen.        1/26

m) A red king.       1/26

n) a black face card.     3/26

o) red cards.       1/2

C) A queen. 1/13

D) Not a queen. 12/13

E) either red or king. 7/13

F) red and a king. 1/26

H) a red face card. 3/26

I) 2 of spades. 1/52

J) Bears a number less than 4. 3/13

K) 10 of a black suit. 1/26

L) a black king. 1/26

M) Either a black card or a king. 7/13

N) black and a king. 1/26

O) King or black card. 7/13

P) a jack, queen or a king. 3/13

Q) neither a heart nor a king. 9/13

R) spade or an ace. 9/13

S) neither an ace nor a king. 11/13

T) A diamond. 1/4

U) An ace or a queen. 2/13

V) A card bearing a number between and including 2 & 6. 5/13

) Numbers 1 to 10 are written on 10 separate slips and kept in a box and mixed up well. One slip is chosen from the box without looking on it. Find the probability of 

a) getting a number 5.       1/10

b) getting a number less than 8.      7/10

c) getting a number greater than 7.     3/10

d) getting a two digit number.           1/10

8) 17 cards number 1,2,3,....17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability the number on the card is :

A) odd. 9/17

B) a prime. 7/17

C) divisible by 3. 5/17

) Prime numbers between 1 and 25 are written on identical slips and put in a box and mixed up . If a slip is drawn at random. What is the probability of getting

a) one digit number. 4/9

b) an even number. 1/9

c) an odd number. 8/9

d) number greater than 11. 4/9

) What is the probability that a number selected from the numbers 1,2,3,..... 25 is a prime number, when each of the given numbers is equally likely to be selected. 9/25

9) Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is:

A) an even number. 1/2

B) a number less than 14. 3/25

C) a number which is a perfect square. 9/100

D) A prime number less than 20. 2/25

15) In a lottery 50 tickets numbered 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number. 3/10

16) What is the probability that a number selected from the numbers 1, 2, 3,.....15 is a multiple of 4 ? 1/5

17) In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize. 2/5

18) if the probability of winning a game is 0.3, what is the probability of loosing it.                      

) There are 35 students in a class, with 20 boys and 15 girls. From these students one is chosen at random. What is the probability that the chosen student is

a) a boy.      4/7

b) a girl.           3/7

) A letter is chosen from the word EQUATION . Find the probability that letter is a consonant.         3/8

) Find the probability of getting vowel in the word EQUATION .         5/8

) Find the probability of getting vowel in the word MATHEMATICS .      4/11

) A bag contains 8 white balls, 5 green balls and 7 red balls . They are mixed throughly and one ball is drawn at random. Find the probability of getting

a) a red ball.          7/20

b) a green ball.        1/4

c) a yellow ball.       0

d) a white ball.         2/5

) A spinner whose vectors are painted in the seven spectrum colours (VIBGYOR) is spun 35 times. How many times do you expect the painter to be at the red colour ? T times

)  The keyboard of mobile has (0 to 9) digits on it. Find the probability of getting

a) even number.      1/2

b) multiple of 5.        1/10

c) factor of 9.       3/10

d) a natural number.      9/10

e) perfect number.     1/10

f) prime number.      2/5

g) composite number.       2/5

h) whole number.           1

) A bag contains 3 white balls, 5 red balls, 7 green balls and 9 yellow balls. Find the probability of getting 

a) red .       5/24

b) yellow balls.      9/24

c) blue balls.           0

6) A bag contains 3 red and 2 blue marbles. A marble is drawn at random of drawing a blue marble. 2/5

) A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is:

A) black. 7/24

B) red. 5/24

C) not Green. 5/6

7) It is know that a box of 600 electric bulbs contains 12 defective bulbs. One bulb is taken out at a random from this box. What is the probability that it is a non defective bulb ? 0.98

10) A child has a block in the shape of a cube with one letter written on each face as shown

 A B C D E A

The cube is thrown once. What is the probability getting

A) A. 1/3

B) D. 1/6

Fill in the blanks:

1) The probability of a sure event is ______.

2) If an event cannot occur than its probability is _____.

3) The probability of selecting P from the word SPECIAL is ____.

4) The probability of an event cannot be more than ______.

5) if a dice is thrown once, the probability getting an even prime number is____.

6) probability of an impossible event is ____

7) The probability of an event (other than sure and impossible event) lies between ___

8) Every elementary event associated to a random experiment has____ probability.

Choose the Correct Option:

) Three coins are tossed simultaneously. What are the possible outcomes?

a) 7 b) 8 c) 9 d) 10

) From 1 to 50, what is the probability getting a multiple of 6 ?

a) 4/25 b) 6/30 c) 2/25 d) 3/30

) Two dice are thrown simultaneously. Find the probability getting an even number as su.

a) 2 b) 5  c) 1/2 d) 2/3

) What is the probability getting vowel from the English alphabet ?

a) 26/5  b) 5/26 c) 1/26 d) 26/1

) A number from 1 to 100 is chosen at random. What is the probability getting an even number ?

a) 1/2 b) 2/3  c) 2/1 d) 3/2

) Two coins are tossed simultaneously. Find the probability of getting two heads.

a) 1/2 b) 4 c) 1/4 d) 2

10) the king, queen and jack of clubs are removed from a deck of 52 playing cards and the well shuffled. One card is selected from the remaining cards. find the probability of getting:

A) a heart. 13/49

B) a king. 3/49

C) a club. 10/49

D) the 10 of heart. 1/49

E) a black card. 1/2

F) 8 of red colour. 1/20

G) A king of black colour. 0

11) A bag contains 3 red, 4 white, 5 blue marbles. All the marbles are identical in shape and size. One marble is drawn at random from the bag. Find the probability that the marble drawn is:

A) red. 1/4

B) Red or white. 7/12

12) A bag contains 3 red and 2 blue marbles. A marble is drawn at random. what is the probability of drawing a blue marble. 2/5

13) A bag contains 5 red balls 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is.

A) black. 7/24

B) red. 5/24

C) not green. 5/6

14) An urn contains 10 red and 8 white balls. One ball is drawn at random. Find the probability that the ball drawn is white. 4/9

15) A bag contains 3 red balls, 5 black balls and 4 white balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is:

A) white ? 1/3

B) red ? 1/4

C) black ? 5/12

D) not red ? 3/4

16) A bag contains 6 red, 8 black and 4 white balls. A ball is drawn at random. What is the probability that the ball drawn is not black ? 5/9

17) A bag contains 5 white 7 red balls. One ball is drawn at random. What is the probability that the ball drawn is white ? 5/12

18) A box contains 5 red, 11 white and 7 black balls. One ball is drawn at random. Find the probability that the ball drawn is a white ball. 11/23

19) A bag contains 6 red, 5 blue, 3 white and 4 black balls. A ball is drawn at random. Find the probability that the ball is red or black. 5/9

20) In a bag there are 6 black, 4 white and 3 yellow balls. A ball is drawn at random. Find the probability of getting a yellow or a white ball. 7/13

21) A box contains 7 red, 5 white and 8 green balls identical in all respects except colour. One ball is drawn at random. Find the probability that it is not white. 3/4

22) find the probability that a leap year selected at random will contain 53 Sundays. 2/7

23) A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is

A) red ? 7/15

B) Black Or White . 8/15

C) not black. 2/5

24) A bag contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. find the probability that the ball drawn is

A) white. 2/5

B) red. 4/15

C) not black. 2/3

D) red or white. 2/3

25) What is the probability that an ordinary year has 53 Mondays? 1/7

26) What is the probability that a leap year has 53 Mondays and 3 sundays? 1/7

27) The probability that two boys do not have the same birthday is 0.394. what is the probability that the two boys have the same birthday? 0.606

28) If P(E)= 0.95, find P(not E). 0.05

29) what is the probability that the number selected from the number 1,2, 3,......25 is a prime number, when each of the given number is equally likely to be selected ? 9/25

30) tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. what is the probability that the ticket has a number which is a multiple of 3 or 7 ? 2/5

31) A bag contains a certain number of blue balls. A ball is drawn. Find the probability that the ball drawn is

A) black. 0

B) blue. 1

32) A bag contains 12 ball out of which x are white.

A) If one ball is drawn at random, what is the probability that it will be white ball ? x/12

B) if 6 more white balls and put in the bag, the probability of drawing a white ball will be double then that in (A). find x. 3

33) A bag contains 6 red, 8 white and x blue balls which are not identical in shape and size. The probability that a ball drawn at random is blue or white is 5/7. Find x. 7

34) it is known that box of 600 electric bulbs contains 12 defective bulbs. One bulb is taken out the random from this box. what is the probability that it is a non defective bulb? 0.98 

35) A box contains 1000 bulbs out of which 25 are defective. It is not possible to just look at the bulb and tell whether or not it is defective. One bulb is taken out at random from the box. Find the probability that the bulb taken out is

A) A good one. 39/40

B) A defective one. 1/40

36) 17 cards numbered 1, 2,3, ...17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is.         

A) odd. 9/17

B) a prime number. 7/17

C) divisible by 3. 5/17

D) divisible by 3 and 2 both. 2/17

37) cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability that number on the card is.

A) an even number. 1/2

B) a number less than 14. 3/25

C) a number which is a perfect square. 9/100

D) A prime number less than 20. 2/25

38) 1000 tickets of a lottery were sold and there are 5 prizes on these tickets. If Ram has purchased one Lottery ticket, what is the probability of winning a prize? 0.005

39) A and B throw a pair of dice. If A throws 9, find B's chance of throwing a higher number. 1/6

40) In a lottery of 50 tickets numbered 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number. 3/10

41) What is the probability that a number selected from the numbers 1, 2, 3,...,15 is a multiple of 4 ? 1/5

42) Tickets numbered 1 to 20 are mixed up and a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 7 ? 2/5

43) In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize. 2/5

44) A bag contains 50 identical cards which are numbered from 1 to 50. If one card is drawn at random from the bag. Find the probability that it bears:

A) A perfect square number. 7/50

B) A number divisible by 4. 6/25

C) A number divisible by 5. 1/5

D) A number divisible by 4 or 5. 2/5

E) A number divisible by 4 and 5. 1/25

F) A number not divisible by 4.

G) A number not divisible by 5.

H) Either divisible by 4 or divisible by 5.

I) neither divisible by 4 nor divisible by 5.

J) Divisible by 4 but not divisible by 5.

K) Divisible by 5 but not divisible by 4.

L) Divisible by 20.

M) not divisible by 20.

45) If the probability of winning a game is 0.3. What is the probability of losing it. 0.7

                       EXERCISE--2

                    -----------------------

1) Probability of a sure event is__.

2) Probability of an impossible event is____.

3) The probability of an event (other than sure and impossible event) lies between____.

4) Every elementary event associated to a random experiment has ___ probability.

DISCOUNT

* Discount is given on the printed price.

* Selling price is determined after deducting discount from the printed price.

   Printed price - Discount = selling price.

** Discount %= (Discount  x 100)/ printed price.

* GST (Goods and Service Tax) has been in effect July 01, 2017. It speaks about the uniform tax throughout the country.

) A fan is marked at Rs600. During the winter session the shopkeeper allows a discount and sells the fan for Rs500. Find the discount percent on the item.        97/6%

) The mark price of a fridge on Amazon.com is Rs25000 and the online company is allowing a discount of 20% on it. Find the selling price of fridge.      Rs20000

) On the eve of Gandhi Jayanti a saree is sold for Rs9720 after allowing 20% discount. What is the marked price ?     Rs12150

) A trader marks is good at 40% above of the cost price and allows a discount of 25%. What is his gain percent ?       5%

) The marked price of a book is Rs225, if the shopkeeper allows a discount of 12% to his customer and gains 25%, find the cost price of the book.      Rs158.40

) A shopkeeper marks his shirt at such a price that after allowing 10% discount for cash payment, he still gains 5%. Find the marked price of shirt that costs him Rs400.   Rs525

) What should be the price on a washing machine whose cost is Rs14400, if the retailer wants to get a profit of 10% after giving a discount of 12%.     Rs18000

) Find single discount equivalent to successive discount of 20% in 10%.     28%

) a shopkeeper gives the customer to choose either of the two option for discount

a) 20% , 10% and 20%  b) 45% 

Which option would you like and why ?       2nd option is better

) Mona purchased a computer for Rs25290 which includes a GST of 8%. Find the list price of computer?      Rs24000

) A mixer grinder is available for Rs14170 inclusive of GST. If the original price of the mixer-grinder is Rs13000, find the rate of GST.    9%

) The price of a computer is Rs27000. The GST charged is 12%. Find the amount to be paid .   Rs30240

) Uma purchased a hair driver for Rs5400 including 8% GST. Find the price before GST was added.        Rs4968

) The marked price of a ceiling fan is Rs1250 and the shopkeeper allows a discount of 6% on it. Find the selling price of the fan.     Rs1175

) A dealer marks a TV set with a price which is 25% more than the cost price and allows a discount of 10% on it. Find his gain or loss percent.     12.5%

) A dealer purchased a washing machine for Rs7670. He allows a discount of 12% on its marked price and still gains 10%. Find the marked price of the machine.   Rs9575

) Divya buys goods worth Rs5500. She gets a discount of 5% on it. After getting this discount if GST is charged at the rate of 5%, find the total amount she has to pay.    Rs5486.25

) The marked price of a bicycle is 2800. If after giving a discount of 8%, GST at the rate of 5% charged , find the selling price of the bicycle.     Rs2704.80

) The list price of a DVD player is Rs13500. If the rate of GST is 11%, find the selling price of DVD?     Rs14985

) Sarita gives a discount of 15% on the sarees sold by her and still gets a profit of 40/3%. What is the cost price of a saree whose marked price is Rs3200?     Rs2400

) A jeweller allows a discount of 16% to his customer and still gains 20%. Find the marked price of a ring which costs the jewellers Rs1190.      Rs1700

) Find the single discount which is equivalent to two successive discount of 20% and 5%.    24%

) Find the rate of discount being given on a shirt whose selling price is Rs546 after deduction of Rs104 on marked price.     16%

) Mohit bought a shirt for Rs17.50 including GST 10%. Find the original price.    Rs15.75

) Divyanjali Gupta bought a TV for Rs27000. If the GST charged at the rate of 8% of the list price, what is the list price of the TV set ?    Rs25000

) The value of a car including GST is Rs382500. If the basic price of the car be Rs340000, find the rate of GST on cars.    25/2%

) The marked price of a sewing machine is Rs2300. It is sold at a discount of 4%. Find the selling price of the sewing machine.     Rs2208

) The list price of a TV set is Rs19750. If the GST is 4%, find the SP of TV set .   Rs20540

) An item is sold for Rs680 by giving a discount of 5% on its marked price. What is the marked price ?    Rs715.79

) On a discount of 20% an article costs Rs596. What was its original price.   Rs745

) Nishtha purchased leather jacket for Rs2550 and 10% GST is added on it. What is the total cost of jacket ?    Rs2805

) A shopkeeper marks on article 60% more than the cost price and allows a discount of 25% on it. Find its gain percent.      20%

) How much percent above the cost price should a shopkeeper marks his goods so that after allowing a discount of 25% on the marked price he gains 20%.     60%

) A mixer grinder is available for Rs14170 inclusive of GST. If the original price of mixer grinder is Rs13000, find the rate of GST.    9%

) Find the single discount equivalent to successive discounts of 20% and 10%.    28%

) The price of a computer is Rs27000. The GST charged at 12%, find the amount one has to pay to buy it.    Rs30240

PYTHAGORAS THEOREM

EXERCISE - A

1) Which of the triangles whose sides are given are right-angled ?

a) 7cm, 24cm, 25cm.

b) 50cm, 80cm, 100cm.

c) 3cm, 8cm, 6cm.

d) 13cm, 12cm, 5cm. 

e) 1.4cm, 4.8cm, 5cm. 

f) 5,12,13.

g) 6,8,10

h) 24,10,26.

i) 8,15,17.

j) 9,10,11.

k) 9,10,41.

l) 36,15,39.

m) 16,63,65

2) The sides of two triangles are given below . Select the right angle triangle out of these 

a) 6, 8 and 10

b)  12, 13 and 15

3) 14) If the sides of a triangle are in 3:4:5, Is the triangle is right angled triangle?

A

EXERCISE - B

1) Foot of a 10m long ladder leaning against a vertical wall is 6m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches. 

2) A guy attached a wire 24m long to a vertical pole of height 18m and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taught? 

3) If a ladder 10m long reaches a window 8m above the ground, then find the distance of the foot of the ladder from the base of the wall.

4) The foot of a ladder is at a distance of 8 m from the wall of a house and the top of the ladder is at the height of 15m from the ground. Find the length of the ladder .

5) A ladder 13m long reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building.

6) A man goes 15m due east and then 8m due North. How far is he from the starting point ?

7) A girl walks 200 towards East and then she walks 150m towards North. Find the distance of the girl from the starting point.

8) A 5m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4m high. If the foot of the ladder is moved 1.6m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.

10) A ladder reaches a window 12m above the ground on one side of the street, keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 9m high. If the length of the ladder is 15m, then find the width of the street.

EXERCISE - C

1) Two poles of height 6m and 11m stand on a plane ground. If the distance between their feet is 12m, find the distance between their tops. 

2) The heights of two vertical pillars are 7m and 12m. The distance between their feet is 12m. Find the distance between their tops.

3) 

EXERCISE - D

1) ABC is an equilateral triangle of side 2a. Find each of its altitude. 

2) 13) In a right angled triangle, if hypotenuse is 20cm and the ratio of the other two sides is 4:3, find the sides. 

3) The hypotenuse of a right angled triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

4) A right angled triangle has hypotenuse of length p cm and one side of length q cm. If p - q=1, find the length of the third side of the triangle.

5) If the sides of a rectangular plot are 15m and 8m, then find the length of its diagonal.

6) If a side of a rhombus is 10cm and one of the diagonals is 16cm, then find the length of the other diagonals.

EXERCISE - E

1) ABC is an isosceles with AB= AC= 12 and BC= 8cm. Find the altitude on BC and hence calculate its area. 

2) Find the area and the perimeter of a square whose diagonals is 10cm long. 

A) 50, 20 B) 50√2, 20 C) 50, 20√2 D) 50√2, 20√2

) In a rhombus, if diagonals are 30cm and 40cm, find its perimeter.

31) In ∆ ABC, BC= 12cm, CA=16cm and AB= 20 cm. Find the value of angle C.

21) In a ∆ABC, AB= 6√3, BC= 6 and AC= 12cm, then angle B is

32) In ∆ ABC, AD perpendicular to BC, AD= BD= 5cm, BC= 17cm. Find the value of AC.

Miscellaneous Exercise:

1) An aeroplane leaves an airport and flies due to North at a speed of 1000km per hour. At the same time, another aeroplane leaves the same airport and flies due to West at a speed of 1200 kmph. How far apart will be the two planes after 3/2 hours?

2) For going to a city B from city A, there is route via city C such that AC perpendicular to CB, AC= 2x km and CB = 2(x+7) km. It is proposed to construct a 26km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of highway. 

3) In ∆PQR PD perpendicular to QR, such that D lies on QR. If PQ= a, PR= b, QD= c and DR= d then find the value of (a+ b)(a - b). 

4) In ∆ABC, AB= AC=x, BC= 10cm and the area of ∆ABC is 60cm². Find x. 

5) In a triangle ABC, AD perpendicular to BC, AB= 25cm, AC = 17cm and AD= 15cm. Find the length of BC.         28cm

6) ∆ABC is a right angled triangle at B. Given that AB= 9cm, and AC = 15cm and D,E are the midpoint of the sides AB and AC respectively, Calculate

i) the length of BC. 

ii) the area of ∆ADE. 

7) In ∆ ABC is right angled at A. AD perpendicular to BC. If BC=1.25m, AB= 1m, Find AD.

DIAGRAMMATIC QUESTIONS:

8) In the following figure, ABC is a right angled triangle at B. AD and CE are the two median drawn from A and C respectively.

If AC= 5cm and AD=3√5/2 cm,. find the length of CE.

PROVE THE FOLLOWING:

1) ABC is a triangle in which Angle A=90° degree. AD Perpendicular to BC has been drawn . 

Prove that BD x DC = AD²

2) In a ∆ ABC, AD perpendicular to BC. If AD²= BD x DC,

prove BAC=90°.

3) ABC is a right angled triangle in which C is a right angle. If p the length of Perpendicular from C on AB and AB = c, BC = a, CA = b, 

show that

a) pc = ab

b) 1/p² = 1/a² + 1/b².

4) Let O be any point inside the ∆ABC and let OD, OE, OF be perpendiculars to the sides BC, CA and AB respectively.

 Prove that

AF²+ BD²+ CE²= AE²+ CD²+ BF²

5) P and Q are the mid points of the sides CA and AB respectively of a ∆ ABC, right angled at C.

Prove that 

a) 4AQ² = 4AC² + BC².

b) 4BP² = 4BC²+ AC².

c) 4(AQ²+ BP²) = 5AB²

6) ABC is an isosceles triangle. The base BC is produced to D and D is joined to A.

Prove that AD² = AC²+ BD.CD

7) ∆ ABC is an isosceles triangle in which AB = AC and BE perpendicular to AC. 

Prove that

a) BC²= 2AC . CE.

b) BE² - CE²= 2CE. AE.

8) ABC is a triangle in which B and C are acute angles. If BE perpendicular to AC and CF perpendicular to AB, 

Prove that BC²= AB. BF + AC. CE.

9) ABCD is a trapezium in which AB||CD.

Prove that AC²+ BD²= AD²+ BC²+ 2AB. CD.

10) ABC is an equilateral triangle in which the side BC has been trisected D.

Prove that 9AD²= 7AB².

11) In a ∆ ABC, angle B = 90° and D is the midpoint of BC.

Prove that AC²= AD²+ 3CD².

12) ABC is a right angled triangle in which Angle B=90°. Let P and Q be any two points AB and BC respectively. 

Prove that AQ²+ CP²= AC² + PQ².