LINEAR EQUATIONS

1. Solve the following equations and check your result.
a) 2(5x-3) -3 (2x -1) = 9 (b) x - (7 – 8x) = 2 c) (x +1)2 = x + 2 /5 d) 9x - 1/2(3 + 4x) = 3

2. When 4 is subtracted from three times a number and the result is divided by 3 more than the number, we get 2/5. Find the number.
3. A number is as much greater than 84 as it is less than 108. Find the number.
4. A number consists of two digits whose sum is 9. If 9 is added to the number, its digits are reversed. Find the number.
5 The digit is unit’s place of a two digit number is three times that in the ten’s place. If the digits are reversed, the new number is 36 greater than the original number. Find the number.
6. A steamer goes down stream form one port to another in 5 hours while it covers the same distance upstream in 6 hours. If the speed of the stream is 3 kmph, find the speed of the steamer in still water.
7. The distance between two station A and B is 290 km. Two motorcyclists start simultaneously form A and B towards each other and the distance between them after 3 hours is 20km. IF the speed of one motorcyclist is less than the other by 10 kmph, find the speed of each motorcyclist.
8. The denominator of a fraction exceeds numerator by 3. IF numerator is doubled and the denominator is increased by 14, the fraction becomes rd of the original fraction. Find the fraction.
9. The length of a rectangle exceeds its breadth by 4 cm. IF length and breadth are each increased by 3 cm, the area of the new rectangle will be 57 cm2 more than that of the original rectangle. Find the length and breadth of the given rectangle.
10. Five years ago, a man was seven times as old as his son. After 5 years, he will be 3 times as old as his son. Find their present ages.

AREA, SURFACE AREA, VOLUME

1 ) The volume of a circular pipe of length 1metre is 3850 cu .cm. Find its diameter.

2 ) A cylinder is open at one end. The external diameter is 10 cm and thickness 2 cm .If height is 8 cm ,find
a) Capacity of the cylinder
b) Volume of metal used

3) Find the number of coins 1.5 cm in diameter and 0.2 cm thick to be melted to form a right circular cylinder of height 16 cm and diameter 6 cm.

4) A rectangular paper of width 30 cm and length 77 cm is rolled along its width ,to form a cylinder.What is the volume of the cylinder formed.

5) A conical tent is 9 m high and radius of its base is 12 m.
a) What is the cost of canvas required to make ir ,if one square metre of canvas costs Rs10
b) How many persons can be accommodated in the tent,if each person requires 2 square metre of ground and 15 m3 of space to breathe in?

6) The diameter of a road-roller is 84Cm and its length is 120cm. It takes 500 complete revolutions to level a playground. Find area of the playground in sq .m.

7) A solid right circular cylinder of diameter 16cm and height 2cm is melted and cast into a right circular cone of height 3 times that of the cylinder. Find the curved surface area of the cone.

8) The dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs.498.96. If the rate of white-wash is Rs.2 per sq. metre, find
a) Inside surface area of the dome.
b) The volume of the air inside the dome.

9) A well is dug 20m deep and has a diameter of 7m. The earth which is spread on a rectangular plot, 22m long and 14m broad. What is the height of the platform that is formed?

10) A conical cup 9cm high, has a circular base of diameter 14cm. The cup is full of water which is now poured into a cylindrical vessel of diameter 10cm. What will be the height of the water in the vessel?

11) Curved surface area of a cone is 308 sq. cm and its slant height is 14cm. Find the total surface area of the cone.

12) The radii of two circles are 5cm and 12cm. The area of a third circle is equal to the sum of the areas of the two circles. Find the radius of the third circle.

13) From a rectangular sheet of sides 30cm and 40cm, a circular sheet as big as possible is cut off. Find the area of the remaining part.

14) The volume of the conical tent is 1232 cu.m and the area of its base is 154 sq.m. Find the length of canvas required to build the tent if canvas is 2m in width.

15) A cylindrical road-roller made of iron is 1m wide. Its inner diameter is 54cm and thickness of the iron sheet rolled into the road-roller is 9cm. Find the weight of roller if 1c.c. of iron weighs 8g.

16) Find the number of coins, 1.5cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder of height 10cm and diameter 4.5cm.

17) A hemispherical bowl is made of steel 0.25cm thick. Inside radius of the bowl is 5cm. Find the volume of steel used in making the bowl.

18) A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base of the cone is 6cm and its height is 4cm. Find the cost of painting the doll at the rate of Rs.5 per thousand sq.cm.

19) The altitude of a triangle is five-thirds the length of its base. If the altitude was increased by 4cm and the base decreased by 2cm, the area of the triangle would remain the same. Find the base and altitude of the triangle.

20) The area of a trapezium is 91 sq.cm and height is 7cm. If one of the parallel sides is longer than the other by 8cm, find the length of the parallel sides.

21) The parallel sides of a trapezium are 25cm and 13cm. Its non-parallel sides are equal, each being 10cm. Find the area of the trapezium.

22) The area of an equilateral triangle is 16√3 sq.cm. Find the length of each side of the triangle.

CUBES AND CUBE ROOTS

1) Find the cubes of the following
a) 35 b) 402 c) 98 d) 11/12 e) 23 8/9 f) 0.01 g) 5.17

2) Examine if the following are perfect cubes
a) 392 b) 106480 c) 1728 d) 400

3) Find the cube root of
a) 91125 b) 531441 c ) - 571787 d) 343/1131
e) -13824 g) 0.008 h) 0.000064 i) 9261/42875

4) What is the smallest number by which 625 must be divided so that the quotient is a perfect cube.Also find the cube root of the quotient.

5) Find the smallest number which when multiplied by 3600 will make the product a perfect cube.Find the cube root of the product.

6) Multiply 137592 by the smallest number so that the product is a perfect square.Also find the cube root of the product.

SQUARES AND SQUARE ROOTS

1) Find the square of
a) 17 b) 431 c) 13/14 d) 8 3/7 e) 0.03 f) 1.02

2) Find the square root by prime factorisation
a) 729 b) 7056 c) 5929 d) 38416

3) Find the square root of
a) 34 15/49 b) 21 2797/3364 c) 2116/15129

4) Find the square root of
a) 37.0881 b) 0.000529 c) 42.25 d) 84.8241

5) Find the square root upto 3 places of decimals of
a) 66 b) 2 1/12 c) 7.234 d) 11 2/3 e) 1

6) What must be added to 452664 to make it a perfect square?

7) Find the least number that should be subtracted from 2361 to make it a perfect square. Also find the square root of the number obtained.

8) 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows.find the number of rows and plants in the garden.

9) Find the smallest number by which 3645 should be divided so as to get a perfect square. Also find the square root of the number obtained.

10) Simplify
( √81 + √0.81 + √0.0081) × √10000

POLYNOMIALS

1) Divide
a) – 20 x⁴ by 10 x
b) 3y ³ by √3 y
c) 4a⁴ by - 2 √2 a²
d) y⁴ - 3y ³ + ½ y² by 3y
e) -4p³ + 4p²+p+4 by 2p

2) Divide by taking factors
a) x²+6x + 8 by x+4
b) y²- y -12 by y – 4
c) x² + 7x +10 by x + 5
d) x⁴+3x²+2 by x² + 2 (Hint: Take x² = y)

3) Divide using by long division method
a) 3y⁴ - y³ + 12 y² + 2 by 3y² - 1
b) 6y – 6 y²+ 5y³ - 1 5y - 1
c) 8q³ - 6q² + 4q – 1 by 2 + 4q
d) a⁴ + a³ + a² by a + 1
e) x( x² - x +1) – ( 9 + 4 x⁴) by 4x - 1

CONSTRUCTION OF QUADRILATERALS

1) Construct a rhombus with side 4.5 cm and one diagonal 6 cm.
2) Construct a parallelogram ABCD in which AB = 3.5 cm ,BC = 4 cm and AC = 6.5 cm.
3) Construct a quadrilateral ABCD in which AB = AD = 3 cm, BC = 2.5 cm, AC = 4 cm and BD = 5 cm.
4) Construct a quadrilateral ABCD in which AC = AD = 6 cm, BC = 7.5 cm, CD = 5 cm and BD = 10 cm.
5) Construct a quadrilateral ABCD in which AB = 3.5 cm, BC = 6.5 cm, and its 3 angles
A =75 degrees, B = 105 degrees and C = 120 .
6) Construct a quadrilateral PQRS in which PQ = 3.5 cm, QR = 6.5 cm, and 3 angles P = 100 degrees, R = 110 degrees and S = 75 degrees.
7) Construct a rectangle with sides 4.5 cm and 6 cm.
8) Construct a parallelogram whose 2 sides and one angle are 4 cm, 5.5 cm and 70 degrees respectively.
9) Construct a PQRS in which PQ = 3.5 cm, QR = 2.5 cm , RS = 4 cm, angles Q = 75 degrees and R = 120 degrees
10) Construct a trapezium ABCD in which AB CD, AB = 8 cm ,BC = 6 cm, CD = 4 cm and
angle B = 60 degrees.( Hint : Find angle C by using the fact that AB CD)

SPECIAL TYPES OF QUADRILATERALS

1) The angles P,Q,R,S of a quadrilateral are in the ratio 1:3:7:9
a) find the measure of each angle.
b) Is PQRS a trapezium? Why?
c) Is PQRS a parallelogram? Why?

2) State True or False
a) Every rectangle is a parallelogram.
b) Every parallelogram is a rhombus.
c) Every rhombus is a parallelogram.
d) Every rectangle is a square.
e) Every square is a rectangle.
f) Every square is a rhombus.
g) Every parallelogram is a rectangle.
h) Every parallelogram is a square.

3) 2 adjacent angles of a parallelogram are equal.Find the measure of each angle of the parallelogram.

4) The ratio of 2 adjacent sides of a parallelogram are in the ratio 2:3.If its perimeter is 48 cm,find the sides of the parallelogram.

5) The diagnols of a quadrilateral are of lengths 10 cm and 24 cm. If the diagnols bisect each other at right angles, find the length of each side of the quadrilateral. What special name will you give to the quadrilateral?

CLASS VIII

PROFIT , LOSS and DISCOUNT

1) Find selling price if
a) Marked price = Rs 1300 and Discount = 1.5%
b) Marked price = Rs 5450 and Discount = 5%

2) Fiind marked price if
a) Selling price =Rs 3430 and discount = 2%
b) Selling price =Rs9250 and discount = 7 ½ %

3) Find Discount per cent when
a) M.P = Rs 625 and S.P = Rs 562.50
s) M.P = Rs 1600 and S.P =Rs1180

4) S.P of 9 articles is equal to the C.P of 15 articles.Find the gain or loss per cent in the transaction.

5) The C.P of an article is Rs 360.If after allowing a discount of 10% , the shopkeeper still earns 25%
profit , find the M.P of the article.

6) A shopkeeper purchases a TV for Rs 2000 and a radio for Rs 750.He sells the TV at a profit of 20%
and radio at a loss of 5%.What is his total loss or gain.

7) The M.P of an article is Rs 500.The shopkeeper gives a discount of 5% and still makes a profit
of 25%.Find the cost price of the article.

8) A person gained 20% after selling an article for Rs 240.At what price should he sell the article to
gain 10%.

9) Harish bought a cradle for Rs 215 and later sold it to Ram at a profit of 5%.He used it for his son for
2 years and then sold it to his servant at a loss of 20%.For how much did the servant get it.

10) A shopkeeper buys pens at the rate of Rs 75 per 100.For much should he sell each pen so as to make a gain of 15%.

11) By selling a pen for Rs 17.50, a shopkeeper suffers a loss of 12 ½ %.What price of the pen would bring the shopkeeper a gain of 20%.

12) By selling an article for Rs 24 , A person loses 20% of his cost.If he sells it for Rs 27 ,what profit or loss would be there for him.

14) A dealer buys an article for Rs 380.At what price must he mark itr so that after allowing a discount of 5% he still makes a profit of 25%.

15) The C.P of 100 mangoes is equal to the S.P of 15 mangoes.Find the gain or loss %.

Studying Mathematics

Studying Mathematics is Different from Studying Other Subjects

• Mathematics is learned by doing problems. Do the homework. The problems help you learn the formulas and techniques you do need to know, as well as improve your problem-solving power.
• 
  A word of warning: Each class builds on the previous ones, all semester long. You must keep up with the Instructor: attend class, read the text and do homework every day. Falling a day behind puts you at a disadvantage. Falling a week behind puts you in deep trouble.
• 
  A word of encouragement: Each class builds on the previous ones, all semester long. You're always reviewing previous material as you do new material. Many of the ideas hang together. Identifying and learning the key concepts means you don't have to memorize as much.
• 
  The higher the math class, the more types of problems: in earlier classes, problems often required just one step to find a solution. Increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece - divide and conquer!
• 
  When you work problems on homework, write out complete solutions, as if you were taking a test. Don't just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don't just do some mental gymnastics to convince yourself that you could get the correct answer. If you can't get the answer, get help.
• 
  The practice you get doing homework and reviewing will make test problems easier to tackle.

Solving an Applied Problem

First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then complete the conversion of the problem into math, i.e., find equations which describe relationships among the variables, and describe the goal of the problem mathematically.

Solve the mathematics problem you have generated, using whatever skills and techniques you need .
As a final step, you should convert the answer of your mathematics problem back into words, so that you have now solved the original applied problem.

Good Test Taking Strategy

Just as it is important to think about how you spend your study time , it is important to think about what strategies you will use when you take a test. Good test-taking strategy can make a big difference to your grade!

Taking a Test

First look over the entire test. You'll get a sense of its length. Try to identify those problems you definitely know how to do right away, and those you expect to have to think about.

Do the problems in the order that suits you! Start with the problems that you know for sure you can do. This builds confidence and means you don't miss any sure points just because you run out of time. Then try the problems you think you can figure out; then finally try the ones you are least sure about.

Time is of the essence - work as quickly and continuously as you can while still writing legibly and showing all your work. If you get stuck on a problem, move on to another one - you can come back later.

Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a 10 point question. Starting with the easy questions will probably put you ahead of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes) on that question, and if you have not almost finished it, go on to another problem. Do not spend 20 minutes on a problem which will yield few or no points when there are other problems still to try.

Show all your work: make it as easy as possible for the Instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the Instructor will assign partial credit based on the work you show.

Never waste time erasing! Just draw a line through the work you want ignored and move on.
Not only does erasing waste precious time, but you may discover later that you erased something useful (and/or maybe worth partial credit if you cannot complete the problem).
You are (usually) not required to fit your answer in the space provided - you can put your answer on another sheet to avoid needing to erase.

Make sure you read the questions carefully, and do all parts of each problem.

Verify your answers - does each answer make sense given the context of the problem?
If you finish early, check every problem (that means rework everything from scratch).