Mathematics For Class 10

Kulachi Hansraj Model School , Ashok Vihar, Ph-III Holiday Homework- Assignments Grade 10 Quadratic Equations 1. Write the general form of a quadratic equation. 2. For what value of k the quadratic equation kx2- 5x+ k = 0 has equal roots? 3. Quadratic equation x2 + 7x + 10 has ___________ roots. 4. Form a quadratic equation whose roots are -1 & 3. 5. If -2 is a root of the equation 3x2-5x+2k=0, then what is the value of k? Find other root also. 6. State the condition so that the equation px2+qx+r=0, p≠0, may have 1) Equal real roots 2) Two different roots 3) No real roots 7. Write the quadratic equation for the following problem: “Square of a number is four less than five times the number" 8. Write the discriminant of the equation 3x2-7x-2=0. 9. Write the condition when equation x2+2x-35=0 has real roots. 10. If the quadratic equation ax2+bx+c=0, a≠0 has equal roots then show that

b2=4ac.

11.The sum of two natural numbers is 8. Determine the numbers if the sum of their reciprocals is 8/15. 12. For what value of p does equation x2-4x+2p=0 has repeated roots? 13. Check whether the equation x2+3x+1=(x-2)2 is a quadratic equation. 14. The sum of two numbers is 15.If the sum of their reciprocals is

, find the numbers.

15. Solve the quadratic equation 9x2-15x+6=0 by the method of completing the square. 16.Find the value of K for which the given equation has real roots:

(1) Kx2 - 6x – 2 = 0 (2) 5x2 – Kx + 1 = 0 17.Some students arranged a picnic. The budget for food was Rs.500. But, 5 of them failed to go and thus the cost of food for each student increased by Rs.5. How many students attended the picnic?

18.If a student had walked 1km/hr faster, he would have taken 15 minutes less to cover a distance of 3 km. Find the rate at which he was walking. 19.In a class test ,the sum of Radha’s marks in Mathematics and English is 30.Had she got 2 marks more in Mathematics and 3 marks less in English, the product of the marks would have been 210.Find her marks in the two subjects. 20. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number. 21. A fast train takes 3 hrs less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speeds of the two trains. 22. Two trains leave a railway station at the same time .The first train travels due west and the second train due north. The first train travels 5km/hr faster than the second train. If after two hours, they are 50 km apart, find the average speed of each train. 23. A peacock is sitting on the top of a pillar, which is 9m high. From a point 27m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal, at what distance from the whole is the snake caught? 24. 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 a river. Find the total number of camels. 25. Two water taps together can fill a tank in 9

hours. The tap of larger diameter takes

10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank. Probability 1. The probability that it will rain tomorrow is 0.85. What is the probability that it will not rain tomorrow? 2. If x is the probability of an event then what will be the probability of its complement?

3. Why is tossing a coin considered to be fair way of deciding which team should choose ends in a game of cricket? 4. A coin is tossed twice .If the second throw results in tail, a dice is thrown .Describe sample space. 5. What is the probability of drawing a blue marble from a bag containing 3 red& 2 blue marbles? 6. What is the probability that a number selected from numbers 1, 2, 3, - - - -, 15 is a multiple of 5? 7. What is the probability that an ordinary year has 53 Sundays? 8. In a lottery, there are 10 prizes & 25 blanks. Find the probability of getting a prize? 9. A number is selected from numbers 1 to 27 .What will be the probability that it is prime? 10. It is known that a box of 600 electric bulbs contains 12 defective bulbs .One bulb is taken out at random from this box .What is the probability that it is a non –defective bulb? 11. The probability of guessing the correct answer to a certain test questions is x/12.If the probability of not guessing the correct answer to this question is 2/3 then what is the value of x? 12. A card is drawn at random from a well shuffled pack of 52 cards .Find the probability that the card drawn is a red queen? 13. A box contains 90 cards which are numbered from 1 to 90. If one card is drawn at random from box .Find the probability that it bears a number divisible by 5? 14. Three unbiased coins are tossed together. Find the probability of getting: (a) All heads (b) Two heads (c) Atmost two heads (d) Getting a head and tail alternately 15. In a single throw of two dice, find the probability that neither a doublet nor a total 9 will appear. 16. Find the probability of 53 Sundays in a leap year. 17. In a single throw of two dice, what is the probability of getting a total of 9 or 10? 18. Is falling of a fan an equally likely outcome? Give reason for your answer. 19. From a well shuffled pack of 52 cards, a card is drawn at random. What is the probability that it is a jack of red suit? 20. A letter is chosen at random from the word "MATHEMATICS". What is the probability

that it is a vowel? 21. Two dice are thrown simultaneously. What is the probability of getting an even doublet? 22. Two die are thrown simultaneously. What is the probability of getting an even prime number? 23. Cards marked with numbers 5 to 50 are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is: a) a prime number less than 10 b) a number which is not a perfect square. 24. All cards of ace, jack and queen are removed from a deck of playing cards .One card is drawn at random from the remaining cards, find the probability that the card drawn is a) A face card b) A red queen Linear Equations in two variables 1. Does the point (1,-2) lie on the line whose equation is 3x-y-5=0? 2. For what value of ‘K’ the pair of equations x-ky+4=0 and 2x-4y-8=0 is inconsistent? 3. Which axis is the graph of equation x=0? 4. How many solutions of the equation 5x-4y+11=0 are possible? 5. What is the value of ‘K’ for which the graph of the equations 2x-3y=9 and kx-9y=18 are parallel lines? 6. Find the value of ‘a’ for which the system of equations 3x+2y-4=0 and ax-y-3=0 will represent intersecting lines? 7. Write a linear equation in two variables which is consistent to equation 5(x-y) =3. 8. What is the solution of 2 x - 5 y=0 and 2 3 x- 7 y=0. 9. Find the co-ordinates of the point where the line 2x-3y=6 meets x-axis and y-axis. 10. Write a linear equation in two variables which is inconsistent to equation 5(x-y) =3. 11. Draw the graph of the following pair of linear equations x+3y = 6 and 2x – 3y= 12. Hence find the area of the region bounded by x = 0, y = 0 and 2x – 3y = 12. 12. Solve the following system of linear equations graphically 3x + y – 12 = 0 and x – 3y + 6 = 0 . Shade the region bounded by these lines and x-axis. 13. Draw the graph of the following pair of linear equations 5x – y = 5 and 3x – y = 3. Determine the coordinates of the vertices of the triangle formed by these lines and y axis.

14. Graphically, find whether the following pair of equations is consistent or inconsistent. 5x – 8y + 1 = 0 3x – 24 y + 3/5 = 0 15. Solve for x and y by (i) Substitution method (ii) Elimination method (iii) Cross multiplication method px + qy = o lx + my = n 16.

Solve for x and y (i) ___1____ + ____5___ = - 3 2 (x+2y) 3 (3x -2y) 2 ___5____ - ____3___ 4 (x+2y) (ii)

5 (3x -2y)

20.

0 (v)

2x –( 3/y) = 9, 3x + (7/y) = 2

iv)

(vi)

2 + 3 = 13

x +1 + y -1 = 8 3

x-1 + y + 1 = 9

19.

60

(iii) x –4 = 4(y +2) 3(x-2) = 2y + 20

2

18.

0

x + y = 5xy 3x + 2y = 13xy, x 0, y

17.

= 61 where x + 2y 0 and 3x – 2y

x

y

0

y

5 – 4 = -2

x, y 0

3 2 x y A number consists of two digits whose sum is five. When the digits are reversed the number becomes greater by nine. Find the numbers. Suresh can row a boat 8 km downstream & return in 1 hr 40 mins. If the speed of the stream is 2km/h, find the speed of the boat in still water. A train travels a distance of 480 km at a uniform speed. If the speed had been 8km/h less, it would have taken 3 hrs more to cover the same distance. Find the speed of the train. A two digit number is four times the sum of its digits and twice the product of the digits. Find the number.

21.

22.

23.

24.

The denominator of a fraction is 4 more than twice the numerator when both the numerator and denominator are decreased by 6, then denominator becomes 12 times the numerator. Determine the fraction. Points A and B are 70 km a part on a highway. A car starts form A and another car starts form B simultaneously, if they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in one hour. Find the speed of the two cars. A man travels 370 km partly by train and partly by car. If the covers 250 km by train and rest by car, it takes him 4 hours. But if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car. It takes 12 hours to fill a swimming pool using two pipes. If the larger pipe is used for 4 hours and the smaller pipe for nine hours, only half the pool is filled. How long would it take for each pipe alone to fill the pool?

Polynomials 1. Find the quadratic polynomial, the sum and product of whose zeroes are 1) ½, -2 2) –3, -7 3) 5+√3 & 5-√2 2. Find the zeroes of the quadratic polynomial p(x) = t 2 – 15 3. How many maximum zeroes can a polynomial of degree two have? 4. What is the zero of the polynomial ax + b=0, a≠0? 5. Find the sum & the product of the zeroes of the polynomial 6x2 -x-2 6. Give examples of polynomials f(x), g(x), q(x) & r(x) which satisfy the division algorithm 1) deg r(x)=0 2) deg f(x)=deg q(x)=2 7. State Division Algorithm 8. Find the quadratic polynomial whose one zero is 5 & product of zeroes is 30. 9. Write general form of a quadratic polynomial. 10. The linear polynomial ax + b, a ≠ 0, has exactly one zero, namely, the__________ of the point where the graph of y = ax + b intersects the x-axis. 11. The graphs of y = p(x) are given in Fig. below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

12.

Find the zeroes of the quadratic polynomial z2 + 5z + 6 and verify the relationship between the zeroes and the coefficient of the polynomial.

13. 14.

Write a quadratic polynomial whose zeroes are 2 + 2 and 2- 2. The sum and product of the zeroes of a quadratic polynomial are ½ and -3 respectively. Write the quadratic polynomial. Find the quotient and the remainder when 3x4 + 5x3 – 7x2 + 2x + 2 is divided by x2 + 3x + 1.

15. 16. 17. 18. 19. 20. 21. 22.

23. 24. 25.

If two zeroes of the polynomial x4 – 6x3 – 26x2 + 138x – 35 are 2±, 3 find other zeroes. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x – 2. What must be added to 8x4 + 14x3 – 2 x2 + 7x – 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x – 2. Check whether g(x) is a factor of p (x). p (x) = 6x3 + x2 – 19 x + 6 and g(x) = 2x – 3. On dividing x 3 – 3x2 + 5x – 3 by a polynomial g(x) the quotient and remainder were x -3 and 7x – 9 respectively. Find g(x). Find the cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and the product of its zeroes 4, -6 and 1 respectively. If α & β are zeroes of the quadratic polynomial x2 – (k+6) x + 2 (2k-1). Find the value of k if α + β = ½ αβ. If one zero of the polynomial p (x) = 3x2 – 8x + 2k + 1 is seven times the other, find the zeroes and the value of k. Verify that 2,1, 1are the zeroes of the cubic polynomial x3 – 4x2 + 5x – 2. Also verify the relationship between the zeroes and the coefficients. Find all the zeroes of x3 -4x, if two of its zeroes are 0 and 2.

Project work : Instructions from C.B.S.E. Every student will be asked to do at least one project, based on the concepts learnt in the classroom. The project should be preferably carried out individually and not in a group. The project may not be mere repetition or extension of the laboratory activities, but should aim at extension of learning to real life situations. Besides, it should also be somewhat open-ended and innovative. The project can be carried out beyond the school working hours. Some sample projects are given in the document but these are only illustrative in nature. The teacher may encourage the students to take up new projects. The weightage of five marks for project work could be further split up as under Identification and statement of the project : 01 mark Design of the project : 01 mark Procedure/processes adopted : 01 mark Write- up of the project : 01 mark Interpretation of result : 01 mark General Instructions: · Each student is required to make a handwritten project report according to the project allotted. Roll Number...............Project Number 1-10.............................1 11-20...........................2 21-30...........................3 31-40...........................4 41-50...........................5 • General lay-out of the project report has the following format Page Number....................Content 1................Your Name, Class, Class Roll No., Board Roll No. (Leave empty space), Title of project. 2................Content – Page description 3................Brief description of project 4-10 (may change)............Procedure (With pictures) 11..............Mathematics used /involved 12...................Conclusion /Result 13. What did you learn? Share your experience. 14....................List of resources (List of encyclopedia ,websites , reference books , journals etc) 15....................Acknowledgement Project 1 Objective To appreciate that finding probability through experiment is different from finding probability by calculation. Students become sensitive towards the fact that if they increase the number of observations, probability found through experiment approaches the calculated probability. Description 1. You may either work individually or in a group of two. 2. Roll a pair of dice (take different colours) 100 times and record your observations. 3. Calculate the probability of following from your observations : · Getting same number on both · Getting sum =10 · Getting sum >8 · Getting prime numbers on both Now compare your result with actual probabilities of these events.

Project 2 Objective :To make mathematical designs and patterns using arithmetic progression. Description 1.In this project you may work individually or in a group of two. 2.Explore Mathematical designs and patterns using the notion of arithmetic progression. 3. Make some designs like http://mathematicsprojects.blogspot.com/2008/12/mathematical-designsusing-ap.html and write a paragraph about each of design created by you explaining Math concept used. Note: The designs are to be created using paper cutting and pasting strategy. Example 1. Take terms in an arithmetic progression say a1, a2, a3. 2. First of all, the starting base shape of the design will be prepared fro m a paper. For that add the given numbers i.e. s = a1 + a2 + a3. Cut a square (of side s cm) and paste it on a sheet. 3. Cut rectangular strips of different colours of size a1× s, a2 × s, a3×s. (Note that a1, a2, a3 are in arithmetic progression.) 4. Paste the rectangular strips adjacent to each other on one side of the square. 5. Cut similar rectangular strips for the remaining 3 sides. 6. Join the gap by straight lines. Right angled triangles are formed. This is how one such design can be easily made where the rectangular strip’s width is changing in an arithmetic progression and the length is the same. 7. You may take any other regular polygon as the starting shape and build newer designs. 8. You may take a circle as the starting shape, take equidistant points on itsCircumference say 1, 2, …., 36, and repeat similar procedure Project 3 Objective:Exploring different proofs of the Pythagoras Theorem Description Make a project report on different ways of proving/visualizing the famous Pythagoras theorem. Useful links http://www.ies.co.jp/math/java/geo/pythagoras.html http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html Project 4 Objective: Exploring Mathematics around Description 1. Look aound yourself in the house,in the garden,in the market,in a bank,in the nature...so on 2. Click photographs using a digital camera/mobile and explore the hidden Mathematics. 3.Write a paragraph about Mathematics observed by you in each picture. Note:Click minimum 10 photographs. You may visit http://mathematicsaround.blogspot.com/ to see pictures. Project 5 Objective:Analysis of a language text, using graphical and pie chart techniques. Description 1.Select any paragraph containing approximately 250 words fromany source. e.g. newspaper, magazine, textbook, etc. 2. Read every word and obtain a frequency table for each letter of the alphabet as follows Letter Frequency A B C D ...XYZ

3. Note down the number of two-letter words, three- letter words, …. so onand obtain a frequency table as follows Number of words Frequency 2 letters 3letters 4 letters.... 4. Select 10 different words from the text which have frequency greater than 1. Giveranks 1, 2, 3, …., 10 in decreasing order of their frequency. Obtain a table as follows Word Frequencyonit....... on two after . . 5. Investigate the followingFrom table 1a) What is the most frequently occurring letter? b) What is the least frequently occurring letter? c) Compare the frequency of vowels d) Which vowel is most commonly used? e) Which vowel has the least frequency? f) Make a pie chart of the vowels a, e, i, o, u, and remaining letters. (The piechart will thus have 6 sectors.) g) Compare the percentage of vowels with that of consonants in the given text. From table 2a) Compare the frequency of two letter words, three letter words, ….. and so on. b) Make a pie chart. Note any interesting patterns. From table 3 a) The relation between the frequency of a word to its rank. b) Plot a graph between the frequency and reciprocal of word rank. What do youobserve? Do you see any interesting pattern? c) Repeat the experiment by choosing text from any other language that you knowand see if any common pattern emerges. In case of any query... Mail to your Math teacher [email protected]