QUESTION BANK FOR PT -2 MATHEMATICS

Transcription

1 QUESTION BANK FOR PT -2 (A Student Support Material) MATHEMATICS Chief Compiler: Mr. K.Srinivasa Rao. (TGT) Co-Compilers: Mr.M.A.Raju(TGT), Mrs.K.Chaitanya (TGT) Mrs.Sreelatha Nair(TGT), Mr.RVA Subramanyam (TGT) NAVY CHILDREN SCHOOL, VISAKHAPATNAM CLASS 10 th

2 A Word to Student An attempt has been made to knit the Practice material. The Question Bank (Student Support Material ) has been prepared by a team of Math teachers, who teach Class 10 in this academic year( ). The Student is expected to revise the Concepts taught in his/her Class-room as well as that are prescribed in the NCERT text book thoroughly well in advance to the Periodic Test-2. Once the concepts given in the NCERT text book are understood, the student may start practicing the model questions given in this material. The Student is expected to list out the definitions, formulae, Trigonometic table values.so that he/she can recapitulate the concepts before attempting an Examination. At the end a Sample paper is provided so that the student can test him/her self by maintaining time-management which enable to score well in the Examination. All the Best K.Srinivasa Rao. Page 2 of 16

3 REAL NUMBERS SECTION A (1mark Qns) 1. Write the product of HCF and LCM of the numbers 24 and If a rational number is expressed as a non terminating recurring decimal expansion, then what can be the prime factorization of the value? 3. State whether has terminating or non terminating repeating decimal expansion. 4. If the number is expressed in the standard form of a rational number then what can be the prime factorization of the denominator? SECTION-B(2 marks Qns) 5. Find the HCF of the numbers 324 and Explain why 3 X 5 X 7 X is a composite number? 7. Find LCM of 144, 180 and Check whether the value of 35 ends with zero or not for any natural number of. 9. Can two numbers have 15 as their HCF and 240 as their LCM? Give reason. 10. Show that square of an odd positive number is in the form of 2 +1 where is a positive integer. SECTION-C(3 marks Qns) 11. Show that cube of a positive integer is in the form of 9, where is a positive integer. 12. Using Euclid s Division Lemma find the HCF of 441, 567 and During a sale colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy? 14. Prove that Prove that is an irrational number. 16. Show that cube of any positive integer is in the form of 4, Find LCM and HCF of 12, 28 and 84. Also show that product of the three numbers is not equal to product of LCM and HCF. SECTION-D (4marks Qns) 18. A bookseller purchased 117 books out of which 45 books are of Maths and remaining are Physics. Each of books has same size. The books Maths and Physics are to be packed in separate bundles such that each bundle should consist same number of books. Find the least number of bundles which can be made for the 117 books. 19. A residential school is to accommodate students,from three different cities X, Y and Z. The number of students from the cities X, Y and Z are 32, 48, and 24 respectively. Find the minimum number of halls required if in each hall the number of students from each city should be same. Also find total number of students in each hall. 20. Show that only one of the numbers n, n+2 and n+4 is divisible by 3. Page 3 of 16

4 POLYNOMIALS SECTION-A(1 mark Qns) 1. The graph of a polynomial ( ) is shown below. Find the number of zeroes of the polynomial. 2. The graph of a polynomial ( ) is shown below. Find the number of zeroes of the polynomial. 3. Write a quadratic polynomial whose zeroes are 5 and If two zeros of a quadratic polynomial are multiplicative inverse to each other and are equal then write the polynomial. How many such quadratic polynomials can you write? 5. Find a quadratic polynomial whose sum and product of its zeros are -2 and -15 respectively. 6. If the product of zeros of a quadratic polynomial 4 + is 3 then evaluate. 7. If the sum of product of two zeros taken together of a polynomial is 2 then what is the value of? 8. Write the zeroes of the polynomial If. ( )=. ( ) then what is the. ( )? ( ( ), ( ) ( ) h, ) 10. If α, β, and γ are the zeros of a polynomial then find the value of α+β+γ. Page 4 of 16

5 (2 marks questions) 11. Find the zeros of quadratic polynomial 8 9 and verify the relationship between the zeros and the coefficients of the polynomial. 12. If α and β are the zeros of a quadratic polynomial 3 +2 then find the quadratic polynomial whose zeros are (α+2) and (β+2). 13. Find the reminder when ( ): 5 +6 is divided by ( ): If α and β are the zeros of a polynomial + +1 then find the value of α + β. 15. Find the zeros of the polynomial (3 marks questions) 16. Find the zeros of the polynomial and verify the relationship between the zeros and coefficients of the polynomial. 17. On dividing by a polynomial ( ) the quotient and the remainder were respectively. Find ( ). 18. Check whether is a factor of If ( + ) , find value of, and hence find the zeros of the polynomial. 20. If one zero of the polynomial ( ):( +4) is reciprocal of the other then find the value of (4 marks questions) 21. Find all the zeros of 4 +4 if one of its zeros is Find all the zeros of the polynomial if one zero is 3 and second zero is additive inverse of the first. 23. If α and β are the zeros of , find the quadratic polynomial whose zeros are α β. 24. If two zeros of a polynomial are zeros., find the other Page 5 of 16

6 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (1 mark questions) 1. What is the value of for which the system of linear equations = =0 has infinitely many solutions. 2. What is the geometrical representation of the system of linear equations +3 5= =0? 3. Write the value of for which the system of linear equations +3 5=0 and =0 has no solution. 4. At what value of the system of linear equations +4 = =5 is consistent? 5. If one of the vertices of the triangle formed by the lines represented by the equations + 5=0, +3=0 is (1, 4) then write the other vertices. (2 marks questions) 6. Solve: = =0 by substitution method. 7. If the system +3 = = has no solution, find the value of. 8. Solve : + 5=0, =0 by substitution method. 9. Solve: 7 +3 =11 3 =5 by elimination method. 10. Solve: = =0.8 (3 marks questions) 11. Solve the system of linear equations = =0 graphically and shade the region bounded by the lines and -axis. 12. Represent the system of linear equations 2 + 6=0 2 +2=0 and find the vertices of the triangle formed by the lines and -axis. 13. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers. Page 6 of 16

7 14. For which values of p and q, will the following pair of linear equations have infinitely many solutions? 4x + 5y = 2 (2p + 7q) x + (p + 8q) y = 2q p Solve the following pair of linear equations: 21x + 47y = x + 21y = Solve the system of linear equations 43x + 67y = 24 and 67x + 43y = The difference between two numbers is 26 and one number is thrice the other. Find the numbers. 18. Solve the pair of linear equations + +1=0 = Solve for : + =3.3. = In a triangle if one angle is 50 and difference of the other two angles is 10, then find the angles of the triangle. (4 marks questions) 21. A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number. 22. A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from th station A to B costs Rs Also, one reserved first class ticket and one reserved first class half ticket from A to B costs Rs Find the full first class fare from station A to B, and also the reservation charges for a ticket. 23. It can take 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately. 24. There are some students in the two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from A to B. If 20 students are sent from B to A, the number of students in A become double the number of students in B. Find the number of students in the two halls. Page 7 of 16

8 25. The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father. 26. Draw the graph of lines which represent the equations 5 =5, +2 = 1 and 6 + =17.Find the vertices of the triangle formed by these three lines. 27. Solve = ( ) ( ) and + =. ( ) ( ) 28. If ( +1) is a factor of , then find the values of, given that 2 3 = Places A and B are 80km apart from each other on a highway. A car starts from A and other from B at the same time. If they move in the same direction, they meet in 8 hours and if they move in opposite direction, they meet in 1hour and 20 minutes. Find the speeds of the two cars. 30. The sum of a two digit number and the number formed by interchanging the digit is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number. Page 8 of 16

9 QUADRATIC EQUATIONS SECTION-A(1 mark Qns) 1. If one root of =0 is 2 then what is the other root? 2. Find the Discriminant of =0. 3. If are the roots of a Quadratic equation + + =0, what will be the value of Write the nature of the roots of the Quadratic equation 1=0. SECTION-B(2 mark Qns) 5. Solve 4 +4 ( )=0 6. Find the values of 2( +1) + =0 has real and equal roots. 7. If are the roots of the equation + + =0,then show that + = Solve for : =0 SECTION-C(3 mark Qns) 9. Solve for : + = 10. Show that the equation 2( + ) +2( + ) +1=0 has no real roots. 11. Sum of two numbers is 15 and sum of their reciprocals is.find the numbers. 12. Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number. SECTION-D(4 mark Qns) 13. In a flight of 600 km an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time of flight increased by 30 minutes. Find the duration of the flight. 14. The sum of ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124.Determine their present ages. 15. Sum of the areas of two squares is 400 sq.cm. If the difference of their perimeters is 16 cm. find the sides of the two squares. 16. A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work. Page 9 of 16

10 SIMILAR TRIANGLES SECTION : A (1 mark Qns) 1. ABC is similar to DEF. If AB = 5cm, DE = 7cm then what is the ratio of their areas? 2. Two poles of height 10m & 15 m stand vertically on a plane ground. If the distance between their feet is 5 3m then what is the distance between their tops. 3. A man goes 24 km in due east and then He goes 10 km in due north. How far is He from the starting Point? 4. Find the value of if DE BC, =, = +3, =3 +1 & = A D E B C SECTION : B (2 marks Qns) 5. D In the adjacent figure = + + C A Show that =90 B 6. ABC and DBC are two triangles on the same base BC where A and D are on either side of BC. If AD intersects BC at O, Prove that ( ) =. ( ) 7. P In the adjacent figure, = =. S Show that PT X QR = PR X ST. Q T R 8. In an equilateral triangle, prove that three times the square of one side is equal to four time the square of its altitude. SECTION : C (3 marks Qns) 9. Two poles of height are apart.prove that the height of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by. Page 10 of 16

11 E CE and DE are equal chords of a circle with 10. Centre O. If =90,find the ratio of areas O CED and AOB. C O D A B 11. In the adjacent figure, ABC is a triangle right angled at C and is the length of the perpendicular from C to AB. If =, =, = then prove that = Prove that ratio of areas of two similar triangles is in the ratio of squares of their corresponding interior angle bisectors. SECTION : D (4 marks Qns) 13. In ABC, =90, AD and CE are two medians of ABC from A and C respectively.. If AC = 5cm and AD = cm, show that length of CE is 2 5 cm. 14. A In the adjacent figure =90, D and E trisect BC. Show that 8 =3 +5. B D E C 15. State and prove Pythagoras heorem. 16. State and prove Basic Proportionality theorem. 17. Prove that ratio of areas of two similar triangles is in the ratio of squares of their corresponding sides. 18. State and prove the Converse of Pythagoras theorem. 19. State and prove the converse of Thales theorem. 20. Prove that the perpendicular drawn from right vertex to hypotenuse of a right angled triangle divides it into two similar triangles and also similar to the given triangle. ******** Page 11 of 16

12 1. If sec A = INTRODUCTION TO TRIGONOMETRY SECTION : A (1 mark Qns) and A+ B =90, then what is the value of cosec A? 2. What is the value of sin40.cos50 +cos40.sin What is the value of. 4. If tan = 3, h h h + 5. If tan = SECTION : B (2 marks Qns), then find the value of. 6. Evaluate the following: - sin²25 + sin²65 + (tan5 tan15 tan30 tan75 tan85 ) 7. If tan θ = then find the value of cos²θ-sin²θ. 8. If 7 sin2a +3 cos2a= 4, show that tana =1/ 3. SECTION : C (3 marks Qns) 9. Prove that tan²θ + cot²θ + 2 = cosec²θ sec²θ. 10. If cosθ + sinθ = 2 cos θ, then show that (cosθ-sinθ) = 2 sinθ. 11. Prove that + = If sec θ + tan θ = p, prove that sinθ = (p²-1) / (p²+1). SECTION : D (4 marks Qns) 13. If sin θ + cos θ = and sec θ+ cosec θ =n, then prove that n ( 1) =2 14. Prove that 1+cos + 1+cos sin =1+ cos. 15. If 5sin +3cos =4, then find the value of 3sin 5cos. ********* Page 12 of 16

13 SOME APPLICATIONS OF TRIGONOMETRY SECTION : A (1 mark Qns) 1. The length of the shadow of a pillar is 3 times its height. What is the angle of elevation of the source of light. 2. A ladder 50m long just reaches the top of a vertical wall. If the ladder makes an angle of 600 with the wall, find the height of the wall. 3. If length of the shadow and height of a tower are in the ratio 1:1. Then find the angle of elevation. SECTION : B (2 marks Qns) 4. The shadow of tower, when the angle of elevation of the sun is 45 is found to be 10m longer than when it is 60. Find the height of the tower. 5. An observer 1.5m tall is 20.5 metres away from a tower 22m high. Determine the angle of elevation of the top of the tower from the eye of the observer. SECTION : C (3 marks Qns) 6. From the top of a hill, the angle of depression of two consecutive kilometer stones due east are found to be 30 and 45. Find the height of the hill. 7. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h m.at a point on the plane, the angles of elevation of the bottom and the top of the flag staff are, respectively. Prove that the height of the tower is 8. A man on the deck of a ship 14m above water level observes that the angle of elevation of the top of a cliff is 60 and the angle of depression of the base of the cliff is 30. Calculate the distance of the cliff from the ship and the height of the cliff. SECTION : D (4 marks Qns) 9. The angle of elevation of a cloud from a point 60m above a lake is 30 and the angle of depression of the reflection of the cloud in the lake is 60. Find the height of the cloud from the surface of the lake. 10. The angle of elevation of a jet fighter from a point A on the ground is 600. After a flight of 15 seconds, the angle of elevation changes to 30. If the jet is flying at a speed of 720km/ hr, find the constant height at which the jet is flying. ******* Page 13 of 16

14 PROBABILITY SECTION : A (1 mark Qns) 1. The probability of getting rotten egg in a lot of 400 is Then find the no. of good eggs in the lot. 2. A girl calculates that the probability of her winning a prize in a lottery is 0.08.If 6000 tickets are sold, how many tickets has she bought. 3. What is the probability of getting a prime number in the first 100 natural numbers? SECTION : B (2 marks Qns) 4. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability of drawing (a) a face card (b) card which is neither a king nor a red card 5. A coin is tossed two times. Find the probability of getting almost one head. 6. A die is thrown once. Find the probability of getting. a) prime number b) A number divisible by 2. SECTION : C (3 marks Qns) 7. Three unbiased coins are thrown simultaneously. Find the probability of getting. i. Exactly two heads. Ii.At least two heads. iii. At most two heads. 8. Find the probability of getting 53 Sundays in a leap year. SECTION : D (4 marks Qns) 9. Red queens and black jacks are removed from a pack of 52 playing card. A card is drawn at random from the remaining card, after reshuffling them. find the probability that the drawn card is: i) King ii) of red colour iii) a face card iv) a queen 10. All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards after reshuffling them. Find the probability that the card drawn is i) of red colour ii) a queen iii) an ace iv) a face card. ******** Page 14 of 16

15 NAVY CHILDREN SCHOOL VISAKHAPATNAM PERIODIC TEST - 2 ( ) CLASS: X SAMPLE Q.PAPER MAX.MARKS:80. SUBJECT: MATHEMATICS TIME: 3Hours. Instructions: a) All Questions are compulsory. b) The Question paper consists of 30 questions divided into four sections: A, B, C and D. SECTION-A (1m X 6 = 6m) 1. State whether has terminating or non terminating repeating decimal expansion. 2. If α, β, and γ are the zeros of a polynomial then find the value of α+β+γ. 3. What is the geometrical representation of the system of linear equations +2 5= =0? 4. Write the nature of roots of the quadratic equation 8 +17=0 5. Write the value of What is the probability of getting a Queen of black heart? SECTION-B (2m X 6 = 12m) 7. Show that square of an odd positive number is in the form of 2 +1 where is a positive integer. 8. Find the zeroes of Solve for : 2 + 6=0 2 +2=0 10. Find the roots of 12 +8=0 using Quadratic formula. 11. In an equilateral triangle, prove that three times the square of one side is equal to four time the square of its altitude. 12. Two coins tossed once and collected all the possible outcomes.find the probability of getting (i) atleast one Head SECTION-C (3m X 10 = 30m) (ii) atmost one Tail 13. Show that cube of any positive integer is in the form of 4, Draw the graph of the linear equations = and + = and determine the coordinates of their intersecting points. 15. Solve the equation. + =, 3, A C E If AB, CD and EF are perpendiculars on the same line B D F then show that + = Page 15 of 16

16 17. Prove that ratio of areas of two similar triangles is in the ratio of squares of their corresponding medians. 18. If ( + ) , find value of, and hence find the zeros of the polynomial. 19. Prove that + = Prove that =1 geometrically. 21. Find the probability of getting 53 Sundays in a leap year. 22. A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h m.at a point on the plane, the angles of elevation of the bottom and the top of the flag staff are, respectively. Prove that the height of the tower is SECTION-D (4m X 8 = 32m) 23. If α and β are the zeros of , find the quadratic polynomial whose zeros are α β. 24. The owner of a taxi cab company decides to run all the cars he has on CNG fuel instead of petrol/diesel. The car hire charges in city comprises of fixed charges together with the charge for the distance covered. For a journey of 12km, the charge paid Rs.89 and for a journey of 20 km, the charge paid is Rs Find (i) What will a person have to pay for travelling a distance of 30 km? (ii). Which values of the owner have been depicted here? 25. State and prove Basic Proportionality theorem. 26. A person on tour has Rs. 360 for his daily expenses. If he exceeds his tour Programme by four days, he must cut down his daily expenses by Rs 3 per day. Find the number of days of his tour Programme. 27. If Cos Ɵ Sin Ɵ = 2 Ɵ, prove that Cos Ɵ + Sin Ɵ = 2Cos Ɵ. 28. In ABC, =90, AD and CE are two medians of ABC from A and C respectively.. If AC = 5cm and AD = cm, show that length of CE is 2 5 cm 29. Show that = 30. A boy, whose eye level is 1.3m from the ground, spots a balloon moving with the wind in a horizontal line at same height from the ground. The angle of elevation of the balloon from the eyes of the boy at an instant is60. After 2 seconds, the angle of elevation reduces to 30, if the speed of the wind at that moment is 29 3 m/s, then find the height of the balloon from the ground. Page 16 of 16

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