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Mathematics with SolutionClass 10 Term 1ISBN 978-81-8137-695-4First Edition 1999Twenty-sixth Latest Revised Edition[R1] PUBLISHED BY 4583 / 15, Daryaganj, New Delhi - 110 002 RACHNA SAGAR PVT. LTD. PO Box 7226 Phone 011 - 4358 5858, 2328 5568 – EDUCATIONAL PUBLISHERS – Fax 011 - 2324 3519, 4311 5858 Email [email protected], where quality speaks for itself... [email protected], [email protected],Offices [email protected] [email protected] Web www.rachnasagar.in IE License No. 0501009426Ahmedabad A 87, Swagat Bungalow - 3, Opp. Swagat Bungalow - 1, Near ICICI Bank, New C.G. Road Chandkheda, 382 424,  Phone  0 99246 45576BENGALURU 90 / 7 & 90 / 8, 1st Floor, 1st Cross, Vittal Nagar, Mysore Road, 560 026 Phone  0 90085 57707, (080) 2674 7475, 2674 7476BHOPAL E 6 / 127, Ground Floor, Arera Colony, 462 016, Phone  0 97525 93355, (0755) 422 3838CHANDIGARH S.C.O. No. 31, Second Floor, Sector 31D, 160 031 Phone  (0172) 262 4882, Fax (0172) 508 6882CHENNAI Old No. 18, New No. 80, Ramar Koil Street (Opp. Chennai Trade Centre), Nandambakkam, 600 089  Phone  0 87545 80792COCHIN Building No. 52 / 1000, Pravada Nivas, Muttathil Lane, Kadavanthara, 682 020 Phone  0 73561 22770DEHRADUN I - 15, Nehru Colony, 248 001,  Phone  0 73889 33938GUWAHATI K.C. Sen Road, H.No. 20, Near Pink Arcade, Paltan Bazar, 781 008 Phone  0 70860 90863HYDERABAD H. No. 1 - 8 - 588 / 02 / 03, Street No. 10, Achaiah Nagar Nallakunta, 500 044 Phone  0 91009 14234JAIPUR Plot No. 89, Shiv Colony, Gali No. 12, Moti Nagar, Queens Road, 302 021 Phone  0 97999 99123KOLKATA 139, Saheed Hemanta Basu Sarani, (Jawpur Road), 700 074,  Phone  0 93301 02176LUCKNOW C 1454, Indira Nagar, 226016,  Phone (0522) 400 4909MUMBAI VAASTU SIDDHI, 003 / 004, A Wing, Ground Floor, Shree Vaastu Enclave (Behind Manish Park), Jijamata Road Pump House, Andheri (East), 400 093 Phone  0 81084 48884, 0 84258 69445PATNA 1st Floor, Annpurna Market, K 54, Hanuman Nagar, Kankarbagh, 800 020 Phone  0 97714 41611, (0612) 235 1127RANCHI D 17 MIG, Harmu Housing Colony, 834 012,  Phone  0 97714 41620© Reserved with All rights reserved. No part of this publication may be reproduced in any form the publishers whatsoever, without the prior written permission of the publishers.Disclaimer The publishers and the author or seller will not be responsible for any damage or loss ofaction of anyone, of any kind, in any manner, therefrom. Every effort has been made to avoid errors oromissions in this publication. In spite of this, some errors might have crept in. Any mistake, error ordiscrepancy noted may be brought to our notice which shall be taken care of in the next edition. Forbinding mistakes, misprints or for missing pages, etc., the publisher’s liability is limited to replacementwithin one month of purchase by similar edition. All expenses in this connection are to be borne bythe purchaser. (2)

6 Practice PaperTime Allowed : 3 hours] [Max. Marks : 90General Instructions — Questions 1–4 (1 Mark each), 5–10 (2 Marks each), 11–20 (3 Marks each), 21–31 (4 Marks each) SECTION – A 1. DSTN and DPQR are similar D′s such that ∠S = 38°, ∠R = 55°. Find ∠T. 2. cot2(3A + 15°) – 1 = 0. If 3A + 15° is acute then find the value of A. 3. ABC is right triangle right angled at C if AC = 3 BC, find ∠B. 4. Write the relationship between mean, median and mode for a moderately skewed distribution. SECTION – B 5. If 2 sin2q – 2 2 sin q + 1 = 0, find the value of sin2q + tan q. 6. Using Euclid’s division algorithm find the HCF of 305 and 793. 7. The number ( 5 + 3)8( 5 − 3)8 is rational or irrational. Justify your answer. 8. For what value of a the given pair of linear equation has no solution: ax + (a – 1)y = 1; (a + 1)x – ay = 1 A 9. In given figure AD =3 cm, AE =5 cm, BD = 4 cm, CE = 4 3 cm 5 cm Ecm, CF = 2 cm and BF= 2.5 cm. Find the pair of parallel D 4 cmline segment. 4 cm 10. Find the mode of the following distribution: B 2.5 cm F 2 cm C Class interval Frequency 0 – 10 2 10 – 20 4 20 – 30 7 30 – 40 9 40 – 50 12 50 – 60 3 60 – 70 2 70 – 80 1542 Together with®  Mathematics—10

27. Find the value of cosec 45° geometrically. Hence, find cos 45° and sin 45°. 28. Show that 2 sec2q – sec4q – 2 cosec2q + cosec4q = cot4q – tan4q. 29. Using the formula s in(A + B) = sin A . cos B + cos A . sin B, find the value of sin 75°. Hence find the value of 1 + cot275°. 30. The distribution of heights (in cm) of 96 children is given below:Height 124 – 128 128 – 132 132 – 136 136 – 140 140 – 144 144 – 148 148 – 152(in cm)Children 7 8 17 24 20 12 8 Draw a more than type cumulative curve for the above data and use it to find the median height of the children. 31. The median of the following frequency distribution is 28.5. Find the values of x and y, if sum of the frequency is 58. Also find 3x + y.Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60Frequency 5 x 20 15 y 3552 Together with®  Mathematics—10

19. Find the mean age in years from the frequency distribution given below: Class intervals Frequency of age in year ( fi) 25 – 29 4 30 – 34 14 35 – 39 22 40 – 44 16 45 – 49 6 50 – 54 5 55 – 59 3 20. Calculate the median for the following data: Class intervals Frequency 130 – 140 5 140 – 150 9 150 – 160 17 160 – 170 28 170 – 180 24 180 – 190 10 190 – 200 7 SECTION – D 21. A rational number in its decimal expansion is 2.387387387 ... . What can you say pabout the prime factor of q, where the expressed in the form q ( p and q are positiveintegers). Give reason. Also express the number in the form of p . q 22. Draw the graphs of the pair of linear equations 3x + 4y =12 and 3x – 4y = –12. Also find the area of the triangle formed by these lines and the x-axis. *23. An NGO decided to distribute books and pencils to the students of a school running by some other NGO. For this they collected some amount from a number of people. The total amount collected is represented by 6x5 + 15x4 + 16x3 + 4x2 + 11x – 34. The amount is equally divided between each student. The number of students who received the amount is represented by 2x3 + 5x2 + 2x – 7. After distribution x + 1 amount is left with the NGO. Find the amount received by each student from the NGO. What values have been depicted by the NGO?*Value-based Question Practice Papers 555

24. If one zero of the quadratic polynomial 3x2 – 2kx + 9 is three times the other, where k > 0, find the zeroes of 2kx2 +17x + 5. 25. In ∆ABC, AD ^ BC, BE ^ AC. A E B DC If Area DADC = 64 cm2, area DBEC = 121 cm2, AD = 5.6 cm. Find BE. 26. State and prove converse of Pythagorous theorem. sec2θ + cosec2θ (sin θ + cos θ) 2 sin θ. cos θ 27. Prove that = − 2. A 28. In ∆ABC, AB = 24 cm AC – BC = 18 cm ∠B = 90° Find tan A and cos C. 29. If cos A + sin A = x, cos A – sin A = y. B C Show that x2 − y2 = tan A 2 cot A x2 + y2 + 30. If the median of the following distribution is 28.5, find x and y. Class intervals Frequency 0 – 10 5 10 – 20 x 20 – 30 20 30 – 40 15 40 – 50 y 50 – 60 5 Total 60 31. The following table gives production yield per hectare of wheat of 100 farms of a village. Production yield (in kg/ha) 50 – 55 55 – 60 60 – 65 65 – 70 70 – 75 75 – 80 Number of farms 2 8 12 24 38 16 Change the distribution to a more than type distribution and draw its ogive.556 Together with®  Mathematics—10

Practice Paper 10Time Allowed : 3 hours] [Max. Marks : 90General Instructions — Questions 1– 4 (1 Mark each), 5 –10 (2 Marks each), 11–20 (3 Marks each), 21– 31 (4 Marks each) SECTION – A 1. If 2 cos q – 1 = 0, find cot q + sec q. 2. Find the minimum value of cos2q + sec2q. 3. If DABC ~ DDEF, area DABC = 81 cm2, AB = 4 cm, DE = 7 cm, find area DDEF. 4. For a given data less than ogive and more than ogive intersect at (28.5, 30). What is the median of the data. SECTION – B 5. Use Euclid’s algorithm to find the HCF of 270 and 450. 6. Express 0.27 as a fraction in simplest form. 7. If 3 tan q = 3 sin q, then find the value of cos2q – sin2q. 8. Find the value of p for which given pair of linear equation will have infinite solutions 2x – py = 9 4x + 6y + 18 = 0 A 9. In the given figure DABC, ∠B = ∠C and BD = CE. Prove thatDE || BC. D E B C 10. Find the unknown entries a, b, c, d, e and f in the following distribution Class intervals Frequency ( f) Cumulative frequency (cf) 0 – 10 12 10 – 20 b a 20 – 30 8 27 30 – 40 d c 40 – 50 e 45 50 – 60 f 50 60 Practice Papers 557

SECTION – C 11. Show that 7 is an irrational number. 12. Solve for x and y 2 + 3 = 13, 5 − 4 =– 2, x, y ≠ 0. x y x y 13. a, b are zeroes of polynomial px2 – 18x + 28. If ab = 14, find the value of 2p2 – 8. 14. Check graphically whether the pair of linear equations 4x – y – 8 = 0 and 2x – 3y + 6 = 0 is consistent. A 15. In given figure, DE is parallel to BC and AD : DB = 2 : 3. D E Determine area DADE : Area DABC. BC 16. In the given figure ABCD is a quadrilateral and P, Q, R and S are points of trisections of the sides AB, BC, CD and DA respectively. Prove that PQRS is a parallelogram. D RC Q S AP B 17. If tan q + 1 = 2 , show that cos q – sin q = 2 sin q. 18. Prove that 1 + cos A + 1 sin A A = 2 cosec A. sin A + cos 19. Find the mode of the following data: Marks Number of students Less than 10 3 Less than 20 8 Less than 30 20 Less than 40 40 Less than 50 80 Less than 60 85 Less than 70 90 Less than 80 100 558 Together with®  Mathematics—10

20. The mean of the following frequency distribution is 25. Find the value of p. xi 10 5p 30 40 50 fi 5 p + 1 6 3 1 SECTION – D *21. A trader was moving along a road selling oranges in a basket. An idler who did not have much work to do, started to get the traders into a words duel. This grew into a fight. The idler pulled the basket with oranges and dashed it on the floor. All the oranges went into a nearby dram. The trader then requested the panchayat to ask the idler to pay for the oranges. The panchayat asked the trader how many oranges were there? He gave the following reply. If counted in pairs, one will remain. If counted in 3, two will remain. If counted in 4, three will remain, if counted in 5, four will remain. If counted in 6, five will remain. If counted in 7, nothing will remain and my basket cannot more than 150 oranges. (i) How many oranges were there. (ii) What is the value shown by the traders in this question. 22. The age of a teacher is equal to the sum of ages of his six students. After 15 years, twice the age of the teacher will be equal to sum of the ages of his six students. Find the present age of the teacher. 23. Divide 2x4 – 9x3 + 5x2 + 4x – 9 by x2 – 4x + 3 and verify the division algorithm. 24. If two zeroes of the polynomial x4 – 6x3 + 6x2 + 10x – 3 are 2 + 3 and 2 – 3 , then find the other zeroes. 25. State and prove the basic proportionality theorem. 26. In the given figure, O is a point in the interior of DABC, OD, OE and OF are the perpendicular drawn to the sides BC, CA and AB respectively. Show that (AF)2 + (BD)2 + (CE)2 = (AE)2 + (CD)2 + (BF)2.­ A F E O*Value-based Question BDC Practice Papers 559

27. If 18 tan2q + 4 sec2q = 26. Find the value of (sin q + cos q)2 + (sec q + cosec q)2. 28. Without using trigonometric table, evaluate: cosisnec222673° °+–sitnan26232°7° + 1 tan 10° . tan 60° . tan 80° – 4 sin 43° 3 cos 47° 29. If x cos3q + y sin3q = sin q . cos q and x cos q = y sin q, show that x2 + y2 = 1. 30. Find mean and modal marks of students for the following distribution. Marks Number of students 0 and above 100 10 and above 95 20 and above 85 30 and above 70 40 and above 60 50 and above 50 60 and above 40 70 and above 35 80 and above 30 90 and above 20 100 and above 0 31. Draw an ogive for the following distribution By ‘less than’ method Class intervals Frequency 0 – 10 7 10 – 20 10 20 – 30 25 30 – 40 51 40 – 50 6 50 – 60 3560 Together with®  Mathematics—10