MODEL TEST PAPER 9 FIRST TERM (SA-I) MATHEMATICS (With Answers) CLASS X llme Allowed, : 3 to 3% Hours] LMaximum Marks : 80 General Instructions : (i) All are compulsory. (ii) The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 10 questions of I mark each, Section B comprises of 8 questions of 2 marks each, Section C comprises of 10 questions of 3 marks each and Section D comprises of 6 questions of 4 marks each. (iii) Question numbers I to 10 in Section A are multiple choice questions where you are to select one correct option out of the given foul: (iv) There is no overall choice. However, internal choice has been provided in I question of two marks, 3 questions of three marks each and 2 questions of four marks each. You have to attempt only one of the alternatiues in all such questions. (v) Use of calculators is not permitted. Question numbers 1 to 10 are of dne mark each. 1. If sec 13 +tan e =p, then the value of see 13 -tan e is 2. figure, AB 11 DE. The length of CD is (a) 3. cm (6) 3.7m - (c) 3 cm (d) 4.2 ern 4 3. If cos 8 = -, then the value of (cot I3 + cosec 11~ is (a) 3 (6) 6 (c) 9 (d) 8
4. 1n figure, the graph of a polynomial p(x) is shown. The number of zeroes ofp(x) is Y (a) 3 (c) 4. If cot 0 = + 1, then the value of cosecz 0 is (a) 2&(& + 1) (b) 2&(& + 1) (c) 3&(6 + 1) (d) &(& + 1) P 6. If a rational number x is expressed as x = -, wherep, q are integers, q s 0 andp, q have a no common factor (except I), then the decimal expansion of x is terminating if and only ifq has a prime factorization of the form : (a) 2". 3n (b) 2m. n (c) Zm. 7n (dl m. 3n 7. If A = 4" and B = 30, then the value of sin A cos B - cos A sin B is is &+I &-I (c)- (dl- 2& 2& 8. If the HCF of 210 and is expressible in the form 210 x + x n, then the value of n (a) - 18 (b) - 17 (c) - 20 (d) - 19 9. If the pair of linear equations 2x + 3y = 7 and 203 + (a + PI. = 28 has infinitely many solutions, then the values of a and p are (a) 4 and (b) 3 and (c) 4 and 7 (d) 4 and 8 10. The mean of first 1 natural numbers is (a) 6 (b) 8 (c) 7 (dl 9
Model Test Papers 287 Question numbers 11 to 18 carry 2 marks each. 11. If a and p are the.zeroes of the quadratic polynomialp(x) = 3x2 - x + 7, then find the value of a2 t p2.. Explain why the number 7 x 11 x 13 t 13 is composite. 13. For what values of a and b, the following pair of linear equations has an infinite number of solutions : Zzt3y=7 (a-b)x+(atb)y=3a+b-2 14. Show that : 2 sec2 E - sec4 E - 2 cosec2 0 t cosec4 0 = cot4 0 - tan4 0 Prove that : 1. In AABC, AD is a median and E is the mid-point ofad. If BE produced meets AC at F, 1' show that AF - -AC. 3-16. Prove that the sum of the squares on the sides of a rhombus is equal to the sum of squafes on its diagonals. 17. The following distribution gives the marks of 0 students of a particular school : - - Marks / 0-10 1 10-20 1 20-30 1 30-40 1 40-0 1 0-60 No. of students Write the above distribution as less than type cumulative frequency distribution. 18. For the following grouped freauencv distribution find the mode : Class / 3-6 1 6-'9 1 9-1 -1 ( 1-18 1 18-21 1 21-24 Frequency 2 6 8 10 1 23 21 13 6. 3 Question numbers 19 to 28 cany 3 marks each. 19. Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m t 8 for some integer m. 20. Prove that - is an irrational.
Prove that & +,6 is an irrational number. 21. The class M students of a certain public school wanted to &ve a farewell party to the outgoing students of class X. They decided to purchase two kinds of sweets, one costing? 70 per kg and the other costing 7 84 per kg. They estimated that 36 kg of sweets were needed. If total money spent on sweets was? 2800, find how many sweets of each kind they purchased. Ram tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as yon will be". Find their present ages. 22. If the sum of the zeroes of the quadratic polynomialp(x) = kx2 + 22 + 3k is equal to their product, find the value of k. 23. Prove that : (sin 0 + sec 0)2 + (cos 0 + cosec 0)' = (1 + sec 0 cosec 0 24. Prove that : sec0-tan0 =1-2sec0tan0+2tan20 sec 0 + tan 0 2. In figure, if PQ (1 BC and PR (1 CD, prove that -=- AR AQ AD AB' B 26. Triangle ABC is right-angled at B and D is the md-point of BC. Prove that. = 4AD2-3 ~ ~ 2 27. From the data eiven below, find the value of p, if the mean is 330., I - Loss per-shop 1 0-100 1 100-200 ( 200-300 1300-400 1400-00 ( 00-600 No. of shops -. Calculate the mean frbm the following data, using step-deviation method : Classes 1 20-2 1 2-30 1 30-3 1 3-40 1 40-4 1 4-0 1 0- Frequency 10 10 1 28. Find the median of the following data : Classes ( 0-10 (10-20 (20-30 (30-40 140-0 (0-60 160-70 1-70-80 Frequency 1 1 3 1 1 1 1 1 6 ( 3 0 1 141 6 1 7 1 3 30 D. 8 1 20 P 11 2 4 20
Question numbers 29 to 34 carry 4 marks each. 29. Prove that : 1 1 1 --=-- 1 cosec 8 - tan 8 sin 8 sin 8 cosec 8 + cot 0 30. Prove that : cos 8 sin 8 t = cos 8 + sin 8 1-tan8 1-cot8 sec 31" + sec2 (90" - 8) - cot2 8 2 cos2 60" tan2 31" tanz 9" Evaluate : + cosec 9" 2 (sin2 3" t sin2 ") 3 (sec2 39' - cot2 Y) 1 + - tan 10" tan 30" tan 40" tan 4" tan 0" tan 80" & 31. Obtain all the zeroes of the polynomialflz) : 3x4 + 6x3-2?c2 - lox -, if two of its zeroes 32. Represent the following system of equations graphically : x-ytl=o 42+3~=24 -" Find the points where the lines meet the x-axis. 33. The following distribution gives the daily income of 0 workers of a factory. - Dailv income (in f) 1 100-0 1 0-140 1 140-160 1 160-180 1 180-200 1 Numbers of workers 14 8 6 10 Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive. Hence, obtain the median income from the graph and verify the result by using the formula. 34. In a triangle, if square of one side is equal to the sum of the squares of other two sides, then the angle opposite the first side is a right angle. Prove. If a line is drawn parallel to one side of a triangle, to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Prove. Section 'A' Section 'B'
17. Upper Cumulative Frequency Limits Frequency (cf) Less than 10 Less than 20 6 11 Less than 30 8 19 1 1 Less than40 / 31 Less than 0 Less than 60 13 6 18. Mode = 14.6 Section 'C' 21. One kind of sweet = 16 kg and other kind of sweet = 20 kg Ram's present age = 42 years and Daughter's present age = years. 44 0 27. p= 2 Mean = 3. 28. Median = 33.33 30. 2 Section 'D' 31. The zeroes off(x) are - $, $,-land-1. 32- (- l,o), (6,O) 33. Daily income (in 7) Less than 0 Less than 140.. Less than 160 Less than 180 Less than 200 Cumulative frequency. 26 34 40 0