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Sample Paper (Based on CBSE CCE SA - 1) (X) MATHEMATICS [Time allowed: 3 hours] [Maximum marks: 80] General Instructions: 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into 4 sections, section A, B, C, and D. 3. Section A contains 10 multiple choice type questions each carry 1 mark. Section B contains 8 ques- tions of 2 marks each, section C contains 10 questions of 3 marks each and section D contains 6 questions of 4 marks each. 4. There is no overall choice. However, internal choice has been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of four marks each. Attempt only one of the alterna- tives in all such questions. 5. Use of calculators is not permitted. SECTION - A 1. The sum of exponents of prime factors in the prime factorisation of 196 is (a) 2 (b) 7 P 120o (c) 4 (d) 5 2. In fig. 1, ∆PQR ~ ∆PST, then ∠S is Q 70 o R (a) 50o (b) 60o T S (c) 70o (d) 120o Fig. 1 3. The equation of a line which is parallel to x-axis and at a distance of 5 units below it is (a) x = –5 (b) x = 5 (c) y = –5 (d) y = 5 tan 20o tan θ p 4. If +2 = , then the value of p is cot 70 o cot ( 90 – θ ) 2 (a) 6 (b) 2 (c) –6 (d) 1 5. The value of k for which x = –1 is a zero of the polynomial x2 – 2k + 3 is (a) –2 (b) 0 (c) 2 (d) –1 1 6. If A and B are acute angles, such that sin A = , tan B = 3 , then cos (A + B) is 2 (a) 1 (b) 0 1 3 (c) (d) 2 2 AVTE/X/M/10-11/Sept./SA-1 1 7. For a data consisting of 50 observations distributed into class intervals of width 10, the assumed mean is 25 and ∑fu i i is 11, then mean is (a) 30 (b) 27 (c) 34 (d) 27.2 8. If tan A = cot B, where A and B are acute angles, then A + B is (a) 0o (b) 90o (c) 30o (d) 45o 9. The polynomial when divided by –x2 + x – 1 gives quotient x - 2 and remainder 3 is (a) x3 – 3x2 + 3x – 5 (b) x3 – 3x2 + 8x (c) –x3 + 3x2 + x + 5 (d) –x3 + 3x2 – 3x + 5 10. The modal class for the following distribution is Class 10-20 20-30 30-40 40-50 50-60 Frequency 15 26 18 27 14 (a) 20 – 30 (b) 40 – 50 (c) 30 – 40 (d) 50 – 60 SECTION - B x y 11. Find the condition for which the system of equations + = c and bx + ay = 4ab is inconsistent. a b 12. Find the zeroes of the given quadratic polynomial x2 – 2x – 8 and hence verify the relationship of zeroes with the coefficients. 13. Prove that sec4 θ – sec2 θ = tan2 θ + tan4 θ. A 14. Check whether 7n can end with the digit 0 for any natural number n. 6 cm 6 cm 15. In fig. 2, ∆ABC is an isosceles triangle with AB = AC = 6 cm, If BC = 8 cm, find sin2 B + sin2 C. B C Fig. 2 16. Find p, if mean of the following distribution is 20. x 15 17 19 20 + p 23 f 2 3 4 5p 6 17. Show that sum and product of two irrational numbers 7 + 5 and 7 − 5 is rational. A C AE DE 18. In fig. 3, AC is parallel to BD. Is = ? Justify your answer. CE BE E B D OR Fig. 3 a x In the fig. 4, express x in terms of a, b, c. 30° 30° b c Fig. 4 AVTE/X/M/10-11/Sept./SA-1 2 SECTION - C 19. ABCD is a trapezium in which AB || DC and its diagonals intersect at O. If AO = (3x – 1) cm, OC = (5x – 3) cm, BO = (2x + 1) cm and OD = (6x – 5) cm, find the value of x. x y x y 20. Solve for x and y: + = a + b; 2 + 2 = 2. a b a b 21. Find all the zeroes of the polynomial 2x4 – 10x3 + 5x2 + 15x – 12, if it given that two of its zeroes 3 3 are and − . 2 2 22. Show that any positive even integer is of the form 8p, 8p + 2, 8p + 4 and 8p + 6 where p is some integer. OR In a seminar the number of participants in Hindi, English and Mathematics are 60, 84, 108 respectively. Find the minimum number of rooms required, if in each room the same number of participants are to be seated and all of them being in the same subject. cos 2 20o + cos 2 70o 23. Find the value of : tan 7o tan23o tan60o tan83o tan67o + . sin 2 59o + sin 2 31o OR 1 + cos A sin A Prove that: + = 2 cos ecA. sin A 1 + cos A 24. The following table shows the age distribution of cases of malaria admitted during an year in a particular hospital. Age(in year) 5-14 15-24 25-34 35-44 45-54 55-64 No. of Cases 6 11 21 23 14 5 Find the modal age. 25. P and Q are points on the sides CA and CB of ∆ABC right angled at C. Prove that AQ2 + BP2 = AB2 + PQ2. OR In ∆ABC, AD is a median, X is a point on AD such that AX : XD = 2 : 3. Ray BX intersects AC in Y. Prove that CY = 3 AY. sec θ − 1 sec θ + 1 26. Prove that + = 2 cosec θ . sec θ + 1 sec θ − 1 27. In ∆MNR, ∠N = 90o, MN = 8 cm, RN – MN = 7 cm, then find the value of sin R, tan R and sec M. AVTE/X/M/10-11/Sept./SA-1 3 28. For the following frequency distribution, find the average daily income of an employee in a factory: Daily income (in Rs.) No. of employees More than 300 0 More than 250 12 More than 200 21 More than 150 41 More than 100 53 More than 50 59 More than 0 60 SECTION - D 29. A person starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was Rs. 6000 after four years of his service and Rs. 7500 after ten years of service, find his initial salary when he started the job and the annual increment. 30. If median of the distribution given below is 28.5. Find x and y, if total frequency is 56. Class interval 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 5 x 20 15 y 5 B+C A B+C A 31. In ∆ABC, show that sin cos + cos sin = 1 . 2 2 2 2 OR P If tanθ + sinθ = m and tanθ - sinθ = n, show that, m - n = ± 4 mn . 2 2 32. State and prove Basic Proportionality Theorem. Use the above A B PA PB Q R theorem and solve: In fig. 5 , AB || QR, If = 3 , find . AQ BR Fig. 5 33. Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Use the above theorem and prove, In an equilateral triangle ABC, AD is the altitude drawn from A on BC. Prove that 3AB2 = 4AD2. 34. A boat travels for 7 hours. If it travels 4 hrs downstream and 3 hrs upstream, then it travels a distance of 116 km. But if it travels 3 hrs downstream and 4 hrs upstream, it covers the distance of 108 km. Find the speed of the boat in still water. OR Draw the graph of x + 3y = 6 and 2x - 3y = 12. Also, find the area of triangle formed by x = 0, y = 0 and 2x - 3y = 12. For answers log on www.avteschool.org or send a mail to avte37@yahoo.co.in AVTE/X/M/10-11/Sept./SA-1 4