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# CBSE Proficiency Test in Mathematics Sample Paper

## Proficiency Test Sample Papers

Section I: Multiple Choice questions This section contains 11 questions. For questions 1 to 11 only one of the four options is correct. You have to indicate your answer by filling the appropriate bubble in the Answersheet. A correct answer will earn 3 marks, a wrong answer will earn (—1) mark, and an unattempted question will earn 0 mark. 1. Three villages A, B and C form a scalene triangle on flat land (see figure below). A well needs to be constructed on the same flat land in such a way that it is equidistant from the three villages. A BC The well should be built at (A) the incentre of AABC. (B) the centroid of AABC. (C) the circumcentre of AABC. (D) the orthocentre of AABC. 2. In a game, a number is chosen at random from the set {1, 2,3,…, 28,29,30}. What is the probability that the number chosen is a product of exactly two different prime numbers? (A)^ v ‘ 30 (B) 6 (OT5 (D) 5 3. Two vertical poles P1 and P2 stand 30 metres apart on the ground (see figure below). M is a point on pole P2 such that the two ends of pole P1 subtend a right angle at the point M and the angle of elevation of the top of pole P1 from the point M is 60°. P1 P2 ‘J’M 30 m The height of the pole P1 , in metres, is (A) 20^3 (B) 40^3 (C) 60^3 (D) 120^3 4. In the coordinate plane, AABC has vertices A(0,0), B(8, —2) and C(8,10). Point D lies on BC, such that BD : DC = 5 : 7. What is the area of AADC? (A) 28 (B) 32 (C) 36 (D) 40 5 5. P, Q, R and S are four points on a circle with centre O(see figure below). PQ and SR intersect at the point T outside the circle. T If ZPOS = 100° and ZROQ = 70° then ZPTS is equal to (A) 20° (B) 25° (C) 15° (D) 18° 6. A cylindrical tennis ball container contains three balls stacked on one another, such that they touch the wall of the container (see figure below). The top and bottom balls also touch the lid and the base of the container respectively. 7. The sum of all the numbers between 1 and 1000, which are divisible by 5 but not by 2 is (A) 101100 (B) 50000 (C) 50050 (D) 10100 8. In AABC, ZA = 25°, ZB = 35°, and AB = 16 units. In APQR, ZP = 35°, ZQ = 120°, and PR = 4 units. Which of the following is true? (A) Area(AABC) = 2 Area(APQR) (B) Area(AABC) = 4 Area(APQR) (C) Area(AABC) = 8 Area(APQR) (D) Area(AABC) = 16 Area(APQR) 9. A quadratic polynomial, f (x), is such that: f (x) > 0, for — 3 < x < 2 < 0, otherwise Which of the following can be the polynomial f (x)? (A) —x2 — x — 6 (B) —x2 + x + 6 (C) —x2 + x — 6 (D) —x2 — x + 6 If the volume of a tennis ball is 160 cm3, then what is the volume of the container? (A) 720 cm3 (B) 840 cm3 (C) 1440 cm3 (D) 480 cm3 6 10. An arithmetic progression is such that the sum of the first 8 numbers is —100 and the common difference is 1. For which n would the sum of the first n numbers be —100 again? (A) 24 (B) 25 (C) 30 (D) There is no such n = 8 11. The sum to 100 terms of (1 — 2 + 3 — 4 + 5 ) is: (A) —50 (B) —500 (C) —100 (D) —1000 Section II: Numerical questions This section contains 4 questions. For questions 12 to 15 the answer is an integer between 0 and 99. You have to indicate the answer by filling bubbles in the appropriate grid provided in the Answersheet. Each question carries 5 marks. 12. A rectangular plot of land is 100 m by 60 m (see figure below). It has a grass-bed of equal width all around 3 it on the boundary (shaded area in the figure). The area of the grass-bed is 5th of the area of the plot. What is the width of the grass-bed in metres? 100 m 60 m 13. A shoe shop keeps a record of the number of pairs of shoes sold daily. The record of actual sales for a week (Monday to Sunday) was lost. The manager could only get the following table which showed how much the sale had increased or decreased over the previous day’s sale: Mon Tue Wed Thu Fri Sat Sun + 1 + 1 +2 — 11 +7 —7 0 If a is the mean of daily sale for the week and b is the mode of daily sale for the week, then what is the value of a b? 14. The annual salary of Mr. Nair for the year 2010 is Rs. 40,000. Every year his salary increases by Rs. 4,000. At the beginning of the year 2010 he had borrowed a sum of Rs. 44,000 from a bank. Given the following information determine the number of years he will take to repay the principal and the interest to the bank. (a) The interest due for payment is Rs. 3,600 at the end of the year 2010 and increases by Rs. 800 every year thereafter and (b) each year he is going to pay the bank 20% of his annual income towards loan repayment. 15. From a point A on a straight road the angle of elevation of the top of a vertical tower situated on the roof of a vertical building on the same road is Q. The angle of elevation of the bottom of the tower from a point B on the road is again Q. The height of the building is 50 metres. If AB : BY is 2 : 5, where Y is the base of the building, what is the height of the tower? 7 a1 a2 b1 b2 a1 a2 Section III: Column-matching questions This section contains 2 questions. For questions 16 to 17 you have to match the options in Column II for each item in Column I. You have to indicate the matches by filling bubbles in the appropriate grid provided in the Answersheet. If all correct options are matched, and no incorrect option is matched, each item in Column I earns 2 marks. 16. Column I lists certain geometric shapes and Column II describes certain properties of geometric shapes. For each item in Column I, choose ALL the correct options in Column II. Column I (i) Parallelogram (ii) Rhombus (iii) Rectangle Column II (A) Diagonals bisect each other. (B) Diagonals are perpendicular to each other. (C) The figure includes at least one pair of equal sides. (D) A circle drawn through any three vertices always passes through the fourth vertex. (E) the pair of lines obtained by joining the mid- points of the opposite sides are always per- pendicular to each other. 17. Consider the equation 3(a1x + b1y — c1 )2 + 2(a2x + b2y — c2)2 = 0 in real variables x and y, where a1, a2, b1, b2, c1, c2 are non-zero real numbers. For each item in Column I, choose ALL the correct options in Column II. Column I (i) Unique solution (ii) No solution (iii) Infinitely many solutions Column II (A) * = b1. a2 b2 (B) ^ = f1 = – d c2′ (C) * = b1 = * a2 b2 c2 (D) – = f1 b1 bi’ (E) a22b21 + a21 b22 > 2a1 a2b1 b2

### Solutions

1. C 2. A 3. B 4. A 5. C 6. A 7. B 8. D 9. D 10. B 11. A 12. 10 13. 5 14. 10 15. 20 16. i. AC ii. ABC iii. ACDE 17. i. AE ii. C iii. B