# DESIGN OF THE QUESTION PAPER Mathematics Class X

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1 SET-I DESIGN OF THE QUESTION PAPER Mathematics Class X Time : 3 Hours Maximum Marks : 80 Weightage and the distribution of marks over different dimensions of the question shall be as follows: (A) Weightage to Content/ Subject Units : S.No. Content Unit Marks. Number Systems 04. Algebra 0 3. Trigonometry 4. Coordinate Geometry Geometry 6 6. Mensuration 0 7. Statistics and Probability 0 Total : 80 (B) Weightage to Forms of Questions : S.No. Form of Marks for each Number of Total Marks Questions Question Questions. MCQ SAR SA LA Total 30 80

2 04 EXEMPLAR PROBLEMS (C) Scheme of Options All questions are compulsory, i.e., there is no overall choice. However, internal choices are provided in one question of marks, three questions of 3 marks each and two questions of 6 marks each. (D) Weightage to Difficulty Level of Questions S.No. Estimated Difficulty Percentage of Marks Level of Questions. Easy 0. Average Difficult 0 Note : A question may vary in difficulty level from individual to individual. As such, the assessment in respect of each will be made by the paper setter/ teacher on the basis of general anticipation from the groups as whole taking the examination. This provision is only to make the paper balanced in its weight, rather to determine the pattern of marking at any stage.

3 DESIGN OF THE QUESTION PAPER, SET-I 05 BLUE PRINT MATHEMATICS CLASS X Form of Question Units Number Systems () - - 4(3) Algebra 3(3) 9(3) 6 0(8) Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmatic Progressions Trigonometry 3 6 (4) Introduction to Trigonometry, Some Applications of Trigonometry Coordinate Geometry 4() 3-8(4) Geometry - 9(3) 6 6(5) Triangles, Circles, Constructions Mensuration (3) Areas related to Circles, Surface Areas and Volumes Statistics & Probability (3) Total 0(0) 0(5) 30(0) 30(5) 80(30)

4 06 EXEMPLAR PROBLEMS Maximum Marks : 80 General Instructions Mathematics Class X Time : 3 Hours. All questions are compulsory.. The question paper consists of 30 questions divided into four sections A, B, C, and D.Section A contains 0 questions of mark each, Section B contains 5 questions of marks each, Section C contains 0 questions of 3 marks each and Section D contains 5 questions of 6 marks each. 3. There is no overall choice. However, an internal choice has been provided in one question of marks, three questions of 3 marks and two questions of 6 marks each. 4. In questions on construction, the drawing should be neat and exactly as per given measurements. 5. Use of calculators is not allowed. SECTION A. After how many decimal places will the decimal expansion of the number 47 3 terminate? 5 (A) 5 (B) (C) 3 (D). Euclid s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a bq + r, where (A) 0 r a (B) 0 < r < b (C) 0 r b (D) 0 r < b 3. The number of zeroes, the polynomial p (x) (x ) + 4 can have, is (A) (B) (C) 0 (D) 3 4. A pair of linear equations a x + b y + c 0; a x + b y + c 0 is said to be inconsistent, if a b a b c a b c a c (A) (B) (C) (D) a b a b c a b c a c 5. The smallest value of k for which the equation x + kx has real roots, is (A) 6 (B) 6 (C) 36 (D) 3 6. The coordinates of the points P and Q are (4, 3) and (, 7). Then the abscissa of a point R on the line segment PQ such that PR 3 PQ 5 is

6 08 EXEMPLAR PROBLEMS 4. The length of a line segment is 0 units. If one end is (, 3) and the abscissa of the other end is 0, then its ordinate is either 3 or 9. Give justification for the two answers What is the maximum value of? Justify your answer. cosecθ SECTION C 6. Find the zeroes of the polynomial p (x) 4 3 x 3 x 3 and verify the relationship between the zeroes and the coefficients. On dividing the polynomial f (x) x 3 5x + 6x 4 by a polynomial g(x), the quotient q (x) and remainder r (x) are x 3, 3x + 5, respectively. Find the polynomial g (x). 7. Solve the equations 5x y 5 and 3x y 3 graphically. 8. If the sum of the first n terms of an AP is 4n n, what is the0 th term and the n th term? How many terms of the AP : 9, 7, 5,... must be taken to give a sum 636? 9. If (, ), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find the values of x and y. 0. The sides AB, BC and median AD of a ABC are respectively propotional to the sides PQ, QR and the median PM of PQR. Show that ABC ~ PQR.. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 7 cm, respectively. Find the sides AB and AC.. Construct an isosceles triangle whose base is 6 cm and altitude 5 cm and then another triangle whose sides are 7 of the corresponding sides of the isosceles 5 triangle. cos θ sinθ+ 3. Prove that sin θ+ cos θ cosec θ cot θ. Evaluate: 3cos 43 cos37 cosec53 sin 47 tan 5 tan 5 tan 45 tan 65 tan85

7 DESIGN OF THE QUESTION PAPER, SET-I In the figure, ABC is a triangle right angled at A. Semicircles are drawn on AB, AC and BC as diameters. Find the area of the shaded region. 5. A bag contains white, black and red balls only. A ball is drawn at random from the bag. The probability of getting a white ball is 3 0 and that of a black ball is 5. Find the probability of getting a red ball. If the bag contains 0 black balls, then find the total number of balls in the bag. SECTION D 6. If the price of a book is reduced by Rs 5, a person can buy 5 more books for Rs 300. Find the original list price of the book. The sum of the ages of two friends is 0 years. Four years ago, the product of their ages in years was 48. Is this situation possible? If so, determine their present ages. 7. Prove that the lengths of the tangents drawn from an external point to a circle are equal. Using the above theorem, prove that: If quadrilateral ABCD is circumscribing a circle, then AB + CD AD + BC. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides. Using the above theorem, do the following : ABC is an iscosceles triangle right angled at B. Two equilateral triangles ACD and ABE are constructed on the sides AC and AB, respectively. Find the ratio of the areas of ΑΒΕ and ACD.

8 0 EXEMPLAR PROBLEMS 8. The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30 and 60, respectively. Find the height of the tower and also the horizontal distances between the building and the tower. 9. A well of diameter 3 m and 4 m deep is dug. The earth, taken out of it, has been evenly spread all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. 30. The following table shows the ages of the patients admitted in a hospital during a month: Age (in years) : Number of patients : Find the mode and the mean of the data given above.

9 DESIGN OF THE QUESTION PAPER, SET-I MARKING SCHEME SECTION A MARKS. (C). (D) 3. (C) 4. (C) 5. (A) 6. (D) 7. (C) 8. (B) 9. (A) 0. (D) ( 0 0) SECTION B. No ( ) 4 n n Therefore, is the only prime number in its prime facorisation, so it cannot end with zero. ( ). 5 th term ( ) 0 will be added in 0 terms (since d ) Therefore, ( ) No ( ) Here, a 3 (odd), d 4 (even) Sum of (odd + even) odd but 44 is even ( ) 3. Yes ( ) Here, PQ 8, PR 6, therefore, area sq. units. ( ) 4. Let ordinate of the point be y. Then (0 ) + (y + 3) 0, i.e.,y + 3 ± 6, i.e., y 3 or 9 ( + ) 5. Maximum value 3 ( )

10 EXEMPLAR PROBLEMS Since 3 cosec 3 sinθ, sinθ, therefore, 3 sinθ 3 ( ) SECTION C 6. p(x) 4 3 x 3 x 3 3 (x x ) 3 (x + ) (x ) Therefore, two zeroes are, Here a 4 3, b 3, c 3 Therefore, α + β +, b a 3 4 3, i.e., α + β b a αβ, c 3 a 4 3, i.e., αβ c a f(x) g(x) q(x) + r(x) Therefore, x 3 5x + 6x 4 g(x) (x 3) + ( 3x + 5) Therefore, g(x) 3 x 5x + 6 x 4+ 3 x 5 x 3 3 x 5x + 9 x 4 x x y 5 3x y 3 x x + 3 x 3 x 3 y y 0 3 6

11 DESIGN OF THE QUESTION PAPER, SET-I 3 For correct graph () Solution is x, y 0 8. S n 4n n. Therefore, t 0 S 0 S 9 (40 00) (36 8) ( ) t n S n S n (4n n ) [4 (n ) (n ) ] ( ) 4n n 4n n + n 5 n a 9, d 8, S n 636 Using S n n [a + (n ) d], we have 636 n [8 + (n ) 8] ( )

12 4 EXEMPLAR PROBLEMS Solving to get n ( ) 9. Let A (, ), B (4, y) and C (x, 6) and D (3, 5) be the vertices. x + The mid-point of AC is,4 ( ) 7 y + 5 and mid-point of BD is, ABCD is a parellologram. Therefore, y + 5 ( ) x + 7,i.e., x 6 4, i.e., 3 0. Given AB BC BD AD PQ QR QM PM [SSS] ( ) Therefore, Β Q. Also, since AB BC,i.e., PQ QR ABC ~ ( ). Let AE (AF) x cm. Area ABC. 4. (AB + BC + AC) s( s a)( s b)( s c)

13 DESIGN OF THE QUESTION PAPER, SET-I 5 i.e., 4 s s( s a)( s b)( s c ) ( ) 6 s (s a) (s b) (s c) ( ) i.e., 6 (5 + x) x. 8. 7, i.e., x 6 Therefore, AB 4 cm and AC 3 cm ( ). Construction of isosceles with base 6 cm and altitude 5 cm Construction of similar with scale factor 7 5 () 3. LHS cos θ sin θ+ sin θ+ cos θ cot θ + cosec θ + cot θ cosec θ cot θ + cosec θ (cosec θ cot θ ) cosec θ+ cot θ (cosec θ cot θ) (cosec θ cot θ) cosec θ+ cot θ (cosec θ cot θ) (cosec θ+ cot θ ) cosec θ cot θ 3cos 43 cos37 cosec 53 sin 47 tan5 tan 5 tan 45 tan 65 tan85 3cos 43 cos37.sec37 cos 43 tan5 tan 5 cot 5 cot 5 () (3) Required area area of semicircle with diameter AB + area of semicircle with diameter AC + area of right triangle ABC area of semicircle with diameter BC

14 6 EXEMPLAR PROBLEMS Required area π.(3) + π (4) π (5) sq. units 4 + π ( ) 4 sq. units 5. P(Red ball) {P(White ball) + P(Black ball )} ( ) Let the total number of balls be y. Therefore, 0,i.e., y y 5 50 ( ) SECTION D 6. Let the list price of a book be Rs x Therefore, number of books, for Rs x ( ) No. of books, when price is (x 5) 300 x 5 ( ) Therefore, 300 x x 5 () 300 (x x + 5) 5x (x 5) 300 x (x 5), i.e., x 5x i.e., 0, x 5 (rejected) Therefore, list price of a book Rs 0 Let the present age of one of them be x years, so the age of the other (0 x) years Therefore, 4 years ago, their ages were, x 4, 6 x years Therefore, (x 4) (6 x) 48 ( )

15 DESIGN OF THE QUESTION PAPER, SET-I 7 i.e., x + 6 x + 4x x 0x + x 0 Here B 4 AC (0) 4() 48 ( ) Thus, the equation has no real solution Hence, the given situation is not possible 7. For correct, given, to prove, contruction and figure () For correct proof () AP AS BP BQ DR DS CR CQ (tangents to a circle from external point are equal) Adding to get (AP + BP) + (DR + CR) (AS + DS) + (BQ + CQ) i.e., For correct,given, to prove, construction and figure () For correct proof () Let AB BC a, i.e., AC a + a a ( ) area ABC area ACD AB a AC a ( ) 8. For correct figure In ABD, AB BD tan 60 3 Therefore, AB 3 BD (I) In ACE, AE AE tan 30 EC BD i.e., 3 (AB 50), i.e., 3 (AB 50) BD BD 3

16 8 EXEMPLAR PROBLEMS Therefore, from (I) AB 3. 3 (AB 50),i.e., AB 3AB 50,i.e., AB 75 m BD 3 (75 50) 5 3 m 9. Volume of earth dug out πr h π (.5) π 3 () Area of circular ring π[r r ] π[(5.5) (.5) ] π(7) (4) 8π Let π 3.5 π Age (in years) Total No. of patients(f ) Class marks(x i ) ( ) f i x i Mean fx i i f i years Modal class is (35 45) ( ) Therefore, Mode l + f f0 f f f 0 h Putting l 35, f 3, f 0, f 4 and h 0, we get Mode years Note: Full credit should be given for alternative correct solution.

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