BOARD OF SECONDARY EDUCATION (AP) SUMMATIVE ASSESSMENT - II TENTH CLASS MATHEMATICS MODEL PAPER

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1 BOARD OF SECONDARY EDUCATION (AP) SUMMATIVE ASSESSMENT - II TENTH CLASS MATHEMATICS MODEL PAPER PAPER I (ENGLISH VERSION) Time: hrs. 45 mins. PART - A & B Maximum Marks: 40 INSTRUCTIONS: i) In the time duration of hrs. 45 mins. 5 minutes of time is allotted to read and understand the question paper. ii) iii) Answer the questions under PART A on separate answer book. Write the answers to the questions under PART - B on the question paper itself and attach it to the answer book of PART A. Time: hrs. PART A Marks: 0 INSTRUCTIONS: i) PART A comprises of three Sections I, II, III. ii) All the questions are compulsory. iii) There is no overall choice. However, there is an Internal Choice to the questions under Section III. INSTRUCTIONS: i) Answer ALL the questions. SECTION I ii) Each question carries ONE Mark. 4 = 4. Find the value of log 8 + log 8.. Write an equation of line geometrically intersect to the line 5x + 6y + = 0.. Find the 6th term in G.P.,,,...? Find the total surface area of sphere with diameter 4 cm. SECTION II INSTRUCTIONS: i) Answer ALL the questions. ii) Each question carries TWO marks. 5 = 0 5. Find the quadratic polynomial whose zeros are 5 and respectively How many three digit numbers are divisible by Solve x y = 40 and 4x y = Find the roots of (x ) 4(x ) + = 0. MP-AP

2 9. Find the volume of largest right circular cone that can be out of a cube whose edge is 7 cm. SECTION III INSTRUCTIONS: i) Answer ALL the questions. ii) Each question carries FOUR marks. iii) Each question has Internal Choice. 4 4 = 6 0. a) Change into the form of logn and find the value of N. i) log + log 5 ii) log 64 log iii) log 5 iv) log 5 + log log 45 b) Write the following into set form and find A B, A B, A B; where A= { x/x is a two digit number whose sum of the digits is 9 } B = {x/x is a two digit number which is a multiple of 6 }. a) On dividing x x + 5x by g(x), the quotient and remainders are (x ) and (7x 9) respectively then find g(x). b) A hemispherical bowl of internal radius 5 cm. contains a liquid is to be filled into cylinderical bottles of diameter 5 cm. and height 6 cm. How many bottles are need to empty the bowl?. a) The sum of the 5 th and 0 th terms of an A.P. is 75 and 8 th and 0th terms is 5. Find first four terms of the A.P. b) A motor boat whose speed is 8 km/hr in still water if takes hour more to go 4 km upstream than to return downstream to the same spot. Find the speed of the stream?. a) Draw the graph for x + 4x 5 and show that the X coordinates of point of intersection of X axis is zeros of that polynomial. b) Solve the pair of equations graphically x + y 6 = 0, 4x y 4 = 0.

3 Time: 0 Minutes PART B Maximum Marks: 0 INSTRUCTIONS: i) Answer ALL the questions. ii) Each question carries mark. iii) Answers are to be written in question paper only. iv) Marks will not be awarded in any case of over writing and re writing or erased answers. v) Write the CAPITAL LETTER (A, B, C, D) showing the correct answer for the following questions in the brackets provided against them. 0 = 0 SECTION IV 4. Which of the following is non terminating repeating decimal ( ) A) B) C) D) H.C.F of and 8 is ( ) A) B) 6 C) 6 D) Decimal form of 5 A) 0.75 B).75 C) D) Which of the following diagram represents A B when A B ( ) I A B II A B A A) I B) II C) III D) IV 8. A is the set of factors of which does not belong to A ( ) A) B) 4 C) 5 D) 9. A = {5, 7, 8}; B = {8, 6, 4} then which of the following set represents {6, 4} ( ) A) A B B) A B C) A B D) B A 0. The graph of the polynomial ax + bx + c (a = 0) represents ( ) A) Parabola B) Straight line C) Circle D) Ellipse ( ). p(x) = g(x).q(x) + r(x); g(x) is a linear polynomial then the degree of r(x) is ( ) A) 0 B) C) D). If α, β are the zeros of a quadratic polynomial p(x) and α = β, then the number of intercepts on X axis made by the graph p(x) are ( ) A) B) C) D) 0 III B IV A& B

4 . The pair of linear equations x + y + k = 0, 6x + 9y + = 0 has an infinite solutions then k =... ( ) A) B) C) 0 D) 4. The condition, if ax + by + c = 0 represents a linear equation in variables x and y is ( ) A) a + b 0 B) a + b 0 C) a + b = 0 D) A, B 5. Sum of the roots of x + 7x + 5 = 0 is ( ) 7 7 A) B) C) 5 D) 7 6. The roots are equal for bx + cx + a = 0 then a =... ( ) b c c b A) B) C) D) 4a 4a 4b 4c 7. The quadratic equation having roots are ( + ), ( ) is ( ) A) x x + 4 = 0 B) x 4x + = 0 C) x + 4x + = 0 D) x + x = 0 8. In an A.P. S n = 4n n, then t 0 =... ( ) A) 400 B) 70 C) 97 D) 7 9. Which term in the G.P.,,,... is 79 ( ) A) B) 6 C) 8 D) 9 0. Sum of 'n' terms in A.P. is ( ) n A) S n = [a + (n )d] B) S n = a + (n )d n C) S n = [a + l] D) A, C. A heap of rice is in the form of a cone, its diameter m. and height 7 m, its volume is... cu.m ( ) A) 64 B) 54 C) 6 D) 5 7. The volume of a hemisphere of radius cm. is... cu.cm A) B) C) D) A cylinder, a cone and a hemisphere are of equal base and have the same height then the ratio of their volumes is... ( ) A) : : B) : : C) : : D) : :

5 . Find the value of log 8 + log 8. A: Given log 8 + log 8 ANSWERS PART A SECTION I = log 8 8 (... loga + logb = log ab) = log 44 = log = log = =. Write an equation of line geometrically intersect to the line 5x + 6y + = 0. a b A: For the linear equations a x + b y + c = 0 and a x + b y + c = 0 if then the lines are a b geometrically intersect each other. We can write any equation of the line geometrically intersect to the line 5x + 6y + = 0 by satisfying above condition. for example, 6x 7y + 5 = 0 is an solution.. Find the 6 th term in G.P.,,,...? 9 7 A: In the given G.P. a, a =a.r = a common ratio r = = = a 6th term a r 5 = ( ) 5 = 4 4. Find the total surface area of sphere with diameter 4 cm. A: Given diameter of the sphere d = 4 c.m. Total surface area of the sphere = 4Πr 4 4 =4 7 = 66 sq.cm. SECTION II 5. Find the quadratic polynomial whose zeros are 5 and respectively. 5 A: Given that zeros of the polynomial are α = 5, β = 5 The polynomial whose zeros α, β are = x x(α + β) + αβ = x x( ( )) ( ) =x x 5 (or) 5x 4x 5

6 6. How many three digit numbers are divisible by 7. A: Three digit numbers are 00, 0,..., 999 we know that 05 is the first digit number multiple of 7 and 994 is the last digit number multiple of 7. Required progression is 05,,..., 994 This is in A.P. ( = 9 = 7) Here a = 05, a =, a n = 994 d = 05=7 a n = a + (n )d 994 = 05 + (n )7 (n )7 = = n = = 7 7 n = 7 + = 8 Number of three digit numbers are divisible by 7 are 8 7. Solve x y = 40 and 4x y = 50. A: Given equations are x y = 40...() 4x y = 50...() 4x y=50 6x =80 x = 0 0 x= = 5 Substitute x = 5 in eq (), we get 5 y = = y 5 = y x = 5, y = 5 8. Find the roots of (x ) 4(x ) + = 0. A: Given equation is (x ) 4(x ) + = 0 Let x = a then the equation is a 4a + =0 a a a + =0 a(a ) (a ) = 0 (a )(a )=0

7 a =0 or a =0 a = or a = substitute a = x x = or x = x= + or x = + 5 x = or x = = 9. Find the volume of largest right circular cone that can be out of a cube whose edge is 7 cm. A: The edge of the cube = 7 cm. if we cut a largest right circular cone from the cube then the diameter of the cone d = 7 cm. 7 Radius of the cone r = cm. Also height of the cone h = 7 cm. volume of the cone = Πr h 7 7 = = 6 = 4.7 cm SECTION III 0. a) Change into the form of logn and find the value of N. i) log + log 5 ii) log 64 log iii) log 5 iv) log 5 + log log 45 A: Given that i) log + log 5 = log + log 5 N = 45 = log 9 5 = log 45 this is in the form of log N ii) log 64 log = log = log = log 8 This is in the form of logn and N = 8 iii) log 5 = log (5) = log (8 ) = log 8

8 This is in the form of logn and N = 8 iv) log 5 + log log 45 = log 5 + log log 45 = log 5 + log 9 log = log 45 = log 5 This is in the form of logn, Here N = 5 b) Write the following into set form and find A B, A B, A B; where A= { x/x is a two digit number whose sum of the digits is 9 } B = { x/x is a two digit number which is a multiple of 6 } A: A= {8, 7, 6, 45, 54, 6, 7, 8, 90} B = {, 8, 4, 0, 6, 4, 48, 54, 60, 66, 7, 78, 84, 90, 96} A B = {, 8, 4, 7, 0, 6, 4, 45, 48, 54, 60, 6, 66, 7, 78, 8, 84, 90, 96} A B = {8, 6, 54, 7, 90} A B = {7, 45, 6, 8}. a) On dividing x x + 5x by g(x), the quotient and remainders are (x ) and (7x 9) respectively then find g(x). A: Let p(x) = x x + 5x Quotient q(x) = x Remainder r(x) = 7x 9 By the division algorithm, p(x) = g(x)q(x) + r(x) x x + 5x = g(x)(x ) + (7x 9) x x + 5x 7x + 9 = g(x)(x ) g(x)(x ) = x x x + 6 x x x + 6 g(x) = x x ) x x x+ 6 (x x x 0 x + 6 x g(x) = x

9 b) A hemispherical bowl of internal radius 5 cm. contains a liquid is to be filled into cylinderical bottles of diameter 5 cm. and height 6 cm. How many bottles are need to empty the bowl? A: Internal radius of hemispherical bowl (r) = 5 cm volume of the liquide in the bowl = Πr = Π 5 cu.cm Diameter of the cylindrical bottle (d) = 5 cm d 5 Radius of the cylindrical bottle (r) = = cm Height of the cylindrical bottle (h) = 6 cm 5 5 volume of a bottle = Π 6 cu.cm. volume of liquid in the bowl No.of bottles required to fill the liquid = volume of a bottle Π = 5 5 Π = = 5 = 80. a) The sum of the 5 th and 0 th terms of an A.P. is 75 and 8 th and 0 th terms is 5. Find first four terms of the A.P. A: Let the first term in the A.P. is 'a' and common difference is 'd' 5 th term t 5 = a + 4d 0 th term t 0 =a + 9d 8 th term t 8 = a + 7d t 5 + t 0 = (a + 4d) + (a + 9d) = a + d By the sum, t 5 + t 0 =75 a + d = 75...() t 8 + t 0 = (a + 7d) + (a + 9d) = a + 6d By the sum, t 8 + t 0 = 5 a + 6d = 5...() () () We get a + 6d = 5 a + d = 75 d=60

10 60 d = = 0 From (), a + 0 = 75 a = a = 85 st term a = nd term a + d = + 0 = rd term a + d = + 0 = th term a + d = + 0 = b) A motor boat whose speed is 8 km/hr in still water if takes hour more to go 4 km upstream than to return downstream to the same spot. Find the speed of the stream? A: Let the speed of the stream be x km/hr The speed of the boat upstream = (8 x) km/hr and the speed of the boat downstream = (8 + x) km/hr. distance 4 The time taken to go upstream = = hr. speed (8 x) 4 similarly, the time taken to go downstream = hr. (8 + x) 4 4 By the sum, = 8 x 8 + x 4(8 + x) 4(8 x) = (8 x)(8 + x) x x = 8 8 x x x + 48x 4 = 0 By the formula 48 ± 48 4 ( 4) x = 48 ± 600 = 48 ± 60 = = or

11 x = 6 (or) 54 Speed cannot be negative x = 6 speed of the stream = 6 km/hr. a) Draw the graph for x + 4x 5 and show that the x coordinates of point of intersection of X axis is zeros of that polynomial. A: Given polynomial is x + 4x 5 let y = x + 4x 5 Table for y = x + 4x 5is x x x y From the graph, solution set is {, 5}

12 b) Solve the pair of equations graphically x + y 6 = 0 and 4x y 4 = 0. A: Given equations are x + y 6 = 0 and 4x y 4 = 0 Table for x + y 6 = 0 (or) y = 6 x is x 0 y 6 0 4x 4 Table for 4x y 4 = 0 (or) y = is x 0 y 0 From the graph the solution set is {, } PART B ANSWERS 4 D; 5 C; 6 A; 7 B; 8 C; 9 D; 0 B; A; C; D; 4 D; 5 B; 6 C; 7 B; 8 D; 9 A; 0 D; A; D; C. Writer: S.V.S. Suryanarayana Murthy

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, a 1. , a 2. ,..., a n CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.