C.B.S.E Class X

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SOLVE PPER with SE Marking Scheme..S.E. 08 lass X elhi & Outside elhi Set Mathematics Time : Hours Ma. Marks : 80 General Instructions : (i) ll questions in both the sections are compulsory.

1 SOLVE PPER with SE Marking Scheme..S.E. 08 lass X elhi & Outside elhi Set Mathematics Time : Hours Ma. Marks : 80 General Instructions : (i) ll questions in both the sections are compulsory. (ii) This question paper consists of 0 question divided into four sections,, and. (iii) Section contains 6 questions mark each. Section contains 6 question of marks each. Section contains 0 questions of marks each. Section contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four question of marks each and questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculator is not permitted. elhi Set ode No. 0/ SETION- Question numbers to 6 carry mark each.. if = is one root of the quadratic equation k 6 = 0, then find the value of k.. What is the HF of smallest prime number and the smallest composite number?. Find the distance of a point P(, y) from the origin. 4. In an P, if the common difference (d) = 4, and the seventh term (a 7 ) is 4, then find the first term. 5. What is the value of (cos 67 sin )? 6. Given ~ PQR, if PQ =, then find ar. ar PQR Question numbers 7 to carry marks each. ( ) 7. Given that is irrational, prove that 5+ SETION- 8. In Fig., is a rectangle. Find the values of and y. + y is an irrational number. 4 cm y 0 cm (Fig. ) 9. Find the sum of first 8 multiples of. 0. Find the ratio in which P(4, m) divides the line segment joining the points (, ) and (6, ). Hence find m.. Two different dice are tossed together. Find the probability : (i) of getting a doublet (ii) of getting a sum 0, of the numbers on the two dice.

2 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x. n integer is chosen at random between and 00. Find the probability that it is : (i) divisible by 8. (ii) not divisible by 8. SETION- Question numbers to carry marks each.. Find HF and LM of 404 and 96 and verify that HF LM = Product of the two given numbers. 4. Find all zeroes of the polynomial ( ) if two of its zeroes are + ( ) and ( ). 5. If (, ), (a, 0), (4, b) and (, ) are the vertices of a parallelogram, find the values of a and b. Hence find the lengths of its sides. If ( 5, 7), ( 4, 5), (, 6) and (4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral. 6. plane left 0 minutes late than its scheduled time and in order to reach of destination 500 km away in time, it had to increase its speed by 00 km/h from the usual speed. Find its usual speed. 7. Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal. If the area of two similar triangles are equal, prove that they are congruent. 8. Prove that the lengths of tangents drawn from an eternal point to a circle are equal. 4sinθ cosθ+ 9. If 4 tan q =, evaluate 4sinθ+ cosθ If tan = cot( 8 ), where is an acute angle, find the value of. 0. Find the area of the shaded region in Fig., where arcs drawn with centres,, and intersect in pairs at midpoints P, Q, R and S of the sides,, and respectively of a square of side cm. [Use p =.4] P S Q R (Fig. ). wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig.. If the height of the cylinder is 0 cm and its base is of radius.5 cm. Find the total surface area of the article. (Fig. ) heap of rice is in the form of a cone of base diameter 4 m and height.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?

3 Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x [. The table below show the salaries of 80 persons : Salary (In thousand `) No. of Persons alculate the median salary of the data. SETION- Question numbers to 0 carry 4 marks each.. motor boat whose speed is 8 km/hr in still water takes hr more to go 4 km upstream than to return downstream to the same spot. Find the speed of the stream. train travels at a certain average speed for a distance of 6 km and then travels at a distance of 7 km at an average speed of 6 km/hr more than its original speed. If it takes hours to complete the total journey, what is the original average speed? 4. The sum of four consecutive numbers in an P is and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 5. Find the numbers. 5. In an equilateral, is a point on side such that =. Prove that 9() = 7(). Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 6. raw a triangle with = 6 cm, = 5 cm and Ð = 60. Then construct a triangle whose sides are 4 of the corresponding sides of the. 7. Prove that : sin sin = tan. cos cos 8. The diameters of the lower and upper ends of a bucket in the from of a frustum of a cone are 0 cm and 0 cm respectively. If its height is 4 cm, find : (i) The area of the metal sheet used to make the bucket. (ii) Why we should avoid the bucket made by ordinary plastic? [Use p =.4] 9. s observed from the top of a 00 m high light house from the sea-level, the angles of depression of two ships are 0 and 45. If one ship is eactly behind the other on the same side of the light house, find the distance between the two ships. [Use = 7] 0. The mean of the following distribution is 8. Find the frequency f of the class 9. lass Frequency 6 9 f 5 4 The following distribution gives the daily income of 50 workers of a factory : aily Income (in `) Number of workers onvert the distribution above to a less than type cumulative frequency distribution and draw its ogive.

4 4 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x SOLUTIONS elhi Set ode No. 0/ SETION-. = is one root of the equation \ 9 6k 6 = 0 k =. The required numbers are and 4. HF of and 4 is.. OP= + y etailed nswer : Origin (0, 0) and point P(, y) + [SE Marking Scheme, 08] ( ) + ( ) \ d = y y d = 0 y 0 ( ) + ( ) d = + y \ istance between point P and origin = + y unit 4. a + 6 ( 4) = 4 a = 8 [SE Marking Scheme, 08] etailed nswer : Given d = 4 and a 7 = 4 We know that n th term of.p. is a n = a + (n )d a 7 = a + (7 )d [n = 7] 4 = a + 6( 4) \ a = = 8 \ First term of an.p. = 8 5. cos 67 = sin \ cos 67 sin = 0 [SE Marking Scheme, 08] etailed nswer : cos 67 sin = cos 67 {sin } = cos 67 {sin(90 67) } = cos 67 {sin(90 67) } = cos 67 cos 67 = 0 [sin(90 q) = cos q] \ cos 67 sin = 0 6. ar == ar PQR PQ etailed nswer : ~ PQ R(Given) and \ = = 9 [SE Marking Scheme, 08] \ PQ = = = QR PR ar = ar PQR PQ ar = ar PQR = 9 ar = ar PQR 9 SETION- 7. Let us assume ( 5+ ) is a rational number \ 5+ = p q (where q ¹ 0 and p and q are integers) = p 5 q q is a rational number as RHS is rational. This contradicts the given fact that is irrational. ( 5+ ) Hence is an irrational number. 8. = and = + y = 0 and y = 4 Solving to get = and y= 8. + [SE Marking Scheme, 08] etailed nswer: is a rectangle + y 4 cm y 0 cm

5 Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x [ 5 \ = and = = + y = 0...(i) = y = 4...(ii) dding equation (i) and (iii) = 44 = 44 = Substitute in equation (i) + y = 0 y = 0 = 8 \ = cm and y = 8 cm + 9. S = = ( ) = 8 9 = 08 [SE Marking Scheme, 08] etailed nswer: First 8 multiples of, 6, 9,, 5, 8,, 4 S = These numbers are in.p. where a =, d = and n = 8 [ ] S n = n a n d + ( ) [ ] S 8 = 8 + ( 8 ) S 8 = 4[6 + ] S 8 = 4 7 = 08 \ Thus, sum of first 8 multiples of of Let P : P =k : \ 6k + = 4 k + Þ k =, ratio is : Hence m = + = 0 (, ) k P (4,m) etailed nswer: P Let P = λ (4, y m) (, ) P ( y ), (, 6 ) [SE Marking Scheme, 08] P = m m 4 = (, 6 ) ( y ), + m + m λ( 6) + ( ) λ + 4l + 4 = 6l + 6l 4l = 4 l = l = \ Ratio in which point P divides the line segment = : P y = my + my m + m m = ( ) + ( ) + m = + = 0 \ m = 0. Total number of possible outcomes = 6 (i) oublets are (, ), (, ), (, ), (4, 4), (5, 5), (6, 6 ) Total number of doublets = 6 \ Probability (getting a doublet) = 6 6 or 6 (ii) Favourable outcomes are (4, 6), (5, 5), (6, 4) i.e., \ Probability (getting a sum 0) =. Total number of outcomes = 98 or 6 (i) Favourable outcomes are 8,6,4,..., 96, i.e., \ Probability (integer is divisible by 8) = 6 98 or 49 6 (ii) Probability (integer is not divisible by 8) = 49 = 4 49 SETION = 0 = 0 96 = = 5 \ HF of 404 and 96 = = 4 LM of 404 and 96 =0 5 = 9696 HF LM = = 8784 lso = 8,784 \ Hence HF LM = Product of 404 and p() = and are zeroes of p() \ p() = ( ) ( + ) g ( ) = ( 4 + ) g () ( ) ( 4 + ) = \ g() = =( + )( )

6 6 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x Therefore other zeroes are = and = (, ) ( 4, b) \ Therefore all zeroes are +,, and 5. is a parallelogram \ iagonals and bisect each other Therefore Mid point of is same as mid point of (, ) ( 4, b) o (, ) ( a, 0) Þ Þ a+, = + + 4, b a + = and b + = Þ a=, b=. Therefore length of sides are 0 units each. + rea of quad = ar + ar rea of = ( 5) ( 5 5) + ( 4) (, ) P ( a, 0) \ Midpoint of diagonal b, = + b, Mid-point of diagonal a + 0+, = a +, Mid point of mid point of diagonal = a + and + b = = a + and + b = \ a = and b = \ a = and b = ( ) + ( ) = y y = ( + ) + ( 0 ) = 9+ = 0 unit (4, 5) ( 5, 7) (5 7) + (4) (7 + 5) = 5 sq units (, 6) ( 4, 5) ( ) + ( ) = y y ( ) + ( ) = 4 0 = 9+ = 0 unit is a parallelogram (Given) \ = = 0 unit = = 0 unit + rea of ar = ( 4) ( 6 5) + ( ) (5 + 5) + 4 ( 5 + 6) = 9 sq units Hence, rea of quad. = = 7 sq units [SE Marking Scheme, 08] etailed nswer: We know that diagonals of parallelogram bisect each other. 6. Let usual speed of the plane be km/hr. \ = = 0 ( + 600)( 500) = 0 ¹ -600, \ = 500 Speed of plane = 500 km/hr. [SE Marking Scheme, 08] etailed nswer: Let the speed of plane be km/hr Time taken to cover 500 km

7 Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x [ 7 Time taken to cover 500 km (t ) = istance speed When speed increased 00 km/hr Given (t ) = 500 hr + 00 = 500 hr t + t = 0 minutes = hr = ( + 00) = = + 00 = = = 0 ( + 600) 500( + 600) = 0 ( + 600)( 500) = 0 If = 0 Þ = 600 speed can not be negative. If 500 = 0 Þ = 500 \ Speed of the plane = 500 km/hr 7. Let the side of the square be 'a' units \ = a + a = a Þ = a units. E a a a a rea of equilateral triangle F= 4 a sq.u a rea of equilateral triangle E = a rea of F = F 4 ( a) = a sq.u r E Let ~ PQR. \ ar = = = ar PQR PQ QR PR Given ar = ar PQR Þ = = = PQ QR PR Þ = PQ, = QR, = PR Þ Therefore PQR. (SSS congruence rule) 8. orrect given, to prove,figure, 4 construction correct proof. [SE Marking Scheme, 08] etailed nswer: Given : P and P are tangents of circle having centre O. P To prove : P = P onstruction Join OP, O and O Proof : OP and OP O = O (Radius of circle) OP = OP (ommon side) ÐOP = ÐOP = 90 (Radius tangent angle) \ OP (RHS congruency rule) \ P = P (PT) Hence proved tan q = O Þ tan q = 4 Þ sin q = 5 and cos q= 4 5 \ 4sinθ cosθ = 5 5 4sinθ+ cosθ = + tan Å = cot( 8 ) Þ 90 = 8 Þ =08 Þ =6 0. Radius of each arc drawn = 6 cm rea of one quadrant = (.4) 6 4 rea of four quadrants =.4 6 =.04 cm rea of square = = 44 cm H ence rea of shaded region = = 0.96 cm

8 8 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x. Total surface rea of article= S of cylinder + S of hemispheres S of cylinder =πrh = = 0 cm Surface area of two hemispherical scoops = = 54 cm SETION-. Let the speed of stream be km/hr. \ The speed of the boat upstream=(8 - )km/hr and speed of the boat downstream=(8+) km/hr s given in the question, = Total surface area of article= 0+54 = 74 cm Radius of conical heap = m Volume of rice =.5 m 7 =58 m rea of canvas cloth required= πrl l = + ( 5. ) =.5 m \ rea of canvas required = 7.5. Salary (in thousand `) = 47.4 m No. of Persons (f) c.f = f N = 80 =40 Median class = 0 5 h Median = N l + f = (40-49) = =.4 Median salary is `.4 thousand or ` 40 (appro) Þ = 0 Þ ( + 54)( 6) = 0 ¹ 54 \ = 6 Speed of the stream = 6 km/hr. Let the original average speed of train be km/hr. Therefore, Þ 9 6 = = + 6 Þ ( 4) ( + ) = 0 ¹ \ = 4 is not equal= \ = 4 Original speed of train is 4km/hr. etailed nswer: Speed of motor boat in still water [SE Marking Scheme, 08] = 8 km/hr Let the speed of stream be km/hr Speed of boat in upstream = (8 ) km/hr in downstream = (8 + ) km/hr Time taken to cover 4 km in up stream (t ) = istance Speed 4 = hr 8 Time taken to cover 4 km in up stream (t ) = t t = hr hr (Given) = = = ( 8 )( 8 + )

9 Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x [ 9 4 = 4 48 = = = 0 ( + 54) 6( + 54) = 0 ( + 54) ( 6) = 0 If + 54 = 0 \ = 54 ut speed can not be negative If 6 = 0 \ = 6 \ Speed of stream = 6 km/hr Let the original speed of train be km/hr Time = istance Speed Total time to complete journey = hr = = ( + 6) = = = = = 0 ( 4) + ( 4) = 0 ( 4)( + ) = 0 If + = 0 Þ = Speed cannot be negative If 4 = 0 Þ = 4 \ Speed of train = 4 km/hr 4. Let the four consecutive terms of.p. be (a d), (a d), (a + d) and (a + d) y given conditions a d + a d + d + a + d = Þ 4a = Þ a = 8 nd ( a d)( a+ d) = 7 ( a d)( a+ d) 5 a 9d = 7 a d 5 d = 4 \ d = ± \ Numbers are, 6, 0 and 4 or 4, 0, 6 and. 5. raw E ^ E E (RHS congruence rule) \ E = E = = Let === E Now E = = = 6 and E=E Now =E +E...() nd =E +E...() From () and () Þ - = 6 Þ = 8 Þ = 6 Þ 9 = =E E Given, to Prove, onstruction and Figure orrect Proof [SE Marking Scheme, 08] etailed nswer: Given : In triangle, Ð = 90 To Prove : = + onstruction : raw ^ Proof : In and Ð = Ð = 90 (y construction)

10 0 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x Ð = Ð (ommon angle) \ ~ ( Similar Test) \ = = = =...(i) In and Ð = Ð (ommon angle) Ð = Ð = 90 (y construction) \ ~ ( Similar test) = = = = From equation (i) and (ii)...(ii) + = + + = ( + ) + = + = 6. orrect construction of. orrect construction of similar to. [SE Marking Scheme, 08] etailed nswer: Step of onstruction : (i) onstruct a in which = 6 cm, = 5 cm and Ð = 60. (ii) raw a ray X such that ÐX = cm ' 6 cm ' 4 (iii) Locate 4 points,, and 4 such that = = = 4. (iv) Join 4. (v) Through draw a line parallel to 4 which meet at ' (vi) Through ' draw a line parallel to which meet at '. (vii) '' is the required triangle. 7. LHS = = sin sin cos cos sin ( sin ) cos ( cos ) sin ( ( cos )) = cos ( cos ) (cos ) = tan (cos ) = tan= RHS 8. Here r = 5 cm, r =5 cm and h= 4 cm (i) rea of metal sheet = S of the bucket + rea of lower end = pl(r+r)+ pr where l = 4 + ( 5 5) =6 cm \ Surface area of metal sheet =.4(6 0+5)cm =7. cm We should avoid use of plastic because it is nondegradable or similar value. 9. Let be the tower and ships are at points and m tan 45 = Þ = Þ = lso tan 0 = = + Þ Þ = + = + Þ + = Þ = ( ) = 00 (.7 ) = 7. m. [SE Marking Scheme, 08] etailed nswer: : Let OP is the light house nd and are the position of ships respectively \ OP = 00 m, Ð = 0 and Ð = 45

11 Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x [ 0. P 00 m In PO, In PO, O tan = OP O tan 45 = = 00 O 00 O O = 00 m tan = OP O tan 0 = 00 O = 00 = = = 00( ) = 00(.7 ) = = 7. m \ istance between two ships = 7. m lass lass mark () Frequency (f) f 0 f f 40+f 704+0f etailed nswer : lass Frequency (f) lass mark () d = h f d = f 0 f f 8 + f Σfd = + h Σf 8 8 = ( 8 8 = f 8) 40 + f f f 0 = f 8 f = 8 umulative frequency distribution table less than type is aily Income (in `) umulative Frequency (c.f.) For åf åf f Mean = 8 = 40 + f Þ 70+8 f = f Þ f = 8 [SE Marking Scheme, 08] aily Income (in `) Number of Workers (f) umulative Frequency (c.f.) 00 0

12 ] Oswaal SE Solved Paper - 08, MTHEMTIS, lass-x Less than aily income in (`) Number of Workers (f) No. of Workers (c f )

[Class-X] MATHEMATICS SESSION:

[Class-X] MATHEMATICS SESSION: [Class-X] MTHEMTICS SESSION:017-18 Time allowed: 3 hrs. Maximum Marks : 80 General Instructions : (i) ll questions are compulsory. (ii) This question paper consists of 30 questions divided into four sections,