KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE G È.G È.G È.. Æ fioê, d È 2018 S. S. L. C. EXAMINATION, JUNE, 2018

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1 CCE PR REVISED & UN-REVISED O %lo ÆË v ÃO y Æ fio» flms ÿ,» fl Ê«fiÀ M, ÊMV fl KARNATAKA SECONDARY EDUCATION EXAMINATION BOARD, MALLESWARAM, BANGALORE G È.G È.G È.. Æ fioê, d È 08 S. S. L. C. EXAMINATION, JUNE, 08» D} V fl MODEL ANSWERS MO : ] MOÊfi} MSÊ : 8-E Date : ] CODE NO. : 8-E Œ æ fl : V {} Subject : MATHEMATICS ( Ê Æ p O» fl / New Syllabus ) ( Æ» ~%} S %/ Private Repeater ) (BMW«ŒÈ Œ M} / English Version ) D [ V Œ V fl : 00 [ Max. : 00 Ans. Key I.. A and B are two sets, such that n ( A ) 7, n ( B ) 6 and n ( A U B ) 5 ; then n ( A I B ) is (A) (B) 6 (C) 4 (D) 5 (A) [ Turn over

2 8-E CCE PR Ans. Key. Geometric mean between and 8 is (A) 6 (B) 6 (C) 4 (D) 4 (C) 4. HCF of any two prime numbers is (A) a prime number (B) a composite number (C) an odd number (D) an even number (C) an odd number 4. If f ( x ) x + x x + 6 then the value of f ( ) is (A) 0 (B) 0 (C) 8 (D) 8 5. (D) 8. In ABC, ABC 90, BD AC if BD 8 cm and AD 4 cm then the length of CD is (A) 6 cm (B) 4 cm (C) 64 cm (D) cm (A) 6 cm

3 CCE PR 8-E Ans. Key sin ( 90 θ ) 6. cos ( 90 θ ) where θ is acute, is equal to (A) sec θ (B) cot θ (C) tan θ (D) cosec θ (B) cot θ 7. The co-ordinates of the mid-point of the line segment joining the points (, ) and ( 4, 7 ) are (A) (, 5 ) (B) (, ) (C) (, 5 ) (D) ( 6, 0 ) (C) (, 5 ) 8. Formula used to find the surface area of a sphere whose radius r units is (A) πr (B) πr (C) πr (D) 4πr (D) 4π r. [ Turn over

4 8-E 4 CCE PR II. Answer the following : 6 6 ( Question Numbers 9 to 4, give full marks to direct answers ) 9. A boy has pants and 4 shirts. How many different pairs of a pant and a shirt can he dress up with? Number of ways of pairing a pant and a shirt Write sample space for the random experiment tossing two fair coins simultaneously once. S { HH, TT, HT, TH }. The given pie chart shows the annual agricultural yield of different crops in a certain place. If the total production is 600 tons, what is the yield of Ragi in tons? Yield of Ragi

5 CCE PR 5 8-E. If ( x + ) is one of the factor of f ( x ) x + 5x + 6, find the other factor. Method : Factor method x + 5x + 6 x + x + x + 6 x ( x + ) + ( x + ) ( x + ) ( x + ) The other factor is ( x + ) Method : Division method x + ) x + 5x + 6 ( x + ( ) x + x x + 6 x + 6 ( ) ( ) 0 The other factor is ( x + ). What are concentric circles? Circles having the same centre but different radii are called concentric circles. 4. Two straight lines are perpendicular to each other. If the slope of one line is, find the slope of the other line. m m m m Slope of the other line. [ Turn over

6 8-E 6 CCE PR III. 5. If A {,, } and B {,, 4, 5 } are the subsets of U {,,, 4, 5, 6, 7, 8 }, verify ( A I B ) l A l U B l. A I B {, } ( A I B ) l U (A I B ) {, 4, 5, 6, 7, 8 }... i) A l { 4, 5, 6, 7, 8 } B l {, 6, 7, 8 } A l U B l {, 4, 5, 6, 7, 8 }... ii) From (i) and (ii) ( A I B ) l A l U B l 6. Find the sum of infinite terms of the geometric series a, r, S? S a r

7 CCE PR 7 8-E 7. Prove that + is an irrational number. Let us assume + is a rational number. p + where p, q z, q 0 q p q q is a rational number p q Q is rational. q But is not a rational number. This leads to a contradiction. Our assumption that + is a rational number is wrong. + is an irrational number. 8. Find the number of diagonals that can be drawn in an octagon. An octagon has 8 vertices n 8 8 Total number of sides and diagonals C 4 n n ( n ) 8 8 ( 8 ) C C lines includes 8 sides. Number of diagonals [ Turn over

8 8-E 8 CCE PR Alternate method : Number of diagonals in a polygon of n sides n ( n ) In an octagon n 8 Number of diagonals 4 8 ( 8 ) Any other correct alternate method may be given marks. 9. Find the sum of all two digit natural numbers which are divisible by 5. Two-digit numbers which are divisible by 5 0, 5, 0, Sum of all two-digit numbers a 0, d 5, T 95 n T a + ( n ) d n ( n ) 5 85 ( n ) 5 ( n ) 7 n 8 Method : Sum of n natural numbers n S n [ a + ( n ) d ] S 8 8 [ 0 + ( 8 ) 5 ] 9 ( ) 9 05 S OR

9 CCE PR 9 8-E n 8, a 0, l 95 n ( a + l ) S n 9 8 ( ) S Alternate method : 5 ( ) 5 ( 9 ) 5 ( 90 ) Find how many 4 digit numbers can be formed by using the digits,,, 4, 5 without repetition? How many of these are less than 000? If OR ( n P ) + 50 n P, find the value of n. Number of 4-digit numbers 5 P digit numbers which are less than Thousand s Hundred s Ten s place Unit place place place P 4 P P P 4 4 numbers. OR [ Turn over

10 8-E 0 CCE PR 4-digit numbers which are less than P 4 4 OR ( n P ) + 50 n P n ( n ) + 50 n ( n ) n n n n 4n n 50 n 50 n 5 n ± 5 n 5. Two unbiased dice whose faces are numbered to 6 are rolled once. Find the probability of getting a sum equal to 7 on their top faces. Total number of possible outcomes Event of getting a sum equal to 7 n ( s ) 6 (, 6 ) A ( 4, ) (, 5 ) ( 5, ) (, 4 ) ( 6, ) n ( A ) 6 n ( A ) Probability of getting the event A P ( A ) n ( S ) 6 6 6

11 CCE PR 8-E. Rationalise the denominator and simplify :. 5 Rationalising factor of 5 is ( ( 5 ) 5 + ) ( ) ( ) ( 0 + ) Simplify ( ) ( 0 + ). ( ) ( 0 + ) ( ) ( ) ( 5 5 ) ( 5 + ) 5 ( 5 + ) 5 ( 5 + ) ( ) 6 ( 5 ) [ Turn over

12 8-E CCE PR 4. Find the quotient and remainder by using synthetic division : ( x x + 7x 5 ) ( x ) OR Verify whether ( x ) is a factor of f ( x ) x x + 6x 0 by using factor theorem Quotient x + 7x + 8 Remainder 79. OR Let f ( x ) x x + 6x 0 If ( x ) is a factor of f ( x ), then f ( ) 0 Now f ( ) ( ) + 6 ( ) f ( ) 0 x is not a factor of x x + 6x 0.

13 CCE PR 8-E 5. In Δ ABC, DE BC, if AD cm, DB 5 cm and AE 4 cm, find AC. In Δ ABC, DE BC AD AE DB EC BPT 4 5 EC EC cm AC AE + EC cm. Alternate method : In Δ ABC, DE BC AD AE Cor. BPT AB AC AC AC cm. [ Turn over

14 8-E 4 CCE PR 6. Draw a circle of radius 4 5 cm and a chord PQ of length 7 cm in it. Construct a tangent at P. r 4 5 cm Chord PQ 7 cm Circle Chord Tangent 7. Find the distance between the co-ordinates of the points (, 4 ) and ( 8, ) by using distance formula. Coordinates of ( x y ) Point A (, 4 ) ( x y ) Point B ( 8, )

15 CCE PR 5 8-E Distance between the points d ( x x ) + ( y y ) ( 8 ) + ( 4 ) In a hockey match team A scored one goal less than twice the number of goals scored by team B. If the product of the number of goals scored by both the teams is 5, find the number of goals scored by each team. Let the goals scored by team A be x and goals scored by team B be y. x ( y ) Product of the goals scored by both teams 5 xy 5 ( y ) y 5 y y 5 0 y 6y + 5y 5 0 y ( y ) + 5 ( y ) 0 ( y ) ( y + 5 ) 0 y If y, then x 6 5 Goals scored by team A 5 Goals scored by team B. [ Turn over

16 8-E 6 CCE PR 9. In the given ABC, θ is acute. Write the values of the following trigonometric ratios related to θ : (a) sin θ (b) cos θ (c) cosec θ (d) sec θ. a) sin θ Opp Hyp BC AC b) cos θ Adj Hyp AB AC c) cosec θ sin θ d) sec θ cos θ. Direct answers may be given marks.

17 CCE PR 7 8-E 0. Draw a plan by using the information given below : ( Scale 0 metres cm ) 0 0 m cm 0 Metre to C to D 90 0 to E to B From A m cm m 4 5 cm 0 40 m 40 7 cm m cm m 4 cm m 5 cm 0 [ Turn over

18 8-E 8 CCE PR. If P {,,, 4 }, Q {,, 4, 5, 6 } are the subsets of U {,,, 4, 5, 6, 7, 8, 9, 0 }, then draw Venn diagram to represent ( P U Q ) l.. Write the formula used to find the following : (a) Sum of first n natural numbers (b) Harmonic mean between a and b ( a > b ). n ( n + ) a) n ab b) Harmonic mean ( H ). a + b. Write the values of the following : (a) 00 P 0 (b) 0 C. a) b) 00 P 0 0 C 0

19 CCE PR 9 8-E 4. Draw a pie chart to represent the survey carried out in the class regarding places of visit for excursion and the number of students who opted each place. Places Mysuru Vijayapura Gokorna Chitradurga Number of students Places No. of students Central angle Mysuru o 40 Vijayapura 6 54 Gokorna 8 Chitradurga Calculation Pie chart [ Turn over

20 8-E 0 CCE PR 5. Find the product of and. LCM of the order of surds. For any other alternative method give marks. 6. Determine the nature of the roots of the equation x 5x 0. a, b 5, c Δ b 4ac ( 5 ) 4 ( ) ( ) Δ > 0, Roots are real and distinct.

21 CCE PR 8-E 7. In Rhombus ABCD, prove that 4AB AC + BD. In AOB, AO B 90 AB AO + BO But AO AC, BO BD AB AC + BD AC BD AC + BD 4 4AB AC + BD. Any correct alternate method may be given marks. 8. Find the remainder when P ( x ) x + x 5x + 8 is divided by ( x ) by remainder theorem. By remainder theorem, the required remainder is P ( ) P ( ) ( ) + ( ) 5 ( ) The remainder P ( ) 47. [ Turn over

22 8-E CCE PR 9. Find the distance between origin and the point ( 8, 5 ). Distance between origin and ( x, y ) x + y Here ( x, y ) ( 8, 5 ) d ( 8 ) d 7 units. 5 sin θ + cos θ 40. If cos θ, find the value of. sin θ cos θ 5 Given cos θ Adj AB Hyp AC In ABC, A B C 90 BC AC AB BC sin θ. Figure Finding opp. side

23 CCE PR 8-E sin θ + cos θ sin θ cos θ Any other alternate method may be given marks. IV. 4. In a harmonic progression 5th term is and th term is 5. Find its 5th term. OR If the third term of a geometric progression is and its sixth term is 96, find the sum of first 9 terms. T 5 and T 5 Reciprocals of HP are in AP. a + 4d... (i) a + 0d 5... (ii) By solving (i) and (ii) a + 0d 5 ( ) a + 4d ( ) 6d d 6 If d, then a + 4 ( ) a + a 0 [ Turn over

24 8-E 4 CCE PR If a 0 and d then T n a + ( n ) d T ( 5 ) T 5 Alternate method : The corresponding T and 5 T of AP are T and 5 T 5 d T p T q p q T 5 T If d then a + 4 ( ) a + a 0 If a 0 and d T n a + ( n ) d

25 CCE PR 5 8-E T ( 5 ) T 5 OR T ar... (i) T 96 6 ar (ii) ar 5 ar 96 8 OR ar ( r ) 96 r r 8 r 8 r 96 If r then a ( ) 4a a If a and r, n 9 then a ( r n ) S n r ( 9 ) S 9 ( 5 ) 5 S 5 9 [ Turn over

26 8-E 6 CCE PR 4. Calculate the variance of the following data : Class-interval Frequency ( f ) 5 4 i) Step deviation method : A i 5 C.I. f x d x A i d f d f d Variance N 5 fd 6 f d f d fd σ i N N ( )

27 CCE PR 7 8-E Direct method : C.I. f x fx x f x Variance N 5 fx 0 f f x f x σ N N x Assumed mean method : Assumed mean A C.I. f x d x A fd d f d N 5 fd 0 f d 550 [ Turn over

28 8-E 8 CCE PR Variance f d f d σ N N Actual mean method : C.I. f x fx d x x d f d Mean N 5 fx 0 f x f x N d 490 Variance f d σ N

29 CCE PR 9 8-E 4. Solve ( x + ) ( x ) + 0 by using formula. OR If one root of the equation x + px + q 0 is four times the other, prove that 4p 5q 0. ( x + ) ( x ) + 0 x ( x ) + ( x ) + 0 6x 4x + 9x x + 5x 4 0 where a 6, b 5. c 4 x b ± b 4 ac a 5 ± ( 4 ) 6 5 ± ± 5 ± 5 + or 5 6 or 6 x or 4. OR [ Turn over

30 8-E 0 CCE PR x + px + q 0 where a, b p, c q If m and n are the roots then m 4n Sum of the roots m + n 4n + n b a p 5n p n p 5... (i) Product of the roots mn a c 4n n q 4 n q... (ii) Substituting (i) in (ii) Then p 4 q 5 4p 5 q 4p 5q 4p 5q 0

31 CCE PR 8-E 44. Prove that The tangents drawn from an external point to a circle are equal. Data : A is the centre of the circle. To prove : BP BQ B is an external point. BP and BQ are the tangents. Construction : AP, AQ and AB are joined. Proof : In APB and AQB, A P B AQ B Radius drawn at the point of contact is perpendicular to the tangent. hyp. AB hyp AB AP AQ Common side Radii of the same circle. APB AQB RHS theorem. BP BQ CPCT. [ Turn over

32 8-E CCE PR 45. In Δ ABC, AB AC. P is a point on BC such that PN AC and PM AB as shown in the figure. Prove that MB. CP NC. BP. OR In Δ ABC, DE BC. If DE BC and the area of Δ ABC is 8 cm, show that the area of Δ ADE is 6 cm. In Δ ABC, AB AC B C angles opposite to equal sides In BMP and CNP B M P M B P P N C right angles N C P equal angles MBP ~ NCP equiangular triangles MB NC BP MP AA - criteria CP NP MB. CP BP. NC. OR

33 CCE PR 8-E Given DE BC DE BC In Δ ADE and Δ ABC, A D E D A E A B C Corresponding angles B AC Common angle Δ ADE ~ Δ ABC Equiangular triangles Area of Area of Δ ADE Δ ABC Area of Δ ADE 8 Area of Δ ADE DE BC cm. 46. Prove that ( + cot A cosec A ) ( + tan A + sec A ). OR From the top of a building 0 m high, the angle of elevation of the top of a vertical pole is 0 and the angle of depression of the foot of the same pole is 60. Find the height of the pole. cos A sin A sin A sin A cos A cos A sin A + cos A sin A cos A + sin A + cos A ( sin A + cos A ) sin A cos A sin A + cos A + sin A cos A sin A cos A [ Turn over

34 8-E 4 CCE PR but sin A + cos A + sin A cos A sin A cos A sin A cos A sin A cos A OR In BED, D B E 0 DE tan 0 BE x 0 BE BE ( x 0 ) In ABC, AC B 60 AB tan 60 AC 0 ( x 0 )

35 CCE PR 5 8-E ( x 0 ) 0 x 60 0 x 80 x m. Height of the pole 6 6 m ( approximate ). V. 47. Solve the equation x + x 6 0 graphically. x + x 6 0 y x + x 6 x 0 4 y Alternate method : Table Drawing parabola Identifying roots 4 x + x 6 0 y x, y 6 x y x x 0 y y 6 x x 0 y Table Drawing parabola Identifying roots 4 [ Turn over

36 8-E 6 CCE PR

37 CCE PR 7 8-E Alternate method : [ Turn over

38 8-E 8 CCE PR 48. Construct a direct common tangent to two circles of radii 4 cm and cm whose centres are 9 cm apart. Measure and write the length of the direct common tangent. R 4 cm, r cm R r 4 cm d 9 cm Length of the tangent EF 8 8 cm

39 CCE PR 9 8-E Drawing AB and marking mid-point Drawing C, C, C Joining DB, EF Measuring and writing the length of the tangent Prove that In a right angled triangle, square on the hypotenuse is equal to sum of the squares on the other two sides. Figure Data To prove Construction Data : In ABC, A B C 90 To prove : Construction : AC AB + BC BD AC drawn. Proof : Comparing ABC and ABD A B C B AC Statement A D B B A D Right angles common angle Reason BAC ~ DAB Equiangular triangles BA AC AA criteria DA AB AB AC. AD... (i) [ Turn over

40 8-E 40 CCE PR Comparing ABC and BDC A B C AC B B D C Right angles common angle B C D BCA ~ DCB Equiangular triangles BC AC AA criteria DC BC BC AC. DC... (ii) By adding (i) and (ii) AB + BC AC AD + AC DC AC ( AD + DC ) Q AD + DC AC AC AC AB + BC AC A solid is in the shape of a cylinder with a cone attached at one end and a hemisphere attached to the other end as shown in the figure. All of them are of the same radius 7 cm. If the total length of the solid is 6 cm and height of the cylinder is 0 cm, calculate the cost of painting the outer surface of the solid at the rate of Rs. 0 per 00 cm. OR

41 CCE PR 4 8-E A solid metallic cylinder of diameter cm and height 5 cm is melted and recast into toys in the shape of right circular cone mounted on a hemisphere as shown in the figure. If radii of the cone and hemisphere are each equal to cm and the height of the toy is 7 cm, calculate the number of such toys that can be formed. Height of the cone Total height of the solid ( height of the cylinder + radius of the hemisphere ) 6 ( ) But 7, 4, 5 are Pythagorian triplets cm. Slant height of the cone l 5 cm. TSA of the solid LSA of the cone + LSA of the cylinder + LSA of the hemisphere π rl + πrh + πr πr ( l + h + r ) 7 ( ) sq.cm sq.cm. Cost of painting at the rate of Rs. 0 per 00 cm Rs [ Turn over

42 8-E 4 CCE PR Alternate method : Height of the cone h 4 cm Slant height of the cone l 5 cm. LSA of the cone πrl π 7 5 sq.cm 75 π sq.cm. LSA of the cylinder πrh π 7 0 sq.cm 40 π sq.cm. LSA of the hemisphere π r π 7 98π sq.cm. TSA of the solid LSA of the cone + LSA of the cylinder + LSA of the hemisphere ( 75 π + 40 π + 98 π ) sq.cm sq.cm. Cost of painting Rs OR

43 CCE PR 4 8-E Volume of the metal cylinder π r h cubic units r 6 cm π 6 5 c.c. h 5 cm Volume of the toy Volume of the cone r cm + h 7 4 cm Volume of the hemisphere π r h + πr πr ( h + r ) π ( 4 6 ) + 0 π Number of toys Volume of the cylinder Volume of the toy π 0 π 8 4 Alternate method : Cylinder Cone Hemisphere r 6 cm r cm r cm h 5 cm Number of toys h 4 cm Volume of the metal cylinder Volume of the toy [ Turn over

44 8-E 44 CCE PR πr h πr h + πr π ( 6 5 ) π ( )