QUANTITATIVE APTITUDE

Transcription

1 QUANTITATIVE APTITUDE. ABC is a right-angled triangle, right angled at A and AD is the altitude on BC. If AB : AC :, what is the ratio of BD to DC? (a) : (b) 9 : 6 (c) : (d) :. ABC and PQR are two triangles such that AB AC PQ PR.. BAC α and. QPR., and α... What is the relation between α and., such that the area of.abc area of.pqr? (a) α.? 80 (b) α.? 90 (c) α.? 0 (d) α.? 0. A solid sphere is cut into four equal parts. If the total surface area of each part now is times that of the sphere, then what is the value of? (a), (b), (c), (d),8. Two spheres of radii 6 cm and cm are inscribed in a right circular cone. The bigger sphere touches the smaller one and also the base of the cone. What is the height of the cone? (a) cm (b) 5 cm (c)856 cm (d)75 cm 5. ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle through the points A, B, C and D. If.? ADC 0, what is the value of.bac? (a) 0 (b) 0 (c) 50 (d) 60 6.Triangles ABC and DEF are similar. If the length of the perpendicular AP from A on the opposite side BC is cm and the length of the perpendicular DQ from D on the opposite side EF is cm, then what is the area of the triangle ABC? (a) One and half times the area of the triangle DEF (b) Four times the area of the triangle DEF (c) Twice the area of the triangle DEF (d) Three times the area of the triangle DEF

2 7. A semi-circular thin sheet of metal of diameter 8 cm is bent and an open conical cup is made. What is the ratio of radius to slant height of the conical cup? (a) : (b) : (c) : (d) : 8. s and t are transversals cutting a set of parallel lines such that a segment of length in s corresponds to a segment of length 5 in t. What is the length of segment in t corresponding to a segment of length in s? (a) 0 (b)65 (c) (d)5 9. Squares ABDE and ACFH are drawn eternally on the sides AB and AC respectively, of a scalene triangle ABC. Which one of the following is correct? (a) BH CE (b) AD AF (c) BF CD (d) DF EH 0. O and O are the centres of the two circles with radii 7 cm and 9 cm respectively. The distance between the centres is 0 cm. If PQ be the transverse common tangent to the circles, which cuts OO at R, what is the length of OR? (a) 0 cm (b) cm (c)5cm (d) 5 cm. In.PQR, QR 0, RP and PQ. D is the mid-point of PR. DE is drawn parallel to PQ meeting QR in E. EF is drawn parallel to RP meeting PQ in F. What is the length of DF? (a) (b) 6 (c) (d) 5. In a locality there are 500 households out of which 60 households have no land and total land possessed by the rest 0 households is 500 acres. What is the mean landholding per household? (a) 0 acres (b) acre (c)5 acres(d) Mean cannot be computed with the given data. The proportions of male students and female students in a class are equal. The average height of male students of the class is 8 cm and average height of all the students of the class is 7 cm. What is the average height of female students of the class? (a) 6 cm (b) 5 cm (c) cm (d) Average cannot be computed with the given data. The length of a rectangle R is 0% more than the side of a square S. The width of the rectangle R is 0% less than the side of the square S. What is the ratio of the area of A B C P E D F Q O B J E C T I V E -T Y P E Q U E S T I O N S

3 R to that of S? (a) : (b) 99 : 00 (c) 99 : 00 (d) 0 : If an angle of a triangle remains unchanged but each of its two including sides is doubled, then by what factor does the area get multiplied? (a) (b) (c) (d) 6 6. Shadow of a person X, when angle of elevation of the sun is α, is equal in length to the shadow of a person Y, when angle of elevation of the sun is ( ) α. Which one of the following is correct? (a) Person X is shorter than person Y (b) Person X is twice as tall as Y (c) Person X is taller than person Y but is not twice as tall as Y (d) Both X and Y are of equal height 7. What is the value of -sin cos cos (sec - cos ) A A A A eca. sin - cos sin cos A A A A? (a) cos A (b) sec A (c) sin A (d) cosec A 8. If cos sin α α cosα, then what is the value of cos - sin α α? (a) cosα (b) sinα (c) (d) 9. What is the value of sin sin sin... sin cos cos cos...cos ????????? (a) 0 (b) (c) (d) 0. How many solutions does the equation sin. cos. have? (a) It has no solution

4 (b) It has only one solution (c) It has two solutions (d) It has infinite number of solutions. What is the value of (tan A cot A sec A) (tan A cot A - sec A)? (a) sin A (b) cosec A (c) cos A (d) sec A. What is the value of the epression tan π π π 6 6 cos - cot - sin π 8 sec π? (a) (b) (c) (d) 5. If a ( cos cos. φ) y b ( cos sin. φ ) and z c ( sin.) then which one of the following is correct? (a) ( - ) aa ( - ) y bb ( - ) z cc (b)aybzc (c) y z a b c (d)( ) ( ) ( ) aay bbz cc. If a cos. sin., y a sin. cos., then which one of the following epressions is independent of.?.( )( ) y y.( ) yy Select the correct answer using the codes given below: Codes: (a) only (b) only (c) Both and (d) Neither nor 5. If sin. sec., what is the value of sin. cos.? Ifsincoscossin... then what is the value of., where 0. π? (a) 0 (b) 5 (c) 60 (d) Let A, B, C be subsets of the universal set represented by the circles. Let A, B and C denote their complements in

5 X. Then which of below corresponds to the shaded portion in the figure given above? (a) A B C 'ν ν (b) A B C ν ν' (c) A B C ν ν' ' (d) A B C ' ' ν ν 8. Match List-I with List-II and select the correct answer using the codes given below the lists: List-I List-II A. ( - ) ( - ') E A E A.. φ B. E A A A A -(( ')-( ')). ν. A C. (E A A A ν. ( - ')). A D. (( - ) ( - ))- E E A φ φ.. E Here A is the complement set of A, E is the universal set and φ is an empty set. Codes: A B C D (a) (b) (c) (d) A B C O B J E C T I V E -T Y P E Q U E S T I O N S 9. man or women or boys can do a piece of work in 55 days. In how many days can man, woman and boy do the same work? (a) 7 days (b) 0 days (c) 6 days (d) days 0. A number when divided by 765 leaves a remainder. What will be the remainder if the number is divided by 7? (a) 8 (b) 7 (c) 6 (d) 5. Match List-I (Product) with List-II (Digit in the Unit Place) and select the correct answer using the codes given below the Lists: List-I List-II (Product) (Digit in the Unit Place) A. (87)6. B. (5)9. C. (59) Codes: A B C (a) (b) (c) 5 (d) 5. Let ( ) ( ) A B A B A B.. ν. Consider the following:. A is a proper subset of B.

6 . B is a proper subset of A. Which of the statements given above is/are correct? (a) only (b) only (c) Both and (d) Neither nor. What least number must be subtracted from 797 so that (797 - ) on being divided by 8, 9 and leaves in each case the same remainder? (a) 0 (b) (c) (d). A miture (0 litres) contains wine and water in the ratio :. To make the ratio 5 :, how much additional amount of water is required? (a) 5 litres (b) litres (c) litres (d) litres 5. A train leaves Amritsar at 6 a.m. and reaches Delhi at Noon. Another train leaves Delhi at 8 a.m. and reaches Amritsar at 5 p.m. At what time do the two trains cross one another? (a) 0 : 00 a.m. (b) 0 : a.m. (c) 0 : 0 a.m. (d) 0 : a.m. 6. What is the first value of n, for which n n is not a prime? (a) (b) 0 (c) 0 (d) 0 7. A dishonest dealer professes to sell his goods at cost price but uses a false weight and thus gains 9 %. For a kg, what is the weight used by him? (a) 960 gm (b) 90 gm (c) 900 gm (d) 990 gm 8. If, 09, 999 is divided by, then what is remainder? (a) 098 (b) 888 (c) 00 (d) 0 9. When a certain number is multiplied by 8, the product consists entirely s. What is the minimum number of such s in the product? (a) 7 (b) 8 (c) 9 (d) 0 0. Which one of the following is divisible by ( a a5) and ( a a5) individually? (a)( )( )( ) a a a a a a (b)( )( )( ) a a a a a a (c)( )( )( ) a a a a a a (d)( )( )( ) a a a a a a. If the difference in roots of the equation

7 - p q 0 is unity, then which one of the following is correct? (a) p q (b) p - q (c) p q - (d) p - q -. What is the value of log log log 5? (a) (b) (c) (d). If a b c a b c where a b c. 0, abc. 0, what is the value of (a b) (b c) (c a)? (a) 0 (b) (c) - (d). If y 7 and y, what is the value of y - y? (a) 8 (b) 0 (c) (d) 6 5. What is the GCD of ( y) and ( - y y)? (a) y y (b) y - y (c) - y - y (d) - y y 6. If sin A mn m n what is the value of tan A? (a) mn m n (b) mn m n (c) m n mn (d) m n m n

8 7. AB is the hypotenuse of a right-angled triangle ABC. If BC and AB AC y, then which one of the following is correct? (a) sin ( ) A y y (b) sin ( ) A y y (c) sin ( ) A y (d) sin ( ) A y y 8. What is the value of sin cos y cos sin y sin sin y cos cos y? (a) 0 (b) - (c) (d) 9. What is the angle of elevation of sun for length of shadow to be same as height of the person? (a) 0 (b) 0 (c) 5 (d) There are 56 farmers in a village. In a season, out of these, 0 farmers did not use any pesticide while the remaining 6 farmers used some pesticides and total ependiture on pesticides by these 6 farmers was Rs 8,780. What is the median ependiture per farmer in the village? (a) 8780

9 6 (b) (c) 0 (d) Median cannot be computed 5. Which one of the following statements is not correct for a bar graph? (a) All bars have different thickness (b) Distance between two consecutive bars is the same (c) The bars can touch each other (d) The thickness has no significance 5. A person made 65 telephone calls in the month of May in a year. There was Friday on st May of the year. The average telephone call on Sundays of the month was 7. What was the average calls per day on the rest days of the month? (a) 65 (b) 5 (c) 7 6 (d) Out of three circles having centres P, Q and R, each circle touches remaining two circles eternally. If PQ cm, QR 6 cm and PR 8 cm, what is sum of the radii of the circles? (a) 8 cm (b) 5 cm (c) cm (d) 9 cm 5. ABC is an isosceles triangle with AB AC. Side BA is produced to D, such that AB AD. Consider the following:..acd is an isosceles triangle...bcd is an isosceles triangle...bcd is a right-angled triangle. Which of the statements given above are correct? (a) and (b) and (c) and (d), and 55. A circle touches all the four sides of a quadrilateral ABCD. Which one of the following is always correct? (a) AB BC CD DA (b) AB AD BC CD (c) AB CD BC DA (d) AB BC CD DA times the diameter of the circle 56. What are the values of A and B respectively, if 5

10 ? A B (a), 0 (b) 0, (c) 0, 0 (d), Directions: The following five (5) items consist of two statements: one labelled as the Assertion (A) and the other as Reason (R). You are to eamine these two statements carefully and select the answers to these items using the codes given below: Codes: (a) Both A and R are individually true and R is the correct eplanation of A (b) Both A and R are individually true but R is not the correct eplanation of A (c) A is true but R is false (d) A is false but R is true 57. Assertion (A): The system of the equations y 0 8y 0 has no solution. Reason (R): The system of equations a by c 0 a by c 0 has no solution, if a a b b. 58. Assertion (A): sin lies between 0 and for 0< < 90. Reason (R): sin increases as increases from 0 to Assertion (A): log0 ( ) log0 log0 log0 Reason (R): log0 (a b c) log0 a log0 b log0 c where a, b, c are positive real numbers.

11 60. Assertion (A): (( ) ) is irrational number. Reason (R): (( ) ) a a a is always irrational if a is irrational. 6. Assertion (A): is a rational epression. Reason (R): Every polynomial is a rational epression. 6. In a triangle ABC, the angle C is obtuse. Which one of the following is correct? (a) AB > AC BC (b) AB < AC BC (c) AC > AB BC (d) BC > AB AC 6. V, V, V and V are the volumes of four cubes of side lengths cm, cm, cm and cm respectively. Some statements regarding these volumes are given below:. V V V < V. V V V < V. (V V) V V Which of the above are correct? (a) and (b) and (c) and (d), and 6. If the total length of diagonals of a cube is cm, what is the total length of edges of the cube? (a) 5 cm (b) cm (c) 6 cm (d) cm 65. A circle of 5 cm radius is drawn and another circle of cm radius is cut out of this circle. What is the radius of a circle which has the same area as the area of the bigger circle ecluding the cut one? (a) cm (b) cm (c) cm (d).5 cm 66. Ecept for one face of a given cube, identical cubes are glued through their faces to all the other faces of the given cube. If each side of the given cube measures cm, then what is the total surface area of the solid body thus formed? (a) 5 cm (b) cm (c) 70 cm (d) 79 cm 67. Two concentric spheres A and B, have radii r and r respectively. A cone is inscribed in the latter so as to

12 circumscribe the former. What is the curved surface area of the cone? (a) πr (b) πr (c) 6 πr (d) 8 πr 68. A spherical iron ball is dropped into a cylindrical vessel of base diameter cm, containing water. The water level is increased by 9 cm. What is the radius of the ball? (a).5 cm (b) 7 cm (c) 9 cm (d) cm 69. The eternal and internal diameters of a hemispherical bowl are 0 cm and 8 cm respectively. What is the total surface area of the bowl? (a) 57.7 cm (b) 86 cm (c) 9 cm (d) 0 cm 70. If in the figure given above, MN, what is the area of the shaded region? (a) π (b) π (c) π (d) π 7. The radius and height of a right solid circular cone are r and h respectively. A conical cavity of radius r and height h is cut out of the cone. What is the whole surface area of the rest of the portion? (a) πr r h r 5 ( ) (b) 5 πr r h (c) πr r h r ( ) (d) 7 πr r h r ( ) 7. The radius of a cylindrical cistern is 0 metres and its height is 5 metres. Initially the cistern is empty. We start

13 filling the cistern with water through a pipe whose diameter is 50 cm. Water is coming out of the pipe with a velocity of 5 m/s. How many minutes will it take in filling the cistern with water? (a) 0 (b) 0 (c) 60 (d) A sphere of radius cm is cut by a plane whose distance from the centre of the sphere is 5 cm. What is the circumference of the plane circular section? (a) 0π cm (b) π cm (c) π cm (d) 6π cm 7. A square of side a cm is cut from each corner of a rectangular sheet of metal of 0 cm by cm. The resulting projections are folded up and the seams welded to construct an open bo. What is the volume of the bo thus obtained? (a) 0a - 8a a (b) 0a 8a a (c) 0a a a (d) 0a - a a 75. A right circular cone is cut by two planes parallel to the base and trisecting the altitude. What is the ratio of the volumes of the three parts: top, middle and bottom respectively? (a) : 7 : 9 (b) : : (c) : 8 : 7 (d) : 7 : Two rectangles A and B are such that the length of A equals the sum of the diagonal and the greater side of B, while the breadth of A equals the difference of the diagonal and the greater side of B. Which one of the following is correct? (a) Area of A is greater than the area of B (b) Area of A is equal to the area of B (c) Area of A is equal to the area of a square with its side equal to the smaller side of B (d) Area of A is equal to the area of a rectangle whose dimensions are those of the diagonal and the shorter side of B 77. What is the least number by which 675 be multiplied so that the product is a perfect square? (a) (b) (c) 5 (d) A sum of money placed at compound interest trebles itself in 5 years. In how many years will it amount to nine times itself? (a) 0 years (b) years (c) 5 years (d) 8 years 79. The area of a circle varies as the square of its radius.

14 If the area of circle of radius 0 cm is 00 cm, then what is the area of circle with radius cm? (a) 60 cm (b) cm (c) cm (d) 5.5 cm 80. A machine is sold at a profit of 0%. Had it been sold for Rs 0 less, there would have been a loss of 0%. What is the cost price of the machine? (a) Rs 75 (b) Rs 00 (c) Rs 50 (d) Rs What is the LCM of ,, and 5 96? (a) 05 (b) 05 (c) 05 8 (d) workers, after working for 0 hours a day assemble some instruments in 5 days. How much number of days is required to assemble the same number of instruments if 5 workers are employed working 8 hours a day? (a) 0 days (b) days (c) 5 days (d) 8 days 8. A train started from a station with certain number of passengers. At the first halt, half of the passengers got down and 0 passengers got in. At the second halt, half of the passengers got down, and then the train left for its destination with 60 passengers. How many passengers were there in the train when it started from the first station? (a) 800 (b) 900 (c) 000 (d) A man can walk uphill at the rate of km/hr and downhill at the rate of km/hr. If the total time required to walk a certain distance up the hill and return to the starting point was hr 6 min, then what was the distance he walked up the hill?

15 (a) 6 km (b) 5 km (c) km (d) km 85. In an election a total of 5,00,000 voters participated. A candidate got,55,000 votes which was 60% of the total valid votes. What was the percentage of invalid votes? (a) 0% (b) % (c) 5% (d) ( )% , y and z are three sums of money such that y is simple interest on and z is simple interest on y for the same time and rate. Which one of the following is correct? (a) yz (b) y z (c) z y (d) yz 87. If one root is common in equations b a 0 and a b 0 and a b., then which one of the following is correct? (a) a b - (b) a b (c) a - b (d) a - b What is the nature of the roots of the equation p - (p - q) - q 0 where p, q are integers and p. 0? (a) Both roots are always integers (b) Both roots are always rationals (c) One root is rational and other is irrational (d) Both roots are always of opposite sign 89. Let l, m and n be real numbers such that l n. m. What is the quotient on dividing l - m n lmn by (l -m n)? (a) l m n - lm - mn - ln (b) l m n lm mn - ln (c) l - m n lm mn - ln (d) l - m n - lm - mn ln 90. Consider the following statements:. an bn is divisible by a b if n k, where k is a positive integer.. an - bn is divisible by a - b if n k, where k is a positive integer. Which of the statements given above is/are correct? (a) only (b) only (c) Both and (d) Neither nor 9. If the roots of the equation 0 are α? and, and the roots of the equation a b 0 are α

16 ? and? α, what is the value of α? (a) (b) (c) - (d) - 9. What is the remainder when is divided by -? (a) 7 (b) 5 (c) 7 (d) 9. If the value of logy 0, then what will be the value of log0 ( y 0 )? (a) 0 (b) 0 (c) - (d) 0 9. If y z 0 and yz. 0, then what is the value of y z y z z y - - -? (a) 0 (b) (c) - (d) 95. If log0 log0 y log0 ( y), then what is the value of log0 - log0 y? (a) log0 ( - y) (b) log0 (y - ) (c) log0 ( - y) (d) - log0 (y - ) 96. If α? and are the roots of the equation a b 0, then what are the roots of the equation - a b 0? (a) α?, - (b) -, α? (c) -, - α? (d) α?, 97. For what value of p is the coefficient of in the product ( - ) ( - k) (p ) equal to 0 and the constant term equal to? (a) (b) 5 (c)

17 5 (d) If m and n are natural numbers such that m - n 960, what is the value of m? (a) 0 (b) (c) 6 (d) Cannot be determined 99. When b is divided by, the quotient is a and remainder is - b. What are the values of a and b respectively? (a), 0 (b) -, 0 (c), (d) -, If y z and y what is the value of yz? (a) (b) - (c) 0 (d) ANSWERS With EXPLANATIONS. (b).abc is a rt.,.a 90 AB : AC :. BC 5 Let BD. DC 5 - AD AB BD AC DC (-) DC 0 BD : DC : 9 : 6. (a) ar.abc ar.pqr. AB.AC. sin α PQ.PR.sin.. sinα sin. But α... α ? α. 80 [Q If sin. sinα then

18 either.α or α. (b) A. πr. π π r r.. πr.. (d).. OAB OCD ~. OA OC AB CD. h h 6 h AP h CP -( ) - 6 h h -( ) h (b) Q AB is a diameter..? ACB 90.ADC 0..ABC BAC (b) As.s are similar. ar ABC ar DEF.. AP DQ ( ) AP

19 DQ ( ) ar.abc (ar.def) 7. (a) d 8 cm. Slant height of open conical cup 8 cm Circumference of base of cone πr π? π? R π. R 7 R : l 7 : : 8. (a) (a).bah 90.BAC.CAE 90.BAC..BAH.CAE AB AE, AH AC.... BAH EAC (SAS).BH CE (C.P.C.T.) 0. (d) OO 0 cm Let OR. O R OPR O QR ~ '. OR O R OP O Q ' ' (d) In.PQRQDE PQ D and D is the mid point of PR. E is the mid point of QR EF PR D. F is the mid point of PQ In.PQR, D and F are the mid points of PR and PQ respectively. DF QR 0 5 cm

20 . (b). (a) M : F : A.T.S. 7 8? - 6. (b) Let the side of a square be a A.T.S. ( ) ( ) a a a??? (c) A ab sin. A B C D A (a) (b) sin.. A ( ab sin.) A A 6. (c) 7. (c) -sin cos cos ( cos - sin ) A A A A A. (sin cos )(sin cos ) (sin cos )(sin cos sin cos ) A A A A A A A A A A sin A sin A cos A a b (a b) (a b - ab) 8. (b) cosα sinα cosα.sinα (cosα) - cosα. sinα ( -) cosα. cosα sin - α

21 . cosα sinα sinα.cosα - sinα sinα 9. (b) sin (90-.) cos.. cos 88 sin cos sin 88 cos (90-.) sin. 0. (a) sin. cos.. Dividing by (sin.) (cos.).sin.. cos 5 cos. sin 5.sin (. 5 ) But - < < sinα. sin( ). 5.. No solution. (b) Given ep (tan A cot A) - sec A tan A cot A tan A cota - sec A tana cota - seca cot A - (seca - tana) (cot A ) - () coseca. (a) Value ( ) ( ) - () - ( ) 8 ( ). (a) ( - ) ( ) ( - ) a a y b b z c c

22 cos cos. φcos sin. φsin. cos (cos sin ) sin. φ φ. cos sin... (c) y a cos sin sin cos.... a a cos sin (cos sin ).... a cos sin.. y acos.. sin ( ) y y a a cos sin cos sin.... a which is independent of. Sly y y a a cos sin cos sin.... a independent of. 5. (b) sin.cos. (sin. cos.) - sin. cos. - (sin. cos.) - ( ) ( sin. sec. cos. sin. cos. ) 6. (a) sin cos

23 .. cos sin.. sin ( cos ) ( cos )sin..... sin cos cos ( cos )sin..... sin cos cos ( cos )sin ( cos ) ( cos )sin.... sin. sin 0.? (b) 8. (a) (A - B) means the set of all elements which belong to A but do not belong to B. 9. (b) M W B M W B B B B B Boys can do the work in 55 days

24 boys can do the work in 55?? 0 days 0. (a) No (5 ) 8. R 8. (d) 7 gives unit s digit. (87) gives unit s digit gives unit s digit gives unit s digit 7. (). gives 7 as unit s digit 9 gives unit s digit (59)0.59 gives 9 as unit s digit A B C 5. (d). (b) L.C.M. of 8, 9, 79. No. div by 8, 9, leaving remainder (d) Milk? 0 0 l. Water 0 l A.T.S (d) Let the distance between Amritsar and Delhi be Time taken by first train 6 hrs. S 6 km/hr and that by second train 9 hrs S 9 km/hr Let the time taken (after 8 a.m.) by the two trains to cross each other be t hrs. A.T.S ( 6 6 t) 9 t.t hrs mins. Trains cross each other at 0. a.m. 6. (d) (c) Gain% Error True Error -? Weight used gms 8. (d)

25 9. (c) 8 9 No. should be div by 9 also Product consists of s only. Min no. of such s 9 Q... 9 times 9 8 which is div. by 9 0. (d). (a) Let the roots be α and α. Sum of roots α α p.α p- Product of roots α (α ) q ( p- ) ( p- ) q.p q. (c) log 5 log 9 9 log 9 9. log log9 log log (logee log log( ) log m n log m log log log. (a) a b c a b c. a b a b c - c b a ab c a b c c a b c ( ). a b ab -( ) ( )

26 a b c a b c. a b ab a b c a b c ( ) 0. (a b) [ c a b c ab abc a b c ( ) ( ) ] 0.(a b) [ca bc c ab] 0 Q abc. 0, a b c.0.(ab) (bc) (ca) 0. (a) y Squaring we get, y y. y - y Again squaring, y y y - y. 7 y - y y.y -y 8 5. (b) y y y - y ( y) - (y).( y - y) ( y y) - y y y - y y ( y) ( -y y) - y ( -y) ( y) ( - y y) - y ( -y) ( y) ( y) [ - y y - y y] ( y) ( y - y). G.C.D. y - y 6. (b) cos A - sin A -( ) mn m n m n m n - tan A sin cos A A mn m n -

27 7. (a) Let AB a. AC y - a AB BC AC. a (y-a).a y y sin A BC AB y y y y 8. (c) sin cosy cos siny sin siny cos cosy sin cosy sin siny cos siny cos cosy sin (cosy siny) cos (siny cosy) sin. cos. (sin.cos. ) 9. (c) tan. P B tan (d) 5. (a) 5. (b) On st May it was Friday. On rd May it was Sunday-, 0, 7,,. In May there were 5 Sundays A B C No. of calls on 5 Sundays Ramaining days 6. Reqd average (d) PQ r r QR r r 6 PR r r 8 Adding, (rrr) 8. r r r 9 5. (b) In. ABC, AB AC... AB AD. AC AD In. ACD, AC AD... In. BCD,. (.. ) (.. ). 80 or BCD is a rt. In. ACD, AC AD.. ACD is an isosceles.

28 () and () true 55. (c) AP AS BP BQ CR CQ DR DS Adding AP BP CR DR AS BQ CQ DS or AB CD (AS DS) (BQ CQ) AD BC 56. (a)( ) ( ) ( )( ) A B 5 On solving, A B 5.A, B (a) a a, b b 8, c c Q a a b b c c.. It has no solution 58. (a) 59. (c) log () log 6 log ( ) log log log But log (m n). log m log n 60. (c) (( ) ) ( ) irrational

29 If a (( ) ) ( ) rational no. is an irrational 6. (a) 6. (a) AB BD AD (in rt. ABD) BD (CD AC) BD CD AC In rt. BCD, CD.AC BC BD CD BC AC CD.AC > BC AC Q CD.AC > 0 6. (d) V, V () 8, V 7, V 6 6. (d) 65. (c) π π r (5 - ) 6 π.r cm 66. (b) T.S.A. of body (a) 5a (a a a) 6a 6 9 cm 67. (c) 68. (b) 8 7 π πr r?. ( ) 69. (b) T.S.A. of hemispherical bowl πr πr ( ) π π R r. πr πr π (R r) 7 ( 5 ) 86 cm 70. (c) LM LN MN Shaded area π[( ) ( ) ( ) ] LM LN MN π ( ) LM LN MN π ( ) MN MN π? π 7. (a).. OAB OCD ~.OB OD AB CD OB. l T.S.A. of rest portion πr π( ) r (π π r r l l )

30 πr (5r l) πr (5r h r 7. (d) D S t h 5 t (length of water) Vol. of a cylinder πr h A.T.S. π( ) 50 00? 5t π(0) 5.t 800 sec 80 min 7. (c) Radius of circular section 5. C r π πcm 7. (a) Length of the remaining sheet - a Length of bo Breadth of bo 0-a, Height of bo a. Vol of bo ( - a) (0 - a)a 0a - 8a a 75. (a) h h, h h r : r : r Let r r, r r, r r h : h : h r : r : r Ratio of vol πr h : ( π π r h r h ) : ( π π r h r h ) r r. : [( ). r r r r ] : [(r) r - (r) r] : 7 : (c) l d l, b d - l Area of A l b (d l) (d-l) d - l b Area of a square with side smaller side of B P

31 Q R A B C r r r A B C D A B C D P Q R S tangents from an et pt to a circle are equal B C A D B C A D l P r/ h/ h O Rr r r r h h h l b B d A d l' b' 77. (b) Reqd no. 78. (a) ( R 00 )5 or ( R 00 )5 9 ( R 00 )n. ( R 00 )n.[( R 00 )5] ( R 00 )n.( R 00 )0 ( R 00 )n.n 0 years 79. (c) A ka, k(0) 00.k A k() () 80. (b) Let C.P. Rs 00. S.P. (I) Rs 0 If Loss 0%, S.P. Rs 90

32 Diff 0-90 Rs 0 If diff Rs 0 then C.P. Rs 00 If diff is Rs 0 then C.P. Rs (a) L.C.M. of ,,, LCM of HCF of ,,,,,, (c) Workers hrs days more workers less days 5 : Less hrs more days 8 : 0.??? (d) Let the reqd no. of passengers be A.T.S (a) d d d km (t D

33 Speed ) 85. (c) Valid votes ,000 %age of invalid votes , -,,? (a) y r t rt y?? 00 00, z y rt rt z y? y 00z y.y z 87. (a) Let the common root be α. α α 0 b a... (i). α α 0 a b... (ii) Solving we get α (, - ) Qa b a b..0. a b 0. a b (b) S p q p -, P q p p, q are integers, p (b) l n.m. l n -m.0 or l -m n.0 l -m n lmn l (-m) n -l(-m)n [l(-m) n] [l (-m) n - l (-m) - (-m) n -nl] (l - m n) (l m n lm mn - nl) Clearly quotient l m n lm mn - nl 90. (c) 9. (a) α? -, α? α?? α

34 α? α? ( )- - α? α? α? a (-) - -?. a a 9. (a) R f() 7 9. (c) logy0 log0y, log0 ( y 0 ) log0y - log00-9. (a) y z 0, y -z Squaring we get, y y z or y - z -y Sly y z - - yz z - y -z Value - - y yz z - ( ) z y yz (a) 96. (c) α? - a, α? b. - - α? a, (-α ) (-?) b Clearly the reqd roots are -α, -? 97. (b) ( -) ( - k) (p ) p -p (k ) (pk -k - ) k Constant term k Coeff of -p (k) - p ( ) 0. p (a) m -n (6 -) 6 ( -) 0-6. m (a) b ( ) ( a ) (- b) Comparing the coeff of and constant terms,

35 a, b (b) y.yz z y. y y (y ) z z or yz z z or yz 0 yz - : : 5 :