Assignment ( ) Class-XI. = iii. v. A B= A B '

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1 Assigmet (8-9) Class-XI. Proe that: ( A B)' = A' B ' i A ( BAC) = ( A B) ( A C) ii A ( B C) = ( A B) ( A C) iv. A B= A B= φ v. A B= A B ' v A B B ' A'. A relatio R is dified o the set z of itegers as: (, y) R + y 5. Epress R & of ordered pairs ad hece fid their respective domais.. Let R is a relatio o the set of itegers defed by. (, y) R y is divisible by i (, y) R y is divisible by a. (, ) R z b. (, ) R ( y, ) R, y z c. (, ) R & ( y, z) R (, z) R, y, z z 4. Let R is a relatio o Q defied by R {( a, b) : a, b Q & a b z} ( a, a) R a Q i ( a, a) R ( b, a) R a, b, Q ii ( a, a) R, ( b, c) R ( a, c) R a, b, c Q =. Show that 5. Let R be a relatio o N defied as R { a b a b N a b } ( a, a) R a N i ( a, a) R ( b, a) R ii ( a, a) R, ( b, c) R ( a, c) R 6. Fid the domai ad rage of the fuctios: = (, ) :, & =. Are the followig true? R as the set f ( ) 9 f ( ) = ii + = i 7. Fid the domai of f ( ) = lg4{ log5( log (8 77) )} f ( ) = iv. + f ( ) = si 8. Fid the domai of f ( ) = ac 9. If αβ, are the solutios of the equatio a ta + b sec = c, the show that ta ( α+ β) = a c

2 . Prove that i cos cos cos 5 cos 7 = 6 si si 4 si 6 si8 = 6 ii iv. cos 6 cos 4 cos66 cos78 = 6 5 si 4 si 6 = 8. Prove that (cosα cos β) (siα si β) 4 cos α β =. If π 4π cosθ y cos θ = + = z cos θ+, the prove that y yz z + + =.. Prove that si 5 si + si = ta cos 5 cos 4. Prove that 5. Prove that π π si + si + + si 4 + = si. 4 π π cos cos cos + 4 π π 6. Prove that ta + ta + ta = ta 7. Prove that ta 8 = π 8. Prove that cot = Prove that ta 4 = + 6. Prove that ( I ay ABC) : i a (cos C cos B) = ( b c) cos 8 a cos A+ b cos B+ cos C= abc A a cos( B C) + b cos ( C A) + C cos A B = abc ii ( )

3 iv. ( ) ( ) b c A c a B a b C cot + cot + ( ) cot = v. b c c a a b si A+ si B+ si C= a b c. Solve that trig. Equatios : cos + si = i 4 si si si 4= si ii si si 4+ si 6= iv. ta + ta + ta = v. cos ec = + cot v si si + si = cos cos + cos. Usig mathematical iductio, prove that θ si + siθ+ si θ+ si θ+ + si θ= si θ, z θ si i si θ siθ+ siθ+ + si ( ) θ=, siθ si α = siα ii cosα cos α cos 4α cos( α) iv. S + 4η 5 is divisible by 576 η N = N 8 + V ( 9 ). If + = + a, b,, y, R, Show that / ( iy) a ib, y + = a b 4 ( a b ) β α 4. If αβare, differet comple umbers with β= the prove that = αβ 5. Solve z = z 6. Fid the modulus of argumet of i= z= π π cos + i si 7. Solve : (7 ) + (8 ) = i i 8. Solve : 6, < + 4 4

4 i ii + + > + iv If the letters of the followig words are arraged as i dictioary the fid the rak of the give word: SERIES i AGAIN ii SURITI iv. ZENITH. How may 4-letter words ca be formed from the letter of the followig word? INTERMEDIATE i MATHEMATICS ii ALLAHABAD. How may words ca be formed from the letters of the word ALLAHABAD. There is o restrictio. i Each word starts with A eds with D ii Vowels are together iv. Vowels occupy odd places. I how may ways 4 cards ca be selected from a pack of 5 cards if. The four cards are of same suit i The four cards are of differet suit. ii The four cards are of same umber. iv. The four cards are face cards. v. Two cards are red & two are black?. I haw may ways a committee of 4 ca be made from 4 me ad wome icludig. at least wome. i at most wome. ii at least I ma ad I woma? 4. If a & b are distict itegers, proe usig biomial theorem that a b is divisible by a b. 5. If rd, 4 th, 5 th terms i the epasio of ( + a) are 84, 8 & 56 respecively, fid the values of a, &. 6. If the 7 th term from the begiig ad ed i + are equal, fid. 7. The sum of three umbers i GP is ad the seem of their squares i 89. Fid the umbers. 8. Sum to terms : (i) (ii)

5 9. If S be the sum, P the profuct ad R the sum of reciprolas of terms of a GP, proe that S = P R 4. If f is a fuctio satisfyig f ( + y) = f ( ). f ( y), y N such that f () = ad f ( ) =, di. 4. If a & b are rests of + p= ad c, d are the roots of Prove that (9 + p) : ( q p) = 7 :5 + = where a, b, c, d are i GP. q 4. Sum to terms: (i) (ii) + ( + ) + ( + + ) +... (iii) ( + ) ( + ) 4. It S, S, S are sum of is + atural umbers their squares ad their cubes, prove that 9 S = S (+ 8 S ) 44. Fid the equatio of the lie passig through (, ) ad cuttig off itercepts o the aes whose sum is of 45. The equatio of the base of a equilateral triagle is + y= ad its verte is (, ), fid the legth ad eudatios of the ather sides. 46. Fid the equatio of the lie passig through itersectio of the lies y= 5& + y= ad (i) makes equal & the itercepts o the aes (ii) whose distace from the origi is Fid the equatio of the circle circumscribig the triagle whose sides are + y =, + y = & + y 5= 48. The lie y+ 6= meets the circle o AB as diameter. + 9= at A & B. fid the equatio of the circle y y 49. Fid the equatio of the parabola of (i) focus is (,) & taget at verte is +y= (ii) laotus rectum is 4, ais is +4y-4=&taget at verte is 4-y+7= (iii) vertees (, ) ad directress is = y. 5. Fid the electricity, vertices, foci, directrices, the legth & equatio of the latus rectum of ellipse: (i) (ii) = y y 5 + 7y 5 9y+ 5= 5. Fid the equatio of ellipse, give.

6 4 (i) Vertices (, ± ), e= 5 (ii) Foci ( ±, ), a= 4 5. Fid the electricity, vertices, foci, divectrices legth & equatio of latus rectum of the hyperbola 6 9y + + 6y 64= 5. If e ad e are eccetricities of a hyperbola ad its cojugate, prove that + e e ' =. 54. Fid the equatio of hyperbola of (i) foci (± ) cojugate ais = 4 (ii) Vertices (± 6) e= 5 / Fid the locus of the poit P if (i) PA+ PB= where A (4,,) & B ( 4,,) (ii) PA + PB = k where A(,4,5) & B (,, 7) 56. Fid the ratio i which the lie joiig A (4, 8,) ad (6,, 8) is divided by (i) y plae (ii) the plae + y+ z = 57. Fid the legths of the melos of the triagle where vertices are A (,, 6) B (, 4, ), & c (6,, ) 58. Evaluate the followig limits lt (i) y ( + y) sec ( + y) sec y lt + cos (ii) si (iii) lt π/4 ta ta π cos + 4 lt (iv) si si + si 5 lt cos cos (v) lt ta + 4 ta ta (vi) ta 59. Fid dy from first priciple if y= d (i) si (ii) si (iii) cos (iv) ta (v) e (vi) log cos (vii) si 6. Differetiate the followig fuctios: (i) sec sec + (ii) si cos si + cos (iii) cos + 4 (iv) e cos + ta

7 6. Prove that followig statemets are true usig (a) Direct method (b) Cotra positive method (c) Method of cotradictio. (i) If & y are odd itegers the y is a odd itere (ii) If is eve the is eve. (iii) If is a real o. such that + 4= the = 6. Fid (a) Mea deviatio about mea (b) Mea deviatio about media (c) Mea, variace & S. Deviatio for the followig data. C.I f A bo cotais red marbles, blue marbles ad gree marbles. If 4 marbles are draw from the bo. What is the probability that. (i) All will be blue. (ii) At least oe will be gree. (iii) There is at least oe faces each with umber. 64. A die has two faces each with umber three faces with umber ad oe face with umbers. It the die is rolled oce, fid (i) P () (ii) P ( or ) (iii) P ( A ) 65. If 4-digit umbers greater tha 5, are radomly fromed from the digits,,, 5, 7 what is the probity of formig umber divisible by 5 whe. (i) the digits are repeated? (ii) the digits are ot repeated? 66. If 4 cards are draw at radom from the pack of 5 playig cards, fid the probability of gettig. (i) all 4 cards of same suit. (ii) all 4 cards of same umber. (iii) all 4 cards beig face cards. (iv) red ad black cards (v) I card from each suet 67. Two dice are throw simultaeously. Fid the probability of gettig. (i) Sum total 8 (ii) Sum total more tha

8 (iii) Eve umber o the fist die & total sum If 5 me ad 4 wome are arraged i a lie at radom. What is the probability that o two wome are together? 69. If letters of the word MARBLES. Fid the probability that (i) both are vowels (ii) both are cosoats (iii) oe is level & oe is cosoat? 7. If cois are tossed simultaeously, fid the probability of gettig (i) heads (ii) at least heads (iii) head o the fist ad tail o the secod. 7. If A ad B are two evets such that P( A) =.54, P( B) =.69 & P( A B) =.5 Fid: (i) P( A B) (ii) P( A B) (iii) P( A B) (iv) P( A B)

JEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018)

JEE(Advanced) 2018 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 20 th MAY, 2018) JEE(Advaced) 08 TEST PAPER WITH SOLUTION (HELD ON SUNDAY 0 th MAY, 08) PART- : JEE(Advaced) 08/Paper- SECTION. For ay positive iteger, defie ƒ : (0, ) as ƒ () j ta j j for all (0, ). (Here, the iverse