Name: _____________________ () Class: Date: Marks:Time:Section A Questions (45 marks)Question-1.By mathematical induction, prove that 6) 13 2 )( 1 () 4 ( 7 3 6 21+ + + + + + + n n nn n !or all positive inte"ers n.Question-2.By mathematical induction, prove that 63 ) 1 2 )( 1 2 )( 3 2 () 1 2 )( 1 2 ( 7 3 3 1+ + + + + + + + n n nn n !or all positive inte"ers n.Question-3.By mathematical induction, prove that) 1 6 3 ( ) 2 3 )( 1 3 ( 11 # # 22+ + + + + + + n n n n n !or all positive inte"ers n.Question-4.By mathematical induction, prove that1221 ) 1 4 )( 3 4 )( 7 4 () 3 4 )( 1 4 ( 14 11 11 7 7 3+ + + + + + + + n n nn n !or all positive inte"ers n.Question-.By mathematical induction, prove that 4) 3 )( 2 )( 1 () 2 )( 1 ( 4 3 2 3 2 1+ + + + + + + + n n n nn n n !or all positive inte"ers n.1Question-6.By mathematical induction, prove that126 ) 2 3 )( 1 3 )( 4 3 )( 7 3 () 4 3 )( 1 3 )( 2 3 ( 13 1$ 7 1$ 7 4 7 4 1+ + + + + + + + + + n n n nn n n !or all positiveinte"ers n.Question-7.By mathematical induction, prove that 2 3121) 2 3 )( 1 3 (311 #3# 3 23+ + + +++ n n n !or all positive inte"ers n.Question-#.By mathematical induction, prove that 1 2 1 ) 2 (11 611 411 212 2 2 2++ +++ nnn !or all positive inte"ers n.Question-%.By mathematical induction, prove that 1) 1 (1) 1 () 1 2 ( ) 1 (4 373 2

2 131 1++ ++ + ++ +n n nnn n !or all positive inte"ers n.Question-1$.By mathematical induction, prove that n nn n 2 ) 1 2 ( 1 2 ) 1 2 ( 2 7 21 31 2 + + + + + + !or all positive inte"ers n.Question-11.By mathematical induction, prove that 1 21 2) 1 2 )( 1 2 (2) 1 2 )( 1 2 (2) 1 2 )( 1 2 (11 113 2 2 + + + + +nnn nn !or all positive inte"ers n.Question-12.By mathematical induction, prove that 211 2) 1 (1 ) 1 (3 2 1+ + + + + ++xx n nxnx x xn nn !or all positive inte"ers n.2Question-13.By mathematical induction, prove that 222) 2 )( 1 (2 42 34 32 23 22 12 1 4 3 2++ ++ ++++ +n n nnn n !or all positive inte"ers n.Question-14.&et 2) 1 (11+ nTn . By mathematical induction, prove that ) 1 ( 223 2 1++ nnT T T Tn !or all positive inte"ers n.Question-1.&et 2) 3 2 (41+ nTn . By mathematical induction, prove that ) 3 2 ( )2 ( 33 2 1++ nnT T T Tn !or all positive inte"ers n.Question-16.&et ) 1 (12+ +n nn nTn . By mathematical induction, prove that 123 2 1+ + + + +nnT T T Tn!or all positive inte"ersn.Question-17.&et ' n n Tn , (heren n 2 1 ' . By mathematical induction, prove that1 )' 1 (3 2 1 + + + + + n T T T Tn !or all positive inte"ers n.Question-1#.&et nnnT21 . By mathematical induction, prove that nnnT T T T2113 2 1+ + + + + !or all positive inte"ersn.Question-1%.&et 1 2 1 22 + +n nTn . By mathematical induction, prove that1 1 23 2 1 + + + + + n T T T Tn!or all positive inte"ers n.3Question-2$.)olve361 2 +n nC C .Question-21.)olve1#222 +n nC C .Question-22.)olven C Cn n321 2 + .Question-23.)olve 32 21212 + +n n nC C C .Question-24.*+pand

21

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+xxin descendin" po(ers o! x.Question-2.*+pand 7) 4 1 ( x +in ascendin" po(ers o! x up to the 4th term.Question-26.*+pand #) 1 3 ( xin ascendin" po(ers o! x up to the 4th term.Question-27.*+pand %) 3 2 ( + xin ascendin" po(ers o! x up to the 4th term.Question-2#.*+pand #) 3 ( y x + in ascendin" po(ers o! x up to the 4th term.Question-2%.4*+pand 1$) 2 1 ( x in descendin" po(ers o! x up to the th term.Question-3$.*+pand %12

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x in descendin" po(ers o! x up to the 4th term.Question-31.*+pand 1243

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xxin descendin" po(ers o! x up to the 3th term.Question-32.*+pand 11 2) 3 ( + xin descendin" po(ers o! x up to the 4th term.Question-33.*valuate ) 3 2 ( lim3x e exx +.Question-34.*valuate,_

+ +xxe ex212 1lim.Question-3.,ind the coe!!icient o! x3y7 in the e+pansion o! 1$)4 ( y x + .Question-36.,ind the coe!!icient o! x4y# in the e+pansion o! 1222

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yx .Question-37.,ind the th term in the e+pansion o! 7421

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+ xin ascendin" po(ers o! x.

Question-3#.,ind the #th term in the e+pansion o! 12242

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xxin ascendin" po(ers o! x.Question-3%.,ind the 6th term in the e+pansion o! %234

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+xxin descendin" po(ers o! x.Question-4$.,ind the %th term in the e+pansion o! 1$232

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x in descendin" po(ers o! x.Question-41.,ind the constant term in the e+pansion o! %212

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xx .Question-42.,ind the constant term in the e+pansion o! #3412

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xx .Question-43.,ind the constant term in the e+pansion o! 12

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+xx .Question-44.,ind the constant term in the e+pansion o! %236

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xx.Question-4.,ind the coe!!icient o! x in the e+pansion o! 4 3)2 ( ) 3 1 ( x x + . 6Question-46.,ind the coe!!icient o! x2 in the e+pansion o! 7 4) 1 2 ( ) 4 1 ( + x x . Question-47.,ind the coe!!icient o! x3 in the e+pansion o! # 2) 1 ( ) 2 ( x x + . Question-4#.,ind the coe!!icient o! x3 in the e+pansion o! 62) 2 3 ( 12xx+

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.Question-4%.*+pand nx ) 1 3 ( in descendin" po(ers o! x up to the 3th term.Question-$.*+pand nax ,_

+3

2in descendin" po(ers o! x up to the 4th term.Question-1.&et Tr -e the coe!!icient o! the rth term in the e+pansion o!

)3 ( x +in ascendin" po(ers o! x. ,ind T4 . T3.Question-2.,or (hat value o! c (ill the coe!!icients o! x3 and x4 in the e+pansion o! #) 1 ( cx +-e e/ual0Question-3.&et n -e a positive inte"er. 1n the e+pansion o! nx) 3 1 ( + , i! the coe!!icient o! x3 is t(ice that o! x2, !ind n.Question-4.1n the e+pansion o! nx) 1 ( + , the coe!!icient o! x

is the arithmetic mean o! the coe!!icients o! x4 and x6. ,ind the possi-le values o! n.7Question-.2iven that n is a positive inte"er and 4th term in the e+pansion o! nxx

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2312in descendin" po(ers o! x is a constant, !ind n and determine the value o! this term.Question-6.1n the e+pansion o! nx ) 3 (2+in descendin" po(ers o! x, (here n is a positive inte"er, the coe!!icient o! the third term is 13. ,ind the value o! n and the coe!!icient o! x4.Question-7.3hen nxx

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+4416is e+panded in ascendin" po(ers o! x, the 4th term o! the e+pansion is a constant.(a) ,ind n.(b) ,ind the constant term o! the e+pansion.Question-#.1t is "iven that + + + +21 )1 ( bx ax xn(a) *+press a and b in terms o! n.(b) 1! b 4 37, !ind a and n.Question-%.1t is "iven that+ + +

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+24$%6#4 bx axxn terms involvin" hi"her po(ers o! x.(a) ,ind n.(b) ,ind a and b.Question-6$.(a) *+pand nx ax ) 2 1 ( ) 1 (

+ in ascendin" po(ers o! x up to the x2 term.(b) 1! the coe!!icients o! x and x2 in (a) are 3 and 1$ respectively, !ind a and n.Question-61.1n the e+pansion o! #) )( 1 ( bx a x + , the coe!!icient o! x

is 5ero. ,ind the value o! ba.#Question-62.1n the e+pansion o! nqx px ) 1 )( 4 1 ( + + , the coe!!icient o! x is 5ero and the coe!!icient o! x2 is 64$. ,ind the values o! p and q.Question-63.(a) *+pand

) 1 ( bx in ascendin" po(ers o! x.(b) 7onsider the e+pansion 3) 1 ( ) 4 ( bx x + .(i) *+press the coe!!icient o! x2 in the e+pansion in terms o! b.(ii) 1! the coe!!icient o! x2 in the e+pansion is 14#12, !ind b.Question-64.&et m -e a positive inte"er.(a) *+pand

) 1 ( )1 ( ax xm +in ascendin" po(ers o! x up to the x2 term.(b) 1! the coe!!icients o! x and x2 in the a-ove e+pansion are 2$ and %$ respectively, !ind a and m.Question-6.1! nx ax x f ) 2 1 ( ) 1 ( ) (4 2+ + , (here n is a positive inte"er and a is a constant, !ind the values o! n and a so that the e+pansion o! f(x) is + + +2222 22 1 x xQuestion-66.1n the !i"ure, OXY is a sector (ith radius 1 cm and its area is 1$8 cm2.(a) ,ind XOY in radian measure.(b) 9ence, !ind the perimeter o! the sector. (&eave your ans(ers in terms o! .)Question-67.:he !i"ure sho(s a sector OAB (ith radius R cm and AOB 4 1.3 rad.:he area o! sector OAB is 26 cm2.%(a) ,ind the value o! R. (&eave your ans(ers in surd !orm.)(b) 9ence, !ind the area o! OAB. (2ive your ans(er correct to 3 si"ni!icant !i"ures.)Question-6#.1n the !i"ure, OAB and OXY are t(o sectors o! radii 1$ cm and 1# cm respectively. OAX and OBY are strai"ht lines. :he area o! the shaded re"ion is 1$$ cm2.(a) ,indXOY in radian measure.(b) 9ence, !ind the di!!erence o! arc AB and arc XY.(2ive your ans(er correct to 3 si"ni!icant !i"ures.)Question-6%.:he !i"ure sho(s a rectan"le ABCD (ith dimensionscm12 cm.ADE is a sector (ith centre A.(a) ,ind DAC in radian measure.(b) 9ence, !ind the area o! the shaded re"ion. (2ive your ans(ers correct to 3 si"ni!icant !i"ures.) Question-7$.1n the !i"ure, OAB and OCD are t(o sectors (ith the same centre O, AC 4 2 cm, OAC and OBD are strai"ht lines. :he ratio o! the len"th o! arc AB to that o! arc CD is 2 . 3.(a) ,ind AO.(b) 9ence, !ind the ratio o! the area o! the shaded re"ion to the areao! the sector OCD.1$Question-71.1n the !i"ure, OAB is a sector (ith radius R cm and rad3 AOB.:he sector OAB is !olded into a ri"ht circular cone (ith radius r cmso that OA and OB ;oin to"ether.(a),ind the radius o! the -ase o! the cone in terms o! R. (b),ind OP in terms o! R. (c)1! the volume o! the cone is 3 2cm3, !ind R. (2ive your ans(ers correct to 3 si"ni!icant !i"ures i! necessary.)Question-72.1! 14sin and$ sec < , !ind cotsec sin .Question-73.1! 1217sec and 23 < ABC is an isosceles trian"le (ith ABC 4 #rad and AB 4 AC. BA is produced to D so that CD is perpendicular to BD. )uppose CD 41.(a),ind CAD and AC. (&eave your ans(ers in surd !orm i! necessary.)(b) ,ind BD and BC. (&eave your ans(ers in surd !orm i! necessary.)(c) 9ence, sho( that(i)tan#4 1 2 ,(ii) cosec#4 2 2 4 + .Question-1$.1t is "iven that6 cosec sec , (here is acute and cos sin > .(a) By e+pandin" 2) cos (sin + , sho( that 32cos sin + .(b) ,ind the value o! cos sin .(c) Asin" the result o! (a) and (-), solve . (2ive your ans(ers correct to 3 si"ni!icant !i"ures.)Question-11.(a) =rove that 2 2cosec ) 1 2 ( B cot cosec ) 1 BC( cot cosec ) 1 C( + + + + + a a a a a a , (here a is a constant.(b) 9ence, solve the !ollo(in" e/uations !or 2 $ . (2ive your ans(erscorrect to 3 si"ni!icant !i"ures i! necessary.)(i) cot41cosec4

cot41cosec4

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(ii)3tan 41sin 4%2 2 16