xf.kr MATHEMATICS Ñi;k tk p dj ysa fd bl iz'u&i= esa eqfnzr iz'u 20 gsaa Please make sure that the printed question paper are contains 20 questions.

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1 CLASS : th (Sr. Secondary) Code No. 6 Series : SS-M/08 Roll No. SET : C f.kr MATHEMATICS [ Hindi and English Medium ] ACADEMIC/OPEN (Only for Fresh/Re-appear Candidates) Time allowed : hours ] [ Maimum Marks : 80 Ñi;k tk p dj ysa fd bl iz'u&i= esa eqfnzr iz'u 0 gsaa GRAPH Please make sure that the printed question paper are contains 0 questions. iz'u&i= esa nkfgus gkfk dh vksj fn;s ;s dksm uecj rfkk lsv dks Nk= mùkj&iqflrdk ds eq[;&i`"b ij fy[ksaa The Code No. and Set on the right side of the question paper should be written by the candidate on the front page of the answer-book. Ñi;k iz'u dk mùkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad vo'; fy[ksaa Before beginning to answer a question, its Serial Number must be written. mùkj&iqflrdk ds chp esa [kkyh iuuk@iuus u NksMsa+A Don t leave blank page/pages in your answer-book. mùkj&iqflrdk ds vfrfjdr dksbz vu; 'khv ugha feysha vr% vko';drkuqlkj gh fy[ksa vksj fy[kk mùkj u dkvsaa Ecept answer-book, no etra sheet will be given. Write to the point and do not strike the written answer. ijh{kkfkhz viuk jksy ua0 iz'u&i= ij vo'; fy[ksaa Candidates must write their Roll Number on the question paper. d`i;k iz'uksa dk mùkj nsus ls iwoz ;g lqfuf'pr dj ysa fd iz'u&i= iw.kz o lgh gs] ijh{kk ds mijkur bl lecu/k esa dksbz Hkh nkok Lohdkj ugha fd;k tk;ska 6/(Set : C) P. T. O.

2 6/(Set : C) ( ) 6/(Set : C) Before answering the question, ensure that you have been supplied the correct and complete question paper, no claim in this regard, will be entertained after eamination. lkeku; funsz'k % (i) bl iz'u-i= esa 0 iz'u gsa] tks fd pkj [k.mksa % v] c] l vksj n esa ck Vs, gsa % [k.m ^v* % bl [k.m esa,d ç'u gs tks cgqfodyih; çdkj ds 6 (i-vi) Hkkksa esa gsa izr;sd Hkk vad dk gsa [k.m ^c* % bl [k.m esa ls rd dqy nl ç'u gsaa çr;sd ç'u vadksa dk gsa [k.m ^l* % bl [k.m esa ls 6 rd dqy ik p ç'u gsaa çr;sd ç'u vadksa dk gsa [k.m ^n* % bl [k.m esa 7 ls 0 rd dqy pkj ç'u gsaa çr;sd ç'u 6 vadksa dk gsa (ii) lhkh ç'u vfuok;z gsaa (iii) [k.m ^n* ds dqn ç'uksa esa vkarfjd fodyi fn;s ;s gsa] muesa ls,d gh iz'u dks pquuk gsa (iv) fn;s ;s zkq-isij dks viuh mùkj-iqflrdk ds lkfk vo'; urfkh djsaa (v) zkq-isij ij viuh mùkj-iqflrdk dk Øekad vo'; fy[ksaa (vi) dsyd;qysvj ds ç;ks dh vuqefr ugha gsa General Instructions : (i) This question paper consists of 0 questions which are divided into four Sections : A, B, C and D : Section 'A' : This Section consists of one question which is divided into 6 (i-vi) parts of multiple choice type. Each part carries mark. Section 'B' : This Section consists of ten questions from to. Each question carries marks. Section 'C' : This Section consists of five questions from to 6. Each question carries marks. Section 'D' : This Section consists of four questions from 7 to 0. Each question carries 6 marks. (ii) All questions are compulsory. (iii) Section 'D' contains some questions where internal choice have been provided. Choose one of them.

3 ( ) 6/(Set : C) (iv) You must attach the given graph-paper along with your answerbook. (v) You must write your Answer-book Serial No. on the graph-paper. (vi) Use of Calculator is not permitted. [k.m v SECTION A. (i) ;fn f ( ) =, vksj g ( ) =, rks fog () gs % (ii) buesa ls dksbz ugha + If f ( ) =, and g ( ) =, then fog () is : None of these cos tan dk eku gs % buesa ls dksbz ugha The value of cos tan is : None of these 6/(Set : C) P. T. O.

4 (iii) lehdj.k + = 0 buesa ls dksbz ugha In the equation X +, 0 = X is : None of these + (iv) ;fn = 0, (v) 6/(Set : C) X esa X dk eku gs % ( ) 6/(Set : C) rks dk eku gs % + If = 0, then value of is : cos dk ds lkis{k vodyt gs % sin sin sin buesa ls dksbz ugha The derivative of cos w. r. t. is : sin sin sin None of these (vi) Qyu f() = cos sin dk vf/kdre ;k U;wure eku gs = ij %

5 ( ) 6/(Set : C) The function f() = cos sin has maima or minima value at =. (vii) θ = ij oø = a(θ sinθ), y = a[ cos θ] dh Li'kZjs[kk dh ço.krk gs % The slope of tangent to the curve = a(θ sinθ), y = a[ cos θ] at θ = is : (viii) cos d dk eku gs % ( sin) + c ( cos) + c ( + sin) + c buesa ls dksbz ugha The value of cos d is : ( sin ) + c ( cos ) + c ( + sin ) + c 6/(Set : C) P. T. O.

6 None of these (i) + d + c log+ tan + c buesa ls dksbz ugha The value of + d is : + c log+ + tan + c None of these () d y dy = + d d ( 6 ) 6/(Set : C) dk eku gs % tan ( ) + c tan ( ) c vodyu lehdj.k dh?kkr gs % buesa ls dksbz ugha The degree of the differential equation (i) vodyu lehdj.k dy d y = + cos + c y = cos + c None of these + sin y = cos + c buesa ls dksbz ugha d d = dk gy gs % y = + dy d is : 6/(Set : C)

7 Solution of the differential equation y = + cos + c y = cos + c y = cos + c ( 7 ) 6/(Set : C) dy = + sin is : d None of these (ii) ;fn P = 0., P = 0.6 vksj P(A B) = 0.8, rks P(A/B) gs % If P = 0., P = 0.6 and P(A B) = 0.8, then P(A/B) is : (iii) vpnh rjg QsaVh ;h rk'k dh M~Mh ls nks iùks ykrkj fcuk çfrlfkkiu ds fudkys ;s gsaa çr;sd ds gqdqe (Spade) dk iùkk gksus dh çkf;drk gs % 7 7 The probability of drawing a spade on each of the two consecutive draws from a well-shuffled pack of cards without replacement is : 7 7 6/(Set : C) P. T. O.

8 ( 8 ) 6/(Set : C) (iv) ;fn A vksj B nks LorU=?kVuk, bl çdkj gsa fd P = vksj P =, rks P(A B) gs % buesa ls dksbz ugha If A and B are two independent events such that P = and P =, then P(A B) is : None of these (v) ;fn a =, = b vksj a. =, rks a vksj b ds chp dk dks.k gs % b 6 If a =, b = and a. b =, then angle between a and b is : 6/(Set : C)

9 ( 9 ) 6/(Set : C) 6 (vi) nks fcunqvksa (,, ) vksj (,, ) dks feykus okyh js[kk dk fnd~-dkst;k gs % 8,, ,, ,, buesa ls dksbz ugha The direction cosine of a line joining the two points (,, ) and (,, ) are : 8,, ,, ,, None of these [k.m c SECTION B. n'kkzb, fd f : N N, fn;k gs % f, ;fn fo"ke gs (,dsdh gsa, ;fn legs + ) = Show that f : N N, given by : +, if is odd f ( ) = is one-one., if is even 6/(Set : C) P. T. O.

10 ( 0 ) 6/(Set : C). fl) dhft, % Prove that :. ;fn =, sec + cosec = sec + cosec = A ( ) =, If A =, f ( ) =, then find f. f rks f Kkr dhft,a. f=hkqt dk {ks=qy Kkr dhft, ftlds 'kh"kz (, ), (, ) vksj (, ) gsaa Find the area of the triangle whose vertices are (, ), (, ) and (, ). 6. sin dk ds lkis{k vodyt Kkr dhft,a Find the derivative of sin w. r. t.. 7. dy Kkr dhft,] tcfd d = Find dy, when = a(θ + sinθ), y = a( cosθ) at θ = d. 6/(Set : C) θ ij = a(θ + sinθ), y = a( cosθ)a 8. eku Kkr dhft, % Evaluate : cos d cos d 9. eku Kkr dhft, % Evaluate : d 9 + d. 9 +

11 dy y y d Solve the differential equation : dy y d 0. vodyu lehdj.k ( ) 6/(Set : C) = dks gy dhft,a = y.,d FkSys esa lqsn] 7 yky vksj 8 dkyh sansa gsaa çfrlfkkiu ds lkfk,d-,d djds pkj san fudkyh ;haa de ls de,d lqsn gksus dh çkf;drk D;k gs \ A bag contains white, 7 red and 8 black balls. Four balls are drawn one by one with replacement. What is the probability that at least one is white? [k.m l SECTION C. fl) dhft, % Prove that :. n'kkzb, fd cot cot + sin + + sin + sin + + sin ;fn ;fn sin sin sin sin =. =. f ( ) = ] = ij larr gs ijurq O;qRik ugha gsa < if Show that f ( ) = is continuous but not derivable at if < =.. n'kkzb, fd ( ) ij vfrijoy; = a 0 b yy 0 = 0, y 0 a b y gsa dh Li'kZjs[kk dk lehdj.k 6/(Set : C) P. T. O.

12 ( ) 6/(Set : C) y Show that the equation of tangent to the hyperbola = b a ( 0, y 0 ) is 0 yy0 = a. b.,d FkSys esa lqsn vksj 6 yky sansa gsaa FkSys ls sansa ;kn`pn;k fudkyh ;h gsaa lqsn sanksa dh la[;k dk çkf;drk cavu Kkr dhft,a An urn contains white and 6 red balls. Four balls are drawn at random from the urn. Find the probability distribution of the number of white balls. at 6. a = iˆ + ˆj kˆ vksj b = iˆ + ˆj + kˆ ij vfhkyec ek=d lfn'k Kkr dhft,a Find a unit vector perpendicular to the a = iˆ + ˆj kˆ b = iˆ + ˆj + kˆ. [k.m n SECTION D and 7. fueu lehdj.kksa dks vko;wg fof/k }kjk gy dhft, % 6 y + z =, + y z =, y + z =. Solve the following equations by matri method : y + z =, + y z =, y + z =. 8. ijoy; y = a vksj = ay, a > 0 ds chp ds {ks= dk {ks=qy Kkr dhft,a 6 Find the area of region included between the parabola = ay, a > 0. eku Kkr dhft, % 6/(Set : C) vfkok OR y = a and

13 Evaluate : ( ) 6/(Set : C) / d sin + cos 0 / d sin + cos 0 9. fcunq (0,, 7) ls js[kk 6 + y = = z ij vfhkyec dk ikn (foot) Kkr dhft,a Find the foot of perpendicular from the point + y = = z. (0,, 7) to the line vfkok OR fcunqvksa (, 6, 6), (, 0, 9) vksj (, 0, 6) ls qtjus okys ry dk lehdj.k Kkr dhft,a Find the equation of the plane passing through the points (, 6, 6), (, 0, 9) and (, 0, 6). 0. fueu L.P.P. dks zkq }kjk gy dhft, % 6 U;wure % Z = + y O;ojks/kksa ds vurzr % + y, + y, + y,, y,, y 0. Solve graphically the following L. P. P. : Minimize : Z = + y subject to constraints : + y, + y, 6/(Set : C) P. T. O.

14 ( ) 6/(Set : C) + y,, y,, y 0. 6/(Set : C)

xf.kr MATHEMATICS Ñi;k tk p dj ysa fd bl iz'u&i= esa eqfnzr iz'u 20 gsaa Please make sure that the printed question paper are contains 20 questions.

xf.kr MATHEMATICS Ñi;k tk p dj ysa fd bl iz'u&i= esa eqfnzr iz'u 20 gsaa Please make sure that the printed question paper are contains 20 questions. CLASS : th (Sr. Secondary) Code No. 0 Series : SS-M/07 Roll No. SET : A f.kr MATHEMATICS [ Hindi and English Medium ] ACADEMIC/OPEN (Only for Fresh Candidates) (Evening Session) Time allowed : hours ]