LHROU-H- NY ‘oll No ____ (To be filled in by the candidate) AATHEMATICS {Academic Sessions 2015 ~ 2017 to 2018 - 2020 ) MPAPER-T( Objective Type) 219-(INTER PART -1) Time Allowed : 30 Minutes GROUP =I Maximum Marks + 20 PAPER CODE = 6195 love ; Four possible answers A, B,C and D to each question are given. The choice which you think i carect, fill that cnolo infront ofthat question with Marker or Pen ink inthe answer-book. Cutting or filling {80 oF more circles will recut in_zero matk in that question TA [If x—a isa factor of polynomial f(x) , then f(a) i @ (By <0 (© >0 @) +0 2 | If "Cs="C, , then n is a 9 B) 7 os @) 5 T| The multiplicative inverse of(1,—2)= = wo Ga o@ dh om G2 © [ot term in te sequence ws os oF o 3 | The contrapositive of ~ p> ~q is : (A) Pq (B) a>p (C) ~q>~P @) ~a>P O [From the identity Sx+4=4(x-1)+B(2+2), then value of B (A) -3 (B) 3 (C) -2 (D) 2 7 | The sum of four 4" roots of 16 is = ao 2 4 D) 16 ele re “AE tien ee: (A) 5 @) -5 © 1 @) 1 9 | The arithmetic mean between V2 and 34/2 is = (A) 3N2 (B) 2v2 © 42 (D) v2 10, 12-27 le 7 3 {A) (®B) 1 (Cy 3 @) oO T1 | Period of cot@ is = (A) = (B) 20 @ ©) = ( Tum Over}CHRIS «9 1-12] Number of signals can be made with 4 flags when one flag is used at atime are = (A) * @B) 6 © D 'G "|The cquation sin?x-seex=7 iscalled: (A) Trigonometric equation (B)_ Linear equation (©) Quadratic equation (D) Quantic equation 141 3sina—4sin'a = : (A) sine (B) sind (C) sinze (D) singer 15] Domain of the function y=sin"'x is : (A) -2exe -lsy¥s Sy ises —Fayst & -Fsxs3 0 cisyst © seat D) ~Faved 16 | Francesco Mourolico devised the method of = (A) Partial fraction (B) induction (C) Logarithms (D)- Binomial 17) If £=35 em and @= 1 rad, thenr= : (A) 38° (B) 350m (©) 35rad (D) 35m 8 | tn any ABC with vsual notations , = (Ayr B) a ©) my Orn 19 | The general term in the expansion of (a+ x)" is @) (Rew @) ("ere © (Rare @) ("}evs a x iG r 20/1f sides of @ AABC are a= 4584, b= 5140 and ¢ ~ 3624, then greatest angle willl be (Ay ow (B) B Or (D) a 24-219-1-(Objective Type)-10250 (6195)LAREN. /9 Roll No __ (Tobe filled in by the candidate) (Academie Sessions 2015 - 2017 to 2018 - 2020 ) MATHEMATICS, 219-(INTER PART —1) Ti PAPER - I( Essay Type ) GROUP ~I Maximum Macks : 80 SECTION -1 2, Write short answers to any EIGHT (8) questions : 16 (i If 2, and 2) are complex numbers then show that 2 +7; = =| +2 Gi) Find out real and imaginary parts of (V3 +i) Gi) Factorize a? +48? (iv). Define power set of set and give en example. (v) Define a bijective function, (vi) Construct truth table and show that the statement ~ (pq } > p is a tautology of not. 2 f- (ii) Find the mateix X if a 5 H 1 5] -21]7 [12 3 1-23 (iliy For the matrix 4=|-2 3 1 | find cofactor Ara 4-32 lz Ber 1 (ix) Without expansion show that | +a 1 jy @4B 1 (&) When x4 +2s3 + 42? +3 is divided by (x—2), the remainder is 1, Find the value of k (xi) If a, B are the roots of ax? +bx-+e=0 , a#0 then find the value of a? + 6? ive number and its square is 380. Find the number. (xif) The sum of a posi ‘Write short answers to any EIGHT (8) questions 6 (i) Define partial fraction. Gi) Inthe identity 74-25 = AQ +4) + B(x+3), calculate values of A and B. Giiy Resolve —1~ into partial fractions. 2n42,0=2 s&s, Gv) Write the frst four terms of the sequence, if a, ~a,. (y) Which term of the arithmetic sequence 5,2, (vi) Find three A.Ms between 3 and 11 (wit) IF Le and} aren G.P, show that common ratiois = /° (vii) Insert two GMs between 2 and 16. (ix) Find the value of n when "Cig «9 Swat 22% pqyemsoinge te 023 (xi) Expand 3 3) upto 4 terms, 2 ) * (Tum Over)HRCA HF o 4. Write short answers to any NINE (9) questions: (Find Lif 0=68°20° , r= 180m 2 sin? : sin? = een a 1itan? 9 Trtan? 6 cost]? +sin 112 cost 1°—sin > 28 (ii) Prove sin? 2 sin’ (iii) Prove cos?@= sin? @ (iv) Prove that tan 56°= (vi) Prove 00820" +.c0s100° + cos140° = 0 (vii) Find the period of tant (vill) In AABC, f= 60°. y =15°, b= V6 . find. (ix) If a*200 , b= 120, =150 (0) Prove that myn = rs? find the area of'a triangle ABC (xi) Prove sin(200s"' x)= 2xV1~x* (xii) Solve 1+e03x=0 3 (il) Find the solutions of sin x =-"> SECTION - IT in [0,22] Note : Attempt any THREE questions. 5. (a) Prove that all 2 x 2 non-singular matrices over the real field form a non-abelian group under multiplication. (b) Find three, consecutive numbers in G.P whose sum is 26 and their produet is 216. 4 2 | by sing row operation. 2 (a) Find the inverse of the matrix =] 3 ' (b) Prove that "C, # "C4 = "IC, {a) Solve the system of equations 12x? ~25xp412y? =0 4x? Ty? 148 U 13/19 ,135 1 3733) 3s bey ))+----- then prove that y?+2p-2=0 (a) Prove that =sec0~tand where 0 isnotan odd multiple of wi / Vit4sind (b) If a B.7 are the angles of a triangle ABC, then show that cor Scar Leo art eo 8 ot (a) The sides of a tiangle are x? +x+1, 2x +1 and x7 ~1, Prove that the greatest angle of the triangle is 120" 135 ao (b) Prove that tan”! = 4 tan (b) Prove that tan! 8 a = tan! ity 24-219-1-(Essay Type)-41000 RLHR, CRete14 (To be filled in by the candidate) cll No. ~~“ Aeademie Sessions 2015 =2017 to 2018-2020 ) MATHEMATICS 2194(INTER PART ~ 1) Time Allowed : 2.30 hours / PAPER -I (Essay Type) GROUP ~ I Maximum Marks : 80 SECTION -1 2. Write short answers to any EIGHT (8) questions : (Prove there ofadition 24.52% 16 ii). Find the rmultiplicative inverse of (5. -v5) (ii). Express the complex number 1+ /3 in polar form. Ais) Write the power set of fa, 2,¢}} (©) Show thatthe statement p> (pv q) is tautology (@i) Prove thatthe identity clement © in a group G is unique, (i) A ; fs -2s (it) if B=| 3-14 «find cofactor Bos -2:1 -2 “tan 2 °) ta emi Gx) If A isa skew-symmetric matrix, then show that 4? is a symmetric matrix (9) Solve x?-10=3x4 (si) Ifa, B are the roots of pr-p-e=0 then prove hat (1+@)(14+8)=1-e (sii) Discuss the nature of roots of the equation x? -5x+6=0 3. Write short answers fo any EIGHT (8) questions : 16 (Define proper fraction. x 10e+13 G@-DG?-5r +6) 46) el it) f — cay (r-a}(x-b)(x-e) 1 and are in harmoni fi pn ™ aa harmonic sequence, find k (9) Find sum of infinite geometric series 2+1+0,5 + —— Gv) If the numbers (¥i) Define geometric mean. (wii) If 5, 8 are two A.Ms between @ and , find a and b (viii) If —,— zm faeinar, show that 12%: Cer Gd Expand (1+. )3 upto 3 terms. (xi) Evaluate ¥30 correct to three places of decimal, (xii) Check whether the statement 5" —2” is divisible by 3 for n-2,3 is true or false. (ix) Prove that "C, (Tum Over)LHR -A2-N4 @ 4. Write short answers to any NINE (9) questions : (vii). Find the period of 3sin. (viti) With usual notations prove that 1-1, Gil) Solve the equation for [0,2] cot? Note : 3. @ ) 6 @) & 7. (@ ) ) % @) ib) (@ Find r , when ¢=56em.0=45° (ii) Find the values of all trigonometric functions for — 15 x Gi) yy) (v) Prove cosa, sing 2 (vi) Find the value of cos10S° without using caleulator. 1 (ix) Define in-cirete of the triangle ABC. (&)_ State the law of tangent. ( any two) (sib Show that cos(2sin x) =1—2x7 3 (si) Solve the equation for 8¢[0, ] 2sind + eos SECTION - IL Attempt any THREE question. If © isa group under the operation“ ®” and a,beG , find the solutions of the equations :(i) a¥x=b (i) xka=6 Wand 10" res of an HP are 5 and respectively find ts 14% tem ate a a ? Ga4l) Show that} a asf a aoa ate Prove that "'C, +" 'C, ="G If c, are the roots of Se?—x-2=0 form the equation whose roots are > and 3 Use mathematical induction to prove that n!>n? for integral values of n> 4, Avnilway tain is running on a circular track of radius S00 meters atthe rate of 30 km ‘per hour. Through what angle wil it tum in 10 sec? Reduce sin*@ to an expression involving only fimction of multiples of @raised to the first power. Prove that qr +n +795 1AGB Prove that tan“! 4-+tan“! B = tan AB 25-219-II-(Essay Type)-43000 18CHR: G2-/1.19 Roll No _ mu (To be filled in by the eandidate) MATHEMATICS - (Academie Sessions 2015 ~ 2017 to 2018 - 2020 ) QPAPER—I( Objective Type) 219-(INTER PART—1) Time Allowed : 30 Minutes GROUP ~ I Maximum Marks : 20 PAPER CODE = 6194 Note + Four posible answers A, B, Cand D to each qutstion ae given. The choice which you thin is comect, fil that circle in front of that question with Marker or Pen ink inthe answer-book. Cutting or fling two or more icles wil result zero mark in that question, iW a iT 3 6) is equal to (ay -sino B) sino © cos (D)_~cose Z_| Probability of impossible cvent is wet ® 1 ©o 2 3 | dtan"!4 equals : a -if_2A (a) wn '(—4, B) ann } rte) owls -if 24 af_4 (C) anf 24 D) tan if ) © wit) © (is 7] Which angle is quadrantal angle (a) 45° (B) 60° ©) 270° (D) 120° 5 | solution of equation tanx= fies in the quadrams (A) Land it (B) Mond = (€) Tand I (D) Tand IV 6 | Middle terms in the expansion of (x+)!' are : a B) 1%, ©) Tah ©) Beh 7 [IF A isthe area of triangle ABC, then with usual notation A= (a). Pbesin (B) Jabsine — (€) Lbesinw —D) Soesina 3 | Range of cotangent function = (a) N () z OR myc 7 T Expansion of (3-5x)’ is vali 3 5 A) [zl<¢ ®) [e1<5 (C) |xf<5 (D) |x]<3 TO] With usual notation R= ’ a « a A) B =* Dp) 4 aor 2siner © Fina) sinh n The sum of the four fourth roots of 81 is (ao (B)_8t () = 81 Dy 3 ( Turn Over)Re GL-1-19 ® riz The property Va,be®, a=b>b=a iscalled (A) Commutative (B) Transitive (C) Symmetric (D)_ Reflexive The value of 41.01.11 is = (A) 0 @) 1 oa (D) 24 14 ‘Asquare matrix A=[a,} inwhich a, =0 for all i> iscalled (A) Upper triangular (B) Lower triangular (©) Symmetric (D) Skew-symmetric as © wm) seed Tir BF -4ac>0 but not a perfect square, then roots are : (A) Equal (B) Complex {C) Rational ——(D). rational 17 No term of geomeitic sequence can be = 0 ® 1 © 2 () 3 TS/IT A and B oretwosels, hen AB= A ace @B) ane ©) vay) nay 191 partial fractions of will be of the form A, Bete aval © Ay Bee 3 3 Tf A=LayInep » then [&A[= (A) [a] (By #4] ) KAl (D) eat 25-219-11-(Objective Type)-10750 (6194)