DEEPAWALI ASSIGNMENT

Size: px

Start display at page:

Download "DEEPAWALI ASSIGNMENT"

Hector Nelson
5 years ago
Views:

1 DEEPWLI SSIGNMENT CLSS & DOPPE FO TGET IIT JEE Get Solution & Video Tutorils online Downlod FEE Study Pckges, Test Series from w ww.tekoclsses.com Bhopl : Phone : (0755) Wishing You & Your Fmily Very Hppy & Prosperous Deepwli QUESTION BNK ON COMPOUND NGLES, QUDTIC EQUTION SOLUTIONS OF TINGLE SEQUENCE & POGESSION TIGONOMETIC EQUTIONS & INEQUTIONS Time Limit : 4 Sitting Ech of 75 Minutes durtion pprox. NOTE: This ssignment will be discussed on the very first dy fter Deepwli Vction, hence come prepred.

2 Q. B. on Compound ngles, QE, Sol n of tringle, Seq. & Prog. & trigo. equtions nd inequtions Select the correct lterntive : (Only one is correct) Q. regulr decgon 0,,... 9 is given in the xy plne. Mesure of the 0 7 in degrees is () (C) 7 (D) 54 Q. If + b + c = then b + bc + c lies in the intervl : (), [, ] (C), (D), Q. If the roots of the cubic x px + qx r = 0 re in G.P. then () q = p r p = q r (C) pq = r (D) pr = q Q.4 In tringle BC, : b : c = 4 : 5 : 6. Then + B = () 4C C C (D) Q.5 n equilterl tringle hs sides cm long. n nt wlks round the tringle, mintining distnce of cm from the tringle t ll time. Distnce trvelled by the nt in one round is () 6 (C) + (D) + Q.6 If P(x) = x + bx + c & Q(x) = x + dx + c, where c 0, then P(x). Q(x) = 0 hs () exctly one rel root tlest two rel roots (C) exctly three rel roots (D) ll four re rel roots. Q.7 The set of ll rel numbers x for which log log log x {x x > c}. The vlue of c is log () 0 (00) 00 (C) (00) 004 (D) 004 is defined s (00) Q.8 In tringle BC, CD is the bisector of the ngle C. If cos C hs the vlue nd l (CD) = 6, then hs the vlue equl to b () 9 (C) 6 (D) none Q.9 The rel vlues of '' for which the qudrtic eqution, x ( + 8 ) x + 4 = 0 possesses roots of opposite signs is given by : () > 5 0 < < 4 (C) > 0 (D) > 7 Q.0 The rithmetic men of the nine numbers in the given set {9, 99, 999, } is 9 digit number N, ll whose digits re distinct. The number N does not contin the digit () 0 (C) 5 (D) 9 Bhopl, Phone (0755) , Diwli ssgn.0 []

3 Q. If x = n, stisfies the eqution sin x cos x = sin x & the inequlity x () n =, 0,, 5 n =,, 4, 5 (C) n = 0,, 4 (D) n =,,, 5, then: 4 Q. With usul nottions, in tringle BC, cos(b C) + b cos(c ) + c cos( B) is equl to bc () bc 4 4bc (C) bc (D) Q. If in BC, sin + sin B + sin C = sin sinb sinc then () BC my be sclene tringle BC is right tringle (C) BC is n obtuse ngled tringle (D) BC is n equilterl tringle Q.4 If, b, c re rel numbers stisfying the condition + b + c = 0 then the roots of the qudrtic eqution x + 5bx + 7c = 0 re : () positive negtive (C) rel & distinct (D) imginry Q.5 If sin x cos x + cos x sin x = 8, then the vlue of sin 4x is () 4 (C) (D) 8 Q.6 With usul nottions in tringle BC, ( I I ) ( I I ) ( I I ) hs the vlue equl to () r r (C) 4 r (D) 6 r Q.7 Consider the pttern shown below ow ow 5 ow 7 9 ow etc The number t the end of row 80, is () (C) 648 (D) 65 Q.8 For ll positive integers n let f (n) = log 00 n. Let N = f () + f () + f(4) which of the following reltions is true? () 0 < N < N = (C) < N < (D) N = Q.9 The roots of (x ) (x ) + K (x ) (x 4) = 0, K > 0 re : () rel rel & equl (C) imginry (D) one rel & one imginry Q.0 n isosceles tringle hs sides of length,, nd x. The vlue of x for which the re of the tringle is mximum, is () (C) (D) Bhopl, Phone (0755) , Diwli ssgn.0 []

4 Q. If x sec + y tn = x sec + y tn =, then sec sec = y () x y y x y x y (C) y x y (D) y Q. Lrgest integrl vlue of m for which the qudrtic expression y = x + (m + 6)x + 4m + is lwys positive, x, is () (C) 0 (D) Q. The generl solution of the trigonometric eqution tn x + tn x + tn x = tn x tn x tn x is n () x = n n ± (C) x = n (D) x = where n I Q.4 With usul nottion in BC K = r r r r r r b c equl to : () 6 (C) 64 (D) 8 where K hs the vlue Q.5 If the sum of the roots of the qudrtic eqution, x + bx + c = 0 is equl to sum of the squres of their reciprocls, then c, b, c re in : b ().P. G.P. (C) H.P. (D) none Q.6 The set of vlues of x stisfying the inequlity log x log 4 x 4 ( x ) () (, ) (, )(, ) (C) (, ) (D) none Q.7 s shown in the figure D is the ltitude on BC nd D produced meets the circumcircle of BC t P where DP = x. Similrly EQ = y nd F = z. If, b, c respectively denotes the sides BC, C nd B then hs the vlue equl to () tn + tnb + tnc (C) cos + cosb + cosc x b y c z is : cot + cotb + cotc (D) cosec + cosecb + cosecc Q.8 In BC, the vlue of cos b cos B c cos C b c is equl to : () r r (C) r (D) r Bhopl, Phone (0755) , Diwli ssgn.0 [4]

5 Q.9 If the sum of the first terms of n rithmeticl progression equls tht of the first 9 terms, then the sum of its first 0 terms, is () equl to 0 equl to (C) equl to (D) non unique Q.0 The sum of ll the vlue of m for which the roots x nd x of the qudrtic eqution x mx + m = 0 stisfy the condition x () 4 x x x (C) 4 9, is (D) 4 5 Q. In n.p. with first term '' nd the common difference d (, d 0), the rtio ' ' of the sum of the first n terms to sum of n terms succeeding them does not depend on n. Then the rtio d nd the rtio ' ', respectively re (), 4, (C), (D), Q. D, BE nd CF re the perpendiculrs from the ngulr points of BC upon the opposite sides. The perimeters of the DEF nd BC re in the rtio : () r r r (C) where r is the in rdius nd is the circum rdius of the BC Q. If cos5 + sin5 = p, then cos50 is () p p p (C) p p (D) p p Q.4 Let r, r nd r be the solutions of the eqution x x + 4x =0 then the vlue of (r + )(r + )(r + ) is () (C) 5050 (D) 5066 Q.5 If p, q, r in H.P. nd p & r be different hving sme sign then the roots of the eqution px + qx + r = 0 re () rel & equl rel & distinct (C) irrtionl (D) imginry (D) r Q.6 In BC if b + c = then cot B cot C hs the vlue equl to : () 4 (C) (D) Q.7 Indicte the correct choice : If log 0. (x ) < log 0.09 (x - ), then x lies in the intervl ; () (, ) (, ) (C) (-, - ) (D) none of these Q.8 Number of roots of the eqution, sin(cos x) = cos(sin x) in [0, ] is () 0 (C) (D) 4 Q.9 The sum of the roots of the eqution (x + ) = log ( x + ) log 4 (980 x ) is () 954 log (C) log 954 (D) indeterminte Bhopl, Phone (0755) , Diwli ssgn.0 [5]

6 n Q.40 Let n be positive integer such tht sin + cos =. Then n n ( ) 6 n 8 4 n 8 (C) 4 n < 8 (D) 4 < n < 8 Q.4 Let f, g, h be the lengths of the perpendiculrs from the circumcentre of the BC on the sides, b nd c respectively. If b f c g h = b c then the vlue of is : f g h () /4 / (C) (D) Q.4 The eqution whose roots re sec & cosec cn be : () x x = 0 x x + = 0 (C) x 9x + 9 = 0 (D) none Q.4 Minimum verticl distnce between the grphs of y = + sin x nd y = cos x is () (C) (D) Q.44 Let C be circle with centre P 0 nd B be dimeter of C. Suppose P is the mid point of the line segment P 0 B, P is the mid point of the line segment P B nd so on. Let C, C, C,... be circles with dimeters P 0 P, P P, P P... respectively. Suppose the circles C, C, C,... re ll shded. The rtio of the re of the unshded portion of C to tht of the originl circle C is () 8 : 9 9 : 0 (C) 0 : (D) : Q.45 If the orthocentre nd circumcentre of tringle BC be t equl distnces from the side BC nd lie on the sme side of BC then tnb tnc hs the vlue equl to : () (C) (D) Q.46 The grph of qudrtic polynomil y = x + bx + c (, b, c ) with vertex on y-xis is s shown in the figure. Then which one of the following sttement is INCOECT? () Product of the roots of the corresponding qudrtic eqution is positive. Discriminnt of the qudrtic eqution is negtive. (C) Nothing definite cn be sid bout the sum of the roots, whether positive, negtive or zero. (D) Both roots of the qudrtic eqution re purely imginry. Q.47 If in tringle BC () 8 cos cosb cosc b then the vlue of the ngle is: b c bc c (C) (D) 4 Q.48 If sin () = & sin () = b (0 <,,< /) then cos () 4 b cos() = () b b (C) + + b (D) b Q.49 Concentric circles of rdii,,...00 cms re drwn. The interior of the smllest circle is coloured red nd the ngulr regions re coloured lterntely green nd red, so tht no two djcent regions re of the sme colour. The totl re of the green regions in sq. cm is equl to () (C) 4950 (D) 55 Bhopl, Phone (0755) , Diwli ssgn.0 [6]

7 Q.50 In BC if b = nd C = 0 0 then the mesure of the ngle is () (C) 75 0 (D) 05 0 Q.5 The number of solution of the eqution e x + e x + e x + e x = (e x + e x ) is () 0 (C) (D) more thn Q.5 If in tringle sin : sin C = sin ( B) : sin (B C) then : b : c () re in.p. re in G.P. (C) re in H.P. (D) none of these Q.5 The number of nturl numbers less thn 400 tht re not divisible by 7 or is () 8 59 (C) 60 (D) 76 9x sin x 4 Q.54 The minimum vlue of the expression for x (0, ) is x sin x 6 () 6 (C) (D) 8 Q.55 5 p 4p In BC, = =, b =, c = such tht p+ = p p p p 5 where p =, then () r = r r = r (C) r = r (D) r = r p Q.56 The sum n n n equls () (C) 4 (D) Q.57 The product of the rithmetic men of the lengths of the sides of tringle nd hrmonic men of the lengths of the ltitudes of the tringle is equl to : () (C) (D) 4 [where is the re of the tringle BC] Q.58 The set of rel vlue(s) of p for which the eqution, x + + x = px + 6 hs more thn two solutions is : () (0, 4] ( 4, 4) (C) {4, - 4, 0} (D) {0} Q.59 If O is the circumcentre of the BC nd, nd re the rdii of the circumcircles of tringles OBC, OC nd OB respectively then () bc b c b c hs the vlue equl to: (C) 4 (D) 4 Bhopl, Phone (0755) , Diwli ssgn.0 [7]

8 Q.60 If for n.p.,,,..., n, = nd = 8 then the vlue of equls () 6 (C) 8 (D) Q.6 If in BC, cos cosb + sin sinb sinc = then, the sttement which is incorrect, is () BC is isosceles but not right ngled BC is cute ngled (C) BC is right ngled (D) lest ngle of the tringle is 4 Q.6 The bsolute term in the qudrtic expression x x k k () (C) 0 (D) / n k F HG IF KJ HG I K J s n is Q.6 The number of roots of the eqution, sin x + sin x = + sin x is : () 0 (C) (D) infinite b c (sin sin B sin C) Q.64 In tringle the expression simplifies to () 8 6 (C) (D) 64 Q.65 Number of rel vlues of x stisfying the eqution x 6x 9 + x 6x 6 = is () 0 (C) (D) more thn Q.66 If the roots of the eqution x px r = 0 re tn, tn nd tn then the vlue of sec sec sec is () p + r + rp + p + r rp + (C) p r rp + (D) None Q.67 If r, r, r be the rdii of excircles of the tringle BC, then () cot B cot cot (C) r r r is equl to : tn (D) tn Q.68 There is certin sequence of positive rel numbers. Beginning from the third term, ech term of the sequence is the sum of ll the previous terms. The seventh term is equl to 000 nd the first term is equl to. The second term of this sequence is equl to () 46 (C) 4 (D) 4 Q.69 If x, y nd z re the distnces of incentre from the vertices of the tringle BC respectively then bc x y z is equl to () tn cot (C) tn (D) sin Bhopl, Phone (0755) , Diwli ssgn.0 [8]

9 Q.70 If the eqution (x ) + b(x x + ) + x = 0 is stisfied for ll x then the number of ordered pirs of (, b) cn be () 0 (C) (D) infinite Q.7 In BC, semicircle is inscribed, whose dimeter lies on the side c. Then the rdius of the semicircle is () b b c Where is the re of the tringle BC. (C) s (D) c Q.7 Number of qudrtic equtions with rel roots which remin unchnged even fter squring their roots, is () (C) (D) 4 Q.7 long rod lies n odd number of stones plced t intervls of 0 m. These stones hve to be ssembled round the middle stone. person cn crry only one stone t time. mn crried out the job strting with the stone in the middle, crrying stones in succession, thereby covering distnce of 4.8 km. Then the number of stones is () 5 9 (C) (D) 5 cos cosb cosc Q.74 If in BC, then the tringle is b c () right ngled isosceles (C) equilterl (D) obtuse Q.75 The qudrtic eqution x + bx + c = 0 hs imginry roots if : () <, 0 < c <, b > 0 <, < c < 0, 0 < b < (C) <, c < 0, b > (D) none Q.76 The equtions x + 5x + px + q = 0 nd x + 7x + px + r = 0 hve two roots in common. If the third root of ech eqution is represented by x nd x respectively, then the ordered pir (x, x ) is : () ( 5, 7) (, ) (C) (, ) (D) (5, 7) Q.77 If cos + cosb + cosc = then the sides of the BC re in ().P. G.P (C) H.P. (D) none Select the correct lterntives : (More thn one re correct) Q.78 If sin = sin then sin equl to () sin sin (C) sin (D) sin Q.79 cos is root of the eqution 5x + 5x = 0, < x < 0, then the vlue of sin is : () 4/5 /5 (C) 4/5 (D) 0/5 Q.80 If sin(x y), sin x nd sin (x + y) re in H.P., then sin x. sec y = () (C) (D) Q.8 Which of the following functions hve the mximum vlue unity? () sin x cos x sin x cos x (C) sin x cos x (D) 6 5 sinx cosx Bhopl, Phone (0755) , Diwli ssgn.0 [9]

10 Q.8 In tringle, the lengths of the two lrger sides re 0 nd 9 respectively. If the ngles re in.p., then the length of the third side cn be : Q.8 I f () 5 6 (C) 5 (D) 6 ± 5 b x log 5 x log b + 6 = 0, where > 0, b > 0 & b, then the vlue of x cn be equl to () log b log b log (C) b logb (D) Q.84 If the roots of the eqution, x + px + qx = 0 form n incresing G.P. where p nd q re rel, then () p+q = 0 p (, ) (C) one of the roots is one (D) one root is smller thn & one root is greter thn. Q.85 The grph of the qudrtic polynomil ; y = x + bx + c is s shown in the figure. Then : () b 4c > 0 b < 0 (C) > 0 (D) c < 0 logb x Q.86 If () log b 5 x log b + 6 = 0, where > 0, b > 0 & b, then the vlue of x cn be equl to log b log (C) b b (D) Q.87 Let be the discriminnt nd be the roots of the eqution x + bx + c = 0. Then + nd cn be the roots of the eqution : () x + bx + b = 0 x bx + b = 0 (C) x + bx b + 6c = 0 (D) x bx b + 6c = 0 Q.88 Which of the following sttement(s) is/re true? () log 0 lies between & 4 log cosec (5/6) cos 5 = (C) e ln (ln) is smller thn (D) log 0 + log 0 + log 0 = log Q.89 Select the sttement(s) which re true in respect of tringle BC, ll symbols hve their usul mening. () The inrdius, circumrdius nd one of the exrdii of n equilterl tringle re in the rtio of : :. bc = 4 rs (C) If r = then the vlue of r + r + r = (D)If the dimeter of ny escribed cricle is equl to the perimeter then the tringle must be right tringle. Q.90 The grph of qudrtic polynomil y = x + bx + c (, b, c, 0) is s shown. Then the incorrect sttement(s) is/re () c > 0 b < 0 (C) product of the roots is negtive (D) sum of the roots is positive log Bhopl, Phone (0755) , Diwli ssgn.0 [0]

11 Q.9 T h e p o i n t s ( x, y ) ; B(x, y ) nd C(x, y ) lie on the prbol y = x. If x, x, x re in.p. nd y, y, y re in G.P. then the common rtio of the G.P. is () + + (C) (D) Q.9 Which of the following when simplified, vnishes? () log log 4 log log + log (C) log 8 log 4 log 6 (D) log 0 cot º + log 0 cot º + log 0 cot º log 0 cot 89º Q.9 If one root of the qudrtic eqution, px + qx + r = 0 ( p 0) is surd where p, q, r ;, b re ll rtionls then the other root is b () b ( b) b (C) ( b) b (D) b b SUBJECTIVE: x y x y x y Q.94 Find whether the digonls of the prllelogrm formed by the lines,, b b b x y nd re perpendiculr to ech other or not. b Q.95 Find the re of the prllelogrm formed by the lines 4y x = 0 y 4x + = 0 4y x = 0 y 4x + = 0 Q.96 Find the vlue of for which the lines x + y = 0, x + y = 0, x + y = 0 re concurrent. Q.97 Find the coordintes of the points on the line y = x +, which re t distnce of 5 from the line 4x y + 0 = 0. Q.98 n equilterl tringle hs its centroid t the origin nd one side is x + y =. Find the other two sides. Q.99 The three lines x + y + = 0, x + y 7 = 0, x y 4 = 0 form the three sides of two squres. Find the eqution to the fourth side of ech squre. Q.00 Ech side of squre is of length 4. The centre of the squre is (, 7) nd one of its digonls is prllel to y = x. Find the coordintes of its vertices. Bhopl, Phone (0755) , Diwli ssgn.0 []

12 Select the correct lterntive : (Only one is correct) Q. D Q. C Q. Q.4 D Q.5 C Q.6 B Q.7 B Q.8 Q.9 B Q.0 Q. B Q. Q. D Q.4 C Q.5 C Q.6 D Q.7 Q.8 D Q.9 Q.0 D Q. B Q. C Q. D Q.4 C Q.5 C Q.6 C Q.7 Q.8 Q.9 Q.0 D Q. C Q. C Q. C Q.4 C Q.5 D Q.6 C Q.7 Q.8 Q.9 B Q.40 D Q.4 Q.4 C Q.4 D Q.44 D Q.45 Q.46 C Q.47 D Q.48 B Q.49 B Q.50 D Q.5 C Q.5 Q.5 C Q.54 C Q.55 D Q.56 C Q.57 B Q.58 D Q.59 C Q.60 D Q.6 C Q.6 Q.6 Q.64 C Q.65 Q.66 B Q.67 C Q.68 B Q.69 B Q.70 B Q.7 Q.7 C Q.7 C Q.74 C Q.75 D Q.76 Q.77 Select the correct lterntives : (More thn one re correct) Q.78,B,D Q.79,C Q.80 B,C Q.8,B,C,D Q.8,D Q.8 B,C Q.84,B,C,D Q.85,B,C,D Q.86 B,C Q.87,C Q.88,B,D Q.89,C,D Q.90,C, D Q.9,D Q.9,B,C,D Q.9,C SUBJECTIVE: Q.94 perpendiculr to ech other Q.95 /7 Q.96 ll vlues of Q.97 (8, 9) nd ( 4, 4) Q.98 y + = (x + ); y + = (x + ) Q.99 x y 4 = 0; x y + 6 = 0 Q.00 (, 5), (, 9), (5, 9), (5, 5) NSWE KEY Bhopl, Phone (0755) , Diwli ssgn.0 []

SOLUTION OF TRIANGLES

SOLUTION OF TRIANGLES SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :