SOLUTION OF TRIANGLES

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1 SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No

2 . 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 : sin = sin (A B) : sin (B ) then : b : c re in A.P. re in G.P. re in H.P. none of these. In tringle AB, : b : c = 4 : : 6. Then A + B = 4 4. In tringle AB the reltion The tringle is cute tn = b c holds good. Which of the following option(s) is/re correct? 7 The tringle is obtuse The ngles A, B, (in some order) re in A.P.. The sides of AB stisfy the eqution, + 4b + c = 4b + c. Then the tringle is isosceles. the tringle is obtuse. Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 7 B = cos A = cos With usul nottion in AB, b sin + c sinb equls bc bc bc bc sina 7. Let AB be tringle such tht AB = 6 nd let,b nd c denote the lengths of the sides opposite to A, B nd respectively. The vlue(s) of x for which = x + x +, b = x nd c = x + is (re) ( ) 4 8. If, b, c re the sides of tringle AB then + b c is lwys negtive positive non-negtive non-positive 9. If sides of tringle AB re, b nd c such tht b = + c then exhustive rnge of c b is,,,, 0. If the ngles A, B nd of tringle re in n rithmetic progression nd if,b nd c denote the lengths of the sides opposite to A, B nd respectively, then the vlue of the expresion c sin + c sin A is. In tringle AB if sin A = sin B nd cos A = cos B then prove tht the tringle is obtuse ngled.. Prove tht tringle AB is possible stisfying ( + b) = c + b nd sin A + sin B + sin = +. A. A. D 4. BD. AD 6. AD 7. B 8. B D VK lsses, -9-40, Indr Vihr, Kot. Mob. No

3 Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 8 B. In AB if b + c = then cot cot hs the vlue equl to : 4. With the usul nottion in ny AB, b c sin A sin B sin = cos A 4 cosb = 4 b cos = 4 c sec A bsec B csec tn A tn Btn = = s(s)(sb)(sc ). In AB, if cos A + cos B = 4 sin, then which of the following hold(s) good? A B A B cot cot cot cot, c, b re in A.P., b, c re in G.P. 4. The bse B of AB is fixed nd the vertex A moves, stisfying the condition cot B + cot = cot A, then b + c = vertex A moves on stright line b + c = vertex A moves on n ellipse. In tringle AB, let = 6, b = nd cos (A B) = 4. Assertion : B = eson (): sin A = 6. olumn I olumn II In sclene tringle AB, if cos A = b cos B (P) 0 then equls (Q) 4 In tringle AB, B = nd A =. The mximum possible () 60 vlue which the A cn hve is (S) 90 In AB B = 7 nd B = AD where AD is the ltitude from A, then equls cos A 7. In ny AB, prove tht c cosb bcos. b c bc 8. A B cot cot cot (bc) In ny AB, prove tht =. b c cota cotb cot 9. In tringle AB if ( + b + c) (b + c ) = bc then prove tht 0 < < In tringle AB, if b sin (b cos + c cosb) = 4, then find the re of the tringle AB.. Prove tht {cot (A/) + cot (B/)} { sin (B/) + b sin (A/)} = c cot (/).... B 4. BD. D 6. S; P; P 0.. VK lsses, -9-40, Indr Vihr, Kot. Mob. No

4 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 9. Are of tringle inscribed in circle of rdius 4, if the mesures of its ngles re in the rtio : 4: is 4( ) 4( ) 4( ) 4( ). AB is isosceles with AB = A nd AB = 06º. Point M is the interior of the tringle so tht MBA = 7º nd MAB = º. The number of degrees in AM is equl to 87º 67º 74º 8º. In AB, the rtio sin A b c is not lwys equl to sinb sin Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04, where is the circumrdius bc, where is the re of the tnngle ( + b + c ) / 4. If in tringle AB ngle B = 90 0 then tn A/ is : b c b c b c (bc) (h h h b c b c ) None. In tringle AB the expnsion 4 ( + b + c) (b + c ) (c + b) ( + b c) is equl to b sin 4 sin sin sin. cos 6. If l, m, n re the perpendiculrs from the ngulr points of t AB upon the opposite sides, b, c respectively then, bl cm n is equl to c b b c b bc c ( b c) 4 4 ( + cosa cosb cos) 7. If the medin of tringle AB through A is perpendiculr to AB then tna tnb / / 8. The bse ngle of re. nd.. The rtio of the bse to the height of the tringle is ( ) + VK lsses, -9-40, Indr Vihr, Kot. Mob. No hs the vlue equl to 9. If, b, c the sides of tringle AB be, 4, respectively nd D, E re the points of trisection of side B, then prove tht tn AE = /8. 0. Let, b, c be the sides of tringle & its re. Prove tht + b + c 4 When does the equlity hold?. In AB, let ngles A, B, re in G.P. with common rtio. If circum rdius of AB is then find the vlue (b + c ).. D. D. 4. B. 6. A B. 0

5 . 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 : 4. In tringle AB, AA nd AA re the medins nd ltitudes respectively. Length A A is equl to- c b b c None of these b c. In tringle AB, H nd M re the lengths of the ltitude nd medin to the bse AB. If = 0, b = 6, c = then length (HM) 7 9 none omprehension onsider tringle AB with b =. Altitude from the vertex B meets the opposite side in D, which divides A internlly in the rtio :. A circle of rdius psses through the point A nd D nd touches the circumcircle of the tringle BD t D. 4. If E is the centre of the circle with rdius then ngle EDA equls sin 4 sin 4 sin 4 sin 6. If F is the circumcentre of the tringle BD then which one of the following does not hold good? FD = sin 4 tringle DF is n isosceles tringle 6. If is the circumrdius of the AB, then equl FD = cos Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 40 Are of ADE = (/4) th of the re of DB 4 omprehension In the figure below, it is given tht = 90, AD = DB, ED is perpendiculr to AB, AB = 0 units nd A = units. 7. Are of tringle AE is 4 sq. units sq. units 6 4 sq. units 8. The vlue of tn () is The vlue of cos () is 4 sq. units B.. 4. A. D B 8. A 9. B VK lsses, -9-40, Indr Vihr, Kot. Mob. No

6 . In AB, if tn A nd tn B re the roots of the eqution b(x + ) = c x, then which of the following hold(s) good? sin + cot = cos A + cos B = tn(a B) = b b VK lsses, -9-40, Indr Vihr, Kot. Mob. No r = b. Assertion : Suppose AB is tringle such tht AB =, B = nd A = 4. D is the midpoint of B, E is the midpoint of AD, F is the midpoint of BE, nd G is the midpoint of DF. Then the re of tringle EFG is /4. eson (): EFG = DEF = BDE = ABD = AB A tringle is inscribed in circle of rdius. The length of the sides of the tringle re 7,8 nd 9 units, Assertion : The rdius hs n irrtionl vlue. eson (): Are of the tringle hs n irrtionl vlue. omprehension In tringle AB, let tn A =, tn B =, tn = nd c =. 4. Are of the tringle AB is equl to. The rdius of the circle circumscribing the tringle AB, is equl to Let denote the re of the tringle AB nd p be the re of its pedl tringle. If = k p then k is equl to olumn - I olumn - II In AB if = 4r then the vlue of 4(cosA + cosb + cos) is equl to (P) A tringle hs sides of lengths, nd 7. If the length of the interni (Q) 7 ngle bisector drwn from the vertex opposite to the side length 7 cn be expressed s rtionl in the lowest form n m then the vlue of(m + n), is Let H be the orthocentre of the tringle AB. () 0 If (AH) + (BH) + (H) + (AB) + (B) + (A) = k then k equls onsider tringle AB nd let, b nd c denote the lengths of the (S) sides opposite to the vertices A, B nd respectively. If, b, c re (T) the roots of t t + 47t = 60, cosa cosb cos then the vlue of 4 is equl to b c. ABD. A. 4. B. A Q, P, S, Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 4

7 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 4. A circle is inscribed in tringle AB, right ngled t. The circle is tngent to the segment AB t D nd the length of segments AD nd DB re 7 nd respectively. Are of tringle AB is equl to r r. Let AB be right tringle with BA = 90º then is equl to sin B sin tn B tn sec B sec cot B cot. A tringle AB hs sides AB of length units, A of length unit nd B of length unit. The ngle bisector I intersects the side AB t the point D. The length AD is. 4. In AB, + r = r r r Assertion : r r eson (): AB is right ngied t A. ( ) VK lsses, -9-40, Indr Vihr, Kot. Mob. No A c. Assertion : In AB, if r = r = r then : b: c = : 4 : eson (): In AB, if xr = yr = zr =(x + y + z)r, then : b : c = y+ z : x + z : x + y. 6. With usul nottions, in AB the vlue of (r r) cn be simplified s: bc tn A ( b c) 4 r ( b c) D 4 r 7. In AB, semicircle is inscribed whose dimeter lies on the side c. If x is the length of the ngle bisector through ngle then the rdius of the semicircle is bc 4 (sin A sinb) x s(s )(s b)(s c) x sin s 8. In AB, if r =, =, s =, then which of the following is/re correct? Are of AB is. Product of the sides of the ABis b + c = 4 Sum of the ex-rdii of AB is 9. In AB if B = /, s = ; s c =, then which of the following hold good? r = = r = = / Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE Select the sttement(s) which is/re true with respect to tringle AB, ll symbols hve their usul mening. The inrdius, circumrdius nd one of the exrdii of n equilterl tringle re in the rtio of : :. bc = rs If r = then the vlue of 4 r r r If the dimeter of n excircle be equl to the perimeter of the tringle then the tringle is right ngle.. A. A. A 4. A. A 6. AD 7. A 8. ABD 9. AD 0. AD B

8 . AB is n cute ngled tringle with circumcentre 'O' orthocentre H. If AO = AH then the mesure of the ngle A is 6 4. Let h, h b, h c re lengths of ltitudes drwn from vertices A, B, to sides B A nd AB respectively.the minimum vlue of h b c h h is equl to b c 4. If the orthocentre of AB lies on its circumcircle then : AB is n obtuse ngled tringle AB is n cute ngled tringle AB could be n cute or obtuse ngled tringle AB is such tht cosa vnishes where denotes the continued product. 4. If the dt given to construct tringle AB is =, b = 7, sin A = /4, then it is possible to construct- only one tringle two tringles infinitely mny tringles no tringles. If two sides, b nd the ngle A be such tht two tringles re formed, then the sum of the two vlues of the third side is b b cos A b sin A b c b c 6. In n isosceles AB if the ltitudes intersect on the inscribed circle then the cosine of the verticl ngle A is : /9 / / none 7. olumn - I olumn - II If 'O' is the circumcentre of the AB nd, nd re the rdii (P) of the circumcircles of tringles OB, OA nd OAB respectively then b c hs the vlue equl to AD, BE nd F re the perpendiculrs from the ngulr points of (Q) 4 AB upon the opposite sides. The perimeters of the DEF nd AB re in the rtio If the incircle of the AB touches its sides respectively t L, M nd N () nd if x, y, z be the circumrdii of the tringles MIN, NIL nd LIM (S) 4 xyz where I is the incentre then the vlue of, is r.. B. D 4. D. B 6. A 7. Q, P, VK lsses, -9-40, Indr Vihr, Kot. Mob. No Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 04 lss XI Dte : Btch : P Time : 60 Min. DPP.No. 4 r r

9 Viny Kumr (B. Tech., IIT Delhi) Trget IIT-JEE 0 lss XI Dte : Btch : V Time : 60 Min. DPP.No. 09. The re of the circle exceeds the re of regulr polygon of n sides nd of equl perimeter in the rtio of tn n : n cos n : n sin n : n cot n : n. If the number of sides of two regulr polygons hoving the some permeter be n nd n, prove tht their res re in the rtio cos : cos. n n. The rdius of the circle circumscribed bout regulr n-gon A A... A n is equl to. Prove tht the sum of ll sides nd of ll digonl of n-gon is equl to n cot. n 4. ABD is qudrilterl with n re of nd BD = 00º, ADB = 0º, AD = BD nd B = D s shown in the figure. The product (A) (BD) is equl to D 0º 00º 4 A B. In the figure, AB is right tringle t. A semicircle with centre 0 is tngent to the side A nd B. If the re of the tringle is, then the rdius of the semicircle is c c A O B c 4 c 6. In tringle AB, if A = 0º b = 0 nd = x, then the vlues of x for which there re possible tringles is given by < x < 0 x < < x < 0 < x < 0 7. In tringle AB, B =, B = 4º, = 60º nd AD is ltitude to side B. Assuming AD s dimeter, circle is drwn which cuts side AB nd A t P nd Q respectively. The length PQ is equl to 4 8. Let ABD be cyclic qudrilterl such tht AB =, B =, B = 0º nd re of qudrilterl = 4. Which of the following is/re correct? The vlue of (A) is equl to 9 The sum of ll possible vlues of product A. BD is equl to The sum of ll possible vlues of (AD) is equl to 9 The vlue of (D) cn be 4 9. Two circles re pssing through vertex A of tringle AB nd one of the circle touches the side B t B nd other circle touches the side B t. If = nd A = 0º then find the product of rdii of two circles. 4. D. 6. A ABD 9. VK lsses, -9-40, Indr Vihr, Kot. Mob. No

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the