(+1) PAPER -A IIT-JEE (2013) (Trigonomtery-1) Solutions

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1 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 IIT-JEE () (Trigonomtery-) Solutions (+) PAPER -A TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 6 MINS MAX. MARKS: 8 MARKING SCHEME In Section I (Total Marks: ), for each question you will be awarded marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one ( ) mark will be awarded. In Section II (Total Marks: 6), for each question you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. There are no negative marks in the section. In Section III (Total Marks : ), for each question you will be awarded marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one ( ) mark will be awarded. In Section IV (Total Marks: 8), for each question you will be awarded 4 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks otherwise, There are no negative marks in this section NAME OF THE CANDIDATE CONTACT NUMBER L.K. Gupta (Mathematics Classes) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

2 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 Section-I (Total Marks: ) (Single Correct Answer Type) This section contains 7 multiple choice questions. Each question has 4 choices.(a), (b), (c) and (d), out of which ONLY ONE is correct Value of tan 4 tan 4 + 7tan 4 is (a) (b) (c) (d) Ans. (c) tana tan A We know tana tan A Putting A 4 and squaring we get ( tan 4 ) tan 4 ( tan 4 ) 6 4 tan 4 tan 4 + 7tan 4. If sin α sin β and cos α cos β, then α + β α + β (a) sin (b) cos α β α β (c) sin (d) co s Ans. (c) sin α sinβ sinα + sinβ α + β α β cos sin...(i) and cosα cosβ cosβ cosα α + β α + β sin sin...(ii) From equations (i) and (ii) we get, α + β sin PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

3 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 π π π 7π. + cos + cos + cos + cos is equal to (a) ½ (b) cos π /8 (c) / 8 (d) + Ans. (c) ( + cosπ/8)( + cosπ/8)( + cosπ/8)( + cos7π/8) cos π/6.cos π/6.cos π/6.cos 7π/6 6(cosπ/6cosπ/6cosπ/6cos7π/6) (cos7π/6cosπ/6)(cosπ/6cosπ/6) cos π/8cos π/8 (cosπ/+ cosπ/8)(cosπ/ + cosπ/8) (cosπ/ + cosπ/4) π 4π 6π The value of cos + cos + cos is equal to (a) (b)- (c) / (d) / Ans. (d) π 4π 6π cos cos cos πi 4πi 6πi Re + + e 7 e 7 e 7 πi 4πi 6πi πi 4πi 6πi e + e + e + e + e + e + + e + e + e + e + e + e + (sumofsevenrootsofunity) + πi 4πi 6πi πi 4πi 6πi PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

4 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 π. If A + B + C, then cosa + cosb + coscis equal to (a) 4cos AcosBcosC (b) 4sinAsinBsinC (c) + cosacosbcosc (d) 4sinAsinBsinC Ans. (d) cosa + cosb + cosc cos(a + B)cos(A B) + cosc + sinccos(a B) sin C sinc{cos(a B) + sinc} sinc{cos(a B) cos(a + B)} sinc{sin AsinB} 4sinAsinBsinC 6. If cosθ + sinθ,then7cosθ + 6sinθequals (a) or (b) or (c) or 4 (d) or 6 Ans. (d) cosθ + sinθ ( tan θ/) (tanθ/) ( + tan θ/) ( + tan θ/) tan θ/ tanθ/ tan θ/ tan (θ/) tan(θ/) tanθ/,tanθ/ θ 9,tanθ/ 7cosθ+ 6sinθatθ 9 is ( tan θ/) tanθ/ ( + tan θ/) ( + tan θ/) and cosθ+ 6sinθ + tanθ/ PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 4

5 L.K. Gupta (Mathematic Classes) MOBILE: 9877, If 4nα π, then the numerical values of tanαtanαtanα... tan(n ) α is equal to (a) - (b) (c) (d) Ans. (c) 4na π π tan(n ) α tan(n α α ) tan α cot α tan(n ) α tan(nα α) π tan α cot α... tanαtanαtan α...tan(n ) α (tanαtan(n ) α)(tanαtan(n ) α)...tannα (...)tannα π tan nα tan 4 Section II (Total Marks : 6) (Multiple Correct Answers Type) This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) out of which ONE or MORE may be correct. 8. If cosα and cos β, then (a) cos( α + β ) (b) sin ( α + β ) α β 6 (c) sin (d) cos( α β ) 6 6 Ans. (b, c, d) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

6 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 cosα 4 sinα and cos β sinβ Alternate. (a): cos ( α + β ) cosαcosβ-sinαsinβ Alternate. (b): sin(α + β) sinαcosβ + cosαsinβ α β cos( α β) Alternate.(c) : sin {cosαcosβ + sinαsin β} Alternate.(d) : cs (α β) cosα cosβ + sinα sinβ PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 6

7 L.K. Gupta (Mathematic Classes) MOBILE: 9877, If sin θ+ sinφ a and cos θ + cosφ b, then θ φ θ φ (a) cos ± ( a + b ) (b) cos ( a b ) ± θ φ 4 a b a + b (c) tan ± (d) cos( θ φ ) a + b Ans. (a, c, d) sinθ + sinφ a and cos θ + cosφ b θ + φ θ φ sin cos a (i) θ + φ θ φ and cos cos b (ii) Squaring and adding Eqs. (i) and (ii), then θ φ 4cos a + b θ φ cos ± ( a + b ) and ( + cos( θ φ ) a + b a + b cos( θ φ ) θ φ tan a + b θ φ + tan By componendo and dividendo method θ φ tan 4 a b a + b θ φ 4 a b tan ± a + b cosa cos sin sin. + B A + B + sina sinb cosa cosb n A B (a) tan (c) n n is equal to (n, even or odd) n A B (b) cot (d) none of these PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 7

8 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 Ans. (b, c) n cosa + cosb sina + sinb n A B n B A + sina sinb cosa cosb cot + cot n A B If n even, cot, if n odd, n + cot76 cot6. is equal to cot 76 +cot6 (a) tan6 (b) cot76 Ans. (c, d) + cot76 cot6 cot 76 +cot6 sin76 sin6 + cos76 cos6 sin( ) + sin76 sin6 cos(76 6 ) sin(9 ) cos6 cos9 + cos6 sin9 cos9 sin9 tan 46 tan(9 44 ) cot44 [Alternate.(d)] (c) tan46 (d) co t44 Section III (Total Marks : ) (Paragraph Type) This section contains two paragraphs. Based upon one of the paragraph, multiple choice questions and based on the other paragraph multiple choice questions have to be answered. Each of these questions has four choices (a), (b), (c) and (d), out of which ONLY ONE is correct. Paragraph for Question Nos. and PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 8

9 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 If Pn sin n θ + cos n θ where n W (Whole number) and θ R (real number) On the basis of above information, answer the following questions. If Pn- P n sin θ cos θ P λ, then the value of λ is (a)n (b)n (c)n (d) n 4 Ans. (d) From Eq. (i), P P sin θcos θ P n n n 4 λ n 4 P7 P. The value of is P P P7 P P (a) (b) (c) P P P Ans. (c) From Eq. (i) Pn Pn sin θcos θ Pn 4 P7 P Put n 7, then sin θcos θ (v) P P P and put n, then sin θcos θ (vi) P From Eqs. (v) and (vi), we get P7 P P P P P P P P P P P 7 P (d) P Paragraph for Question Nos. 4 to 6 PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 9

10 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 The measure of an angle in degrees, grades and radians be D, G and C respectively, then the Relation between them D G C c 8 but 9 π π 7,7',44.8''or 66'' 7 On the basis of above information, answer the following questions 4. The value of cos c is (a) < (b) (c) > (d) Ans. (c) c 7,7'44.8' < 6 cos > cos6 c Hence,cos c >. The angle of triangle are in AP and the number of grades in the least is to the Number of radians In the greatest as 4 : π then the angles in degrees are (a) 4,6,7 (b),6, (c) Ans. (b) Let the angle of the triangle be Then (a d) + a + (a + d) 8 a 8 a 6 (a d) a and(a + d),6,9 (d) 4,6,8 Thus the angles are (6 d),6 and(6 + d) Number of grades in the least angle (6 d) (6 d) 9 9 π π And number of radians in the greatest angle (6 + d) (6 + d) 8 8 PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

11 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 (6 + d) 4 According to question, 9 π (6 + d) π 8 π 4 (6 d) (6 + d) 9 8 π d 4 Hence the angle of the triangle are (6 4),6,(6 + 4),i.e,6, 6. The angles of a triangles are in the A.P. and the number of degrees in the least is to the number of radians in the greatest as 6 : π. Then the angles in degrees are (a) 4,6,7 (b),6, (c),6,9 (d) 4,6,8 Ans. (c) Let the angles are the triangle be (a d),a and(a + d) Then,(a d) + a + (a + d) 8 a 8 a 6thus, the angles are (6 d),6 and(6 + d) number of degrees in the least angle (6 d) π And number of radians in the greatest angle (6 + d) 8 Section IV (Total Marks : 8) (Integer Answer Type) This section contains 7 questions. The answer to each of the questions is a single-digit integer, ranging from to 9. The bubble corresponding to the correct answer is to be darkened in the Answer sheet. 7. If 4 cos 6 + cot 7 n + n + n + n4 + n + n6, then the value of must be. PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 6 n i i 9

12 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 Ans. cot cos sin ( + + )( + ) and 4 cos Hence, 4 cos ο Hence, 4cos6 + cot n +, n, n, n4 4, n and n ni n n n n4 n n6 i i i n 9 ο 4 8. If 4 sin 7 α+ β, then the value of ( α + β αβ + ) 4 must be.. Ans. (cos7 + sin7 ) + sin4 + cos6 + + cos7 sin7 ( cos6 ) ( LHS>) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

13 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 ο ο ο Also, cos7 + sin7 ( cos6 ) ο ο ( cos7 > sin7 ) ο ο ο sin7 ( + cos6 ) ( cos6 ) On comparing, we get α +, β 4sin7 ( ) ( ) α + β 8, αβ α + β αβ + 4 ( α + β αβ + ) 4 4 ( α + β αβ + ) 4 9. The three angles of a quadrilateral are value of λ 9 must be Ans. 4 ο 9 First angle 6 ο, second angle 6 ο 6 4ο π 8 ο Third angle 6 π ο ο ο ο Fourth angle 6 ( ) 96 ο λ ο (given) λ 96 λ 9 4 ο 6,6 g and π, if fourth angle is λ then the 6. If <A < π and sina+cosa+tana+cota+seca+coseca 7 and sin A and cos A are the roots of the equation 4x x + a, then the value of a must be.. Ans. sin A and cos A are the roots of the equation 4x x + a, ten sin A +cos A a,sinacosa (i) 4 4 Also, PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

14 L.K. Gupta (Mathematic Classes) MOBILE: 9877, 4677 sina + cos A + tan A + cot A + sec A + cosec A 7 sina cosa (sina + cos A) cosa sina + + cosa sina (sina + cos A) + + (sin A + cos A ) 7 sinacosa sinacosa a a a 7 7 a 4 4 a 8 a.. If sec A tan B+tan A sec B, then the value of must be.. Ans. (secasecb + tanatan B) (secatanb + tanatan B ) sinasinb sina sinb + + cos AcosB cos A cosb + sin Asin B sin A sin B cos Acos B ( + sin A)( sin B) cos Acos B cos Acos B cos Acos B (seca secb + tan A tanb) (seca tanb + tan A secb) + () + 8 (secasecb + tanatan B ) 8 (secasecb + tanatan B ) 8 π. If cot ( θ α ), cot θ,cot( θ + α ) are in AP and θ is not an integral multiple of, then 46sin θ the value of must be. sin α Ans. cot ( θ α ), cot θ,cot( θ + α ), are in AP 6cot θ cot( θ + α ) + cot ( θ + α ) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH 4

15 L.K. Gupta (Mathematic Classes) MOBILE: 9877, cos θ sin θ sinθ sin( θ + α)sin( θ α) 6cos θ(sin θ sin α} sin θcosθ (sin θ sin α ) sin θ ( cosθ ) sin θ sin α or sin sin θ α 46sin θ sin θ 8 sin α sin α 46sin θ sin α 8 8. If sinθ + sin θ + sin θ, then the value of Ans. 4 We have, sin θ ( + sin θ ) sin θ sin θ( cos θ ) cos θ Squaring both sides, we get 4 sin θ( cos θ ) cos θ 4 4 ( cos θ)(4 4cos θ + cos θ ) cos θ cos θ + cos θ 8cos θ + 4 cos θ 6 4 cos θ 4cos θ + 8cos θ cos θ 4cos θ + 8cos θ must be PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O., SECTOR 4 D, CHANDIGARH

IIT JEE (2013) (Trigonometry and Algebra)

IIT JEE (2013) (Trigonometry and Algebra) L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER B IIT JEE () (Trigonometry and Algebra) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE