IIT JEE (2013) (Trigonomtery 1) Solutions

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1 L.K. Gupta (Mathematic Classes) MOBILE: , 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 60 MINS MAX. MARKS: 76 MARKING SCHEME I Sectio I (Total Marks: ), for each questio you will be awarded marks if you darke ONLY the bubble correspodig to the correct aswer ad zero marks if o bubble is darkeed. I all other cases, mius oe () mark will be awarded. I Sectio II (Total Marks: 6), for each questio you will be awarded marks if you darke ALL the bubble(s) correspodig to the correct aswer(s) ONLY ad zero marks otherwise. There are o egative marks i this sectio. I Sectio III (Total Marks: ), for each questio you will be awarded marks if you darke ONLY the bubble correspodig to the correct aswer ad zero marks otherwise. There are o egative marks i this sectio. I Sectio IV (Total Marks: 6), for each questio you will be awarded marks for each row i which you have darkeed ALL the bubble(s) correspodig to the correct aswer(s) ONLY ad zero marks otherwise. Thus, each questio i this sectio carries a maximum of 8 marks. There are o egative marks i this sectio. NAME OF THE CANDIDATE CONTACT NUMBER L.K. Gupta (Mathematics Classes) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

2 L.K. Gupta (Mathematic Classes) MOBILE: , 677 Sectio- I (Total Marks : ) (Sigle Correct Aswer Type) This sectio cotais 8 multiple choice questios. Each questio has four choices (a), (b), (c), (d) out of which ONLY ONE is correct.. si si si9 If k = ta 7 ta ad k =, the cos cos9 cos 7 (a) k = k (b)k = k + (c) k = k (d) oe of these As. (a) Sol: we have, k ta7θ taθ (ta7θ ta9θ) (ta9θ taθ) (taθ taθ) siθ siθ Now, taθ taθ cosθcosθ cosθ siθ si9θ Similarly, ta9θ taθ ad ta7θ ta9θ cos9θ cos7θ si9θ siθ siθ k k cos7θ cos9θ cosθ. If 0 0 < < 80 0, the... cos, the beig umber of s, is equal to, (a) cos As. (a) Sol: (b) cos... ( cosθ) ( umber of s)... cosθ/) (( ) umber of s).. cos (θ/ ) { cos (θ/ ) } cos(θ/ ) (c) cos (d) oe of these PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

3 L.K. Gupta (Mathematic Classes) MOBILE: , 677. If ta / ad ta/ are the roots of the equatio 8x 6 x + 5 =0, the cos ( +) is equal to 67 (a) (b) (c) (d)oe of these As. (a) Sol: 6 ta / ta /, 8 5 ta / ta / 8 ta ta ta ta / ta/ ta 6 cos( ) 5 7 ta The value of cos cos cos cos cos cos cos is equal to (a) / 6 (b) / 7 (c) / 8 (d) oe of these As. (b) Sol: π π π π 5π 6π 7π cos cos cos cos cos cos cos π π π 8π π 5π 6π cos cos cos cos π cos cos cos π π π 8π π π π cos cos cos cos cos cos cos π π π 8π cos cos cos cos PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

4 L.K. Gupta (Mathematic Classes) MOBILE: , 677 π si π.si 5 π si π π si 5 5. If si x six = 6 m c cos x where c,c,c,...,.c are costats,the m 0 6 m0 (a) c c c c = 0 (b) c c c 6 (c) c c 0 (d) c c As. (a) Sol: si x six si x(si x si x) si x( si x) ( cos x) ( ( cos x)) ( cos x cos x)(cos x ) so that Co,C 6,C 9,C6 ad C C C5 0 C C C C cos x 9cos x 6cos x 6. ta7 is equal to ( ) (a) As. (a) Sol. (b) (c) (d) ta7 = ta 5 ta5 ta 5 or PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

5 L.K. Gupta (Mathematic Classes) MOBILE: , 677 ta7 = ( ) ( ) If si ad 50 50, the si( / ) is equal to 65 7 (a) (b) 5 5 (c) 5 (d) 5 Sol: (a) si ad si( / ) cos( / ) 8. If 57 cos x y z, the x y z is equal to cos cos cos (a) (b) 0 (c) (d) oe of these As. (b) Sol x y z k say cos cos x y z k cos cos cos PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 5

6 L.K. Gupta (Mathematic Classes) MOBILE: , 677 k cos cos cos k cos cos 0 Sectio II (Total Marks : 6) (Multiple Correct Aswer (s) Type) This sectio cotais multiple choice questios. Each questio has four choices (a), (b), (c), ad (d) out of which ONE or MORE may be correct. 9. If x sec ta ad y = cosec cot, the y y (a) x (b) x (c) y y As. (c, d) Sol. si cos We have, x =, y cos si ( si )( cos ) Multiplyig, we get xy cossi si cos sicos sicos cossi si cos cossi ( si ) si cos ( cos ) ad x y cossi si cos ( xy ) cossi Thus, xy x y 0 y x ad y = x y x x y (d) xy x y 0 x PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 6

7 L.K. Gupta (Mathematic Classes) MOBILE: , Let f ta sec sec sec. sec, the (a) f (b) f f f As. (a, b, c, d) Sol r f ta / sec r 0 r r r cos ta / r 0 cos r cos ta / r cos r 0 ta / r 0 cos r cos. ta /. cos /. si. si.. si. cos ta r 0 cos cos Alterate. (a) : f ta 6 Alterate. (b) : f ta Alterate. (c) : f ta 6 Alterate (d) : f 5 ta 8 r. If x cos A, where A B, the y cosb A B x ta A y tab (a) ta x y PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 7

8 L.K. Gupta (Mathematic Classes) MOBILE: , 677 A B x ta A y tab (b) ta x y sia B y si A xsib (c) si A B y si A xsib (d) xcos A y cos B 0 As. (a, b, c) Sol x cos A y cosb x y say cos A cosb x cos A,y cosb Alterate. (a) : RHS = x ta A y tab x y cos A ta A cosb tab cos A cosb si A sib cos A cosb A B A B si cos A B A B cos cos A B ta LHS Alterate. (b) : RHS = x ta A y tab x y cos A ta A cosb tab cos A cosb si A sib cos A cosb A B A B cos si A B A B cos cos PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 8

9 L.K. Gupta (Mathematic Classes) MOBILE: , 677 A B ta LHS Alterate. (c) : RHS = y si A x sib y si A x sib cosb si A cos AsiB cosb si A cos A sib sia B LHS si A B Alterate. (d) : LHS = x cos A + y cos B = cos A cos B 0. If cos, si are the roots of the equatio x b x 0 ad cos, si are the roots of the equatio x x 0, the b is equal to (a) (b) (c) (d) As. (b, c) Sol We have, cos si cos si cos b b b, b b 0 Sectio III (Total Marks : ) (Iteger Aswer Type) This sectio cotais 6 questios. The aswer to each of the questios is a sigle-digit iteger, ragig from 0 to 9.The bubble correspodig to the correct aswer is to be darkeed i the ANSWER SHEET. PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 9

10 L.K. Gupta (Mathematic Classes) MOBILE: , 677. If cos A =, the si A 5 A si. Fid the value of λ. As. 0 Sol. si 5A si A = 6 (cos A cos A ) 6(cos A cos A cos A) = = λ = 0.. If P cos x si x, the P 6 P... As. 0 Sol. 6 6 P6 P (cos x si x) (cos x si x) = [(cos x si x) cos x si x(cos x si x) [(cos x si x) cos x si x) ( cos x si x) ( cos x si x) = If acos bcos 6cos 9cos is a idetity, the a =.. As. Sol. 6 Give, a cos bcos 6cos 9cos 6 9cos 6cos a(cos cos ) bcos 6 = a(6cos cos cos ) bcos 6 9a cos ( b a)cos 6acos Comparig, we get 9a 9 a ; b a 0 b =a = a =. 6. If sec k is 5 0, the value of 7 5cot is. The the value of 0 9 (sec ) k PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 0

11 L.K. Gupta (Mathematic Classes) MOBILE: , 677 As. Sol. 5 sec ta (sec ) cot So that k =. 9 (sec ) Cos cos cos cos cos cos equals. The fid the value of k 68 k As. Sol. 6 6 si. 65 si Required result = 65 si 65 si si 65 6si 65 6si 65 6si k =. 8. If 0 cos90 0 si50 As. Sol. Here, cos, the the value of cos(70 0 ) si ad si 50 si(70 0 ) cos0 The give expressio 0 0 si0 cos0 0 cos si0 si60 cos si60 cos0 cos60 si0 = si0 cos0 si si(60 0 ) = 0 si must be PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

12 L.K. Gupta (Mathematic Classes) MOBILE: , The, = = =. Sectio IV (Total Marks : 6) (Matrix-Match Type) This sectio cotais questios. Each questio has four statemets (a, b, c ad d) give i Colum I ad five statemets (p, q, r, s ad t) i Colum II. Ay give statemets i Colum I ca have correct matchig with ONE or MORE statemets(s)give i Colum II. For example, if for a give questio, statemet B matches with the statemets give i q ad r, the for the particular questio, agaist statemet B, darke the bubbles correspodig to q ad r i the ANSWER SHEET. 9. (a) (b) Colum If maximum ad miimum values 7 6ta ta of for all real values ( ta ) of are ad respectively, the If maximum ad miimum values of 5 cos cos for all real values of are ad respectively, the Colum II (P) (Q) 6 (c) If maximum ad miimum values of si cos for all real values of are ad respectively, the (R) 6 (S) 0 (T) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

13 L.K. Gupta (Mathematic Classes) MOBILE: , 677 As. (a)r, s ; (b)r, t ; (c) p, q. Sol (A) Let y = 7 6ta ta ta 7 cos 6 si cos si cos cos 7 si si cos y 8 8, 6, 0 R, S (B) Let y = 5 cos cos / = 5cos cos si cos si cos si 7 y 7 y 0 0, 6, R, T (C) Let y = + si cos cos cos = + cos cos = + cos PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

14 L.K. Gupta (Mathematic Classes) MOBILE: , 677 cos cos cos y,, 6 P, Q 0. Observe the followig colums : Colum I Colum II (A) I a Δ ABC (P) A B C si A B C si cos, si si si (B) I a (Q) A B C cos A B C cos cos Δ ABC, si si si (C) I s Δ ABC, (R) A B C si A B C si si cos cos cos (S) A B C cos cos si (T) A B C cos cos si As. (a)r, t ; (b)p, s; (c) q. Sol: si A si B si C A B C si si si si A B A B C C si cos cos si C A B C si cos cos PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH

15 L.K. Gupta (Mathematic Classes) MOBILE: , 677 A B C C C A B C B A si si si 8 8 C B A si si si A B C si si si (R) A B C cos cos si A B C cos cos si (T) (B) si A si B si C A B C si si si si A B A B C si cos cos si C C A B C si cos cos A B C C C A B C B A si cos cos 8 8 C B A si cos cos A B C cos cos si S A B C si si cos A B C si si cos (P) (C) cos A cos B cos C PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 5

16 L.K. Gupta (Mathematic Classes) MOBILE: , 677 A B C cos cos cos cos A B A B C C cos cos cos cos C A B C cos cos cos A B C C C A B C B A cos si si 8 8 C B A cos si si C B A cos si si C B A cos cos cos A B C cos cos cos (Q) PIONEER EDUCATION (THE BEST WAY TO SUCCESS): S.C.O. 0, SECTOR 0 D, CHANDIGARH 6

L.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:

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