SOLUTION CLASS-XII / (CBSE)

Transcription

1 CBSE XII EXAMINATION-6 SOLUTION--6 CBSE th Board MATHEMATICS SET- CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

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3 CBSE XII EXAMINATION-6 cos sin Ans. A sin cos < < A + A T I cos sin cos sin sin cos sin cos cos cos cos cos Ans. A k A A 7 A k 7 Ans. for unique solution A k C C C ; C C C k epansion along R k k + k Ans. r i j k ˆ ˆ in Cartesian form + + ˆ CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

4 CBSE XII EXAMINATION-6 ˆ / Intercept cutt of on the aes,, a b a c a + b + c / b c Ans. i j k i j k iˆ ˆ 9 ˆ iˆ ˆj ˆ kˆ 9 ˆ 9 jˆ 7 kˆ 9 ˆ b eq n & 7 9 value satisf the eq n So 7, 9 Ans 6. a iˆ ˆj kˆ, b iˆ ˆj kˆ a b iˆ jˆ kˆ ˆ i ˆj kˆ 6iˆ jˆ kˆ unit vector parallel to a b a b a b 6ˆ i ˆj kˆ 6 9 6iˆ ˆj kˆ 9 6 iˆ jˆ kˆ Ans 7. tan + tan + tan + tan tan + tan + tan tan tan tan CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

5 CBSE XII EXAMINATION-6 ; ±, ± OR L.H.S. tan tan tan tan tan tan tan 6 8 tan L.H.S. R.H.S. 8 8 Ans 8. [ ] [] [ ] [8] b solving the equations we get, but Tpist charge Rs. Per Page from a Poor student sham amount taken b sham Rs. but from another person, he take for CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

6 CBSE XII EXAMINATION-6 Pages 7 Rs. amount differ b 7 6 Rs. Less. smpath are reflect this problem Ans 9. at function is continuous, lim f lim f f R.H.L. b h b h lim f lim f h lim h h b h lim lim f L.H.L. bh h bh bh lim h bh b bh b lim f lim f h h lim h lim h lim h a+ + sin a h sin h h sin a h sin h h sin a h sin h + h h a + b a b Ans. cos a+ cos cos cos a d cos a sin cos sin a d cos a CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

7 CBSE XII EXAMINATION-6 d sin cos a cos sin a d cos a sin A B sin A cos B cos A sin B d sina d cos a d sina d d sina d d sina d sin sin d So d cos a sina d cosa + sin a+ d d + cosa + sin a+ d d + sin a+ d 6 6 sin sin p sin q sin p q q p p q sin sin d. d d d OR Ans. + d d C +... eq n of tangent : m... is to + + m d d C + CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 6

8 CBSE XII EXAMINATION-6 m m + ± if P, 8 P, 6 eq n of tangent at P, eq n of tangent at P, Ans. I e d e e d e e d Let f whose, t t t dt I e e t dt e e t f t f t e e t f t + C e e C d dt CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 7

9 CBSE XII EXAMINATION-6 e C OR I d Let A B B C C A A + B... B + C... C + A... A C... From B C... Put and in we get. C + C C + C C C A C A B C A C B / + I d / I d B C CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 8

10 CBSE XII EXAMINATION-6 Ans. I log d d I log log tan C I d... b b Using Propert f d f a b a a I d d I d... Add and, we get I d d I [ ] [ 8 8 ] 6 I 8 Ans. I d d Let + d CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 9

11 CBSE XII EXAMINATION-6 + I d d d d d d d / + d / + 7 d / sin +C 7 Ans. d cos d sin d cos +... d sin sin This is a linear differential equation with cos P, Q sin sin cos I.F. d e sin e log+sin + sin Multipling both sides of i b I.F. +sin, we get d + sin d + cos Integrating with respect to, we get + sin d C C sin... Given that when CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

12 CBSE XII EXAMINATION-6 C C... Put in, we get sin Ans 6. e / d + e / d d d / e e / The given D.E. is a homogeneous differential equation. Put v dv v + d dv d dv d d dv v + d d v ve v e v ve v v e v e e v dv d e v dv d e v dv d e v log + log C e v log e / log c c Given that at,. e c log C e e / log e CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

13 CBSE XII EXAMINATION-6 log e / + e e / Ans 7. A,, B,, C, 9, D,, AB iˆ 6 ˆj kˆ AC iˆ ˆj kˆ AD 8iˆ ˆj kˆ AB AC AD Four points A, B, C, D are coplannar. Ans 8. A -, 8, P P'a,b,c eq n of Line BC general coordinates of P P,, D.R of AP +, 9, AP BC CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

14 CBSE XII EXAMINATION-6 P,, 7 Coordinates of foot of,, 7 Coordinaties of image of A is P a, b, c is a, a b 8, b 6 c 7, c P, 6, Ans 9. bag A white, black bag white, black E first bag selected E second bag selected PE PE PA / E PA / E PE / A PE /A P E P A / E P E P A / E P E P A / E OR P win 6 P lose P Awins a r... b using formula of infinite G.P. CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

15 CBSE XII EXAMINATION-6 Ans. P Awins X larger of of three numbers X,,, 6 P 6 6 P P 6 6 P Xi Pi PiXi PiXi Mean P i X i. 67 P i X i i Pi X i VarX P X i Ans. a, b * c, d a + c, b + d i Commutative a, b * c, d a+c, b+d c, d * a, b c+a, d+b for all, a, b, c, d R * is commulative on A CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

16 CBSE XII EXAMINATION-6 ii Associative : a, b, e, d, e, f A { a, b * c, d } * e, f a + c, b+d * e, f a c e, b d f a c e, b d f a*b * c+d, d+f a * b { c, a * e, f } is associative on A Let, be the identit element in A. then, a, b *, a, b for all a,b A a +, b+ a, b for all a, b A a + a, b + b for all a, b A,, A, is the identit element in A. Let a, b be an investible element of A. a, b * c, d, c, d * a, b a+c, b+d, c+a, d+b a + c b + d a c b d c a d b a, b is an invertible element of A, in such a case the inverse of a, b is a, b sin Ans. cos d d d d cos cos sin cos 8cos cos sin cos 8cos cos cos cos cos cos cos cos d for increasing > d, cos + cos alwas greator than CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE #

17 CBSE XII EXAMINATION-6 So, d is increaring on d, Volume of cone dv d r h sin cos OR sin cos l [ sin + sin cos cos ] l sin sin + cos for maimum or minimum dv d l sin sin + cos sin cos sin tan tan A l v B cos cos again diff. w.r.t., we get d v d l cos 7 tan at cos d v d V is maimum when cos or cos CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 6

18 CBSE XII EXAMINATION-6 Ans., C B, A,- Equation of line AB : - + Equation of line BC : - + Equation of line CA : ar ABC 7 d d 6 d sq.units ˆ Ans. r i j k r iˆ ˆj kˆ ˆ ˆ ˆ r i j k r ˆ i ˆj kˆ r iˆ jˆ kˆ ˆ ˆ CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 7

19 CBSE XII EXAMINATION-6 + Equation of the required plane + + Vector eq n of the required Plane r iˆ jˆ kˆ Ans. Z ,,,,,,,,,. +.9 P, P, P, P, When A invest & B invest CLASS-XII / CBSE Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 PAGE # 8

20 CBSE XII EXAMINATION-6 Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 CLASS-XII / CBSE PAGE # 9 Ans 6. L.H.S. Multipiing R, R and R b,, respectivel take common,, from C, C, & C C C C and C C C 7 c taking common ++ from C & C + + R R R + R + + C C, C C C C + C C C + C

21 CBSE XII EXAMINATION-6 Corporate Office : CG Tower, A-6 &, IPIA, Near Cit Mall, Jhalawar Road, Kota Raj.- PCCP Head Office: J-, Jawahar Nagar, Main Road, Kota Rajasthan- Contact : -7-77, 8878 Toll Free : CIN: U8RJ7PLC9 CLASS-XII / CBSE PAGE # taking and common from R & R epanasion along R OR A A AA 8 8 A A.A A 6A + 7A + KI k k