Solution - Career Point

36.[B] The general term in the expansion of (x 1 + x 2 +n...x n ) n p1p2pgiven ...... x1x2.... xmm ,p p ...37.[B]1 2 pnp 1 + p 2 + p 3 .... + p m = nNow in (1 + x + y – z) 9 , coefficient of x 3 y 4 z= coeft of u 0 x 3 .y 4 z 1 in (u + x + y – z) 99=× (–1) 1 = –2 . 9 C 2 . 7 C 30 3 4 1e x= B 0 + B 1 x + B 2 x 2 + .....1−x⇒ e x = B 0 – B 0 x + B 1 x – B 1 x 2 + B 2 x 2 – B 2 x 2 +....⇒ e x = B 0 + (B 1 – B 0 )x + (B 2 – B 1 )x 2 + .....⇒ 1 + x +x 2 +2x 3 +3x 4 + ....4= B 0 + (B 1 – B 0 )x + (B 2 – B 1 )x 2 + ...∴ B n – B n –1 is coeff. of x nOn comparing coeff. of x n =38.[A] x + y + z + 12 = 0x, y, z are negative integersLet x = – a, y = –b, z = – c,a, b, c are +ve integer then required number ofpoints (x, y, z)= Number of positive integral solution ofa + b + c = 12= 12–1 C 3–1 = 11 C 2 = 5539.[A] p 1 =pp1240.[C] 2f(x) =2 6 612p 2 =2 6 6= × 11 = 112 5 6f (x)1f (1/ x)511f (1/ x) − f (x)⎛ 1 ⎞ ⎛ 1 ⎞= f(x) . ⎜ ⎟ – f ⎜ ⎟ + f(x)⎝ x ⎠ ⎝ x ⎠⎛ 1 ⎞ ⎛ 1 ⎞⇒ f(x) + f ⎜ ⎟ = f(x) . f ⎜ ⎟⎝ x ⎠ ⎝ x ⎠⇒ f(x) = 1 ± x nf(2) = 17⇒ 1 ± 2 n = 17 ⇒ ± 2 4 = 16∴ +ve sign will be take⇒ 2 n = 16 ⇒ n = 4Now, ∴ f(x) = 1 + x 4⇒ f(5) = 5 4 + 1 = 62661n41.[C] A is idenpotent ⇒ A 2 = AA 2 ⎡1x⎤⎡1x⎤⎡13x⎤= ⎢ ⎥⎣02⎢ ⎥ =⎦ ⎣02⎢ ⎥ ≠ A⎦ ⎣04 ⎦∴ not possible for any x42.[A] for any a ∈z ⇒ a = 2 0 a⇒ a R a ∀ a ∈ z∴R is reflexivea R b ⇒ a = 2 k b, k ∈ z ⇒ b = a.2 –k , – k ∈ z⇒ b R a∴ R is symmetrickLet a R b, b R c ⇒ a = 2 1kb , b = 2 2 cka = 21 k2 2 kc = 21+k2c , k 1 + k 2 ∈ z⇒ a R c∴ R is transitiveHence R is an equivalence Relation.43.[D] Q A.m ≥ G.m∴1 ⎛ a b ⎞⎜ + ⎟ ≥2 ⎝ b a ⎠b a.a b= 1 ⇒ ba + ab ≥ 2Similarly cb + bc ≥ 2 & ca + ac ≥ 2Adding we geta b b c a c + + + + + ≥ 6b a c b c ab + c c + a a + b⇒ + +a b c∴ minimum value is 6.1 144.[A] f(x) = ⇒ f ´(x) = – x2xf (b) − f (a)∴= f ´(x 1 )b − a⇒≥ 61 1 ⎛– = (b – a)b a ⎟ ⎞⎜1 − 21;⎝ x ⎠a < x 1 < b⇒ x 1 2 = ab ⇒ x 1 =45.[C]∫cot 4 x dx =∫cot 2 x (cosec 2 x –1)dxab=∫cot 2 x cosec 2 x dx –∫(cosec x − 1)dx= – 31 cot 3 x + cotx + x + c∴f(x) = – 31 cot 3 x + cotx + x + c + 31 cot 3 x – cotx= x + c2XtraEdge for IIT-JEE 120APRIL 2010