May 2011 - Career Point

`tà{xÅtà|vtÄ V{tÄÄxÇzxá1 SetThis section is designed to give IIT JEE aspirants a thorough grinding & exposure to varietyof possible twists and turns of problems in mathematics that would be very helpful in facingIIT JEE. Each and every problem is well thought of in order to strengthen the concepts andwe hope that this section would prove a rich resource for practicing challenging problems andenhancing the preparation level of IIT JEE aspirants.By : Shailendra MaheshwariSolutions will be published in next issueJoint Director Academics, Career Point, KotaPassage :A bag contains ‘n’ cards marked 1, 2, 3, ......, n. ‘X’draws a card from the bag and the card is put backinto the bag. Then ‘Y’ draws a card. The probabilitythat ‘X’ draws.1. The same card as ‘Y’ is –11(A) (B) n 2n12(C)(D)2nn2. a higher card than ‘Y’ is –n −1n −1(A)(B)n2nn −1n −1(C)(D)22n2n3. a lower card than ‘Y’ is –n −1n −1(A)(B)n2nn −1n − 1(C)(d)22n2n4. Evaluate : 228∫∫1010xx2524(1 − x)(1 − x)5049dxdx= ?5. Find the minimum value of⎛2(x 1 – x 2 ) 2 1+ (17 2 )( 2 13)20 ⎟ ⎞⎜x− − x x −⎝⎠where x 1 ∈ R + and x 2 ∈ (13, 17).26. Let f (x) = a 1 tan x + a 2 tan 2x + a3 tan 3x + ................... + a n tan nx , where a1 , a 2 , a 3 , ... a n ∈ R andn ∈ N. If | f (x) | ≤ | tan x | for ∀ x ∈ ⎛ π π ⎞⎜−, ⎟ , Prove⎝ 2 2 ⎠nthat ∑i=1aii≤ 17. Let az 2 + bz + c be a polynomial with complexcoefficients such that a and b are non zero. Prove thatthe zeros of this polynomial lie in the region.b c| z | ≤ + + a b8. Find the fifth degree polynomial which leavesremainder 1 when divided by (x – 1) 3 and remainder–1 when divided by (x + 1) 3 .9. A quadrilateral ABCD is inscribed in a circle ofradius R such that AB 2 + CD 2 = 4R 2 . Using vectormethod prove that its diagonals are at right angle.10. Through a focus of an ellipse two chords are drawnand a conic is described to pass through theirextremities, and also through the centre of the ellipse.Prove that it cuts the major axis in another fixedpoint.Honesty• To be persuasive, You must be believable.To be believable, You must be credible.To be credible, You must be truthful.• An honest man is the noblest work of God.• If I am honesty in all my dealings, I can neverexperience fear.• Prefer a loss to a dishonest gain; one brings painfor the moment, the other for all time.XtraEdge for IIT-JEE 38 MAY 2011