IIT-JEE-Mathematics-Mains–2000

MAINS

Time : two hours                                                                  Max. Marks : 100 
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General Instructions : 
1. There are ten questions in this paper. Attempt all Questions. 
2. Answer each question starting on a new page. The corresponding question number must be written in the left margin. Answer all the parts of a question at one place only. 
3. Use only Arabic numerals (0, 1, 2 ………….9) in answering the questions irrespective of the language in which your answer. 
4. Use of logarithmic tables is not permitted. 
5. Use of calculator is not permitted. 
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                                           PART- B 

1. (a) The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that resulting sum is the square of an integer.                                                           (4) 

 

2. (a) If α, β are the roots of ax2+bx+c=0,(a≠0) and α+d, β+d are the roots of Ax2+Bx+C=0,(A≠0)for some constant d, then prove that 
(b2-4ac)/a2 =(B2-4AC)/A2                                                              (4) 

(b) For every positive integer, prove that 
               √(4n+1)  <  √n + √(n+1)  <   √(4n+2) 
Hence or otherwise, prove that [√n+√(n+1) = [√(4n+1]), where [x] denotes the greatest integer not exceeding x.                                                    (6) 

3. (a) In any triangle ABC, prove that 
Cot A/b + cot B/2 + cot C/2 = A/2 cot B/2 cot C/2                             (3) 

(b) Let ABC be a triangle with incentre I and inradius r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB respectively. If r1, r2 and r3 are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that 
r1/ (r - r1 ) + r2 / (r - r2) + r3  / (r - r3) = (r1 r2 r3 ) / (r - r1) (r-r2) (r-r3) 

4. For points P = (x1, y1) and Q = (x2, y2) of the coordinate plane, a new distance d(P, Q) is defined by d (P, Q) = │x1–x2│+│y1–y2│. Let O = (0, 0) and A=(3, 2). Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from O and A consists of the union of line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

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