IITPrep

Code : MTRG1

MATHEMATICS

TRIGONOMETRY


  1. In an acute angled triangle, Prove that (Ð A + Ð B) (Ð B + Ð C) (Ð C + Ð A) £ 4.(p/Ö 3)3R/s
  2. Perpendiculars are drawn from vertex of the obtuse angle of a rhombus to its side. The length of each ^r is equal to ‘a’ units. The distance between their feet being equal to ‘b’ units. Determine area of rhombus.
  3. From any point P within a DleABC, ^rs PA', PB', PC' are dropped on BC, CA, AB: and circles are described about PA'B', PB'C', PC'A'. Show that area of D formed by joining the centers of these circles is 1/4 D ABC.
  4. tan (p/4 + y/2) = tan3(p/4 + x/2), Prove that: -1 £ sin x . (3+sin2x)/(1+3sin2x) £ 1
  5. The incircle of DABC touches BC, CA, AB at D, E, F respectively. X is a point inside DABC such that the incircle of DXBC touches at D also, and touches CX and XB at Y and Z respectively. Prove that EFZY is a cyclic quadrilateral.
  6. ABCD is a convex quadrilateral inscribed in a circle G. Assume that A, B, and G are fixed and C, D are variable, so that the length of the segment CD is a constant. X, Y, are the points on the rays AC and BC respectively, such that AX = AD and BY = BD. Prove that the distance between X and Y remains constant.
  7. Show that if a , b , g are angles of any arbitrary triangle, then: sin a/2 . sin b/2 . sin g/2 < 1/4
  8. Prove, for the angles a , b , g , the following relation: å sin a .sin b .sin (a - b ) + Õ [sin(a - b )] = 0
  9. Determine all values of x in the interval 0 £ x £ 2p which satisfy the inequality: 2cosx £ ½ Ö(1 + sin 2x) - Ö(1 - sin 2x)½ £ Ö2
  10. Let a, b, c be the lengths of the sides of a triangle and a , b , g respectively, the angles opposite these sides. Prove that if a+b = tan g /2(a tan a + b tan b ) Then the triangle is isosceles.
  11. In triangle ABC, ÐB = 3ÐA. Prove that ac2 = (b2 - a2)(b-a)
  12. In triangle ABC the bisector of ÐB meets AC at D and the bisector of ÐC meets AB at E.These bisectors intersect at O and the lengths of OD and OE are equal. Prove that either ÐBAC=60° or ABC is isosceles.
  13. For the angles A, B, C of a triangle, PT 1 £ å cos A £ 3/2
  14. [Figure for question 14]In the diagram, P is the midpoint of line segment AB, ÐBAC = 60°, and ÐABD = 120°. X is any point in AC such that XP extended meets BD at Y. Prove that the length XY ³ length AB.
  15. If sin a + sin b = 1/2 and cos a + cos b = 5/4, find the value of tan a + tan b .
  16. The lengths of the sides of a triangle are in AP and the greatest angle exceeds the least by 60°. Find a : b : c.
  17. If a, b, c and k are constant quantities and a, b, g are variables subject to the relation a.tan a + b.tan b + c.tan g = k, find the minimum value of å tan2a.
  18. If x.cos a + y.cos 3a = c.sin b and x.sin 2a + y.sin 3a = c.cos b , then show that x.cos (2a +b ) + y.cos (3a +b ) = 0. If x = y then show that (5a +b) is an odd multiple of p.
  19. If tan-1 Ö[ (a2-x2)/(a2+x2) ] + tan-1Ö[ (b2-y2)/(b2+y2)] = a/2, Prove that x4/a4 - 2.(x2y2/a2b2)cos a + y4/b4 = sin2a.
  20. If a.sin2q + b.cos2q = c; b.sin2j + a.cos2j = d, and a.tanq =b.tanj, find the value of a-1+b-1-c-1-d-1.
  21. In a triangle ABC, the length of altitude from A is not less than BC and the length of altitude from B is not less than AC. Find the angles of the triangle.
  22. In the acute angled triangle ABC, AH is the longest altitude (H lies on BC), M is on the midpoint of AC, and CD is an angle bisector (with D on AB). (a) If AH £ BM, prove that the angle ABC £ 60. (b) If AH = BM = CD, prove that ABC is equilateral.
  23. A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets other two sides at X and Y. What is the maximum length XY, if the triangle has perimeter p?