Maths-1A Prasad 2022-2023 PDF
Document Details
Uploaded by Deleted User
Sri Chaitanya Educational Institutions
2023
Tags
Summary
This document is a 2022-2023 past paper for Maths-1A, covering topics such as Algebra (functions, mathematical induction, matrices), Vector Algebra, and Trigonometry. The material includes important questions, categorized by chapter and question type (Long Answer, Short Answer, Very Short Answer).
Full Transcript
SRI CHAITANYA EDUCATIONAL INSTITUTIONS., A.P Mangalagiri MATHEMATICS -IA IMPORTANT QUESTIONS 2022 ---- 2023 Student Name: ……………………………………… Section : …………………, Adm No: ……………… Page | 1 MAT...
SRI CHAITANYA EDUCATIONAL INSTITUTIONS., A.P Mangalagiri MATHEMATICS -IA IMPORTANT QUESTIONS 2022 ---- 2023 Student Name: ……………………………………… Section : …………………, Adm No: ……………… Page | 1 MATHEMATICS-IA S.NO NAME OF THE CHAPTER WEIGHT AGE MARKS ALGEBRA 01 FUNCTIONS 11 (7+2+2) 02 MATHEMATICAL INDUCTION 07 (7) 03 MATRICES 22 (7+7+4+2+2) VECTOR ALGEBRA 04 ADDITION OF VECTORS 08 (4+2+2) 05 PRODUCT OF VECTORS 13 (7+4+2) TRIGONOMETRY 06 TRIGONOMETRY UPTO TRANSFORMATIONS 15 (7+4+2+2) 07 TRIGONOMETRIC EQUATIONS 04 (4) 08 INVERSE TRIGONOMETRIC FUNCRTIONS 04 (04) 09 HYPERBOLIC FUNCTIONS 02 (2) 10 PROPERTIES OF TRIANGLES 11 (7+4) TOTAL MARKS 97 QUESTION BANK ANALYSIS LAQ SAQ VSAQ TOTAL S.NO TOPIC NAME *** ** * *** ** * ALGEBRA 01 FUNCTIONS 8 3 - - - - 28 19 02 MATHEMATICAL INDUCTION 13 3 1 - - - - 17 03 MATRICES 13 3 - 14 4 3 30 67 VECTOR ALGEBRA 04 ADDITION OF VECTORS - - - 6 7 - 16 29 05 PRODUCT OF VECTORS 6 5 2 16 10 5 29 73 TRIGONOMETRY TRIGONOMETRY UPTO 06 TRANSFORMATIONS 14 3 6 12 7 4 37 83 07 TRIGONOMETRIC EQUATIONS - - - 11 8 0 - 19 INVERSE TRIGONOMETRIC 08 FUNCRTIONS - - - 11 9 2 - 22 09 HYPERBOLIC FUNCTIONS - - - - - - 15 - 10 PROPERTIES OF TRIANGLES 11 8 3 12 6 5 26 71 SUB TOTAL 65 25 12 82 51 19 181 435 TOATL 102 152 Page | 1 FUNCTIONS(7M) LONG ANSWER QUESTIONS ***1. Let f : A B, g : B C be bijections. Then show that gof : A C is abijection. 1 ***2. Let f : A B, g : B C be bijections. Then show that gof f 1og 1 1 ***3. Let f : A B be a bijection. Then show that fof I B and f 1of IA ***4. Let f : A B , I A and I B be identity functions on A and B respectively. Then show that foI A foI A f I B of. ***5. Let f : A B be a bijection. Then show that f is a bijection if and only if there 1 exists a function g : B A such that fog : I B and gof I A and in this case, g f ***6. I) If f : R R, g : R R are defined by f x 4 x 1 and g x x2 2 then find a 1 i) gof x ii) gof iii) fof x iv) go f 0 f 0 4 1 ***7. Let f 1, a , 2, c , 4, d , 3, b and g 2, a , 4, b , 1, c , 3, d , then show that 1 gof f 1og 1. ***8. If f : Q Q is defined by f x 5 x 4 x Q then show that f is a bijection and find f 1. **9. Let f : A B, g : B C and h : C D. Then show tht ho gof hog of. x 2, x 1 **10. If the function f is defined by f x 2, 1 x 1 , then find the values of x 1, 3 x 1 a) f 3 b) f 0 c) f 1.5 d) f 2 f 2 e) f 5 3x 2 x 3 **11. If the function f is defined by f x x2 2, x 2 then find the values of 2 x 1, x 3 f 4 , f 2.5 , f 2 , f 4 , f 0 , f 7 Page | 2 VERY SHORT ANSWER QUESTIONS(2+2=4M) **12. Find the domain of the following real valued functions 1 1 i) f x ii) f x x2 1 6x x2 5 x 2 3x 2 1 1 iii) f x iv) f x x 2 | x| x log10 1 x 3 x 3 x v) f x vi) f x 4x x2 x vii) f x log x 2 4x 3 viii) f x x2 25 1 ix) f x log x x x) f x 2 x 1 x 3 1 1 xi) f x xii) f x a 0 log 2 x x 2 a2 *13. If f 1, 2 , 2, 3 , 3, 1 then find i) 2 f ii) 2 f iii) f 2 iv) f 14. If f 4,5 , 5, 6 , 6, 4 and g 4, 4 , 6,5 , 8,5 then find i) f g ii) f g iii) 2 f 4g iv) f 4 v) fg vi) f / g vii) | f | viii) f ix) f 2 x) f 3 *15. If f and g are real valued functions defined by f x 2 x 1 and g x x 2 then f find i) 3 f 2g x ii) fg x iii) iv) f g 2 x g 16. If f : R R, g:R R are defined by f x 3 x 1, g x x 2 1 then find i) fof x 1 ii) fog 2 iii) gof 2a 3 x2 4 2 17. Find the range of the following real valued functions i) log | 4 x | ii) x 2 2 18. If f x 2, g x x , h x 2 x for all x R , then find fo goh x. **19. Find the inverse of the following functions i) If a, b R, f : R R defined by f x ax b a 0 ii) f : R 0, defined by f x 5x iii) f : 0, R defined by f x log 2 x **20. If A 0, , , , and f : A B is a surjection defined by f x cos x then 6 4 3 2 find B. *21. If A 2, 1, 0,1, 2 and f : A B is a surjection defined by f x x2 x 1 then find B. x 1 22. If f x x 1 then find i) fofof x ii) fofofof x x 1 *23. Find the domain and range of the following real valued functions x 2 x i) f x ii) f x 9 x2 iii) f x | x | |1 x | iv) f x 1 x2 2 x Page | 3 3x 3 x 24. If the function f : R R defined by f x , then show that 2 f x y f x y 2f x f y *25. If f : R R, g : R R defined by f x 3 x 2, g x x 2 1 , then find 1 i) gof 2 ii) gof x 1 iii) fog 2 26. Define the following functions and write an example for each i) One-One (Injection) ii) Onto (Surjection) iii) Even and Odd iv) Bijection *27. If f : N N is defined as f x 2 x 3 , is ' f ' onto? Explain with reason. x 1 *28. f :R R defined by f x , then this function is injection or not? Justify. 3 1 x2 29. i) If f : R R is defined by f x , then show that f tan cos 2 1 x2 1 x ii) If f : R R is defined by f x log | | then show that 1 x 2x f 2f x 1 x2 1 1 1 x 30. If f x cos log x , then show that f f f f xy 0 x y 2 y 1 31. If f x , g x x for all x 0, then find gof x x 1 32. If f : R 0 R is defined by f x x3 , then show that f x f 1/ x 0 x3 x x 33. Prove that the real valued function f x x 1 is an even function on e 1 2 R 0. x2 x 1 34. If A 1, 2,3, 4 and f : A R is a function defined by f x , then find x 1 the range of ‘f’. 35. If the function f : 1,1 0, 2 , defined by f x ax b is a surjection, then find a and b. cos 2 x sin 4 x 36. If f x x R then show that f 2012 1. sin 2 x cos 4 x 37. If f : R R is defined as f x y f x f y x, y R and f 1 7 , then find n f r r 1 38. If 2x + 2y = 2, then find the domain of ‘x’ x 1 39. If f ( x) 2 x 1, g ( x) for all x R then find i) (gof) (x) ii) (fog) (x) 2 Page | 4 MATHEMATICAL INDUCTION(7M) LONG ANSWER QUESTIONS 2 2 2 2 2 2 2 n n 1 n 2 ***1. Show that 1 1 2 1 2 3... upto n terms , n N 12 13 13 23 13 23 33 n ***2. Show that... upto n terms 2n 2 9n 13 1 1 3 1 3 5 24 1 1 1 n ***3. Show that n N,... upto n terms 1.4 4.7 7.10 3n 1 n n2 6n 11 ***4. Show that 2.3+3.4+4.5+…upto n terms n N 3 1 1 1 1 n ***5. Show that... , n N. 1.3 3.5 5.7 2n 1 2n 1 2n 1 n n 1 n 2 n 3 ***6. Show that 1.2.3 2.3.4 3.4.5... upto n terms , n N 4 ***7. Prove by Mathematical induction, for all n N a a d a 2d... upto n n terms 2a n 1d 2 ***8. Prove by Mathematical induction, for all n N 2 a rn 1 a ar ar... upto n terms ,r 1 r 1 ***9. Show that 49n 16n 1 divisible by 64 for all positive integers n. ***10. Show that 4n 3n 1 divisible by 9 for all positive integers n. ***11. Show that 3.52 n 1 2n3n 1 is divisible by 17, n N ***12. Use mathematical induction 2.42 n 1 33n 1 is divisible by 11. ***13. Use mathematical induction to prove the statement, 3 5 7 2n 1 2 1 1 1... 1 n 1 1 4 9 n2 n 2 n 2n 1 2n 1 **14. Use mathematical induction to prove the statement, 2k 1 k 1 3 for all n N **15. Use mathematical induction to prove the statement, 43 83 123... upto n terms 2 16n 2 n 1 **16. Use mathematical induction to prove the statement 2 3.2 4.22... upto n terms n.2n , n N. *17. i) Using mathematical induction, show that x m y m is divisible by x y , if ‘m’ is an odd natural number and x, y are natural numbers. ii) If x and y are natural numbers and x y , using mathematical induction, show that x n y n is divisible by x y, n N. Page | 5 MATRICES LONG ANSWER QUESTIONS (7+7M=14M) a1 b1 c1 ***1. If A a2 b2 c2 is non – singular matrix , then show that A is invertible and adjA A1 det A a3 b3 c3 bc ca ab a b c ***2. Show that c a ab bc 2 b c a ab bc ca c a b a a 2 1 a3 a a2 1 ***3. If b b 2 1 b3 0 and b b 2 1 0 then show that abc=-1 c c2 1 c3 c c2 1 a b c 2 2bc a 2 c2 b2 2 ***4. Show that b c a c2 2ac b 2 a2 a 3 b3 c 3 3abc c a b b2 a2 2ab c 2 1 a2 a3 ***5. Show that 1 b 2 b3 (a b)(b c)(c a )(ab bc ca ) 1 c2 c3 abc 2a 2a ***6. Show that 2b bca 2b ( a b c )3 2c 2c cab a b 2c a b ***7. Show that c b c 2a b 2(a b c)3 c a c a 2b bc ca ab ***8. Show that a b b c c a a 3 b3 c 3 3abc a b c a 2 2a 2a 1 1 Show that 2a 1 a 2 1 a 1 3 ***9. 3 3 1 a b c ***10. Show that a 2 b 2 c 2 abc a bb c c a a3 b3 c3 1 a a 2 bc ***11. Show that 1 b b 2 ca 0 1 c c 2 ab Page | 6 ***12. Solve the following simultaneous linear equations by using Cramer’s rule , Matrix inversion and Gauss – Jordan method (i) 3x 4 y 5 z 18, 2 x y 8 z 13,5 x 2 y 7 z 20 (ii) x y z 9, 2 x 5 y 7 z 52, 2 x y z 0 (iii) 2 x y 3z 9, x y z 6, x y z 2 ***13. Examine whether the following system of equations is consistent or inconsistent. If consistent find the complete solutions. i) x y z 4, 2 x 5 y 2 z 3, x 7 y 7 z 5 ii) x y z 3, 2 x 2 y z 3, x y z 1 ii) x y z 6, x y z 2, 2 x y 3z 9 1 2 1 **14. If A 0 10 1 then find A3 3 A2 A 3I 3 1 1 2 a ab ca **15. Show that a b 2b b c 4(a b)(b c)(c a ) ca cb 2c **16. By using Gaus – Jordan method, show that the following system has no solution 2 x 4 y z 0, x 2 y 2 z 5,3x 6 y 7 z 2 SHORT ANSWER QUESTIONS(4M) cos sin **17. If A then show that for all positive integers ' n ' , sin cos cos n sin n An sin n cos n 3 4 1 2n 4n ***18. If A then for any integer n 1show that An 1 1 n 1 2n cos 2 cos sin cos 2 cos sin ***19. If , then show that 0 2 cos sin sin 2 cos sin sin 2 1 2 2 ***20. If 3 A 2 1 2 then show that A1 AT 2 2 1 1 2 2 1 2 If A , B 3 0 then verify that AB B1 A1 1 ***21. 1 3 4 5 4 yz x x ***22. Show that y zx y 4 xyz z z x y Page | 7 x2 2x 3 3x 4 ***23. Find value of X, if x 4 2x 9 3 x 16 0 x 8 2 x 27 3 x 64 1 a a2 ***24. Show that 1 b b 2 (a b)(b c)(c a) 1 c c2 2 1 2 ***25. Show that A 1 0 1 is adjoint and find A1. 2 2 1 1 2 1 ***26. If A 3 2 3 then find A1 1 1 2 1 0 0 1 ***27. If I and E then show that (aI bE )3 a3 I 3a 2bE 0 1 0 0 7 2 2 1 ***28 If A 1 2 and B 4 2 then find AB1 and BA1 5 3 1 0 3 3 4 ***29. If A 2 3 4 , then show that A1 A3. 0 1 1 bc bc 1 ***30. Show that ca c a 1 a bb c c a ab a b 1 1 4 7 3 4 0 If A ,B then prove that A B AT BT T **31. 2 5 8 4 2 1 2 4 4 9 20 22 **32. If A then prove that A A1 and AA1 5 3 9 6 22 34 **33. If A and B are invertible then show that AB is also invertible and ( AB)1 B 1 A1 1 2 2 **34. If A 2 1 2 then show that A2 4 A 5I O. 2 2 1 *35. For any nxn matrix. A prove that A can be uniquely expressed as a sum of a symmetric matrix and a skew symmetric matrix. *36. Show that the determinant of skew – symmetric matrix of order 3 is always zero. 1 2 2 *37. If A 2 1 2 then show that the adjoint of A is 3 A1. Find A-1 2 2 1 Page | 8 VERY SHORT ANSWER QUESTIONS 1 2 3 8 ***38. If A B 7 2 and 2 X A B then find X. 3 4 1 2 3 3 2 1 *39. If A and B find 3B-2A 3 2 1 1 2 3 x 3 2 y 8 5 2 40. If ,find X,Y,Z and a z 2 6 2 a 4 1 1 2 2 *41. Define trace of a matrix and find the trace of A, if A 0 1 2 1 2 1 2 **42. Define symmetric matrix and skew-symmetric matrix 1 2 3 *43. If A 2 5 6 is a symmetric matrix, find x 3 x 7 0 2 1 *44. If A 2 0 2 is a skew – symmetric matrix, find the value of x 1 x 0 1 0 0 **45. If A 2 3 4 and det A = 45, then find x 5 6 x 1 2 46. Find the inverse of the matrix 3 5 47. Define symmetric matrix. Give one example of order 3x3 12 22 32 *48. Find the determinant of 22 32 42 32 42 52 2 1 *49. If is a complex (non – real) cube root of unity, then show that 2 1 0 2 1 2 1 2 3 1 *50. If A 5 0 and B ,then find 2 A BT and 3BT A. 4 0 2 1 4 1 4 7 3 0 4 If A and B then show that A B AT BT T *51. 2 5 8 4 2 1 cos sin *52. If A then show that AA1 A1 A I sin cos Page | 9 cos sin *53. Find the adjoint and the inverse of the matrix sin cos **54. If 2 4 and A2 0 find the value of k. A 1 k i 0 *55. If A ,find A 2 0 i i 0 56. If A , then show that A2 = - I (i2 = -1) 0 i 57. Slove the following system of homogeneous equations x – y + z = 0, x + 2y – z = 0, 2x + y + 3z = 0 58. Define triangular matrix *59. Find the rank of each of the following matrices 1 1 1 1 4 1 1 2 1 1 2 0 1 i) 1 1 1 ii) 2 3 0 iii) 1 0 2 iv) 3 4 1 2 1 1 1 0 1 2 0 1 1 2 3 2 5 1 2 3 1 0 4 v) vi) 2 3 4 2 1 3 0 1 2 a ib c id 60. If A , a 2 b 2 c 2 d 2 1 then find the inverse of A. c id a ib 1 *61. Construct a 3 X 2 matrix whose elements are defined by aij i3j 2 *62. For any square matrix A, show that AA1 is symmetric. 63. Write the definitions of singular and non- singular matrices and give examples. 64. A certain book shop has 10 dozen chemistry books, 8 dozen physics book, 10 dozen Economics books. Their selling prices are Rs.80, Rs.60 and Rs.40 each respectively using matrix algebra, find the total value of the books in the shop. 3 2 1 3 1 0 65. If A 2 2 0 , B 2 1 3 and X=A+B then find X. 1 3 1 4 1 2 1 2 66. If A then find AA1 0 1 2 0 1 1 1 0 67. If A , B 0 then find (ABT)T. 1 1 5 1 2 Page | 10 ADDITION OF VECTORS SHORT ANSWER QUESTIONS ***1. Let A B C D E F be a regular hexagon with centre ' O '. Show that AB AC AD AE AF 3 AD 6 AO. ***2. In ABC , if ' O ' is the circumcentre and H is the orthocenter , then show that i ) OA OB OC OH ii ) HA HB HC 2 HO ***3. If the points whose position vector are 3i 2 j k , 2i 3 j 4k , i j 2k and 146 4i 5 j k are coplanar , then show that 17 ***4. a,b,c are non-coplanar vectors prove that the following four points are coplanar i ) a 4b 3c ,3a 2b 5c , 3a 8b 5c , 3a 2b c ii ) 6a 2b c , 2a b 3c , a 2b 4c , 12a b 3c iii ) 5a 6b 7c, 7 a 8b 9c,3a 20b 5c ***5. If i, j , k are unit vectors along the positive directions of the coordinate axes, then show that the four points 4i 5 j k , j k ,3i 9 j 4k and 4i 4 j 4k are coplanar. ***6. In the two dimensional plane, prove by using vector method, the equation of the x y 1 line whose intercepts on the axes are ' a ' and ' b ' is a b **7. Show that the line joining the pair of points 6a 4b 4c, 4c and the line joining the pair of points a 2b 3c, a 2b 5c intersect at the point 4c when a,b,c are non – coplanar vectors **8. If a , b , c are non coplanar find the point of intersection of the line passing through the points 2a 3b c ,3a 4b 2c with the line joining the points a 2b 3c , a 6b 6c **9. Find the vector equation of the plane passing through point 4i 3 j k ,3i 7 j 10k and 2i 5 j 7k and show that the point i 2 j 3k lies in the plane **10. Find the vector equation of the line parallel to the vector 2i j 2k and passing through the point A whose position vector is 3i j k. If P is a point on this line such that AP = 15 then find the position vector of P. **11. Let a, b be non – collinear vectors. If x 4 y a 2 x y 1 b and y 2 x 2 a 2 x 3 y 1 b are such that 3 2 then find x and y. **12. If a b c d , b c d a and a, b, c are non-coplanar vectors, then show that abcd 0 **13. If a,b,c are non-coplanar then test for the collineararity of the following points whose position vectors are given by i ) a 2b 3c, 2a 3b 4c, 7b 10c ii ) 3a 4b 3c, 4a 5b 6c, 4a 7b 6c iii ) 2a 5b 4c, a 4b 3c, 4a 7b 6c Page | 11 VERY SHORT ANSWER QUESTIONS **14. i) Find the unit vector in the direction of vector a 2i 3 j k. ii) Let a 2 i 4 j 5k , b i j k and c j 2k. Find the unit vector in the opposite direction of a b c 15. Show that the points whose position vectors are -2a+3b+5c,a+2b+3c,7a-c are collinear when a,b,c are non-coplanar vectors. *16. If the position vectors of the point A,B and C are 2 i j k , 4 i 2 j 2k and 6 i 3 j 13k respectively and AB AC , then find the value of **17. If the vectors 3i 4 j k and i 8 j 6k are collinear vectors, then find and **18. If a 2i 5 j k and b 4i m j nk are collinear vectors then find the values of m and n **19. If OA i j k , AB 3i 2 j k , BC i 2 j 2k and CD 2i j 3k , then find the vector OD *20. OABC is a parallelogram. If OA a and OC c , then find the vector equation of the side BC. *21. Find the equation of the plane which passes through the point 2i 4 j 2k , 2i 3 j 5k and parallel to the vector 3i 2 j k *22. Find the vector equation of the line joining the points 2 i j 3k and 4 i 3 j k **23. Find the vector equation of the line passing through the point 2i 3 j k and parallel to the vector 4i 2 j 3k. 24. Find the vector equation of the plane passing through the points. i 2 j 5k , 5 j k and 3i 5 j *25. If a, b, c are the position vectors of the vertices A,B and C respectively of ABC then find the vector equations of the medians through the vertex A. 26. Is the triangle formed by the vectors 3i 5 j 2k , 2i 3 j 5k and 5i 2 j 3k equilateral? 27. Find the vector equation of the plane passing through the points 0, 0, 0 , 0,5, 0 , and 2, 0,1. 28. ABCD is a pentagon, If the sum of the vectors AB, AE , BC , DC , ED and AC is AC then find the value of 29. If , and are the angles made by the vector 3i 6 j 2k with the positive directions of the coordinate axes then find cos , cos and cos. Page | 12 PRODUCT OF VECTORS LONG ANSWER QUESTIONS ***1. i) Find the shortest distance between the skew lines r (6i 2 j 2k ) t (i 2 j 2k ) and r (4i k ) s(3i 2 j 2k ) where s, t are scalars ii) If A (1, 2, 1), B (4, 0, 3), C (1, 2, 1) and D (2, 4, 5), find the distance between AB and CD. ***2. Let a, b, c be three vectors. Then show that i ) a b c a.c b b.c a ii ) a b c a.c b a.b c ***3. Find the equation of the plane passing to the points A (2,3, 1), B (4,5, 2) and C (3, 6,5). ***4. A line makes angles 1 , 2, 3 and 4 with the diagonals of a cube. Show that 4 cos 2 1 cos 2 2 cos 2 3 cos 2 4 3 ***5. Show that in any triangle the altitudes are concurrent. ***6. Find the vector equation of the plane passing throught the intersection of the planes r. i j k 6 and r. 2i 3 j 4k 5 and the point 1,1,1 **7. If a i 2 j 3k , b 2 i j k , c i j 2k then find | (a x b )x c | and | a x( bx c ) | **8. If a i 2 j k , b 2 i j k , find a x (b x c ) and | (a x b ) x c ) **9. If a 2i j 3k , b i 2 j k , c i j 4k and d i j k , then compute | a b c d | **10. a,b,c are non – zero vectors and a is perpendicular to both b and c. If 2 a 2, b 3, c 4 and (b, c) , then find a b c 3 **11. If b c d c a d a b d a b c , then show that the points with position vectors a,b,c and d are coplanar. *12. For any four vectors a, b, c and d , prove that a b c d acd b bcd a and a b c d abd c abd d *13. Show that the volume of a tertrahedron with a , b and c as coterminous edges is 1 | [a b c ] | 6 SHORT ANSWER QUESTIONS ***14. Prove that the smaller angle between any two diagonals of a cube is given by 1 cos 3 ***15. Find the unit vector perpendicular to the plane passing through the points (1, 2,3), (2, 1,1) and (1, 2, 4). ***16. Find the area of the triangle whose vertices are A(1, 2,3), B(2,3,1) and C (3,1, 2). Page | 13 ***17. Find a unit vector perpendicular to the plane determined by the points P(1, 1, 2), Q(2, 0, 1) and R(0, 2,1) ***18. If a 2i 3 j 4k , b i j k and c i j k ,then compute a b c and verify that it is perpendicular to a. ***19. Find the volume of the tetrahedron whose vertices are (1, 2,1), (3, 2,5), (2, 1, 0) and (1, 0,1). ***20. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors 2i 3 j k , i j 2k and 2i j k. ***21. Determine , for which the volume of the parallelepiped whose coterminous edges i j ,3i j and 3 j k is 16 cubic units. ***22. Find the volume of the tetrahedron having the edges i j k , i j and i 2 j k ***23. If a i 2 j 3k , b 2i j k and c i 3 j 2k , verify that a b c a b c ***24. a 3i j 2k , b i 3 j 2k , c 4i 5 j 2k and d i 3 j 5k , then compute the following i ) a b c d and ii ) a b c a d.b ***25. Find. In order that the four points A(3,2,1), B (4, ,5) , C(4,2,-2) and D (6,5,-1) be coplanar ***26. If a 2i j k , b i 2 j 4k , c i j k then find a b. b c ***27. Show that angle in a semi circle is a right angle ***28. If a b c 0, a 3, b 5 and c 7, then find the angle between a and b. ***29. Let a 4i 5 j k , b i 4 j 5k and c 3i j k. Find the vector which is perpendicular to both a and b whose magnitude is twenty one times the magnitude of c. 2 2 2 2 2 **30. Show that for any two vectors a and b , a b a.a b.b a.b a b a.b **31. Show that the points 5, 1,1 , 7, 4, 7 , 1, 6,10 and 1, 3, 4 are the vertices of a rhombus by vectors. **32. Let a and b be vectors, satisfying a b 5 and a, b 450. Find the area of the triangle having a 2b and 3a 2b as two of its sides **33. Find the vector having magnitude 6 units and perpendicular to both 2i k and 3j ik For any three vectors a,b,c prove that b c c a a b a b c 2 **34. **35. Let a, b and c be unit vectors such that b is not parallel to c and a b c b. 1 2 Find the angles made by a with each of b and c. Page | 14 **36. A 1, a, a 2 , B 1, b, b 2 and C 1, c, c 2 are non-coplanar vectors and a a 2 1 a3 b b 2 1 b3 0 , then show that a b c + 1 =0 c c2 1 c3 **37. a, b and c are non – zero and non – collinear vectors and 0, is the angle between b and c. If a b c 1 3 b c a, then find sin **38. If a 2i j 3k , b i 2 j k , c i j 4k and d i j k then compute a b c d **39. For any two vectors a and b then show that 1 a 2 1 b 2 2 1 ab a b a b. 2 *40. Show the points 2i j k , i 3 j 5k and 3i 4 j 4k are the vertices of a right angled triangle. Also find the other angles. a.c a.d *41. Show that for any four vectors a, b, c and d a b. c d and in b.c b.d 2 2 2 2 particular a b a b a.b *42. Show that in any triangle, the perpendicular bisectors of the sides are concurrent. *43. If a, b,c are unit vectors such that a is perpendicular to the plane of b,c and the angle between b and c is , then find a b c 3 *44. If a 1, 1, 6 , b 1, 3, 4 and c 2, 5,3 , then compute the following i) a.b c ii) a b c iii) a b c VERY SHORT ANSWER QUESTIONS **45. If a i 2 j 3k and b 3i j 2k , then show that a b and a b are perpendicular to each other 46. If the vectors i 3 j 5k and 2 i j k are perpendicular to each other, find 2p *47. If 4i j pk is parallel to the vector i 2 j 3k , find p 3 **48. Find the angle between the vectors i 2 j 3k and 3i j 2k. 49. Find the Cartesian equation of the plane through the point A 2, 1, 4 and parallel to the plane 4 x 12 y 3z 7 0 50. Find the angle between the planes r. (2i j 2k ) 3 r. 3i 6 j k 4 **51. Find the area of the parallelogram having 2i 3 j and 3i k as adjacent sides 52. Let a i j k and b 2i 3 j k find i) The projection vector of b and a and its magnitude ii) The vector components of b in the direction of a and perpendicular to a Page | 15 53. a i j k and b 2 i 3 j k then find the projection vector of b on a 54. If a 2i 2 j 3k , b 3i j 2k , then find the angle between 2a n and a 2b 55. If a 2, b 3 and c 4 and each of a,b,c is perpendicular to the sum of the other two vectors , then find the magnitude of a+b+c *56. Find the unit vector perpendicular to the plane determined by the vectors a 4i 3 j k , b 2i 6 j 3k *57. If a 2i j k and b i 3 j 5k , then find a b. 58. If a 2i 3 j 5k , b i 4 j 2k then find a b 59. Let a 2i j k and b 3i 4 j k. If is the angle between a and b, then find sin 2 2 2 2 60. For any vector a ,show that a i a j a k 2 a 6 , then find p q 2 61. If p 2, q 3 and p, q 62. Compute a b c b c a c a b 63. Find the area of the parallelogram having a 2 j k and b i k as adjacent sides 64. Find the area of the parallelogram whose diagonals are 3i j 2k and i 3 j 4k 65. If the vectors a 2i j k , b i 2 j 3k and c 3i pj 5k are coplanar, then find P 66. Show that i a i j a j k a k 2a for any vector a. 67. Prove that for any three vectors a,b,c,[b+c c+a a+b]=2[a b c] 68. Compute i j j k k i 69. Let b 2i j k , c i 3k. If a is a unit vector then find the maximum value of [a b c] 1 70. If e1 e2 sin where e1 and e2 are unit vectors including an angle ,show that 2 1 . 2 71. Find the distance of a points (2, 5, – 3) from the plane r (6 i 3 j 2k ) 4 72. If a 2 i 3 j k and b ai 4 j 2k then find (a b )x (a - b) 73. Find the equation of the plane passing through the point (3,-2,1) and perpendicular to the vector (4,7, –4). Page | 16 TRIGONOMETRY UPTO TRANSFORMATIONS LONG ANSWER QUESTIONS A B C A B C ***1. In triangle ABC , prove that cos cos cos 4 cos cos cos 2 2 2 4 4 4 ***2. If A,B,C are angles of a triangle ,then prove that A B C A B C sin 2 sin 2 sin 2 1 2 cos cos sin 2 2 2 2 2 2 A B C A B C ***3. If A B C , then prove that cos 2 cos 2 cos 2 2 cos cos sin 2 2 2 2 2 2 ***4. If A,B,C are angles in a triangle, then prove that A B C cos A cos B cos C 1 4 cos cos sin 2 2 2 ***5. If A,B,C are angles in a triangle , then prove that A B C A B C sin sin sin 1 4sin.sin.sin 2 2 2 4 4 4 A B C A B C ***6. If A B C 1800 ,then prove that cos 2 cos 2 cos 2 2 1 sin sin sin 2 2 2 2 2 2 ***7. If A,B,C are angles in a triangle , then prove that A B C cos A cos B cos C 1 4sin sin sin 2 2 2 A B C A B C ***8. In triangle ABC , prove that cos cos cos 4 cos cos cos 2 2 2 4 4 4 ***9. If A+B+C+S , then prove that x R ***10. If A B C 2S , then prove that SA SB C cos( S A) cos( S B) cos C 1 4 cos cos cos 2 2 2 ***11. If A B C 2S , then prove that A B C cos( S A) cos( S B) cos( S C ) cos S 4 cos cos cos 2 2 2 ***12. Suppose is not an odd multiple of , m is a non zero real number such that 2 sin( ) 1 m m 1 and . Then prove that tan m.tan cos( ) 1 m 4 4 3 ***13. If A B C , prove that Cos 2 A cos 2 B cos 2C 1 4sin A sin B sin C 2 ***14. If none of A,B, A+B is an integral multiple of , then prove that 1 cos A cos B cos A B A B tan cot 1 cos A cos B cos A B 2 2 A B C A B C **15. In triangle ABC, prove that sin sin sin 1 4 cos cos sin 2 2 2 4 4 4 **16. If If A,B,C are angle in a triangle, then prove that sin 2 A sin 2B sin 2C 4cos A sin B cos C Page | 17 **17. If A+B+C=1800 then prove that cos2A + cos2B + cos2C = – 4 cosAcosBcosC –1 *18. If A B C 900 then show that (i) sin 2 A sin 2 B sin 2 C 1 2sin A sin B sin C (ii) sin 2 A sin 2B sin 2C 4cos A cos B cos C *19. If A B C 00 then prove that cos 2 A cos 2 B cos 2 C 1 2 cos A cos B cos C *20. If A,B,C are angles in a triangle,then prove that A B C sin A sin B sin C 4 cos cos cos 2 2 2 *21. If A B C 270 , then prove that cos 2 A cos 2 B cos 2 C 2 cos A cos B sin C 0 *22. If A B C D 3600 , prove that cos 2 A cos 2 B cos 2C cos 2 D 4 cos( A B) cos( A C ) cos( A D) *23. If A+B+C=0 prove that sin2A+sin2B+sin2C=–4 sinAsinBsinC SHORT ANSWER QUESTIONS ***24. If A B 450 , then prove that i ) 1 tan A1 tan B 2, ii ) cot A 1 cot B 1 2 3 iii) If A B , then show that 1 TanA1 TanB 2 4 Tan sec 1 1 sin ***25. Prove that Tan sec 1 cos 3 7 9 1 ***26. Prove that 1 cos 1 cos 1 cos 1 cos 10 10 10 10 16 sin16 A ***27. If A is not an integral multiple of , prove that cos A.cos 2 A.cos 4 A.cos8 A 16sin A 2 4 8 16 1 and hence deduce that cos.cos.cos.cos 15 15 15 15 16 ***28. Let ABC be a triangle such that cot A cot B cot C 3. then prove that ABC is an equilateral triangle ***29. Prove that tan 700 tan 200 2 tan 500 1 ***30. For A R, prove that i) sin A.sin(60 A) sin(60 A) sin 3 A 4 1 ii ) cos A.cos(60 A) cos(60 A) cos 3 A 4 3 2 3 4 1 iii ) sin 200 sin 400 sin 800 iv) cos cos cos cos 16 9 9 9 9 16 ***31. If 3A is not an odd multiple of , prove that tan A.tan(60 A).tan(60 A) tan 3 A 2 and hence find the value of tan 60 tan 420 tan 660 tan 780 3 5 7 3 ***32. Prove that sin 4 sin 4 sin 4 sin 4 8 8 8 8 2 2 3 9 Show that cos 2 cos 2 cos 2 cos 2 2 10 5 5 10 Page | 18 2 4 8 1 ***33. Prove the following i) cos.cos.cos 7 7 7 8 2 3 4 5 1 ii ) cos.cos.cos.cos.cos 11 11 11 11 11 32 2 3 4 5 ***34. Prove that sin sin sin sin 5 5 5 5 16 ***35. If A is not an integral multiple of then prove that 2 i) tanA+cotA=2cosec2A and ii) cotA – tan A=2cot2A ***36. If none of the denominators is zero then prove that n A B cos A cos B sin A sin B n n 2 cot 2 ; n is even sinA sin B cos A cos B 0; n is odd **37. If sec sec 2sec and cos 1, then show that cos 2 cos 2 4 2 **38. If cos x cos y and cos x cos y then prove the value of 5 7 x y x y 14 tan 5cot 0 2 2 3 **39. Prove that cos 2 760 cos 2 160 cos 760 cos160 4 **40. Prove that 3 cos ec 200 sec 200 4 **41. If A is not integral multiple of , then prove that 2 i) tan A cot A 2 cos ec 2 A ii) cot A tan A 2 cot 2 A **42. If none of 2A and 3A is an odd multiple of , then prove that 2 tan 3 A tan 2 A tan A tan 3 A tan 2 A tan A 24 4 *43. If 0 A B ,sin A B , cos A B , find the value of tan 2A 4 25 5 5 1 5 1 *44. Prove that i ) sin180 ii ) cos 360 4 4 1 Prove that sin 2 45 sin 2 15 sin 2 15 0 0 0 *45. 2 *46. If cos n 0 and cos 0, then show that 2 sin n 1 sin n 1 tan cos n 1 2 cos n cos n 1 2 Page | 19 VERY SHORT ANSWER QUESTIONS **47. If cos sin 2 cos , prove that cos sin 2 sin *48. If 3sin 4 cos 5, then find the value of 4sin 3cos 3 5 7 9 *49. Prove that cot.cot.cot.cot.cot 1 20 20 20 20 20 ***50. Find the period of the following functions 4x 9 i) f ( x) tan 5 x ii) f ( x) cos iii) f ( x) sin x 5 x x iv) f ( x) cos 4 x v) f ( x) 2sin 3cos vi) f x cos 3x 5 7 4 3 vii) f x tan x 4 x 9 x ... n 2 x (n is any positive integer) 1 51. Prove that cos120 cos840 cos1320 cos1560 2 1 52. Prove that cos1000 cos 400 sin1000 sin 400 2 *53. Find the value of i) cos 42 cos 78 cos1620 0 0 *54. Find the value of i) sin 340 cos 640 cos 40 *55. Find the maximum and minimum values of the following functions over R i) f ( x) 7 cos x 24sin x 5 ii) f ( x) sin 2 x cos 2 x iii) cos x 2 2 sin x 3 iv) f ( x) 13cos x 3 3 sin x 4 3 3 v) f x 3sin x 4 cos x 56. Find the value of 0 0 1 0 1 0 1 0 1 1 1 i) sin 82 sin 22 2 0 2 ii) cos 112 sin 52 2 2 iii) sin 52 sin 22 2 2 2 2 2 2 2 2 1 3 57. Prove that 4 sin10 cos100 0 If sec tan , find the value of sin and determine the quadrant in which 2 58. 3 lies 1 59. Show that cos 4 2 cos 2 1 1 sin 4 sec 2 Prove that tan cot sec 2 cos ec 2 sec 2 .cos ec 2 2 60. 2sin 1 cos sin 61. If x, find the value of 1 cos sin 1 sin tan 6100 tan 7000 1 p 2 62. i) If tan 200 p, then prove that tan 5600 tan 4700 1 p 2 Tan1600 Tan1100 1 2 ii) If tan 200 , then prove that 1 Tan1600 Tan1100 2 63. i) Draw the graph of y tan x in between 0, 4 ii) Draw the graph of y cos x in 0, iii) Draw the graph of y sin 2 x in 0, 2 iv) Draw the graph of y = sinx between - and taking four values on X-axis. Page | 20 64. If is not an integral multiple of ,prove that 2 tan 2 tan 2 4 tan 4 8cot 8 cot 65. Prove that 4 cos 660 sin 840 3 15 3 1 66. Prove that cos 200 cos 400 sin 50 sin 250 4 67. If A,B,C are angles of a triangle and if none of them is equal to ,then prove that 2 tan A tan B tan C tan A tan B tan C 1 *68. If sin and does not lie in the third quadrant. Find the value of cos . 3 *69. Find the cosine function whose period is 7. 2 70. Find a sine function whose period is 3 cos 9 sin 9 0 0 *71. Prove that cot 360 cos 9 sin 9 0 0 3 *72. If sin , where , evaluate cos 3 and tan 2 5 2 5 73. If cos and then find sin 2. 13 2 2 tan x 74. For what values of x in the first quadrant is positive? 1 tan 2 x 75. If 0 , show that 2 2 2 2 cos 4 2 cos 8 2 sin 2 0 1 76. Prove that tan and hence deduce the values of tan150 and tan 22. 1 cos 2 2 77. If cos t (0 t 1) and does not lies in the first quadrant find the values of sin and tan 4 78. If sin and is not in the first quadrant the find the value of cos 5 3 4 79. Prove that cos 480.cos120 8 80. Eleminate , from x a cos3 , y b sin 3 81. Find the value of S in 3300.C os1200 Cos 2100 Sin3000 82. Find the extreme value of Cos 2 x Cos 2 x 83 Prove that Sin500 Sin700 Sin100 0 Page | 21 TRIGONOMETRIC EQUATIONS SHORT ANSWER QUESTIONS ***1. Solve the following and write the general solution i ) 2 cos 2 3 sin 1 0 ii ) 2(sin x cos x) 3 iii ) tan 3cot 5sec 1 ***2. If tan( cos ) cot( sin ), then prove that cos 4 2 2 ***3. If tan p cot q , and p q then show that the solutions are in A.P , with common difference. pq ***4. If 1 , 2 are solutions of the equation a cos 2 b sin 2 c, tan 1 tan 2 and a c 0, then find the values of i) tan 1 tan 2 , ii ) tan 1.tan 2 iii ) tan 1 2 ***5. If , are solutions of the Equation a cos b sin c a, b, c R and a 2 b 2 0, Cos cos ,sin sin , then show 2bc c2 a2 that i) sin sin ii) sin sin a b2 2 a 2 b2 2ac c2 b2 iii) cos cos iv) cos cos a b2 2 c2 b2 ***6. Solve i ) sin 2 x cos 2 x sin x cos x ii ) sin x 3 cos x 2 iii) 1 sin 2 3sin cos 1 ***7. If 0 , solve cos .cos 2 cos 3 4 ***8. Solve the equation cot 2 x 3 1 cot x 3 0 0 x 2 81 cos x cos x ..... 43 2 ***9. Find all values of X in , satisfying the equation ***10. Solve 4sin x sin 2 x sin 4 x sin 3 x ***11. Solve the equation 3 sin cos 2 **12. If x is acute and sin x 100 cos 3 x 680 find x. **13. Find the general solution of the equations cos ec 2, cot 3 **14. Solve Tan sec 3, 0 2. 3x x **15. Solve cos 3x cos 2 x sin sin 0 x 2. 2 2 **16. Solve and write the general solution of the equation 4 cos 2 3 2 3 1 cos 2 3 **17. If x y and sin x sin y then find x and y. 3 2 **18. Solve sin 3 4sin sin x sin x where n , n z **19. Given P q show that the solution of cos p cos q 0 from two serieseach of which is in A.P. Also find the common difference of eanch A.Ps Page | 22 INVERSE TRIGONOMETRIC FUNCTIONS SHORT ANSWER QUESTIONS 1 1 1 ***1. Prove that i) tan 1 tan 1 tan 1 2 5 8 4 3 3 8 ii) tan 1 tan 1 tan 1 4 5 19 4 4 5 16 ***2. Prove that i ) sin 1 sin 1 sin 1 5 13 65 2 4 1 ii ) sin 1 2 tan 1 5 3 2 3 8 36 iii ) sin 1 sin 1 cos 1 5 17 85 3 5 323 iv) 2sin 1 cos 1 cos 1 5 13 325 4 2 ***3. Find the value of tan cos 1 tan 1 5 3 4 7 117 Prove that sin 1 sin 1 sin 1 125 ***4. 5 25 ***5. If sin 1 x sin 1 y sin 1 z , then prove that x 1 x 2 y 1 y 2 z 1 z 2 2 xyz ***6. If cos 1 p cos 1 q cos 1 r , then prove that p 2 q 2 r 2 2 pqr 1 ***7. i) if Tan 1 x Tan1 y Tan1 z , then prove that x y z xyz ii ) If Tan 1 x Tan 1 y Tan 1 z , then prove that xy yz zx 1 2 p q p 2 2 pq q2 ***8. If Cos 1 Cos 1 , then prove that 2 .cos 2 sin 2 a b a ab b ***9. Solve the following equations for X. 2x 1 1 x 2 2x i ) 3sin 1 4 cos 2Tan 1 1 x 2 1 x 2 1 x 2 3 x 1 x 1 ii ) Tan 1 tan 1 x2 x2 4 x 1 Prove that cos Tan 1 sin cot 1 x 2 2 ***10. x 2 ***11. Show that sec 2 Tan 1 2 cos ec 2 Cot 1 2 10 1 **12. Find the value of 2Tan 1 5 4 1 a 1 a 2b **13. Prove that tan Cos 1 tan Cos 1 4 2 b 4 2 b a 3 12 33 **14. Prove that sin 1 cos 1 cos 1 5 13 65 Page | 23 41 **15. Prove that Cot 1 9 Co sec 1 4 4 1 1 **16. Prove that cos 2Tan 1 sin 4Tan 1 7 3 2x 1 1 x 2 **17. Prove that sin cot 1 cos 2 1 1 x 2 1 x 1 1 2 **18. Solve tan 1 tan 1 tan 1 2 2 x 1 4 x 1 x 4 3 27 **19. Prove that cos 1 sin 1 tan 1 5 34 11 13 2 **20. Show that cot sin 1 sin tan 1 17 3 *21. If sin 1 x sin 1 y sin 1 z then prove that x4 y 4 z 4 4x2 y 2 z 2 2 x2 y 2 y 2 z 2 z 2 x2 5 12 *22. i) Solve are sin arcsin x 0 ii) Solve sin 1 x sin 1 2 x x x 2 3 HYPERBOLIC FUNCTIONS VERY SHORT ANSWER QUESTIONS 5 ***1. If cosh x , find the value of (i ) cosh 2 x and (ii ) sinh 2 x 2 3 ***2. sinh x , find cosh 2x and sinh 2x 4 *3. If cosh sec then prove that tanh 2 x / 2 tan 2 / 2 4. For x, y R i) sinh x y sinh x cosh y cosh x sinh y ii) cosh x y cosh x cosh y sinh x sinh y i ) cosh x sinh x cosh nx sin nx , for any n R n **5. Prove that ii ) cosh x sinh x cosh nx sinh nx , for any n R n 6. For any x R , prove that cosh 4 x sinh 4 x cosh 2 x x log cot , prove sinh x tan 2 4 4 7. If , and that and 4 cosh x sec 2 8. If u log e tan and if cos 0 , then prove that cosh u sec . 4 2 tanh x tanh y 9. Prove that tanh x y 1 tanh x.tanh y cosh xx sinh x 10. Prove that sinh x cosh x, for x 0 1 tanh x 1 coth x Page | 24 1 1 x Theorem : for x 1,1 , tanh 1 x log e 1 x 11. 2 1 1 ***12. Show that tanh 1 log e 3 2 2 13. If sinh x 5, show that x log e 5 26 14. If sinh x=3 then show that x log(3 10) 15. For any x R then show that Cosh2x = 2 Cosh2x – 1 PROPERTIES OF TRIANGLES LONG ANSWER QUESTIONS 65 21 ***1. If a 13, b 14, c 15, show that R , r 4, r1 , r2 12 and r3 14 8 2 ***2. i ) If r1 2, r2 3, r3 6 and r 1 ,prove that a 3, b 4 and c 5 ii) In ABC , if r 8, r 12, r 24, find a, b, c 1 2 3 r1 r2 r3 1 1 ***3. Show that bc ca ab r 2R ***4. i) Show that r r1 r2 r3 4 R cos C ii) Show that r r3 r1 r2 4 R cos B ***5. In ABC ,prove that r1 r2 r3 r 4 R ***6. If P1 , P2 , P3 are the altitudes drawn from vertices A,B,C to the opposite sides of a 1 1 1 1 (abc) 2 triangle respectively , then show that i ) ii ) p1 p2 p3 P1 P2 P3 r 8R3 ab r1r2 bc r2 r3 ca r3r1 ***7. Show that r3 r1 r2 A B C r ***8. Show that cos 2 cos 2 cos 2 2 2 2 2 2R ***9. If r : R : r1 2 : 5 :12, then prove that the triangle is right angled at A ***10. Prove that a 3 cos B C b3 cos C A c 3 cos A B 3abc A B C ***11. Show that a cos 2 b cos 2 c cos 2 s 2 2 2 R 4R r1 r2 Show that i) a r2 r3 rr1 **12. ii) r1r2 r2 r3 r1 r2 a b c **13. In ABC show that 2 R where R is the circumradius. sin A sin B sin C 3 **14. If cos A cos B cos C , then showthat the triangle is equilateral. 2 A B C cot cot cot **15. Prove that 2 2 2 a b c 2 cot A cot B cot C a 2 b 2 c 2 Page | 25 r1 r2 r3 **16. Prove that a r1r2 r2 r3 r3r1 **17. If a 2 b2 c 2 8R 2 , then prove that the triangle is right angled **18. In ABC ,show that i ) b 2 c 2 a 2 2ca cos B ii) c 2 a 2 b 2 2ab cos C iii) a 2 b2 c 2 2bc cos A **19. The angle of elevation of the top point P of the vertical tower PQ of height h from point A is 450 and from a point B is 60 , where B is a point at a distance 30 meters from the point A measured along the line AB which makes an angle 300 with AQ. Find the height of the tower. *20. A lamp post is istuated at the middle point M of the side AC of a triangular plot ABC with BC=7cm , CA=8m and AB=9m. Lamp post subtend an angle 150 at the point B Find the height of the lamp post. 3 3 *21. The upper th portion of a vertical pole subtends an angle tan 1 at apoint in the 4 5 horizontal plane through its foot and at a distance 40m from the foot. Give that the vertical pole is at a height less than 100m from the ground, find its height. *22. AB is vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60. He moves away from the pole along the line BC to a point D such that CD=7cm.From D, the angle of elevation of the point A is 450.Find the height of the pole.s SHORT ANSWER QUESTIONS 2 bc A ***23. If i ) a b c sec , prove that tan sin bc 2 2 bc A ii ) a b c cos , Prove that sin cos bc 2 a 2 bc A iii ) sin , prove that cos cos bc bc 2 a 2 b2 c2 ***24. cot A cot B cot C 4 cos A cos B cos C a 2 b 2 c 2 ***25. Show that a b c 2abc 1 1 1 1 a 2 b2 c2 ***26. Show that r 2 r12 r2 2 r32 4 r ***27. Show that cos A cos B cos C 1 R b 2 c 2 sin B C ***28. In ABC , show that a2 sin B C A B C s2 ***29. Prove that cot cot cot 2 2 2 Page | 26 1 1 3 ***30. In ABC , if , show that C 60 ac bc abc a b b a ***31. If C 60 , then show that i ) 1 ii ) 2 2 2 0 bc ca c a c b 2 ***32. Show that in ABC , a b cos C c cos B B C bc A ***33. Show that in ABC , tan cot 2 bc 2 A A Show that b c cos 2 b c sin 2 a 2 2 2 ***34. 2 2 abc **35. Show that a 2 cot A b 2 cot B c 2 cot C R **36. If p1 , p2 , p3 are the altitudes of the vertices A ,B ,C of a triangle respectively, show 1 1 1 cot A cot B cot C that 2 2 p12 p2 p3 **37. If a : b : c 7 :8 : 9 , find cos A : cos B : cos C A B C **38. If cot , cot , cot are in A.P., then prove that a, b, c are in A.P. 2 2 2 **39. If r2 r1 r3 r1 2r2 r3. Show that A 900 a 2 b2 sin C **40. If , prove that ABC are is either isosceles or right angled. a b 2 2 sin A B *41. Show that b 2 sin 2C c 2 sin 2 B 2bc sin A A B C *42. Show that a b c tan tan 2c cot 2 2 2 *43. Prove that A B C bc ca ab s 2 ta n ta n ta n 2 2 2 A B C *44. If cot : cot : cot 3 : 5 : 7 , show that a : b : c 6 : 5 : 4 2 2 2 A B C *45. If sin 2 ,sin 2 ,sin 2 are in H.P., then show that a, b, c are in H.P. 2 2 2 ***** Page | 27
