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Question and answers prepared by K.SRINIVAS MATHS LECTURER CHAPTER 1 CHAPTERWISE IMPORTANT QUESTIONS IN IA FUNCTIONS 7 MARKS www.studentsmilestone.comwww.studentsmilestone.com 1. Let f: A B and g : B C be bijection. Then gof: AC is also a bijection 2. Let f: A B and g: BC bebijection. Then (gof)-1 = f-1og-1. 3 Let f: A B and I A and I B be the identity functions on A and B respectively. Then foI A I B of f 4. Let f: A B be a bijection. Then fof 1 = I B and f 1of = I A 5 A and B be two non empty sets If f: A-B is a bijection then f -1:BA is also a bijection 6 If f:A-B and g:B-A two functions such that gof=IA and fog=IB then g= f -1 2 MARKS www.studentsmilestone.com 7 Find the domains of the following real valued functions i) f ( x) 1 x2 ii) f ( x) 2 x 1 ( x 2)( x 3) iv) f ( x) 9 x 2 vi) f ( x) 1 x 1 ( x 3) 2 v) f ( x) x 2 3x 2 1 4 x iii) f ( x) 2 ix) f ( x) x 2 25 x) f ( x) vii) f ( x) log( x 2 4 x 3) viii) f ( x) 4 x x 2 3 x 3 x x xi) f ( x) 2 x 2 x x 8. If the functions f:RR g:RR are defined by f(x) = 3x-2 and g(x) = x2 + 1 Then find the following 9 i) ( gof 1 )( 2) ii) (gof) (x – 1) Let f = {(1,a), (2,c) , (4, d), (3,b)} and g-1 = {2,a), (4,b), (1,c), (3,d)}. Then find (gof)-1 and f 1 o g 1 www.studentsmilestone.com 10 A function f is defined as follows 3 x 2, if x 3 f ( x) x 2 2, if 2 x 2 2 x 1, if x 3 Then find the values of f(4), f(2.5), f(-2) , f(-4), f(0) , f(-7) 11 If f: R R, g: RR are defined by f(x) = 4x – 1 and g(x) = x2 +2 . a 1 Then find i) (gof) (x) ii) ( gof ) 4 12 14 iv) go( fof )(0) Find inverse functions of a) f(x) = 3x – 7 b) f(x) = d) f ( x) e 4 x 7 e) f ( x) log2 x c) f ( x) 5 x 13 iii) ( fof )( x ) 4x 7 3 If f(x)=5x+4 prove that it bijection& find it’s Inverse If f(x)= x 1 find x 1 fofof(x), fofofof(x), x 2 x 1 15 .if f(x)= find range of ‘f ’ f: A R where A= 1,2,3,4 x 1 16 If f= (4,5), (5,6), (6,4),g= (4,4), (6,5), (8,5) i)f+g ii)f-g iii)2f+4g iv) f+4 v) fg vi) f/g vii) f2 viii) f3 ix) f x) f 17 if A 0, , , , and f:A B is surjection defined by f(x)=cosx 6 4 3 2 then find B 18 If A={ -2,-1,0,1,2} and f:A-B defined by f(x)=x2-x+1 then find B 19 If f: R R, g: RR are defined by f(x) = 2x2 +3 and g(x) = 3x-2 . Then find i) (fog) (x) ii) (gof) (x) iii) ( fof ) 0 iv) go( fof )(3) 20 If f: R R, g: RR are defined by f(x) = 3x-1 and g(x) = x2+1 . Then find i) (fof) (x2+1) ii) (gof) (2) iii) ( gof ) 2a 3 iv) fog(x) v)gof(x) 21 If f: and g: are defined by f(x) = 2x-1 and g(x) = x2 Find i) (3f-2g)(x) ii) (fg)(x) f iii) ( x ) iv) (f+g+2)(x) g 22 If f= (1,2),(2, 3),(3, 1) , find i) 2f ii)f+2 iii) f 2 iv) f 23 Find range of following functions i) f(x)= x2 4 x ii)f(x)= 9-x 2 iii) f(x)= x 2 9 iv)f(x)= x2 3-2x CHAPTER 2 MATHEMATICAL INDUCTION 7 MARKS 1. Show that 2 + 3.2 + 4.22 +…….. up to n terms = n.2n for all values of n N. 2. 1.3 + 3.5 + 5.7 + ……. Up to n terms = 3.a) Show that b) Show that n(4n 2 6n 1) 3 1 1 1 n for all n N ........ up to n terms = 1.4 4.7 7.10 3n 1 1 1 1 n ........ up to n terms = 1.3 3.5 5.7 2n 1 4. 13 13 2 3 13 2 3 33 n .......... up to n terms = [2n 2 9n 13] 24 1 1 3 1 3 5 5. 12 (12 2 2 ) (12 2 2 32 ) .......... up to n terms = 6 a) Show that 49 n 16n 1 is divisible by 64 for all positive integers of n. b) n(n 1) 2 (n 2) 12 Show that 2.42n 1 33n1 is divisible by 11 c) Show that 3.52n + 1 + 23n + 1 is divisible by 17 for all n Nwww.studentsmilestone.com d)Show that 6n + 2 + 72n + 1 is divisible by 43 for all n N. 7 Show that 1.2.3+2.3.4+3.4.5………. up to n terms= n(n+1)(n+2)(n+3)/4 8 a) .Show that a+ (a+d)+a+2d)+…………..= b)Show that a+ a(r n 1) r 1 ar+ar2…………..= 9. 2.3 + 3.4 + 4. 5 + ……. Up to n terms = 10 n(n 2 6n 11) 3 1.3 + 2.4+ 3.5 + ……. Up to n terms = CHAPTER 3 n 2a (n 1)d 2 n(n 1)(2n 7) 6 MATRICES 7 MARKSwww.studentsmilestone.com 1 a2 2 Show that 1 b .1 1 c2 b) 4 a b 2c a b c b c 2a b 2(a b c)3 Show that c a c a 2b x 2 2 x 3 3x 4 Find the value of x if x 4 2 x 9 3x 16 0 x 8 2 x 27 3x 64 a a2 1 a3 a a2 1 If b b 2 1 b 3 0 and b b 2 1 0 then show that abc=-1 c 5 c3 a bc 2a 2a 2b bca 2b (a b c) 3 a) Show that 2c 2c c a b 2 3. a3 b 3 (a b)(b c)(c a )( ab bc ca) c2 1 c3 a b c Show that b c a c a b c 2 c2 1 2bc a 2 c2 b2 c2 2ac b 2 b2 a2 a2 2ab c 2 (a 3 b 3 c 3 3abc) 2 6. a 2 2a 2a 1 1 a 2 1 (a 1) 3 Show that 2a 1 3 3 1 0 1 1 b+c c-a b-a 1 7. A 1 0 1 andB= c-b c+a a-b then show that ABA -1 is a diagonal matrix 2 1 1 0 b-c a-c a+b 8 3 -3 4 If A= 2 -3 4 then show that A -1 =A3 0 -1 1 9 Solve the following simultaneous liner equations byCramer’s rule AND Matrix inversion method’ i) x+y+z=9,2x+5y+7z=52,2x+y-z=0 iii) 3x+4y+5z=18; ii) x+y+z=1,2x+2y+3z=6,x+4y+9z=3 2x-y + 8z=13; 5x-2y+7z=20. iv ) 5x-6y+4z=15, 7x+4y-3z=19, 2x+y+6z=46. 10 Solve the following simultaneous liner equations byGauss Jordan method .i) x+y+z=9,2x+5y+7z=52,2x+y-z=0 ii) x+y+z=1,2x+2y+3z=6,x+4y+9z=3 iii) 2x-y+3z=8,-x+2y+z=4,3x+y-4z=0 IV) 3x+4y+5z=18; 2x-y + 8z=13; 5x-2y+7z=20. V ) 5x-6y+4z=15, 7x+4y-3z=19, 2x+y+6z=46. -1 -2 -2 11 . a) If A = 2 1 -2 then show that adjoint of A is 3A T . Find A -1 2 -2 1 1 2 2 b) If 3A= 2 1 -2 then show that A -1 =A’ -2 2 - 1 12.Show that following system of equations are consistent and solve them completely i) x+y+z=1,2x+y+z=2,x+2y+2z=1 ii) x-3y-8z=-10 3x+y-4z=0 2x+5y+6z=13 iii) x+y+z=6,x+2y+3z=10,x+2y+4z=1 iv) x+y+z=3: 2x+2y- z=3; x+y-z=1. 4 marks www.studentsmilestone.com 13 bc ca ab Show that a b b c c a a 3 b 3 c 3 3abc . a b c 14 yz Show that y z a1 b1 15 a) If A= a2 b2 a3 b3 x x zx y 4 xyz . z x y c1 Adj. A c2 is a non-singular matrix then A is invertible and A 1 det A c3 b) Find the inverse of diag [abc] c) Findadjoint and inverse of cos sin cos n n A then show that A sin n sin cos for all integers of n 16 If 17 1 2 2 2 If A= 2 1 2 , then show that A 4 A 5I 0. 2 2 1 18 1 2 1 3 2 If A= 0 1 1 , then find A 3 A A 9 I 0. 3 1 1 19 If 1 3 3 1 4 3 1 3 4 sin n cos n 1 0 0 1 3 3 2 I and E then show that (aI bE) a I 3a bE 0 1 0 0 20 cos 2 If , then show that 2 cos sin 21 If n is a positive integer and A= cos sin cos 2 sin 2 cos sin 3 4 then show that 1 1 cos sin =0 sin 2 1 2n 4n An 1 2n n 7 2 2 1 1 2 and B 4 2 T T A= then find AB and BA . 5 3 1 0 22 If 23 1 a a2 Show that 1 b b 2 (a b)(b c)(c a ) 1 c 24 c2 bc ca ab a b c Show that c a a b b c 2 b c a ab bc ca c a b 25 A certain bookshop has 10 dozen chemistry books , 8 dozen physics books , 10 dozen economics books . Their selling prices are Rs 80,Rs 60,Rs40 each respectively. using matrix algebra find total value of books in shop 26A trust fund has to invest Rs 30,000 in different types of bonds. The first bond pays 5% interest per year and the second bond pays 7% interest per year. Using matrix multiplication determine how to divide Rs 30,000 among the two types , if the trust fund must obtain annual total interest of i) Rs 1800 ii) Rs 2000 2 MARKSwww.studentsmilestone.com 27If 28 1 2 3 8 A , B and 3 4 7 2 2X A B then find 0 2 1 a) If A 2 0 2 is a skew symmetric matrix, then find x. 1 x 0 b) Construct a 3X2 matrix whose elements are defined by 29 X. cos sin If A 1 i 3j 2 sin T T , show that AA A A I 2 . cos x - 3 2y - 8 5 = 6 -2 z + 2 30 .i) If a ij 2 , then find the value of x, y, z and a. a - 4 x -1 2 5 - y 1 2 3 z -1 7 0 4 7 then find the values of x,y,z and a ii) If 0 1 0 a - 5 1 0 0 x -1 2 y - 5 1-x 2 -y iii) z 0 2 2 0 2 then find the values of x,y,z and a 1 -1 1+a 1 -1 1 1 3 -5 31. Find the trace of A, if A = 2 -1 5 32. If A = 1 0 1 i 0 0 -i , B = 0 -1 1 0 and C = 0 i i 0 then show 2 thati) A 2 = B2 = C 2 = -I ii) AB = -BA = -C,(i = -1and I is the unit matrix of order 2) 33If A = 2 4 2 -1 k and A = 0 then find the value of ‘k’ 34. If A = 2 0 1 -1 1 0 and B = -1 1 5 0 1 -2 then find AB T T 12 22 32 2 2 2 35 .i)FIND DETERMINANT OF 2 3 4 .ii) If A = 32 42 52 . 1 0 0 2 3 4 and det A = 45 then find x. 5 -6 x 2 -3 4 6 36 .Find the adjoint and the inverse matrices of the following matrix 37 1 a) 1 1 Find RANK of the following 1 1 1 1 1 1 b) 0 1 0 0 0 1 0 1 1 c) 3 0 -2 2 4 3 0 1 2 -1 1 2 d) 2 5 0 2 3 1 3 4 2 CHAPTER 4 ADDITION OF VECTORS www.studentsmilestone.com4 MARKS 1. If ABCDEF is a regular AB + AC + AD+ AE + AF = 3AD = 6AO 2. In ABC, if O is the circumcenter and H is the orthocenter, show that i) OA + OB + OC =OH hexagon with centreO then prove that ii) HA + HB + HC = 2.HO 3) If G is the centroid of ABC show that GA + GB + GC = O 4. a) if a,b be non collinear vectors =(x+4y)a+(2x+y+1)b and =(y-2x+2)a+(2x-3y-1)b such that 3 =2 then find x,y b) In two dimensional plane prove by vector method equation of line with intercepts “a”, and ”b” is given by x/a+y/b=1 5 If A =(2,4,-1) , B = (4,5,1) and C = (3,6, -3) are the vertices of ABC, find the length of the sides and show that it is a right angled triangle. Find the direction cosines of AB,BC,CA 6. Show that the points with position vectors 2i 3 j 6k , 6i 2 j 3k , 3i j 2k form an equilateral triangle. 7 a,b,c are non coplanarfind the point of intersection of straight line passing through the points 2a + 3b – c, 3a + 4b – 2c with the straight line passing through the points a – 2b + 3c, a- 6b + 6c 8. If the points whose position vectors are 3i-2j-k,2i+3j-4k,-i+j+2k and 4i+5j+λk are coplanar then show that λ= -146/7 9 Write the vector equation of the straight line passing through the points (2i j 3k ), (4i 3 j k ) also find Cartesian form of the line. 10 Find vector equation of plane passing through points 4i-3j-k,3i+7j -10k and 2i+5j-7k and show that the point i+2j-3k lies on it 11 Find the equation of plane passing from points i-2j+5k, -5j-k, -3i+5j www.studentsmilestone.com 2 MARKS 12 Find unit vector in direction of SUM of vectors If a=2i+4j-5k ,b=i+j+k, c=j+2k 13 If a=2i+4j-5k ,b=i+j+k, c=j+2k find unit vector in opposite direction of a+b+c 14 If a=2i+5j+k b=4i+mj+nk are collinear then find m, n 15 If a=i+2j+3k b=6i+j find unit vector in direction of a+b 16 If vectors -3i+4j+λk and µi+8j+6k are collinear then find λ, µ 17 The position vectors of two vectors A,B are a,b . If C is point on AB such that AB=5AC Then find position vector of C 18 If the position vectors of the points A,B,C are -2i+j-k,-4i+2j+2k and 6i-3j-13k respectively And AB=λ AC then find the value of λ 19 If OA= i+j+k AB=3i-2j+k BC=i+2j-2k and CD =2i+j+3k then find the vector OD 20 if , , are angles made by vector 3i-6j+2k with positive axis then find cos ,cos ,cos 21 If position vectors of points 2i+j+k,6i-j+2k,4i-5j-pk are collinear find “p” 22 Find unit vector in the direction of AB if position vectors of A,B are 2i-4j+3k,5i+3j+k 23 show that position vectors -2a+3b+5c,a+2b+3c,7a-c are collinear 24Find vector equation of line passing from points 2i+3j+k,and parallel to 4i-2j+3k 25 OABC is parallelogram . If OA=a and OC=c find vector equation of side BC 26 ABCDE is pentagon If the sum of vectors AB,AE,BC,DC,ED and AC is λAC then find λ 27 Find vector equation of line passing from points 2i+j+3k, -4i+3j-k CHAPTER 5 MULTIPLICATION OF VECTORS 7 MARKSwww.studentsmilestone.com 1 Find the equation of the plane passing through the points A = (2,3-1), B =(4, 5 , 2) and C = (3,6, 5) in Cartesian form 2 3 If A =(-1, 2, -3) , B = (-16, 6, 4) , C =(1,-1,3) and D = (4, 9, 7), find the distance between the lines If A=(1,-2,-1)B=(4,0-3)C=(1,2,-1)and D=(2,-4,-5) find distance between AB and CD 4 Find shortest distance between the skew lines r = (6i+2j+2k)+t(i-2j+2k) and r=(-4i-k)+s(3i-2j-2k) 5 If a = 2i + j – 3k , b = i – 2j + k, c = -i + j – 4k and d = i + j + k , compute (a xb)x(cxd) . 6.Find the value of , for which a = i - j + k; b= 2i + j – k and c = i – j - k are coplanar. 7.If a =(1, -1, 6)b=(1, -3, 4) &c=(2, -5, 3) value of a.(b x c ), a x ( b x c)&( a x b) x c 8 Find the Cartesian equation of plane passing through the points (-2,1,3) and perpendicular to the vector 3i+j+5k 9 Find the Cartesian equation of plane passing through the points (-2,-1,-4) and parallel plane 4x-12y-3z-7=0 to the 10 Find volume of the tetrahedron whose vertices are (1,2,1) (3,2,5),(2,-1,0) and (-1,0,1) 11 Prove that the smaller angle θbetween any two diagonal of a cube is given by Cosθ= 1/3 or θ= cos-1(1/3) 12 a)a,b,c are coplanar vectors .Prove that the following four points are coplanar 6a+2b-c,2a-b+3c,-a+2b-4c,-12a-b-3c b) Find λ in order that the four points A(3,2,1)B(4,λ ,5) C(4,2,-2)D(6,5,-1) be coplanar 13 a) In any triangle altitudes are concurrent b)In any triangle the perpendicular bisectors of sides are concurrent c) show that points (5,-1,1) (7,-4,7),(1,-6,10),(-1,-3,4) are vertices of Rhombus 4 MARKSwww.studentsmilestone.com 14 Show that 2i – j + k, i – 3j – 5k and 3i – 4j - 4k are the vertices of a right angled triangle. Find the other angles of that triangle. 15 If a + b + c = 0, a = 3, b = 5, c = 7, show that the angle between a and b is 3 16.If a = 2i 3 j 5k , b i 4 j 2k . Find the unit vectors perpendicular to both a and b. 17 Find the vector area of the triangle having vertices (1,2,3) , (2,5, -1), (-1, 1, 2). What is the magnitude of the area of that triangle.. 18 Let a and b be two vectors, satisfying a = b = 5 and (a,b) = 45. Find the area of the triangle having a-2b and 3a + 2b as sides.. 19. Find the unit vector perpendicular to the plane with the points (1,2,3), (2,-1,1) and (1,2,-4) 20 If a = 2i j k , b = i 2 j 4k and c i j k , find (a x b). (b x c) 21 Find the area of the parallelogram whose diagonals are 3i + j – 2k and i – 3j + 4 22 If a = i + j + k and b = 2i + 3j + k i) Find the length of the projection of b on a & length of the projection of a on b. ii) Find the vector components of b along a and perpendicular to a. 23 if a=2i-3j+5k and a-b b= -i+4j+2k then find (a+b)x(a-b) and unit vector perpendicular to both a+b 24 Find the unit vector perpendicular to the plane with the points P(1,-1,2), Q(2,0,-1) and R(0,2,1) 25 Find the λ if volume of the parallelepiped having i+j ,3i - j and 3j +λ k as co-terminus edges is 16 . 2 MARKSwww.studentsmilestone.com 26 Find angle between vectors i+2j+3k,3i-j+2k 27 if 4i 2p j+pk is parellel to the vector i+2j+3k then find "p" 3 28 If a=i+2j-3k b=3i-j+2k then show that a+b and a-b are perpendicular to each other 29 if a,b non zero non collinear vectors and a b = a b then find angle between a,b 30 If the vectors 2i+λj+k and4i-2j+2k are perpendicular to each other find λ 31 If a=2i+2j-3k b=3i-j+2k then find angle between 2a+b,a+2b 32 If 33 a= 2i-j+k and b= 3i+4j-k if is angle between a,b find sin p 2 q 3 (p,q)= then find p q 34 2 a 13 b 5 a.b 60 then find a b 6 35 Find the volume of tetrahedron with coterminous edges as i+j+k,i-j,i+2j+k 36 if a=2i-j+k b=i+2j-3k and c=3i+pj+5k are coplanar find “P” 37 Find the volume of the parallelepiped having 2i – 3j , i + j –k and 3i – k as co-terminus edges. 38Find the area of the parallelogram whose adjacent sides 39 are 2i-3jand 3i-k. If the sum of two unit vectors is another unit vector, show that the magnitude of their difference is 3 www.studentsmilestone.com CHAPTER 6 TRIGONOMETRIC RATIOSAND TRANSFORMATION 7 MARKS In triangle ABC prove the following i.e.A+B+C=180 A B C A B C sin sin 1 4 cos cos sin 2 2 2 4 4 4 1. Prove that sin 2 Prove that cos 2 A cos 2B cos 2C 4 cos A cos B cos C 1 3 Prove that sin A sin B sin C 4 cos 4. Prove that cos A cos B cos C 1 4 cos A B C cos cos 2 2 2 A B C sin cos 2 2 2 5 .Prove that sin A sin B sin C 2(1 cos A cos B cos C ) 2 6 a)Prove that sin 2 2 A B C A B C sin sin 1 4sin sin sin 2 2 2 4 4 4 b) Prove that cos c) Prove that cos A B C A B C cos cos 4cos cos cos 2 2 2 4 4 4 A B C A B C cos cos 4 cos cos cos 2 2 2 4 4 4 7if A+B+C=270 provea) cos 2 A cos 2B cos 2C 1 4 sin Asin B sin C b) sin 2 A sin 2 B sin 2C 4 sin A sin B cos C 8 If A+ B +C = 2S, prove that cos( S A) cos( S B) cos( S C ) cos S 4 cos 9 A B C cos cos 2 2 2 cos cos 2 2 2 P.T cos cos cos cos( ) = 4 cos cos cos 2 2 2 If + + = 0 S.T. 1+ cos + cos +cos 4 cos 10 If A,B,C are triangles then prove that sin 2 A B C A B C sin 2 sin 2 1 2cos cos sin 2 2 2 2 2 2 11 .Prove that sin 2 A sin 2 B sin 2 C 2 sin A sin B cos C 12 If A+B+C=180 sin A sin B sin C A B cot cot sin A sin B sin C 2 2 4 MARKSwww.studentsmilestone.com 13 14 15 If A + B = 135 P.T (1+cotA) (1 + cotB) = 2. Hence deduce that cot 67 If (a b) sin( ) (a b) sin( ) S.T a tan b tan S.T sinA sin( 60 A) sin( 60 A) deduce that sin 16 1 2 1 2 9 sin 1 sin 3 A 4 2 3 4 3 sin sin 9 9 9 16 S.T 4 cos cos(60 ) cos(60 ) cos 3 cos 18 cos Hence deduce that 3 5 7 3 cos cos 18 18 18 16 17 tan A tan( 60 A) tan( 60 A) tan 3 A Deduce that tan6.tan42.tan66tan78 = 1 18 prove that 1 cos 19 If sin x sin y 3 5 7 1 cos 1 cos 1 cos 8 8 8 8 1 8 7 x y 3 ; cot( x y ) 24 2 4 1 1 ; cos x cos y 4 3 S.T tan 20.If A+ B = 450prove that (1+tanA) (1+tanB) = 2 and hence deduce that tan22 21 1 2 1 2 sin( 2 4 x) If sin x 0, prove that cos x cos 2 x cos 4 x cos 8 x 4 2 sin x 22 If Sin(A B) 24 4 ; Cos(A-B) find the value of Tan 2A 25 5 23 If Sin(A B) 24 3 TanA 25 4 3 5 7 3 Sin 4 Sin 4 8 8 8 8 2 2 3 4 5 b) prove that sin sin sin sin 5 5 5 5 16 24 a) prove that Sin 4 Find CosB Sin 4 25 prove thatwww.studentsmilestone.com CosA CosB SinA SinB n A B 2Cot if n Even SinA SinB CosA CosB 2 0 if n Odd n 26 If A+B=225 Then prove that n CotA CotB 1 1 CotA 1 CotB 2 27 Prove that Tan70-Tan20=2tan50 28 Prove that Tan50-Tan40=2tan10 29 Find the value of tan10+tan35+tan10tan35 30 Find the value of tan100+tan125+tan100tan125 31 a) sin A sin 1 24 A A and 90 < A < 180 then find sin , cos b)If cos 25 2 2 4 270 360 Find , cos , tan www.studentsmilestone.com 2 2 2 32 Prove that cot(15 A) tan(15 A) 4cos 2 A 1 2sin 2 A 33 Prove that the roots of quadratic equation 16 x 2 12 x 1 0 are sin 218 and cos 2 36 34 Prove that cos 2 76 + cos 216 cos 76cos16 35 Prove that sin 21cos9 cos84cos 6 3 4 1 4 36 Prove that cos12 cos84 cos132 cos156 2 MARKS 37 1 2 www.studentsmilestone.com 3x ii) f(x)=cosec(6-5x) 2 i) sin a) Find period of iii) f(x)= Tan5x 5x 4x+9 iv) f(x)=cos v) f(x)=sin 2 5 vi) tan ( x 2 x 3x .... nx) vii) tan ( x 2 4 x 2 .... n 2 x 2 ) viii) sin 4 x cos 4 x b) Prepare the functions Sinx, Cosx ,Tanx with period “5” 3 5 7 9 .cot cot cot 1 20 20 20 20 38 Prove that cot 39 2 2 2 2 x r cos cos : y r cos sin z r sin S.T x y z r 20 cot 40 Find the value of cos 2 45 sin 2 15 41If 42 If tan A 43 S.T sin 44 P.T tan 45 S.T 46. if sin = 47 i) Find maximum minimum values of cos sin S.T a cos 2 b sin 2 a a b 8 find sin 2 A, cos 2 A, tan 2 A 25 sin 3 Hence find sin15 1 2 cos 2 sin 2 1 Hence find tan15, tan 22 1 cos 2 2 sin 3 cos 3 2 sin cos -4 is not in 3rd quadrant find other ratios www.studentsmilestone.com 5 5cos x 12sin x 13 5cos x 12sin x 13 ii) Find maximum minimum values of iii) Find maximum and minimum values of 5cos x 3cos( x Find extreme values of cos x cos 48 If cos sin 2 cos ,prove 49 If tan 20 0 , then show that i) tan 160 0 tan 110 0 1 2 2 1 tan 160 0. tan 110 0 50 Prove 53 3 )8 x cos x 3 3 iv) 52 1 3 4 0 sin 10 cos10 0 that cos sin 2 sin ii) 51 tan 250 0 tan 340 0 1 2 tan 200 0 tan 110 0 1 2 show that cos100cos40+sin100sin40= 1 2 Find the value of cos42+cos78+cos162 www.studentsmilestone.com Prove that sin50 sin70 sin10 0 54 cos55 cos65 cos175 0 Prove that 55 If 3sin 4cos 5 find the value of 4sin -3cos sin 2 4 6 9 sin 2 sin 2 10 10 10 56 Find the value of sin 2 57 Find the value of sin 330 cos120 cos 210 sin 300 10 58 If A,B,C are angles of triangle ABC then prove that A 2 B 3C AC cos cos 0 2 2 -4 59 If tan = and is not in 4 th quadrant prove that 3 5sin +10cos +9sec +16cosec +4cot 0 tan 6100 tan 7000 1 p 2 60 If tan 20 p , then show that tan 5600 tan 4700 1 p 2 0 61 Prove that 63 Show that cos9 sin 9 =cot3662 Show that Cos42+Cos78+Cos162=0 cos9 sin 9 cos340 cos 40 sin 200 sin140 2 64 Find the value of i) sin 82 1 1 sin 2 22 2 2 1 2 2 ii) cos 112 65Show that sin 600 cos330 sin120 sin150 1 1 1 sin 2 52 2 2 CHAPTER 7 TRIGONOMETRIC EQUATIONS 1 Solve i) 2cos 2 4 MARKS 11sin 7 ii)2sin 2 x 3cos x 3 0 iii) cot 2 x ( 3 1) cot x 3 0 Solve 3 tan 4 10 tan 2 3 0 3) 2) Solve 3 cos sin 2 4 Let , be solutions of the a cos b sin c , where a,b,c are real constants. then show that i) cos cos 2ac , 2 a b2 cos cos c 2 b2 a 2 b2 and 2bc c2 a2 , sin sin 2 ii) sin sin 2 5. Solve sin 3 cos 1 0 a b2 a b2 1 sin cot cos , then sin 4 2 2 2 6 Prove that if tan 7 Solve sin 3 cos 1 0 8 Solve 9solve the equation 3cos2 2 7sin 11solve tan 3cot 5sec 13solve 2(sin x cos x) 3 10 solve sin x 3 cos x 2 12 solve 1 sin 2 3sin cos 2cos2 3sin 1 0 14 Find all values of x in (- , ) satisfying the equation 81+cosx+cos x............ 43 2 15 If 1 , 2 are roots of the equation acos2 +bsin2 =c then find the values of i)tan1 tan 2 ii)tan1 tan 2 and hence find Tan(1 2 ) 16 SOLVE 4sinxsin2xsin4x=sin3x CHAPTER 8 INVERSE TRIGONOMETRIC FUNCTIONS 4 MARKS 1 3 5 8 1 77 sin 17 85 1 1 a) Show that sin sin 3 5 12 1 33 cos 13 65 1 1 b) Show that sin cos 1 1 2 Prove that i ) 2 arctan arctan 3 7 4 3 If cos1 p cos1 q cos 1 r 3 5 ii) arcsec then prove 34 arc cosec 17 5 4 that p 2 q 2 r 2 2 pqr 1 5 3 5 323 27 tan 1 ii ) 2 sin 1 cos 1 cos 1 5 13 325 11 34 4 Prove i) sin 1 cos 1 4 4 44 iii ) 2 cos 1 sin 1 tan 1 5 5 117 5 If sin 1 x sin 1 y sin 1 z , prove that tan 1 x tan 1 y tan 1 z 6 i) If .ii) If tan 1 x tan 1 y tan 1 z 7 P rove that tan 1 2 x 1 x 2 y 1 y 2 z 1 z 2 2 xyz then show that x+y+z=xyz then show that xy+yz+zx=1 3 3 8 tan 1 tan 1 4 5 19 4 1 1 2 9 P rove that tan 1 tan 1 tan 1 0 7 13 9 1 1 1 8 P rove that tan 1 ( ) tan 1 tan 1 2 5 8 4 4 1 2 tan 1 5 3 2 4 1 or tan 1 2 tan 1 3 3 2 10 P rove that sin 1 11 Solve for 'x' if x 1 x 1 8 i) tan 1 tan 1 ii) tan 1 ( x 1) tan 1 ( x 1) tan 1 x2 x2 4 31 12 cos 1 p q p2 2 pq q2 cos 1 then prove that 2 cos 2 sin 2 a b a ab b 2 2p 2x 1 1 q 13 If sin cos tan 1 2 2 1 p 1 q 1 x2 1 show that x= p-q 1+pq 14 If sin 1 x sin 1 y sin 1 z , prove that x 4 y 4 z 4 4 x 2 y 2 z 2 2( x 2 y 2 y 2 z 2 z 2 x 2 ) 3 12 15 a) Find value of sin cos 1 cos 1 5 13 5 3 b) Find value of tan sin 1 cos 1 5 34 4 2 c)Find value of tan cos 1 tan 1 5 3 CHAPTER 9 HYPERBOLIC FUNCTION 2 MARKS 3 , find cosh(2x) and sinh(2x). 4 Ifsinhx=3, then show that x log( 3 10 ) Prove that sinh(x+y)=sinhxcoshy+coshxsinhy Prove that cosh(x+y)=coshxcoshy+sinhxsinhy cosh x sinh x n cosh( nx) sinh( nx) Prove that 1 If sinh x 2 3 4 5 1 1 2 6 Show that tanh 8 If cosh x 1 log e 3 2 7 If sinh x 3 , find cosh(2x) and sinh(2x). 9 2 CHAPTER 10 5 , find cosh(2x) and sinh(2x). 2 Prove that cosh x sinh x n cosh(nx) sinh(nx) PROPERTIES OF TRIANGLES 7 MARKS 1 In ABC, a) if r1 8, r2 12, r3 24, find a,b,c,R b)In ABC, if c) 2 a) In r1 36, r2 18, r3 12, find a,b,c.,R ABC, if If a=26, b=30, cos c r1 2, r2 3, r3 6, find a,b,c.,R 63 65 , prove that R , r 3, r1 16, r2 48, r3 4. 65 4 b) If a=13, b=14, c 15, prove that R that r1 r2 r3 1 1 bc ca ab r 2R 65 21 , r 4, r1 , r2 12, r3 14. 3 8 2 Show 4. 5.S.T 6. ab r1r2 bc r2 r3 ca r3 r1 r3 r1 r2 S.T. r1r r2 r3 r2 r r3 r1 r3 r r1r2 bc ca ab Or. a.(rr1 r2 r3 ) b(rr1 r3 r ) c(rr3 r1r2 ) abc Prove that in ABC a) Show that r r3 r1 r2 4R cos B. b) prove that r r1 r2 r3 4R cos C c) Show that r1 r2 r3 r 4R 7. If p1 , p2 , p3 are altitude from vertices of A,B,C to opposite sides of triangle P.T (abc) 2 83 i) p1 p 2 p3 abc 8R 3 iv) ii) 1 1 1 1 1 1 1 1 iii ) p1 p 2 p3 r3 p2 p3 p1 r1 1 1 1 cot A cot B cot C 2 2 2 p1 p2 p3 A B C cot cot 2 2 2 2 (a b c) cot A cot B cot C a 2 b 2 c 2 cot 8 NOTE IN FINAL EXAM NUMERATOR OR DENOMINATOR WILL BE GIVEN FOR”4” MARKS www.studentsmilestone.com Show that a 2 cot A b 2 cot B c 2 cot C 9 abc R 10 show that i)(r1 -r)(r2 r )(r3 r ) 4 Rr 2 ii) (r1 +r2 )(r2 r3 )(r3 r1 ) 4 Rs 2 11 Prove that a cos( B C ) b cos(C A) c cos( A B) 3abc 3 3 3 12 If cos A cos B cos C 1 then show that it is right angle triangle 2 13 2 If a 2 b2 c 2 8R2 then show that it is right angle triangle 4MARKS www.studentsmilestone.com 14 In ABC , if 15 16 2 1 1 3 , then show that c=60. ac bc abc S.T a cos 2 If cot A B c b cos 2 c cos 2 s 2 2 2 R A B C : cot : cot 3 : 5 : 7, show that a:b:c=6:5:4. 2 2 2 17 . cos A cos B cos C 3 , show that the triangle is equilateral. 2 If (r2 r1 )( r3 r1 ) 2r2 r3 show that A=90 0 18 19 Show that 20 1 1 1 1 a2 b2 c2 r 2 r1 2 r2 2 r3 2 2 If C = 60 S.T a b 1 bc ca If r : R : r1 2 : 5 :12 21 S.T the triangle is right angle triangle 22 i) If a : b : c = 7 : 8 : 9 find cosA : cosB : cosC ii) If b + c : c + a: a + b = 11 : 12 : 13 P. T cos A: cosB :cos C = 7 : 19 : 25www.studentsmilestone.com 23 Show that cos A cos B cos C a 2 b2 c 2 a b c 2abc 24 a) In any Triangle ABC show that tan a (b c)sec prove that tan = b) If sin c) If d) BC bc A cot 2 bc 2 If a 2 bc A prove that cos = cos . bc b+c 2 a (b c)cos prove that sin = 25 If a,b,c are in A.P. then show that 3tan C A 3b 26 If a cos 2 ( ) c cos 2 ( ) 2 2 2 27 If cot 32 2 bc A cos . b+c 2 A C tan 1 2 2 show that a,b,c in A.P A B C ,cot , cot are in A.P. show that a,b,c in A.P. 2 2 2 28 Prove that 30 2 bc A sin . b-c 2 1 1 1 1 r1 (r2 r3 ) a 29 In ABC , prove that r1 r2 r3 r r1r2 r2 r3 r3r1 Show that rr1 r2 r3 2 31If tan If a = 4 : b = 5: c = 7 find cos A 5 C 2 : tan find relation between a,b,c 2 6 2 5 B 33In ABC (a + b + c) (b + c – a) = 3bc 2 Find ‘A’ In ABC a = 3 : b =4 :sinA= 3 find ‘B’ 4 34 If a = 6 ; b = 5; c = 9 find ‘A’35 36 If length of sides of triangle is 3.4.5 find the Circum Diameter 37 If 38 P.T a(b cos C C cos B) b c 39 a) If in a triangle angles are in ratio 1 : 5 : 6.Find side ratios. a b c S.T ABC is equilateral triangle. cos A cos B cos C 2 2 b)If in a triangle angles are in ratio 1 :2 : 7.Find side ratios. r R 40 In equilateral triangle find value of 41 The perimeter of ABC is 12cm and In radius is 1cm. Find area of triangle 42 If a = 18 : b : 24 : c = 30 find r1 44 if b=4 A=45 B=30 46 43If in a triangle perimeter is 30 cm and ‘A’ is right angle find r1 find a & c 45 if A=30 C=90 c=7 3 find a & b Show that (b c)cos A (c a)cos B (a b)cos C a b c C B 2 2 2 47 b cos 2 ( ) c cos 2 ( ) s 48 Show that 2 bc cos A ca cos B ab cos C a b c 2 2 49 If b+c=3a then find the value of cot 51 In ABC express A r cot 2 1 B 2 C 50 2 Show that rr1 cot A 2 in terms of s 52 Show that a cos A b cos B c cos C 54 Express cot 2 R 53 Prove that (b-aCosC)SinA=aCosASinC C A a sin 2 ( ) c sin 2 ( ) in terms of s,a,b,c 2 2 A A 55 Show that (b c) 2 cos 2 ( ) (b c) 2 sin 2 ( ) a 2 2 2 HEIGHTS AND DISTANCE PROBLEMSwww.studentsmilestone.com 56the angle of elevation of the top point P of vertical tower PQ of height “h” from a point A is 450 and from point B is 600 where B is a point at distance 30 meters from the point A measured along line line AB makes an angle 300 with AQ . find the height of tower 57 Two trees A and B on same side of a river. From a point C in the river the distance of the A and B are 250 mts and 300 mts respectively. If the angle C is 450 find distance between the trees ( use √2=1.414) 58 A lamp post is situated at the middle point M of side AC of triangle plot ABC with BC= 7m CA =8m AB=9m lamp post subtends an angle 150 at the point B. Find the height of lamp post 59 Two ships leave port at same time. One goes 24 km/hr in the direction N450E and other travels 32km/hr in direction S750E . Find distance between them at the end of 3hrs. 60 A tree stands vertically on slant of hill . From a point A on the ground 35 mts down the hill from the base of tree ,the angle of elevation of the top of tree is 600 If the the angle elevation of the foot of tree from A is 150 then find height of tree 61 The upper 3/4th portion of vertical pole subtends Tan 1 3 at point in horizontal plane 5 through its foot and at a distance of 40mts from foot. given that the pole is at a height less than 100mts from the ground find its height 62 AB is a vertical pole with B at the ground level and A at top. A man finds that the angle of elevation of point A from a certain point C on the ground 600. He moves away from the pole along the line BC to a point D such that CD=7m. From D, the angle of elevation of point A 45 0. Find the height of the pole 63 let an object be placed at some height h cm and let P and Q be two points of observations which are at distance 10cm apart on a line inclined angle 150 to the horizontal. if angles of elevation of object from P and Q are 300 and 600 respectively then find hwww.studentsmilestone.comwww.studentsmilestone.com