Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
```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: AC is also a bijection
2.
Let f: A B and g: BC 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:BA 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
x2
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:RR g:RR 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: RR 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: RR 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: RR 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)=
x2
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  33n1 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 bc
2a
2a
2b
bca
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
bc ca ab
Show that a  b b  c c  a  a 3  b 3  c 3  3abc .
a
b
c
14
yz
Show that y
z
 a1 b1

15 a) If A= a2 b2

 a3 b3
x
x
zx
y  4 xyz .
z
x y
c1 
c2  is a non-singular matrix then A is invertible and A 1 
det A
c3 
b) Find the inverse of diag [abc]
 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
bc ca ab
a b c
Show that c  a a  b b  c  2 b c a
ab bc ca
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
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 
 AC 
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)tan1  tan  2 ii)tan1 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
cos1 p  cos1 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
x2
x2 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 83

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.
ac bc abc
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
bc ca
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)
BC bc
A

cot
2
bc
2
If
a
2 bc
A
prove that cos =
cos .
bc
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
```
Related documents