section i [ very short answer questions] - FIITJEE Hyderabad
`K U K A T P A L L Y
CENTRE
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN
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INTERMEDIATE PUBLIC EXAMINATION, MARCH 2017 Total No. of Questions - 24
Reg.
Total No. of Printed Pages - 2
No.
Part - III MATHEMATICS, Paper-I (A) (English Version) Time : 3 Hours]
[Max. Marks : 75
10 2 = 20 M
SECTION - A I.
Very Short Answer Type questions:
1.
If A
2.
If f x If A
x2 , h x
2, g x 3
3.
and f : A
0, , , , 6 4 3 2 2
2
2
1
3
2x for all x R , then find fo goh
3
1 0 , B
2
1
4
B is a surjection defined by f x
cos x then find B . x .
1 0 3 and X 1 2
1
A
B then find X .
4.
If A
1 2 then find AA' . 0 1
5.
a
5j
6.
Find the vector equation of the line passing through the point 2 i
2i
vector 4i
k and b 2j
4i
nk are collinear vectors then find m and n .
mj
Find the angle between the vectors i
8.
If sin
9.
Prove that cos 48 cos12
10.
If cosh x
2j
3k and 3i
j
2k .
is not in the first quadrant, find the value of cos .
3
5 8
.
5 , find the value of (i) cosh 2x and (ii) sinh 2x . 2
SECTION – B II.
k and parallel to the
3k .
7.
4 and 5
3j
5 4 = 20 M
Short Answer Type questions: (i) Attempt any five questions (ii) Each question carries four marks
1 a a2 11.
Show that 1 b b 2
a
b b
c c
a .
1 c c2 12. 13.
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 . i 2 j 4 k and c i j k then find a b b c . If a 2 i j k , b
14.
(i) Find the range of 13cos x (ii) Evaluate sin 2 82
1 2
o
3 3 sin x 4 .
sin 2 22
1o . 2
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sin 2
15.
Solve 1
16.
Show that cot sin
17.
In
ABC , if
cos .
3sin
13 17
1
1 a
sin tan
1 c
b
c
3 b
a
1
2 . 3
, show that C
c
60 .
5 7 = 35 M
SECTION – C III.
Long Answer Type questions: (i) Attempt any five questions (ii) Each question carries seven marks
18.
(i) If f : Q
Q is defined by f x
(ii) If f 19.
4 , x Q , show that f is a bijection and find f
5x
and g
4,5 , 5,6 , 6, 4
4, 4 , 6,5 , 8,5
then find f
1
.
g and fg .
Using mathematical induction, prove 1 2 3 2 3 4 3 4 5 ...... upto n terms n n 1 n 2 n 3 , n N. 4 Solve the following system of equations by using Cramer’s rule: 2 x y 3z 9; x y z 6; x y z 2
20.
1 2 1
21.
(i) Show that A
3 2 3 is non-singular and find A
1
.
1 1 2 1 2 2 2 1 2 then show that A2
(ii) If A
4A
0.
5I
2 2 1
22.
Find the shortest distance between the skew lines: r
23.
6i
If A cos S
24.
2j
2k
2j
2k
and r
4i
k
s 3i
2j
2k .
2S , then prove that
B C A
t i
cos S
Show that in a
B
ABC ,
cos C 1 r2
1 r12
1 1 r22
4 cos 1 r32
S
A 2
a
2
cos
b
2
S
B 2
c
2
2
cos
C . 2
.
BLUE PRINT (MATHS-IA) S.No.
Chapter Name ALGEBRA
1. 2. 3.
Functions Mathematical induction Matrices & Determinants
4. 5.
Addition of vectors Product of vectors
6. 7. 8. 9. 10.
TRIGONOMETRY Trigonometry upto transformations Trigonometric equations Inverse trigonometric functions Hyperbolic functions Properties of triangles
Weightage Marks 11 (7 + 2 + 2) 7 (7) 22 (7 + 7 + 4 + 2 + 2)
VECTOR ALGEBRA 8 (4 + 2 + 2) 13 (7 + 4 + 2) 15 (7 + 4 + 2 + 2) 4 (4) 4 (4) 2 (2) 11 (7 + 4)
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VERY SHORT ANSWER QUESTIONS
1. A.
If A
B.
If f : R
C.
If f : R \ 0
D.
Find the domains of the following real valued functions:
B is a surjection defined by f x
R is defined by f x
1
R is defined by f x
a)
f x
2
x
c)
f x
x
2
1
1 log
x2
x
2
1
g) f x
6x
x
3
x
i) f x
2
x
1 1
5
3 x x
cos x then find B .
2x x , then show that f x 1 x2
1 then prove that f x x
f x
log
3
2
2f x .
f x2
f 1 .
x x
10
1
d) f x
1 x
10
log
b)
x
1
e) f x
E.
and f : A
0, , , , 6 4 3 2
x
x
f) f x
log x
h) f x
x2
j) f x
1 log 2
x 1
1
x2
3x
2
x
Find the ranges of the following real valued functions: a)
x
sin
x
x
b)
R is defined by f x
1
x
2
F.
If f : R
0
G.
If f : R
R is defined by f x
H.
If A
2, 1,0,1,2 and f : A
A.
If f x
x x
B.
If f : 4, 5 , 5,6 , 6, 4
c)
x2 4 x 2 x3
d) 1 x
3
x2
9
e) log 4
x2
1 x
0
, then show that f x
1 x2 , then show that f tan 1 x2
f
cos2
B is a surjection defined by f x
x2
x
1 , then find B .
2.
a) f C.
1 x 1
b) fg
4
If f : R i) fof x 2
R, g:R
1
1 then find fofof x and fofofof x .
c)
and g : 4, 4 , 6, 5 , 8, 5
then find
f d) f 2
R are defined by f x
ii) fog 2
3x
x2
1, g x
iii) gof 2 a
1 , then find
3
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x
x 2
D.
Prove that the real valued function f x
E.
Determine whether the following functions are even or odd. a)
F.
f x
x2
log x
If f : R
R and g : R
b) f x
1
e
x
x
1
ex ex
1 is an even function on R \ 0 .
1 1 2x 2
R are defined by f x
3 and g x
3x
2 , then find
i) fog x , ii) gof x , iii) fof o , iv) go fof 3 . G.
Find the inverse of the following functions: a) If a, b
R, f :R
b) f : R
R defined by f x
defined by f x
0, x x
1 , x 1
H.
If f x
I.
If the function f : R f x
J.
y
f x
ax
5x
1 , show that fof
b a
0
c) f : 0, 1
x
R defined by f x
R defined by f x
log 2 x .
x
3x
3
x
, then show that
2
2f x f y .
y
R defined by f x
If the function f : R
4x 4x
2
, then show that f 1 x
1
f x .
3. A.
Construct 3 2 matrix whose elements are defined by aij
B.
Define Trace of matrix and give an example.
C.
If A
2
4
and A2
1 k
1 D.
Find the trace of 2 2 1 2
, B
1 i 2
3j
O , then find the value of k.
3 1 0
3 8
5 5 1
E.
If A
F.
Define symmetric and skew-symetric matrix and give an example.
G.
If A
H.
If A
3 4
2
4
5
3
i
0
0
i
7
2
and 2X
then find A
A
B then find X.
A' and AA' .
, find A 2 . 5
I.
Find the products of i)
1 4 2
1 3
ii)
2 6
1
4
2 3
1 2 1
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x
3 2y
z
2
8
5
2
J.
If
K.
A certain bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 respectively. Find the total amount the bookshop will receive by selling all the books, using matrix algebra.
6
2 a
then find the values of x , y , z and a .
4
4. A.
Define singular matrix and give an example.
B.
Define rank of a matrix.
C.
If A
1
0
0
2
3
4 and det A = 45 find x .
5 D.
If A
6 x
1
5
3
2
4
0
3
1
2
and B
1 0 2 5 then find 3A 4B' . 2 0
0 1
5
1 4 E.
Find the Rank of the Matrix 2 3 0 1 7
F.
If A
2
1
2
and B
2
5
0 2
1
4
3
1
2
1
then find AB’ and BA’.
0
3 0 0
G.
If A
0 3 0 , then find A 4 . 0 0 3 cos
If A
H.
sin
sin
cos
, show that AA’ = A’A = I.
5. A.
ABCDE is a pentagon. If the sum of the vectors AB, AE, BC, DC, ED and AC is the value of .
B.
If the position vectors of the point A, B and C are respectively and AB
C.
,
If
and
AC , then find the value of
be the angles made by the vectors 3i
coordinates axes, then find cos , cos
k,
AC , then find
2 k and 6i
2i
j
6j
2 k with the positive directions of the
4i
2j
3j
13 k
.
and cos .
D.
If OA
i
E.
a
5j
F.
Find the angles made by the straight line passing through the points 1, 3,2 and 3, 5,1 with
2i
k , AB
3i
2j
k , BC
k and b
4i
mj
nk are collinear vectors, then find m and n .
j
i
2j
2 k and CD
2i
j
3k , then find vector OD.
the coordinate axes. G.
Let a of a
2i
b
4j
5k , b
i
j
k and c
j
2 k . Find the unit vector in the opposite direction
c.
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6. A.
Find the vector equation of plane passing through the points i 3i
5k ,
5j
and
.
2j
k , and
5j .
B.
If the vectors
C.
If OA
i
j
k , AB
D.
Let a
i
j
k and b
3i
k and i
4j 3i
k , BC
2j
2i
6 k are collinear vectors, then find
8j i
2 k & CD
2j
2i
3k , then find the vector OD .
j
k find
3j
(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 . 2
2
2
E.
For any vector a , show that a
F.
If p
G.
Find the distance of a points 2,5, 3 from the plane r. 6i
H.
Find the vector equation of the line passing through the point 2 i vector 4i
a
j
a k
2a .
2
3 and p , q
2, q
i
6
, then find p q . 3j
2k
4. k and parallel to the
3j
3k .
2j
I.
OABC is a parallelogram. If OA= a and OC
c , find the vector equation of the side BC.
J.
If a , b , c are the position vectors of the vertices A, B and C respectively of
ABC , then find the
vector equation of the median through the vertex A. K.
Find the vector equation of the line joining the points 2 i
L.
Find the vector equation of the plane passing through the points 0,0,0 , 0,5,0 , and 2,0,1 .
M.
Find the vector equation of the plane passing through the points. i
N.
If the vectors
i
O.
Show that i
a i
P.
Compute i
j j
5k and 2 i
3j
j a j
k
j
a k
3k and
j
4i
2j
k.
3j
5k ,
k.
5j
k are perpendicular to each other, find
.
2 a for any vector a .
i .
k k
7. 11, b
23 and a b
30, then find the angle between the vectors a , b and a
A.
If a
B.
If a
C.
If P , Q , R and S are points whose position vectors are i
i
j
k and b
2i
3j
b.
k , then find the projection vector of b on a and its magnitude.
k,
i
2 j , 2i
3k and 3i
2j
k
respectively, then find the component of RS and PQ . D.
Find the angle between the planes r. 2i
E.
Let e 1 and e 2 be unit vectors containing angle . If
F.
Show that i
G.
Prove that for any three vectors a , b , c b
a i
j
a j
k
k a
j
2 k and r. 3i 1 e1 2
6j
e2
k
sin
4.
, then find
.
2 a for any vector a.
c c
a a
b
2 abc .
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H.
If a
i
3k and b
2j
3i
2 k , then show that a
j
b and a b are perpendicular to each
other. I.
Let a and b be non-zero, non-collinear vectors. If a
a b , then find the angle between
b
a and b .
J.
If a
K.
Find the area of the parallelogram for which the vectors a
2i
3k , b
2j
3i
2 k , then find the angle between 2a
j
2i
b and a
2b .
3 j and b
3i
k are adjacent
sides. L.
If a
M.
Let a
N.
Find the area of the triangle whose vertices are A 1,2,3 , B 2,3,1 and C 3,1,2 .
O.
For any three vectors, a , b , c , prove that b c c a a b
i
2j 2i
3k and b
3i
5j
k are two sides of a triangle, then find its area.
k and b
3i
4j
k . If
j
is the angle between a and b , then find sin .
2
abc .
8. A.
If 3sin
5, then find the value of 4sin
B.
If cos
C.
If tan 20 o
D.
Prove that 3 sin
E.
Prove that cot
F.
Prove that tan
cot
G.
Prove that
tan tan
sec sec
H.
Prove that 1
I.
Prove that sin
J.
Prove that cot
4cos
2 cos , prove that cos
sin
, then show that
16
.cot
cot
20
4
cos
1 1
6 sin
cos ec
2
1
2 sin
tan 160 o tan 110 o 1 tan 160 o.tan 110 o
sec 2
cos ec
.cot
sin
2
cos
2 3 7 .cot ....cot 16 16 16 2
3cos .
2
1 2
4 sin 6
cos6
13
1
cos ec 2
sec 2 .cos ec 2 .
sin . cos
1
tan
cos
sec
sec
2
2 tan 2
3 5 7 9 .cot .cot .cot 20 20 20 20
cot 2
7
1
9. cos2 x in 0,
A.
Draw the graph of y
.
B.
Find the period of the function defined by f x
tan x
C.
Find the extreme values of 5cos x
8 over R.
3cos x
3
4x
9x
...
n2 x
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D.
Find the periods of the functions: a) f x
b) f x
tan 5x
cos
9
c) f x
5
E.
Find the range of
F.
Prove that
1 sin 10o
3 cos10 o
4
G.
Prove that tan 70 o
tan 20 o
2 tan 50 o
H.
Prove that sin 2 52
I.
Draw the graph of y
J.
If A
K.
Prove that
L.
Draw the graph of the tanx between 0 and
M.
Draw the graph of the cos2x in the interval 0,
N.
Find the minimum and maximum values of (i) 3cos x
O.
If
P.
Prove that 4 cos66o
Q.
Prove that cos
B
(i) 7 cos x
4x
1 2
o
24sin x
sin 2 22
o
1 2
3
sin 2 x in
sin 9o sin 9o
3 3 sin x 4
1
4 2
.
,
45o , then prove that (i) 1
cos9o cos9o
5 (ii) 13cos x
sin x
tan A 1
tan B
2 (ii) cot A 1 cot B 1
2
cot 36o
is not an odd multiple of
and if tan
2
sin 84o
3 o
cos 120
4
. . (ii) sin2x
4sin x
1 , then show that
1 1
sin 2 sin 2
cos2x
cos 2 cos 2
tan .
15 o
cos 240
0
10. 4sinh 3 x
A.
Prove that, for any x R , sinh 3x
B.
If cosh x
5 , find the values of (i) cosh 2x and (ii) sinh 2x 2
C.
If sinh x
5, show that x
D.
Show that tanh
E.
If sinh x
F.
Prove that
1
1 2
If sinh x
5
e
26
1 log 3 e 2
3 , find cosh 2x and sinh 2x 4
(i) cosh x (ii) cosh x
G.
log
3sinh x
sinh x sinh x
3 then show that x
n
cosh nx
n
log
cosh nx e
3
sinh nx , for any n R sinh nx , for any n R
10
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SHORT ANSWER QUESTIONS
11. 1 2 2 A.
2 1 2 then show that A2
If A
4A
5I
0
2 2 1 B.
If 3 A
1
2
2
1
2 2 then show that A
2 2 C.
D.
If
3 4
2
3 4 then show that A
0
1 1
E.
If I
2
0 1
a a2
H.
i)
If b b
2
c c
2
0 1
and E
1
cos 2
, then show that
1 0
A' .
1
3
If A
1
cos sin
sin
2
then show that aI
0 0
cos 2
cos sin
cos sin
sin 2
cos sin
1
a3
a a2
1
1
b
3
0 and b b
2
1
c
3
c c
2
1
1
A3 .
bE
3
a3 I
0
3 a 2 bE .
0 then show that abc
1
ii) without expanding the determinant, prove that 1 bc b a) 1 ca c 1 ab a a a2
1 a a2
c a b
1 b b
2
1 c c
2
bc
1 a2
a3
2
ca
1 b
2
3
c c2
ab
1 c2
c) b b
b
c3
ax
by
cz
a
b
c
2
2
2
x
y
z
e) x 1
y
1
z
1
cos
b
c c
a a
b
b) Show that a
b b
c c
a
a3
a
b b
a
c 1
d) Show that ca c ab a
a 1
y
f) Show that
3abc
c c
a
b 1
z y
c3
c
bc b
yz zx xy sin
b
b3
x z
z
x x
y
z
x
4xyz y cos n
then show that for all the positive integers n, An
sin n
I.
If A
J.
If A
K.
If A and A are invertible then show that AB is also invertible and AB
L.
For any nxn matrix A prove that A can be uniquely expressed as sum of a symmetric matrix and skew symmetric matrix.
M.
Show that the determinant of skew-symmetric matrix of order 3 is always zero.
sin
cos
3
4
1
1
then for any integer n
1 show that An
1
2n n
sin n
cos n
4n 1
2n 1
B 1A
1
.
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12. A.
Let ABCDEF be a regular hexagon with centre ‘O’. Show that AB+AC+AD+AE+AF=3AD=6AO.
B.
In
ABC , if ‘O’ is the circumcentre and H is the orthocentre, then show that
i) OA + OB + OC = OH C.
a , b , c are non-coplanar vectors. Prove the following four points are coplanar.
a) D.
E.
ii) HA + HB + HC = 2 HO
a
3c , 3a
4b
2b
5c ,
b) 6a
2b c , 2 a b
3c ,
If
a
b
c
c
a
b
c
d
d,
b
3a a
d
8b
2b a
5c ,
4c ,
3a
c.
2b
12a b 3c
and
are non-coplanar vectors, then show that
a, b , c
0.
Find the equation of the line parallel to the vector 2 i point A whose position vectors is 3i
2 k , and which passes through the
j
k . If P is a point on this line such that AP
j
15 , find
the position vector of P . F.
x
Let a , b be non-collinear vectors. If y
2x
2 a
2x
3y
4y a
2x
1 b are such that 3
G.
Is the triangle formed by the vectors 3i
H.
If the points whose position vectors are 3iˆ
2 ˆj
1 b and
2 then find x and y .
2 k , 2i
5j
y
5 k and
3j
k ,2iˆ
5i
4kˆ , iˆ
3 ˆj
2j
3 k equilateral?
2 kˆ and 4iˆ
ˆj
5 ˆj
kˆ are
146 . 17
coplanar, then show that I.
In the two dimensional plane, prove by using vector method, the equation of the line whose x y 1. intercept on the axes are ' a ' and ' b ' is a b
J.
Show that the line joining the pair of points 6a points
a
2b
3c , a
2b
c , 4c and the line joining the pair of
4b
5c intersect at the point
4c when a , b , c are non-coplanar
vectors. K.
If a , b , c , are non coplanar find the point of intersection of the line passing through the points 3c , a
6b
6c .
Find the vector equation of the plane passing through points 4i
3 ˆj
kˆ , 3iˆ
2a L.
3b
2iˆ
5 ˆj
A.
If a
b
B.
If a
c ,3a
2c with the line joining the points a
4b
7 kˆ and show that the point iˆ
2b
7 ˆj
10 kˆ and
3kˆ lies in the palne.
2 ˆj
13. 0, a
c 2, b
3, b
3 and c
5 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 C.
b
Let a and b be vectors, satisfying a having a
D.
7 , then find the angle between a and b .
2b and 3a
c. 45o . Find the area of the triangle
5 and a, b
b
2b as two of its sides.
For any two vectors a and b , show that 1
a
2
1
b
2
1
a.b
2
a
b
2
a b .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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E.
If a
F.
a , b , c are non-zero vectors and a is perpendicular to both b and c . If a
i
2i
j
k and c
i
2 k . Find the a
3j
b c 2, b
3, c
4
2 , then find a b c 3
and b , c G.
3k , b
2j
G is the centroid of Prove that a2
b2
ABC and a , b , c are the lengths of sides BC , CA and AB respectively. c2
3 OA2
OB2
OC 2
2
a.a b.b
9 OG
2
2
where 'O ' is any point. a2 b 2
2
H.
For any two vectors a and b , a b
I.
Find the vector having magnitude
J.
Find unit vector perpendicular to the plane passing through the points 1, 2, 3 , 2, 1,1 and
a.b
a.b
6 units and perpendicular to both 2i
k and 3 j
i
k.
1,2, 4 . K.
If a , b and c represent the vertices A, B and C respectively of a b
L.
b c
c a is twice the area of
ABC .
If A, B,C and D are four points, then show that AB CD area of
M.
ABC , then prove that
BC AD CA BD is four times the
ABC .
If b c d
c a d
a b c , then show that the points with position vectors
a b d
a , b , c and d are coplanar
N.
If a , b , c are the position vectors of the points A, B and C respectively, then prove that the vector
a b
b c
c a is perpendicular to the plane of
ABC .
O.
Find the volume of the tetrahedron whose vertices are 1,2,1 , 3,2,5 , 2, 1,0 and
P.
Find
Q.
For any three vectors a , b , c prove that b c c
R.
Let a , b and c be unit vectors such that b is not parallel to c and a
1,0,1 .
, in order that the four points A 3,2,1 , B 4, ,5 , C 4,2, 2 and D 6,5, 1 be coplanar. aa b
2
abc . b c
1 b . Find the 2
angle made by a with each of b and c . S.
a , b and c are non-zero and non-collinear vectors and If a b
0,
is the angle between b and c .
1 b c a , then find sin . 3
c
If a , b , c are unit vectors such that a is perpendicular to the plane of b , c and the angle between
T.
b and c is
3
, then find a
b
c .
2 4 8 .cos .cos 7 7 7
1 8
14. A.
Prove that cos
B.
Find the value of 2 sin 6
C.
If cos
0 , tan
sin
cos6
3 sin 4
m and tan
sin
cos 4
n , then show that m2
n2
4 mn
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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D.
E.
Eliminate
from the following:
i) x
a cos3 , y
b sin 3
iii) x
a sec
tan , y
b sec
tan
ii) x
a cos 4 , y
iv) x
cot
tan and y
sec
cos
Find the maximum and minimum values of i) 3sin x
ii) cos x
4cos x
2 2 sin x
3
F.
If sec
G.
If x , y , z are non zero real numbers and if x cos
2sec and cos
sec
then show that xy
yz
H.
Prove that 1
I.
For A R , prove that a)
cos
10
zx
1
3
3
1 , then show that cos 2 3
y cos
2 cos
2
4 3
z cos
. for some
R,
0
cos
3 1 10
cos
7 1 10
cos
9 10
1 16
3 16
sin 20 o sin 40 o sin 60 o sin 80 o
b) sin A.sin 60
A sin 60
A
1 sin 3 A 4
c) cos A.cos 60
A cos 60
A
1 cos 3 A and hence deduce that 4
2 3 4 d) cos cos cos cos 9 9 9 9
J.
b sin 4
1 16
Let ABC be a triangle such that cot A
3 , then prove that ABC is an equilateral
cot B cot C
triangle. K.
If A is not an integral multiple of a) tan A
cot A
2
, prove that b) cot A tan A
2cos ec2 A
L.
If
is not an integral multiple o f
M.
If 3A is not an odd multiple of
2
2
, prove that tan
2tan2
, prove that tan A.tan 60
2cot 2 A 4tan 4 A .tan 60
8cot8 A
cot
tan 3 A and hence
find the value of tan 6 o tan 42 o tan 66 o tan 78o . N.
If
,
are the solutions of the equation a cos
b sin
c ( a , b , c are non-zero numbers) then
show that i) sin O.
sin
2bc a2
b2
If A is not an integral multiple of deduce that cos
ii) sin .sin
c2 a2
a2 b2
, prove that cos A.cos 2 A.cos 4 A.cos8 A
2 4 8 16 .cos .cos .cos 15 15 15 15
sin 16 A and hence 16sin A
1 16
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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P.
If sin x a) tan
x
y 2
1 , then show that 3 7 b) cot x y 24
1 and cos x 4
sin y
cos y
3 4
Q.
Prove that 4 cos 12 o cos 48o cos72 o
R.
If cos x
S.
If a cos
T.
Prove that sin 4
U.
Prove that cos 2 76
V.
If 0
4 and cos x 5
cos y
b cos
A
B
4
3 8
sin 4
cos 2 16
, sin A
x y 2 , find the value of 14 tan 7 2
cos y
0 , then prove that a
, cos
sin 4
8
cos 36 o
5 8
sin 4
7 8
cos76 cos16
B
24 , cos A 25
cos 3x
b tan
5cot
x
y 2
a b cot .
3 2 3 4
B
4 , find the value of tan2A . 5
15. A.
If x is acute and sin x
10o
B.
Solve 7 sin 2
4
C.
Solve 2cos2
D.
Solve 4sin x sin2x sin 4x
E.
If 0
F.
Find all values of x in
G.
Solve the following equations and write general solution. i)
3 sin
1
0 sin3x
, solve cos .cos 2 .cos 3
4 cos2
iii) 1 H.
3 cos 2
sin 2 x
If tan p
tan
3
2
sin 3x
ii) 6 tan 2 x
2 cos 2 x
cos 2 x
2
iv) 2 sin 2 x
sin 2 2 x
2
cos 3x
43
q then show that the solutions are in A.P. with C.D
p
q
. I. Solve
5sec
sin 2
J.
Solve 1
K.
Solve
L.
Solve sin2x
M.
Find the general solution of tan x
N.
If tan
O.
Find the common roots of the equations cos2x
P.
Solve the equation
3sin cos
2 sin x
cos
cos cos 2 x ....
1 cos
cot, q , and p
3cot
1 4
satisfying the equation 81
,
3
68o find x .
cos x
cos2x
cot
3
sin x cos x 1 3
,sec x
2 3 1
sin , then prove that cos
6
cos x
7 sin 2 x
cos x
4
sin2x
2 2
cot x and 2 cos 2 x
cos 2 2 x
1.
0.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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1 and x cos x
Q.
If tan x
tan x
R.
If x
2 and sin x 3
S.
Solve sin 3
y
0,2 , find the values of x . 3 then find x and y . 2
sin y
4sin sin x
n ,n z .
where
sin x
16. 4 5
2 tan
1
y
sin
1
z
1 3
A.
Prove that sin
1
B.
If sin
1
1
C.
Solve arc sin
D.
Prove that cos tan
E.
If cos
1
p
cos
1
q
cos
1
r
, then prove that p2
F.
If sin
1
x
sin
1
y
sin
1
z
, then prove that x 1 x2
G.
Solve sin
1
x
sin
H.
If
1
I.
Prove that sin
1
J.
Prove that sin
1
K.
Prove that cot
1
L.
Prove that cos 2 tan
M.
If cos
N.
If a , b , c are distinct non-zero real numbers having the same sign, prove that cot
x
sin
tan
1
p a
1
If sin
P.
i) If tan
1
ii) If tan
p
1
x
1
x
1
2
12 x
2 1
sin cot
1
2x
1 x2
1
2
2
x
1 x
4 5
sin
1
4 5
sin
1
9
cos ec
1
1
q b
tan
1
x
x2
1
2
2
x
1
sin
1
5 13
sin
1
41 4
1
1 7
1
y
q2
r2
2 pqr
2 x2 y 2
y 2 z2
z2 x 2
. 1
y 1 y2
z 1 z2
2xyz
cot
q
tan tan
1
1
tan
2
z
1
x
sin 2
117 125
16 65
2
4
sin 4 tan
1 q2 1
y
4x 2 y 2 z 2
, then prove that x 2
7 25
bc 1 b c
1
z4
0
1
1 3
, then prove that
cos tan
. x
y4
3
x2
cot
2p
1
arc sin
, then prove that x 4
1
cos
ab 1 a b
O.
5 x
2
p2 a
2
ca 1 c a 1
b
sin 2
2
.
or 2
2x , then prove that x 1 x2
, then prove that x 2
q2
2 pq .cos ab
, then prove that xy
y
z yz
p q . 1 pq
xyz zx
1
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Q.
Solve the following equation for x : cos
R.
Prove that (i) Tan
S.
Solve tan
1
1 2
1
1
2x
1 5
Tan
1 4x 1
1
tan
1
1
Tan
1
1
x
1 8
tan
1
sin
4
x 2
(ii) Tan
6 1
3 4
Tan
1
3 5
Tan
1
8 19
4
2 x2
1
17. (I) A 2
b
c
A.
If cot
B.
In
C.
Show that r r1 r2 r3
D.
In
ABC , show that
E.
In
ABC , if a
F.
If a
G.
In
H.
Prove that a b cos C
I.
In
ABC , if
J.
In
ABC , show that
K.
Show that a2 cot A
a
, find angle B .
1 r1
ABC , prove that
4, b
1 r3
b
c b
c cos A
c
C 2
ABC , find b cos 2
1
1 b
B . 2
c cos 2
c cos B c
a
b2
c2 a
2
2s
3bc , find A .
a
7 , find cos
c
1 r
2
b
5, c
a
1 r2
B . 2
b2
c2
3 b
c
, show that C
sin B C sin B
b 2 cot B
60
c 2 cot C
C
abc R
(II) A.
If r : R : r1
2 : 5 : 12 , then prove that the triangle is right angled at A.
B.
In an equilateral triangle, find the value of
C.
If A
90 o , show that 2 r
D.
If cot
A B C : cot : cot 2 2 2
E.
If b
c
F.
If a
R
b
r . R
c
3 : 5 : 7 , show that a : b : c
6:5: 4
B C 3a, then find the value of cot cot . 2 2
b c sec , prove that tan
2 bc A sin . b c 2
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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r1 r2
r3
G.
Prove that
H.
A B cot 2 2 Prove cot A cot B
I.
cot A
J.
Prove that cot
K.
If C
L.
Show that in
M.
If p1 , p2 , p3 are the altitudes of the vertices A, B,C of a triangle respectively, show that
r1r2
r2 r3
C 2 cot C
cot
cot B
cot
a2
cot C A 2
a
r3r1
cot
a a2
b2 4
B 2
1
1
1
p12
p22
p32
cot A
N.
If a : b : c
O.
If cot
P.
If r2
Q.
If
R.
Show that b 2 sin 2C
S.
If sin 2
c
b2
2
c2
c2
cot
s2
C 2
60 , then show that (i)
ABC , a
b
a b
b c
b cosC cot B
.
c
a
b
1 (ii)
c
2
a a
2
c
2
b2
0
c cos B
cot C
.
7 : 8 : 9 , find cos A : cos B : cosC
A B C , cot , cot are in A.P., then prove that a , b , c are in A.P. 2 2 2
a2 a2
r1 r3 b2 b2
2r2 r3 . Show that A
r1
sin C , prove that sin A B c 2 sin 2 B
90 .
ABC is a right angled.
2 bc sin A
A B C ,sin 2 ,sin 2 are in H.P., then show that a , b , c are in H.P. 2 2 2
LONG ANSWER QUESTIONS
18. 1
1
Let f : A
B, g : B
B.
Let f : A
B , I A and I B be identity functions on A and B respectively. Then foI A
C.
Let f : A
Let
gof
E.
f 1
f
og
.
f
IBof .
B be a function. Then f is a bijection if and only if there exists a function g : B
such that fog D.
C be bijections. Then gof
1
A.
I B and gof
I A and, in this case, g
1, a , 2, c , 4, d , 3, b
f
1
og
1
g
3x
2,
2
2,
2
2x
1,
x
.
2, a , 4, b , 1, c , 3, d
,
then
show
that
.
If the function f is defined by f x
f 4 , f 2.5 , f
1
and
1
f
A
2 ,f
4 ,f 0 ,f
x
x
3 x
2 . Then find the values of 3
7
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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F.
Let f : A
B be a bijection. Then fof
G.
Let f : A
B, g : B
C and h : C
1
1
I B and f
IA .
of
D . Then ho gof
hog of , that is, composition of functions
is associative. H.
If f : R
R are defined by f x
R, g : R
i) gof x I.
Let A
K.
1,2,3 , B
If f : Q
1
(iii) fof x
4
p, q , r . If f : A
a, b , c , C
1, a , 2, c , 3, b , g
f
a
(ii) gof
a, q , b , r , c , p
Q is defined by f x
C are defined by
then show that f
x
2, x
b) f 0
(iv) go fof 0
B, g : B
2,
If the function f is defined by f x a) f 3
2 then find
1
g
1
g f
1
.
4 x Q then show that f is a bijection and find f
5x
x
L.
x2
1 and g x
4x
c) f
3
1 , then find the values of
x x
1
d) f 2
1,5
.
1
1
1,
1
f
2
e) f
5
19. Show that 49n
16n
B.
1 1.3
1 3.5
...
C.
3.52 n
1
D.
1.2.3
A.
1 5.7 2 3n
2.3.4
2n
1 2n
n 2n 1
1
is divisible by 17.
3.4.5
nn
...upto n terms 3
2 3
F.
12
12
22
12
22
G.
Use Mathematical Induction to prove the formula
2 1 3
...
3 5
1 4.7
1 7.10
1
2
13 n
2
12
....upto n terms
Show that, n N ,
I.
If x and y are natural numbers and x
K.
9n
n.2n , n N .
H.
divisible by x
n n
....upto n terms
upto n terms 1 1.4
n 2 n2 24
....upto n terms
32
3
4
1 1
1
3
2 n
1 1
4.22
3
1 n
E.
3.2
3
1
3
2
3
1
1 is divisible by 64 for all positive integers n .
n 3n 1
y , using mathematical induction, show that x n
y n is
y , for all n N
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. L. M.
Use mathematical induction to prove that 2.4
2.3
3.4
4.5
...upto n terms
n n2
6n
2n 1
3
3n 1
is divisible by 11, n N .
11
3
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N.
Prove by Mathematical induction, for all n N a
O.
Show that 4 n
P.
Use mathematical induction to prove the statement, 1
3 1 1
ar 2
..... upto n terms
a rn r
1 1
,r
1
1 divisible by 9 for all positive integers n .
3n
5 1 4
ar
7 ...... 1 9
2n 1 n2
n
1
2
20. A.
b
c c
a a
b
Show that c a
a a
b b
c
b b
c c
a
a B.
b
Show that
c
2b
b
D.
c
a
b
2x
3
3x
4
Find the value of x if x x
4
2x
9
3x
16
27 3x
64
a b c Show that b c a c a b
2
2
2
c2 b
2
2
a
b2
c2
3
3
3
b
2
c
a2
2
2 ab
a3 c
b3
c3
3abc
2
.
2
1
.
3
1 .
a
1
abc a
b b
c c
a
1 1 then find A3
1 1
3 A2
A
3I .
1
1 a2
a3
Show that 1 b 2
b3
2
3
1 c
Show that
1
3
Show that a2
a
b2 b2
a
1
c
3
0
c2
2
1 1
b
0
b
2 ac
2a 2a
a
If A
a2
2bc
3
2 then show that the adjoint of A is 3A' . Find A 1
Show that 2 a
1
8 2x
c
2
1
a
J.
2b
2
3
I.
a
x
a2
H.
2a a
2
G.
a b
c
If A
F.
c
2c
1 E.
a.
2b c
2a
2c C.
a b c
b c c
c
a
2c
b b
c c
a b
c a
a ab
bc
ca
b 2a
b c
a
2 a
b
3
c .
2b
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2a
K.
Show that a c
L.
If A
If A
b c
b
2b
a c
a
b
c
b
4 a
2 3
0
1 4 then find A ' 2
b b
a .
c c
2c
1 2
M.
a
1
1
a1
b1
c1
a2
b2
a3
b3
c 2 is a non-singular matrix, then show that A is invertible and A c3
1
adj A . det A
21. A.
Solve the following equations by Gauss-Jordan method 3x
4y
5z
18 ,
2x
y
13 , 5x
8z
2y
7z
20
B.
Solve 3x
4y
5z
18 , 2 x
y
8z
13 , 5x
2y
7z
20 by using matrix inversion method.
C.
Solve 3x
4y
5z
18 , 2 x
y
8z
13 , 5x
2y
7z
20 by using Cramer’s Rule.
D.
Apply the test of rank to examine whether the following equations are consistent. 2x
E.
y
8 ,
3z
x
2y
4 , 3x
z
y
4z
0 and if consistent find the complete solution.
Find the nontrivial solutions if any, for the following system of equations: 2x
F.
5y
6z
0, x
3y
0 , 3x
8z
y
4z
0
Solve the following equations by Gauss-Jordan method 5x
G.
6y
4z
15 ,
7x
4y
19 , 2 x
3z
y
6z
46
Solve the following system of equations by Gauss – Jordan method. x
H.
y
3 , 2x
z
2y
3, x
z
y
z
1.
By using Gauss-Jordan method, show that the following system has no solution. 2x
4y
0, x
z
2y
2z
5 , 3x
6y
7z
2.
I.
Examine whether the following systems of equations are consistent or inconsistent and if consistent find the complete solution. x y z 4 , 2 x 5y 2 z 3 , x 7 y 7 z 5 .
J.
Show that the following system of equations is consistent and solve it completely: x
y
3 , 2x
z
2y
3, x
z
y
z
1
22. A.
Find the shortest distance between the skew lines r r
B.
4i
If A
k
s 3i
1, 2, 1 , B
2j
6i
2j
2k
t i
2j
2k
and
2k
4,0, 3 , C
1,2, 1 and D
2, 4, 5 , find the distance between AB
and CD. C.
Let a , b , c be three vectors. Then prove that i) a b
c
a.c b
b.c a
ii) a
b c
a.c b
a.b c
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D.
Let a
E.
A line makes angles cos2
j, b
i
cos2
1
k, c
j
cos2
2
i . Find unit vector d such that a.d 1, 2, 3
cos2
3
4
and 4 . 3
Let b
G.
For any four vectors a , b , c and d a b . c d a b
2
j
a2 b 2
a.b
i
2
b c d .
0
with the diagonals of a cube. Show that
4
F.
2i
k, c
k
3k . If a is a unit vector then find the maximum value of a b c . a.c
a.d
and in particular
b.c b.d
.
H.
Find the volume of the parallelopiped whose coterminus edges are represented by the vectors 2 i 3 j k , i j 2 k and 2i j k .
I.
For any four vectors a , b , c and d a b a b
c d
a b dc
c d
b c d a and
a c db
a b cd
J.
Find the equation of the plane passing to the points A
K.
Show that in any triangle the altitudes are concurrent.
4,5,2 and C
2,3, 1 , B
3,6,5 .
23. A.
If A, B, C are angles in a triangle, then prove that sin2 A sin2B sin2C
B.
If A, B, C are angles in a triangle, then prove that sin A
C.
If A
D.
In triangle ABC, prove that cos
E.
If A
B C
2S , then prove that sin S
F.
If A
B C
0 , then prove that cos2 A
G.
If A
B C
2S , then prove that
, then prove that cos2
B C
cos(S
A)
cos(S
B)
A 2
cos(S C )
sin B sin C
A 2
cos2
B 2
cos2
B 2
cos
C 2
4 cos
cos
cos S
A
sin S
cos 2 B
4 cos
C 2
B
cos2 C
If A, B, C are angles in a triangle, then prove that sin 2 A
I.
If A
B
J.
If A
B C
3 , then prove that cos 2 A 2
K.
If A
B C
0 , then prove that sin2 A sin2B sin2C
L.
If A
B C
D
sin A sin B
2
, then prove that cos2 A
A 4
cos
sin C 1
sin
4sin
A B C sin sin 2 2 2 B
4
A B C sin cos 2 2 2
C
cos
4 cos
4
S
A 2
cos
S
B 2
sin
C 2
2 cos A cos B cos C
A B C cos cos 2 2 2
H.
C
2 1
4cos Asin BcosC
cos2B cos2C cos 2 B
sin 2 B
1
cos 2 C
sin 2 C
2 sin A sin B cos C
4sin Asin Bsin C . 2 cos A cos B sin C
4sin Asin Bsin C
2 , then prove that A B A C A D sin C sin D 4 cos sin cos 2 2 2
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24. A 2
A.
Show that a cos2
B.
Prove that a3 cos B C
C.
If a 2
D.
Show that cos A
E.
In
F.
Show that cos 2
G.
If a
13, b
H.
If r1
2, r2
I.
Show that b
J.
Prove that
K.
If cos A
L.
Show that
1 r2
1 r12
M.
Show that
r1 bc
r2 ca
N.
Show that r
O.
Show that
P.
Prove that r12
Q.
If p , p , p
b2
c2
s
A
c 3 cos A B
R 3abc
8R 2 , then prove that the triangle is right angled.
cos B
A 2
cos C
a bc
ab
r
cos 2
C 2
2
4R .
r . 2R
2
65 ,r 8
15 , show that R
A 2
cos 2
cos A a
cos B b
r3
1 r
21 ,r 2 2
1
4 and c
3, b
12 and r
3
14 .
5.
a2 cos C c
b2
c2
2
1 2R 4 R cos B .
r2 r3 r1
r32
a2
1 r3 2
r2
bc
r2 2
c ab
4, r
3 , then show that the triangle is equilateral. 2
r3 ab
r1r2
A 2
2
c sin 2
b
1 r2 2
r1
1, prove that a
b ca
cos C
r3
B 2
r R
1
r3
6 and r
3, r3
c
r2
cos 2
14, c
cos B
2
C 2
c cos2
b3 cos C
ABC , prove that r1
1
B 2
b cos2
r2
ca r3r1 r2 16R2
a2
b2
c2
are altitudes drawn from vertices A, B,C to the opposite sides of a triangle
3
respectively, then show that
1 i) p
1
R.
1 p
2
1 p
Show that i) a
1 ii) p
1 r
3
r2
1
r3
rr1 r2 r3
ii)
1 p
2
r1r2
1 p
3
1 r
iii) p p p
3
1
2
abc 3
8R 3
2
8 3 abc
4R r1 r2 r1 r2
wish you all the best
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CENTRE
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN
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INTERMEDIATE PUBLIC EXAMINATION, MARCH 2017 Total No. of Questions - 24
Reg.
Total No. of Printed Pages - 2
No.
Part - III MATHEMATICS, Paper-I (A) (English Version) Time : 3 Hours]
[Max. Marks : 75
10 2 = 20 M
SECTION - A I.
Very Short Answer Type questions:
1.
If A
2.
If f x If A
x2 , h x
2, g x 3
3.
and f : A
0, , , , 6 4 3 2 2
2
2
1
3
2x for all x R , then find fo goh
3
1 0 , B
2
1
4
B is a surjection defined by f x
cos x then find B . x .
1 0 3 and X 1 2
1
A
B then find X .
4.
If A
1 2 then find AA' . 0 1
5.
a
5j
6.
Find the vector equation of the line passing through the point 2 i
2i
vector 4i
k and b 2j
4i
nk are collinear vectors then find m and n .
mj
Find the angle between the vectors i
8.
If sin
9.
Prove that cos 48 cos12
10.
If cosh x
2j
3k and 3i
j
2k .
is not in the first quadrant, find the value of cos .
3
5 8
.
5 , find the value of (i) cosh 2x and (ii) sinh 2x . 2
SECTION – B II.
k and parallel to the
3k .
7.
4 and 5
3j
5 4 = 20 M
Short Answer Type questions: (i) Attempt any five questions (ii) Each question carries four marks
1 a a2 11.
Show that 1 b b 2
a
b b
c c
a .
1 c c2 12. 13.
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 . i 2 j 4 k and c i j k then find a b b c . If a 2 i j k , b
14.
(i) Find the range of 13cos x (ii) Evaluate sin 2 82
1 2
o
3 3 sin x 4 .
sin 2 22
1o . 2
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sin 2
15.
Solve 1
16.
Show that cot sin
17.
In
ABC , if
cos .
3sin
13 17
1
1 a
sin tan
1 c
b
c
3 b
a
1
2 . 3
, show that C
c
60 .
5 7 = 35 M
SECTION – C III.
Long Answer Type questions: (i) Attempt any five questions (ii) Each question carries seven marks
18.
(i) If f : Q
Q is defined by f x
(ii) If f 19.
4 , x Q , show that f is a bijection and find f
5x
and g
4,5 , 5,6 , 6, 4
4, 4 , 6,5 , 8,5
then find f
1
.
g and fg .
Using mathematical induction, prove 1 2 3 2 3 4 3 4 5 ...... upto n terms n n 1 n 2 n 3 , n N. 4 Solve the following system of equations by using Cramer’s rule: 2 x y 3z 9; x y z 6; x y z 2
20.
1 2 1
21.
(i) Show that A
3 2 3 is non-singular and find A
1
.
1 1 2 1 2 2 2 1 2 then show that A2
(ii) If A
4A
0.
5I
2 2 1
22.
Find the shortest distance between the skew lines: r
23.
6i
If A cos S
24.
2j
2k
2j
2k
and r
4i
k
s 3i
2j
2k .
2S , then prove that
B C A
t i
cos S
Show that in a
B
ABC ,
cos C 1 r2
1 r12
1 1 r22
4 cos 1 r32
S
A 2
a
2
cos
b
2
S
B 2
c
2
2
cos
C . 2
.
BLUE PRINT (MATHS-IA) S.No.
Chapter Name ALGEBRA
1. 2. 3.
Functions Mathematical induction Matrices & Determinants
4. 5.
Addition of vectors Product of vectors
6. 7. 8. 9. 10.
TRIGONOMETRY Trigonometry upto transformations Trigonometric equations Inverse trigonometric functions Hyperbolic functions Properties of triangles
Weightage Marks 11 (7 + 2 + 2) 7 (7) 22 (7 + 7 + 4 + 2 + 2)
VECTOR ALGEBRA 8 (4 + 2 + 2) 13 (7 + 4 + 2) 15 (7 + 4 + 2 + 2) 4 (4) 4 (4) 2 (2) 11 (7 + 4)
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VERY SHORT ANSWER QUESTIONS
1. A.
If A
B.
If f : R
C.
If f : R \ 0
D.
Find the domains of the following real valued functions:
B is a surjection defined by f x
R is defined by f x
1
R is defined by f x
a)
f x
2
x
c)
f x
x
2
1
1 log
x2
x
2
1
g) f x
6x
x
3
x
i) f x
2
x
1 1
5
3 x x
cos x then find B .
2x x , then show that f x 1 x2
1 then prove that f x x
f x
log
3
2
2f x .
f x2
f 1 .
x x
10
1
d) f x
1 x
10
log
b)
x
1
e) f x
E.
and f : A
0, , , , 6 4 3 2
x
x
f) f x
log x
h) f x
x2
j) f x
1 log 2
x 1
1
x2
3x
2
x
Find the ranges of the following real valued functions: a)
x
sin
x
x
b)
R is defined by f x
1
x
2
F.
If f : R
0
G.
If f : R
R is defined by f x
H.
If A
2, 1,0,1,2 and f : A
A.
If f x
x x
B.
If f : 4, 5 , 5,6 , 6, 4
c)
x2 4 x 2 x3
d) 1 x
3
x2
9
e) log 4
x2
1 x
0
, then show that f x
1 x2 , then show that f tan 1 x2
f
cos2
B is a surjection defined by f x
x2
x
1 , then find B .
2.
a) f C.
1 x 1
b) fg
4
If f : R i) fof x 2
R, g:R
1
1 then find fofof x and fofofof x .
c)
and g : 4, 4 , 6, 5 , 8, 5
then find
f d) f 2
R are defined by f x
ii) fog 2
3x
x2
1, g x
iii) gof 2 a
1 , then find
3
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x
x 2
D.
Prove that the real valued function f x
E.
Determine whether the following functions are even or odd. a)
F.
f x
x2
log x
If f : R
R and g : R
b) f x
1
e
x
x
1
ex ex
1 is an even function on R \ 0 .
1 1 2x 2
R are defined by f x
3 and g x
3x
2 , then find
i) fog x , ii) gof x , iii) fof o , iv) go fof 3 . G.
Find the inverse of the following functions: a) If a, b
R, f :R
b) f : R
R defined by f x
defined by f x
0, x x
1 , x 1
H.
If f x
I.
If the function f : R f x
J.
y
f x
ax
5x
1 , show that fof
b a
0
c) f : 0, 1
x
R defined by f x
R defined by f x
log 2 x .
x
3x
3
x
, then show that
2
2f x f y .
y
R defined by f x
If the function f : R
4x 4x
2
, then show that f 1 x
1
f x .
3. A.
Construct 3 2 matrix whose elements are defined by aij
B.
Define Trace of matrix and give an example.
C.
If A
2
4
and A2
1 k
1 D.
Find the trace of 2 2 1 2
, B
1 i 2
3j
O , then find the value of k.
3 1 0
3 8
5 5 1
E.
If A
F.
Define symmetric and skew-symetric matrix and give an example.
G.
If A
H.
If A
3 4
2
4
5
3
i
0
0
i
7
2
and 2X
then find A
A
B then find X.
A' and AA' .
, find A 2 . 5
I.
Find the products of i)
1 4 2
1 3
ii)
2 6
1
4
2 3
1 2 1
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x
3 2y
z
2
8
5
2
J.
If
K.
A certain bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 respectively. Find the total amount the bookshop will receive by selling all the books, using matrix algebra.
6
2 a
then find the values of x , y , z and a .
4
4. A.
Define singular matrix and give an example.
B.
Define rank of a matrix.
C.
If A
1
0
0
2
3
4 and det A = 45 find x .
5 D.
If A
6 x
1
5
3
2
4
0
3
1
2
and B
1 0 2 5 then find 3A 4B' . 2 0
0 1
5
1 4 E.
Find the Rank of the Matrix 2 3 0 1 7
F.
If A
2
1
2
and B
2
5
0 2
1
4
3
1
2
1
then find AB’ and BA’.
0
3 0 0
G.
If A
0 3 0 , then find A 4 . 0 0 3 cos
If A
H.
sin
sin
cos
, show that AA’ = A’A = I.
5. A.
ABCDE is a pentagon. If the sum of the vectors AB, AE, BC, DC, ED and AC is the value of .
B.
If the position vectors of the point A, B and C are respectively and AB
C.
,
If
and
AC , then find the value of
be the angles made by the vectors 3i
coordinates axes, then find cos , cos
k,
AC , then find
2 k and 6i
2i
j
6j
2 k with the positive directions of the
4i
2j
3j
13 k
.
and cos .
D.
If OA
i
E.
a
5j
F.
Find the angles made by the straight line passing through the points 1, 3,2 and 3, 5,1 with
2i
k , AB
3i
2j
k , BC
k and b
4i
mj
nk are collinear vectors, then find m and n .
j
i
2j
2 k and CD
2i
j
3k , then find vector OD.
the coordinate axes. G.
Let a of a
2i
b
4j
5k , b
i
j
k and c
j
2 k . Find the unit vector in the opposite direction
c.
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6. A.
Find the vector equation of plane passing through the points i 3i
5k ,
5j
and
.
2j
k , and
5j .
B.
If the vectors
C.
If OA
i
j
k , AB
D.
Let a
i
j
k and b
3i
k and i
4j 3i
k , BC
2j
2i
6 k are collinear vectors, then find
8j i
2 k & CD
2j
2i
3k , then find the vector OD .
j
k find
3j
(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 . 2
2
2
E.
For any vector a , show that a
F.
If p
G.
Find the distance of a points 2,5, 3 from the plane r. 6i
H.
Find the vector equation of the line passing through the point 2 i vector 4i
a
j
a k
2a .
2
3 and p , q
2, q
i
6
, then find p q . 3j
2k
4. k and parallel to the
3j
3k .
2j
I.
OABC is a parallelogram. If OA= a and OC
c , find the vector equation of the side BC.
J.
If a , b , c are the position vectors of the vertices A, B and C respectively of
ABC , then find the
vector equation of the median through the vertex A. K.
Find the vector equation of the line joining the points 2 i
L.
Find the vector equation of the plane passing through the points 0,0,0 , 0,5,0 , and 2,0,1 .
M.
Find the vector equation of the plane passing through the points. i
N.
If the vectors
i
O.
Show that i
a i
P.
Compute i
j j
5k and 2 i
3j
j a j
k
j
a k
3k and
j
4i
2j
k.
3j
5k ,
k.
5j
k are perpendicular to each other, find
.
2 a for any vector a .
i .
k k
7. 11, b
23 and a b
30, then find the angle between the vectors a , b and a
A.
If a
B.
If a
C.
If P , Q , R and S are points whose position vectors are i
i
j
k and b
2i
3j
b.
k , then find the projection vector of b on a and its magnitude.
k,
i
2 j , 2i
3k and 3i
2j
k
respectively, then find the component of RS and PQ . D.
Find the angle between the planes r. 2i
E.
Let e 1 and e 2 be unit vectors containing angle . If
F.
Show that i
G.
Prove that for any three vectors a , b , c b
a i
j
a j
k
k a
j
2 k and r. 3i 1 e1 2
6j
e2
k
sin
4.
, then find
.
2 a for any vector a.
c c
a a
b
2 abc .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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H.
If a
i
3k and b
2j
3i
2 k , then show that a
j
b and a b are perpendicular to each
other. I.
Let a and b be non-zero, non-collinear vectors. If a
a b , then find the angle between
b
a and b .
J.
If a
K.
Find the area of the parallelogram for which the vectors a
2i
3k , b
2j
3i
2 k , then find the angle between 2a
j
2i
b and a
2b .
3 j and b
3i
k are adjacent
sides. L.
If a
M.
Let a
N.
Find the area of the triangle whose vertices are A 1,2,3 , B 2,3,1 and C 3,1,2 .
O.
For any three vectors, a , b , c , prove that b c c a a b
i
2j 2i
3k and b
3i
5j
k are two sides of a triangle, then find its area.
k and b
3i
4j
k . If
j
is the angle between a and b , then find sin .
2
abc .
8. A.
If 3sin
5, then find the value of 4sin
B.
If cos
C.
If tan 20 o
D.
Prove that 3 sin
E.
Prove that cot
F.
Prove that tan
cot
G.
Prove that
tan tan
sec sec
H.
Prove that 1
I.
Prove that sin
J.
Prove that cot
4cos
2 cos , prove that cos
sin
, then show that
16
.cot
cot
20
4
cos
1 1
6 sin
cos ec
2
1
2 sin
tan 160 o tan 110 o 1 tan 160 o.tan 110 o
sec 2
cos ec
.cot
sin
2
cos
2 3 7 .cot ....cot 16 16 16 2
3cos .
2
1 2
4 sin 6
cos6
13
1
cos ec 2
sec 2 .cos ec 2 .
sin . cos
1
tan
cos
sec
sec
2
2 tan 2
3 5 7 9 .cot .cot .cot 20 20 20 20
cot 2
7
1
9. cos2 x in 0,
A.
Draw the graph of y
.
B.
Find the period of the function defined by f x
tan x
C.
Find the extreme values of 5cos x
8 over R.
3cos x
3
4x
9x
...
n2 x
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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D.
Find the periods of the functions: a) f x
b) f x
tan 5x
cos
9
c) f x
5
E.
Find the range of
F.
Prove that
1 sin 10o
3 cos10 o
4
G.
Prove that tan 70 o
tan 20 o
2 tan 50 o
H.
Prove that sin 2 52
I.
Draw the graph of y
J.
If A
K.
Prove that
L.
Draw the graph of the tanx between 0 and
M.
Draw the graph of the cos2x in the interval 0,
N.
Find the minimum and maximum values of (i) 3cos x
O.
If
P.
Prove that 4 cos66o
Q.
Prove that cos
B
(i) 7 cos x
4x
1 2
o
24sin x
sin 2 22
o
1 2
3
sin 2 x in
sin 9o sin 9o
3 3 sin x 4
1
4 2
.
,
45o , then prove that (i) 1
cos9o cos9o
5 (ii) 13cos x
sin x
tan A 1
tan B
2 (ii) cot A 1 cot B 1
2
cot 36o
is not an odd multiple of
and if tan
2
sin 84o
3 o
cos 120
4
. . (ii) sin2x
4sin x
1 , then show that
1 1
sin 2 sin 2
cos2x
cos 2 cos 2
tan .
15 o
cos 240
0
10. 4sinh 3 x
A.
Prove that, for any x R , sinh 3x
B.
If cosh x
5 , find the values of (i) cosh 2x and (ii) sinh 2x 2
C.
If sinh x
5, show that x
D.
Show that tanh
E.
If sinh x
F.
Prove that
1
1 2
If sinh x
5
e
26
1 log 3 e 2
3 , find cosh 2x and sinh 2x 4
(i) cosh x (ii) cosh x
G.
log
3sinh x
sinh x sinh x
3 then show that x
n
cosh nx
n
log
cosh nx e
3
sinh nx , for any n R sinh nx , for any n R
10
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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SHORT ANSWER QUESTIONS
11. 1 2 2 A.
2 1 2 then show that A2
If A
4A
5I
0
2 2 1 B.
If 3 A
1
2
2
1
2 2 then show that A
2 2 C.
D.
If
3 4
2
3 4 then show that A
0
1 1
E.
If I
2
0 1
a a2
H.
i)
If b b
2
c c
2
0 1
and E
1
cos 2
, then show that
1 0
A' .
1
3
If A
1
cos sin
sin
2
then show that aI
0 0
cos 2
cos sin
cos sin
sin 2
cos sin
1
a3
a a2
1
1
b
3
0 and b b
2
1
c
3
c c
2
1
1
A3 .
bE
3
a3 I
0
3 a 2 bE .
0 then show that abc
1
ii) without expanding the determinant, prove that 1 bc b a) 1 ca c 1 ab a a a2
1 a a2
c a b
1 b b
2
1 c c
2
bc
1 a2
a3
2
ca
1 b
2
3
c c2
ab
1 c2
c) b b
b
c3
ax
by
cz
a
b
c
2
2
2
x
y
z
e) x 1
y
1
z
1
cos
b
c c
a a
b
b) Show that a
b b
c c
a
a3
a
b b
a
c 1
d) Show that ca c ab a
a 1
y
f) Show that
3abc
c c
a
b 1
z y
c3
c
bc b
yz zx xy sin
b
b3
x z
z
x x
y
z
x
4xyz y cos n
then show that for all the positive integers n, An
sin n
I.
If A
J.
If A
K.
If A and A are invertible then show that AB is also invertible and AB
L.
For any nxn matrix A prove that A can be uniquely expressed as sum of a symmetric matrix and skew symmetric matrix.
M.
Show that the determinant of skew-symmetric matrix of order 3 is always zero.
sin
cos
3
4
1
1
then for any integer n
1 show that An
1
2n n
sin n
cos n
4n 1
2n 1
B 1A
1
.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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12. A.
Let ABCDEF be a regular hexagon with centre ‘O’. Show that AB+AC+AD+AE+AF=3AD=6AO.
B.
In
ABC , if ‘O’ is the circumcentre and H is the orthocentre, then show that
i) OA + OB + OC = OH C.
a , b , c are non-coplanar vectors. Prove the following four points are coplanar.
a) D.
E.
ii) HA + HB + HC = 2 HO
a
3c , 3a
4b
2b
5c ,
b) 6a
2b c , 2 a b
3c ,
If
a
b
c
c
a
b
c
d
d,
b
3a a
d
8b
2b a
5c ,
4c ,
3a
c.
2b
12a b 3c
and
are non-coplanar vectors, then show that
a, b , c
0.
Find the equation of the line parallel to the vector 2 i point A whose position vectors is 3i
2 k , and which passes through the
j
k . If P is a point on this line such that AP
j
15 , find
the position vector of P . F.
x
Let a , b be non-collinear vectors. If y
2x
2 a
2x
3y
4y a
2x
1 b are such that 3
G.
Is the triangle formed by the vectors 3i
H.
If the points whose position vectors are 3iˆ
2 ˆj
1 b and
2 then find x and y .
2 k , 2i
5j
y
5 k and
3j
k ,2iˆ
5i
4kˆ , iˆ
3 ˆj
2j
3 k equilateral?
2 kˆ and 4iˆ
ˆj
5 ˆj
kˆ are
146 . 17
coplanar, then show that I.
In the two dimensional plane, prove by using vector method, the equation of the line whose x y 1. intercept on the axes are ' a ' and ' b ' is a b
J.
Show that the line joining the pair of points 6a points
a
2b
3c , a
2b
c , 4c and the line joining the pair of
4b
5c intersect at the point
4c when a , b , c are non-coplanar
vectors. K.
If a , b , c , are non coplanar find the point of intersection of the line passing through the points 3c , a
6b
6c .
Find the vector equation of the plane passing through points 4i
3 ˆj
kˆ , 3iˆ
2a L.
3b
2iˆ
5 ˆj
A.
If a
b
B.
If a
c ,3a
2c with the line joining the points a
4b
7 kˆ and show that the point iˆ
2b
7 ˆj
10 kˆ and
3kˆ lies in the palne.
2 ˆj
13. 0, a
c 2, b
3, b
3 and c
5 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 C.
b
Let a and b be vectors, satisfying a having a
D.
7 , then find the angle between a and b .
2b and 3a
c. 45o . Find the area of the triangle
5 and a, b
b
2b as two of its sides.
For any two vectors a and b , show that 1
a
2
1
b
2
1
a.b
2
a
b
2
a b .
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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E.
If a
F.
a , b , c are non-zero vectors and a is perpendicular to both b and c . If a
i
2i
j
k and c
i
2 k . Find the a
3j
b c 2, b
3, c
4
2 , then find a b c 3
and b , c G.
3k , b
2j
G is the centroid of Prove that a2
b2
ABC and a , b , c are the lengths of sides BC , CA and AB respectively. c2
3 OA2
OB2
OC 2
2
a.a b.b
9 OG
2
2
where 'O ' is any point. a2 b 2
2
H.
For any two vectors a and b , a b
I.
Find the vector having magnitude
J.
Find unit vector perpendicular to the plane passing through the points 1, 2, 3 , 2, 1,1 and
a.b
a.b
6 units and perpendicular to both 2i
k and 3 j
i
k.
1,2, 4 . K.
If a , b and c represent the vertices A, B and C respectively of a b
L.
b c
c a is twice the area of
ABC .
If A, B,C and D are four points, then show that AB CD area of
M.
ABC , then prove that
BC AD CA BD is four times the
ABC .
If b c d
c a d
a b c , then show that the points with position vectors
a b d
a , b , c and d are coplanar
N.
If a , b , c are the position vectors of the points A, B and C respectively, then prove that the vector
a b
b c
c a is perpendicular to the plane of
ABC .
O.
Find the volume of the tetrahedron whose vertices are 1,2,1 , 3,2,5 , 2, 1,0 and
P.
Find
Q.
For any three vectors a , b , c prove that b c c
R.
Let a , b and c be unit vectors such that b is not parallel to c and a
1,0,1 .
, in order that the four points A 3,2,1 , B 4, ,5 , C 4,2, 2 and D 6,5, 1 be coplanar. aa b
2
abc . b c
1 b . Find the 2
angle made by a with each of b and c . S.
a , b and c are non-zero and non-collinear vectors and If a b
0,
is the angle between b and c .
1 b c a , then find sin . 3
c
If a , b , c are unit vectors such that a is perpendicular to the plane of b , c and the angle between
T.
b and c is
3
, then find a
b
c .
2 4 8 .cos .cos 7 7 7
1 8
14. A.
Prove that cos
B.
Find the value of 2 sin 6
C.
If cos
0 , tan
sin
cos6
3 sin 4
m and tan
sin
cos 4
n , then show that m2
n2
4 mn
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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D.
E.
Eliminate
from the following:
i) x
a cos3 , y
b sin 3
iii) x
a sec
tan , y
b sec
tan
ii) x
a cos 4 , y
iv) x
cot
tan and y
sec
cos
Find the maximum and minimum values of i) 3sin x
ii) cos x
4cos x
2 2 sin x
3
F.
If sec
G.
If x , y , z are non zero real numbers and if x cos
2sec and cos
sec
then show that xy
yz
H.
Prove that 1
I.
For A R , prove that a)
cos
10
zx
1
3
3
1 , then show that cos 2 3
y cos
2 cos
2
4 3
z cos
. for some
R,
0
cos
3 1 10
cos
7 1 10
cos
9 10
1 16
3 16
sin 20 o sin 40 o sin 60 o sin 80 o
b) sin A.sin 60
A sin 60
A
1 sin 3 A 4
c) cos A.cos 60
A cos 60
A
1 cos 3 A and hence deduce that 4
2 3 4 d) cos cos cos cos 9 9 9 9
J.
b sin 4
1 16
Let ABC be a triangle such that cot A
3 , then prove that ABC is an equilateral
cot B cot C
triangle. K.
If A is not an integral multiple of a) tan A
cot A
2
, prove that b) cot A tan A
2cos ec2 A
L.
If
is not an integral multiple o f
M.
If 3A is not an odd multiple of
2
2
, prove that tan
2tan2
, prove that tan A.tan 60
2cot 2 A 4tan 4 A .tan 60
8cot8 A
cot
tan 3 A and hence
find the value of tan 6 o tan 42 o tan 66 o tan 78o . N.
If
,
are the solutions of the equation a cos
b sin
c ( a , b , c are non-zero numbers) then
show that i) sin O.
sin
2bc a2
b2
If A is not an integral multiple of deduce that cos
ii) sin .sin
c2 a2
a2 b2
, prove that cos A.cos 2 A.cos 4 A.cos8 A
2 4 8 16 .cos .cos .cos 15 15 15 15
sin 16 A and hence 16sin A
1 16
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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P.
If sin x a) tan
x
y 2
1 , then show that 3 7 b) cot x y 24
1 and cos x 4
sin y
cos y
3 4
Q.
Prove that 4 cos 12 o cos 48o cos72 o
R.
If cos x
S.
If a cos
T.
Prove that sin 4
U.
Prove that cos 2 76
V.
If 0
4 and cos x 5
cos y
b cos
A
B
4
3 8
sin 4
cos 2 16
, sin A
x y 2 , find the value of 14 tan 7 2
cos y
0 , then prove that a
, cos
sin 4
8
cos 36 o
5 8
sin 4
7 8
cos76 cos16
B
24 , cos A 25
cos 3x
b tan
5cot
x
y 2
a b cot .
3 2 3 4
B
4 , find the value of tan2A . 5
15. A.
If x is acute and sin x
10o
B.
Solve 7 sin 2
4
C.
Solve 2cos2
D.
Solve 4sin x sin2x sin 4x
E.
If 0
F.
Find all values of x in
G.
Solve the following equations and write general solution. i)
3 sin
1
0 sin3x
, solve cos .cos 2 .cos 3
4 cos2
iii) 1 H.
3 cos 2
sin 2 x
If tan p
tan
3
2
sin 3x
ii) 6 tan 2 x
2 cos 2 x
cos 2 x
2
iv) 2 sin 2 x
sin 2 2 x
2
cos 3x
43
q then show that the solutions are in A.P. with C.D
p
q
. I. Solve
5sec
sin 2
J.
Solve 1
K.
Solve
L.
Solve sin2x
M.
Find the general solution of tan x
N.
If tan
O.
Find the common roots of the equations cos2x
P.
Solve the equation
3sin cos
2 sin x
cos
cos cos 2 x ....
1 cos
cot, q , and p
3cot
1 4
satisfying the equation 81
,
3
68o find x .
cos x
cos2x
cot
3
sin x cos x 1 3
,sec x
2 3 1
sin , then prove that cos
6
cos x
7 sin 2 x
cos x
4
sin2x
2 2
cot x and 2 cos 2 x
cos 2 2 x
1.
0.
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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1 and x cos x
Q.
If tan x
tan x
R.
If x
2 and sin x 3
S.
Solve sin 3
y
0,2 , find the values of x . 3 then find x and y . 2
sin y
4sin sin x
n ,n z .
where
sin x
16. 4 5
2 tan
1
y
sin
1
z
1 3
A.
Prove that sin
1
B.
If sin
1
1
C.
Solve arc sin
D.
Prove that cos tan
E.
If cos
1
p
cos
1
q
cos
1
r
, then prove that p2
F.
If sin
1
x
sin
1
y
sin
1
z
, then prove that x 1 x2
G.
Solve sin
1
x
sin
H.
If
1
I.
Prove that sin
1
J.
Prove that sin
1
K.
Prove that cot
1
L.
Prove that cos 2 tan
M.
If cos
N.
If a , b , c are distinct non-zero real numbers having the same sign, prove that cot
x
sin
tan
1
p a
1
If sin
P.
i) If tan
1
ii) If tan
p
1
x
1
x
1
2
12 x
2 1
sin cot
1
2x
1 x2
1
2
2
x
1 x
4 5
sin
1
4 5
sin
1
9
cos ec
1
1
q b
tan
1
x
x2
1
2
2
x
1
sin
1
5 13
sin
1
41 4
1
1 7
1
y
q2
r2
2 pqr
2 x2 y 2
y 2 z2
z2 x 2
. 1
y 1 y2
z 1 z2
2xyz
cot
q
tan tan
1
1
tan
2
z
1
x
sin 2
117 125
16 65
2
4
sin 4 tan
1 q2 1
y
4x 2 y 2 z 2
, then prove that x 2
7 25
bc 1 b c
1
z4
0
1
1 3
, then prove that
cos tan
. x
y4
3
x2
cot
2p
1
arc sin
, then prove that x 4
1
cos
ab 1 a b
O.
5 x
2
p2 a
2
ca 1 c a 1
b
sin 2
2
.
or 2
2x , then prove that x 1 x2
, then prove that x 2
q2
2 pq .cos ab
, then prove that xy
y
z yz
p q . 1 pq
xyz zx
1
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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Q.
Solve the following equation for x : cos
R.
Prove that (i) Tan
S.
Solve tan
1
1 2
1
1
2x
1 5
Tan
1 4x 1
1
tan
1
1
Tan
1
1
x
1 8
tan
1
sin
4
x 2
(ii) Tan
6 1
3 4
Tan
1
3 5
Tan
1
8 19
4
2 x2
1
17. (I) A 2
b
c
A.
If cot
B.
In
C.
Show that r r1 r2 r3
D.
In
ABC , show that
E.
In
ABC , if a
F.
If a
G.
In
H.
Prove that a b cos C
I.
In
ABC , if
J.
In
ABC , show that
K.
Show that a2 cot A
a
, find angle B .
1 r1
ABC , prove that
4, b
1 r3
b
c b
c cos A
c
C 2
ABC , find b cos 2
1
1 b
B . 2
c cos 2
c cos B c
a
b2
c2 a
2
2s
3bc , find A .
a
7 , find cos
c
1 r
2
b
5, c
a
1 r2
B . 2
b2
c2
3 b
c
, show that C
sin B C sin B
b 2 cot B
60
c 2 cot C
C
abc R
(II) A.
If r : R : r1
2 : 5 : 12 , then prove that the triangle is right angled at A.
B.
In an equilateral triangle, find the value of
C.
If A
90 o , show that 2 r
D.
If cot
A B C : cot : cot 2 2 2
E.
If b
c
F.
If a
R
b
r . R
c
3 : 5 : 7 , show that a : b : c
6:5: 4
B C 3a, then find the value of cot cot . 2 2
b c sec , prove that tan
2 bc A sin . b c 2
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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r1 r2
r3
G.
Prove that
H.
A B cot 2 2 Prove cot A cot B
I.
cot A
J.
Prove that cot
K.
If C
L.
Show that in
M.
If p1 , p2 , p3 are the altitudes of the vertices A, B,C of a triangle respectively, show that
r1r2
r2 r3
C 2 cot C
cot
cot B
cot
a2
cot C A 2
a
r3r1
cot
a a2
b2 4
B 2
1
1
1
p12
p22
p32
cot A
N.
If a : b : c
O.
If cot
P.
If r2
Q.
If
R.
Show that b 2 sin 2C
S.
If sin 2
c
b2
2
c2
c2
cot
s2
C 2
60 , then show that (i)
ABC , a
b
a b
b c
b cosC cot B
.
c
a
b
1 (ii)
c
2
a a
2
c
2
b2
0
c cos B
cot C
.
7 : 8 : 9 , find cos A : cos B : cosC
A B C , cot , cot are in A.P., then prove that a , b , c are in A.P. 2 2 2
a2 a2
r1 r3 b2 b2
2r2 r3 . Show that A
r1
sin C , prove that sin A B c 2 sin 2 B
90 .
ABC is a right angled.
2 bc sin A
A B C ,sin 2 ,sin 2 are in H.P., then show that a , b , c are in H.P. 2 2 2
LONG ANSWER QUESTIONS
18. 1
1
Let f : A
B, g : B
B.
Let f : A
B , I A and I B be identity functions on A and B respectively. Then foI A
C.
Let f : A
Let
gof
E.
f 1
f
og
.
f
IBof .
B be a function. Then f is a bijection if and only if there exists a function g : B
such that fog D.
C be bijections. Then gof
1
A.
I B and gof
I A and, in this case, g
1, a , 2, c , 4, d , 3, b
f
1
og
1
g
3x
2,
2
2,
2
2x
1,
x
.
2, a , 4, b , 1, c , 3, d
,
then
show
that
.
If the function f is defined by f x
f 4 , f 2.5 , f
1
and
1
f
A
2 ,f
4 ,f 0 ,f
x
x
3 x
2 . Then find the values of 3
7
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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F.
Let f : A
B be a bijection. Then fof
G.
Let f : A
B, g : B
C and h : C
1
1
I B and f
IA .
of
D . Then ho gof
hog of , that is, composition of functions
is associative. H.
If f : R
R are defined by f x
R, g : R
i) gof x I.
Let A
K.
1,2,3 , B
If f : Q
1
(iii) fof x
4
p, q , r . If f : A
a, b , c , C
1, a , 2, c , 3, b , g
f
a
(ii) gof
a, q , b , r , c , p
Q is defined by f x
C are defined by
then show that f
x
2, x
b) f 0
(iv) go fof 0
B, g : B
2,
If the function f is defined by f x a) f 3
2 then find
1
g
1
g f
1
.
4 x Q then show that f is a bijection and find f
5x
x
L.
x2
1 and g x
4x
c) f
3
1 , then find the values of
x x
1
d) f 2
1,5
.
1
1
1,
1
f
2
e) f
5
19. Show that 49n
16n
B.
1 1.3
1 3.5
...
C.
3.52 n
1
D.
1.2.3
A.
1 5.7 2 3n
2.3.4
2n
1 2n
n 2n 1
1
is divisible by 17.
3.4.5
nn
...upto n terms 3
2 3
F.
12
12
22
12
22
G.
Use Mathematical Induction to prove the formula
2 1 3
...
3 5
1 4.7
1 7.10
1
2
13 n
2
12
....upto n terms
Show that, n N ,
I.
If x and y are natural numbers and x
K.
9n
n.2n , n N .
H.
divisible by x
n n
....upto n terms
upto n terms 1 1.4
n 2 n2 24
....upto n terms
32
3
4
1 1
1
3
2 n
1 1
4.22
3
1 n
E.
3.2
3
1
3
2
3
1
1 is divisible by 64 for all positive integers n .
n 3n 1
y , using mathematical induction, show that x n
y n is
y , for all n N
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. L. M.
Use mathematical induction to prove that 2.4
2.3
3.4
4.5
...upto n terms
n n2
6n
2n 1
3
3n 1
is divisible by 11, n N .
11
3
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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N.
Prove by Mathematical induction, for all n N a
O.
Show that 4 n
P.
Use mathematical induction to prove the statement, 1
3 1 1
ar 2
..... upto n terms
a rn r
1 1
,r
1
1 divisible by 9 for all positive integers n .
3n
5 1 4
ar
7 ...... 1 9
2n 1 n2
n
1
2
20. A.
b
c c
a a
b
Show that c a
a a
b b
c
b b
c c
a
a B.
b
Show that
c
2b
b
D.
c
a
b
2x
3
3x
4
Find the value of x if x x
4
2x
9
3x
16
27 3x
64
a b c Show that b c a c a b
2
2
2
c2 b
2
2
a
b2
c2
3
3
3
b
2
c
a2
2
2 ab
a3 c
b3
c3
3abc
2
.
2
1
.
3
1 .
a
1
abc a
b b
c c
a
1 1 then find A3
1 1
3 A2
A
3I .
1
1 a2
a3
Show that 1 b 2
b3
2
3
1 c
Show that
1
3
Show that a2
a
b2 b2
a
1
c
3
0
c2
2
1 1
b
0
b
2 ac
2a 2a
a
If A
a2
2bc
3
2 then show that the adjoint of A is 3A' . Find A 1
Show that 2 a
1
8 2x
c
2
1
a
J.
2b
2
3
I.
a
x
a2
H.
2a a
2
G.
a b
c
If A
F.
c
2c
1 E.
a.
2b c
2a
2c C.
a b c
b c c
c
a
2c
b b
c c
a b
c a
a ab
bc
ca
b 2a
b c
a
2 a
b
3
c .
2b
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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2a
K.
Show that a c
L.
If A
If A
b c
b
2b
a c
a
b
c
b
4 a
2 3
0
1 4 then find A ' 2
b b
a .
c c
2c
1 2
M.
a
1
1
a1
b1
c1
a2
b2
a3
b3
c 2 is a non-singular matrix, then show that A is invertible and A c3
1
adj A . det A
21. A.
Solve the following equations by Gauss-Jordan method 3x
4y
5z
18 ,
2x
y
13 , 5x
8z
2y
7z
20
B.
Solve 3x
4y
5z
18 , 2 x
y
8z
13 , 5x
2y
7z
20 by using matrix inversion method.
C.
Solve 3x
4y
5z
18 , 2 x
y
8z
13 , 5x
2y
7z
20 by using Cramer’s Rule.
D.
Apply the test of rank to examine whether the following equations are consistent. 2x
E.
y
8 ,
3z
x
2y
4 , 3x
z
y
4z
0 and if consistent find the complete solution.
Find the nontrivial solutions if any, for the following system of equations: 2x
F.
5y
6z
0, x
3y
0 , 3x
8z
y
4z
0
Solve the following equations by Gauss-Jordan method 5x
G.
6y
4z
15 ,
7x
4y
19 , 2 x
3z
y
6z
46
Solve the following system of equations by Gauss – Jordan method. x
H.
y
3 , 2x
z
2y
3, x
z
y
z
1.
By using Gauss-Jordan method, show that the following system has no solution. 2x
4y
0, x
z
2y
2z
5 , 3x
6y
7z
2.
I.
Examine whether the following systems of equations are consistent or inconsistent and if consistent find the complete solution. x y z 4 , 2 x 5y 2 z 3 , x 7 y 7 z 5 .
J.
Show that the following system of equations is consistent and solve it completely: x
y
3 , 2x
z
2y
3, x
z
y
z
1
22. A.
Find the shortest distance between the skew lines r r
B.
4i
If A
k
s 3i
1, 2, 1 , B
2j
6i
2j
2k
t i
2j
2k
and
2k
4,0, 3 , C
1,2, 1 and D
2, 4, 5 , find the distance between AB
and CD. C.
Let a , b , c be three vectors. Then prove that i) a b
c
a.c b
b.c a
ii) a
b c
a.c b
a.b c
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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D.
Let a
E.
A line makes angles cos2
j, b
i
cos2
1
k, c
j
cos2
2
i . Find unit vector d such that a.d 1, 2, 3
cos2
3
4
and 4 . 3
Let b
G.
For any four vectors a , b , c and d a b . c d a b
2
j
a2 b 2
a.b
i
2
b c d .
0
with the diagonals of a cube. Show that
4
F.
2i
k, c
k
3k . If a is a unit vector then find the maximum value of a b c . a.c
a.d
and in particular
b.c b.d
.
H.
Find the volume of the parallelopiped whose coterminus edges are represented by the vectors 2 i 3 j k , i j 2 k and 2i j k .
I.
For any four vectors a , b , c and d a b a b
c d
a b dc
c d
b c d a and
a c db
a b cd
J.
Find the equation of the plane passing to the points A
K.
Show that in any triangle the altitudes are concurrent.
4,5,2 and C
2,3, 1 , B
3,6,5 .
23. A.
If A, B, C are angles in a triangle, then prove that sin2 A sin2B sin2C
B.
If A, B, C are angles in a triangle, then prove that sin A
C.
If A
D.
In triangle ABC, prove that cos
E.
If A
B C
2S , then prove that sin S
F.
If A
B C
0 , then prove that cos2 A
G.
If A
B C
2S , then prove that
, then prove that cos2
B C
cos(S
A)
cos(S
B)
A 2
cos(S C )
sin B sin C
A 2
cos2
B 2
cos2
B 2
cos
C 2
4 cos
cos
cos S
A
sin S
cos 2 B
4 cos
C 2
B
cos2 C
If A, B, C are angles in a triangle, then prove that sin 2 A
I.
If A
B
J.
If A
B C
3 , then prove that cos 2 A 2
K.
If A
B C
0 , then prove that sin2 A sin2B sin2C
L.
If A
B C
D
sin A sin B
2
, then prove that cos2 A
A 4
cos
sin C 1
sin
4sin
A B C sin sin 2 2 2 B
4
A B C sin cos 2 2 2
C
cos
4 cos
4
S
A 2
cos
S
B 2
sin
C 2
2 cos A cos B cos C
A B C cos cos 2 2 2
H.
C
2 1
4cos Asin BcosC
cos2B cos2C cos 2 B
sin 2 B
1
cos 2 C
sin 2 C
2 sin A sin B cos C
4sin Asin Bsin C . 2 cos A cos B sin C
4sin Asin Bsin C
2 , then prove that A B A C A D sin C sin D 4 cos sin cos 2 2 2
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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24. A 2
A.
Show that a cos2
B.
Prove that a3 cos B C
C.
If a 2
D.
Show that cos A
E.
In
F.
Show that cos 2
G.
If a
13, b
H.
If r1
2, r2
I.
Show that b
J.
Prove that
K.
If cos A
L.
Show that
1 r2
1 r12
M.
Show that
r1 bc
r2 ca
N.
Show that r
O.
Show that
P.
Prove that r12
Q.
If p , p , p
b2
c2
s
A
c 3 cos A B
R 3abc
8R 2 , then prove that the triangle is right angled.
cos B
A 2
cos C
a bc
ab
r
cos 2
C 2
2
4R .
r . 2R
2
65 ,r 8
15 , show that R
A 2
cos 2
cos A a
cos B b
r3
1 r
21 ,r 2 2
1
4 and c
3, b
12 and r
3
14 .
5.
a2 cos C c
b2
c2
2
1 2R 4 R cos B .
r2 r3 r1
r32
a2
1 r3 2
r2
bc
r2 2
c ab
4, r
3 , then show that the triangle is equilateral. 2
r3 ab
r1r2
A 2
2
c sin 2
b
1 r2 2
r1
1, prove that a
b ca
cos C
r3
B 2
r R
1
r3
6 and r
3, r3
c
r2
cos 2
14, c
cos B
2
C 2
c cos2
b3 cos C
ABC , prove that r1
1
B 2
b cos2
r2
ca r3r1 r2 16R2
a2
b2
c2
are altitudes drawn from vertices A, B,C to the opposite sides of a triangle
3
respectively, then show that
1 i) p
1
R.
1 p
2
1 p
Show that i) a
1 ii) p
1 r
3
r2
1
r3
rr1 r2 r3
ii)
1 p
2
r1r2
1 p
3
1 r
iii) p p p
3
1
2
abc 3
8R 3
2
8 3 abc
4R r1 r2 r1 r2
wish you all the best
FIITJEE KUKATPALLY CENTRE: # 22-97, Plot No.1, Opp. Patel Kunta Huda Park, Vijaynagar Colony, Hyderabad - 500 072. Ph.: 040-64601123
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