NIMS GROUP OF SCHOOLS, U.A.E.
NIMS
U.A.E
GRADE: XII KBGROUP
SUBJECT: OF SCHOOLS,
MATHEMATICS
CHAPTER/UNITS: (NEW
Functions,Inverse
trigonometric
functios, Matrices ,
INDIAN MODEL
SCHOOL,SHARJAH)
Determinants ,Continuity and Defferentiability.
CLASS: XII KB
Date :
TOPICS/UNITS:
Name of the student:
SUBJECT : MATHEMATICS
Estimated Time for completion:
Relations,functions, Inverse trigonometric functions,
Date of submission:
matrices,Determinants , continuity and defferentiability
Answer the following question with needed steps
RELATIONS, FUNCTIONS
If  is a binary operation on R defined by
1.
a  b=
a
4
+
b
7
for
a, b  R, find the value of ( 2  5 )  7 .
2. Consider a binary operation  on the set {1, 2, 3, 4, 5} given by
the
following multiplication table:

1
2
3
4
5
(i)
(ii)
3
1
2
3
1
1
1
1
2
1
1
1
3
1
2
1
1
1
1
Compute 2  3  4 and 2  3  4
Compute 2  3  4  5
4
1
2
1
4
1
5
1
1
1
1
5
Which of the following graphs of relations defines a transitive
relation In A ={1,2,3,4}
i)
ii)
4
R1 = { (1,2),(3,4)(2,3),(2,4)}
R2 = { (1,2),(3,4),(2,4)}
Check whether the relation R in the set R of real numbers defined
as
i)
ii)
5
R= {(a,b):a≤b3 } is an equivalence relation.
R= {(a,b):b = a+1 } is reflexive , symmetric or
transitive.
Are the following functions invertible in their respective domains?.
If
so , find the inverse in each case
i)
ii)
F(x)= x+1
𝑥−1
F(x) =𝑥+1
1
INVERSE TRIGONOMETRIC FUNCTIONS
6.
7
8.
3π
(i)
Find the principal value of sin-1(sin( 5 ))
ii)
Solve tan-1(2x) + tan-1(3x) =
iii)

4
5
 16 
Prove that sin-1  5  + sin-1  13  + sin-1  65  = 2
 
 
 
(a)
Evaluate cosec −1 {cosec (− 4 )}
(b)
Solve for x : cos−1 {x2 +1}+2 tan−1 {1−x2 } = 3
11.
12
4
π
x2 −1
1
2x
2π
If cos 1 x  cos 1 y  cos 1 z   ; show that x 2  y 2  z 2  2 xyz  1
 1  x 2  1 x 2
Pr ove that tan 
 1  x 2  1  x 2
1
9.
10.
π
  1
   cos 1 x 2
 4 2
Solve for x. tan-1(x+1) + tan-1( x) + tan-1(x-1) = tan-13
Write the following in the simplest form.
i)
cos-1(√1 − 𝑥 2 )
ii)
tan-1{1+𝑠𝑖𝑛𝑥}
iii)
tan-1{
𝑐𝑜𝑠𝑥
𝑐𝑜𝑠𝑥−𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑥
}
Find the value of the following
i)
cot-1(tan-1x + cot-1x)
ii)
cos(sec-1x + cosec-1x). IxI>1.
iii)
Tan
-1
−1
2
(1) +sin-1( )
2
MATRICES
13.
14.
3
0 4
]
−2 −1 x
(a)
Find x , if
(b)
sec 
Simplify : tan  
 tan 
10 x [
tan  
+ sec
 sec  
 sec  
.
tan  
2
1
4
15. Solve the equation for x, y, z and t , if 2  x
 y
If A =
1
2
2
17.
 tan 
  sec 

Using elementary transformation find the inverse of the matrix
5 1
5 2
0 5
16
=0
2
1
2
z   3 1
t 
0
 1  3 3
2 4
5
6
2
2
prove that A2 – 4A – 5I = 0 Hence find A-1
1
1 2  3
Find A , where A  2 3
2  . Hence, Solve the system of linear
3  3  4
1
equations :
x + 2y - 3z = -4,2x + 3y + 2z = 2, and
3x - 3y - 4z = 11
18.
If the matrix
19.
 a a  b  c  9
5
a
1  is a skew-symmetric matrix, find a, b and c.

9 a  b  c a 
For two matrices A and B which of the following is not true?
(a) (A + B) (A – B) = A2 – B2
(b) (A + B)2 = (A + B) (A + B)
(c) (A + B)T = AT + BT
(d) (A B)T = BTAT ]
20.
A is a matrix of order 2 X 2. If |kA| = |A|. What is the value of k?
3
DETERMINANTS
21.
22
23.
Using properties of determinants show that
a bc
2a
2a
2b
bca
2b
2c
2c
cab
qr
r p
pq 2 p q r
yz
zx
x y
x
y
z
Using properties of determinants, prove that :
ab
ac
ab
ac
1 b
2

 1  a 2  b2  c 2
bc
1 c
bc
a
bc
24. Pr ove that a  c b
a b ba
cb
c  a  (a  b  c)( a 2  b 2  c 2 )
c
25. With out exp anding Pr ovethat
Show that
a
yz
zx
z
x
y
1
1
1
0
a+b a+2b
a+2b
a+b
Prove that

2
x y
27.
3
Using properties of determinants, prove that
bc ca ab
a b c
1 a2
26.
 a  b  c 
a
a+2b
a+b
= 9b2(a+b)
a
a  b  2c
a
b
c
b  c  2a
b
c
a
c  a  2b
4
 2(a  b  c )3
CONTINUITY AND DIFFERENTIABILITY
28.
Discuss the continuity of the function
1−cos2x
F(x) ={
29
30
31
, x ≠ 0 , at x= 0
5
,x = 0
Find the constants a and b , so that the function ‘f’ defined
below is continous
1
,x ≤ 3
F(x) ={ ax + b , 3 < x < 5
7
, x≥5
1
Discuss the continuity of the function f(x)=3x−4 at x= 5.
4: x  5


If the function f ( x)  ax  b;5  x  7 is continuous at x = 5 and
 11; x  7

x = 7, then
32
find the value of a & b
Differeniate with respect to ‘x’
first principles
the following function
using
sin x
x
i)
ii)
33
x2
sec(2x+1)
Diferentiate the following with respect to x
sin x  x cos x
x sin x  cos x 
i)
ii)
x
3

 1 x  2
x2
Rubrics
5
3
Understanding
Problem solving skill
Presentation
5
1