AAYCEZY

II PU MATHEMATICS

RELATIONS AND FUNCTIONS

FOUR MARK QUESTIONS
tONTINUITYAND DIFFERENTIABILITY 1. Let : RR be given by f(x) = 4x +3. Show
) Find the value of k that f is Invertible and find the inverse of f.
2. Let f:NYbe given by f(x) 4N3, where
ij x i
ifx=: wcontinuous at Y= (y y= 4x+3 rE N). Show that
invertible and find the inverse of f.
s
3 Verify whether the functian N-N oef nea ry
2 Find the value ot K so that the functian I()= rs one cneonto and ciectiv
at x = 5is a 4. Check the injectivity and surjectivity of the
fr=3r-5. fx>5
continucous function.
function f:R Rdefined by
Fund the value of k, f(x) = 3-4x.Is it abijective function?
5 Verify whether the function fRRaefined by
i r. iscontinuous at x
Cosx ()=x² is one one, onto and bijective
4) Find the value of K,
6. Itf:R Rdefined by /(x) =1+r, then show
that is neither oneone nor onto.
if xs2,s continuous at 7. LetA R-(3) and B = R-(1) consider the
l3 fx>2
hunction A Bdefined by f ) = s f
3) ind the relationship berWeen 'a' and 'b' so that
i n e d by ( ) one-one and onto funct1on?
B. Snowt that the function R R defined by
Ibr +3 fx>3s continuous at x=3 f(u) = l is neither one-one nor onto
6 Find the values of a and b such that (x) = 9. Let . R R be a function defined by
( 5 if xs2
axb if 2<r< 10 is a continuous f)=x. (3 isa greatest integer function
21 ifx2 10 then show that is neither one-one nor onto.
function. Y MATRICES
r=0
S contin uous at a=0
X=0 6 71 [o
8) For what value of Ais the function defined by
(a(r-2x) ifx s0
1) #4 =-6 o .-1li o2 olzand C=
l7 -8 o
f)=&y 41 ynContinuous at -2 Calculate AC, BC and (A + B)C Also,
I= 0?
OF verify that (A + B)C= AC + BC.
N9 itA =(then prove that A²-44+I= 2 -31
owherel= &0- . using this A 0 2B=
1
equation find A 3 2
3 1
4 2 5 and C=0 3 2 then
10 If=
A t h e n prove that A-SA + l2 0 31 l1 -2 31
compute (A + B)and (B-Ch Also vernfy
I=0.find A-using this equation where A+ (B B]-C

3) IfA=3-2 then shw that A'

l4 2 1
23A401 = 0.
FIVE MARKS QUESTION

Most expected questlons for FIRST TEST

 l PU MATHEMATICS

4 If A
1. Ify = (tan x)* then prove that
4 and B= [1 3 -6). verify that (a'+1)y, + 2r(t+l)y, =2
AR 2. ty Ae Be" then prove that
-(n n + mny =0

3. If y 3et+2el then prove that

4 53 -11 s+6y = 0
6 calculate AB, AC and A(B + C)
|-1 2 3 4. Ify =3 cos(log x) +4sin(log x) then prove
thatay.,
AlDerity that A(8 + C) = AB + 4C +y =0
0 2 S. y = sin' then prove that (1- r)y:
zy, 0
e A=0 2 1 then prove that A'-64 + 6. If y=co then prove that
3/
(1-)*--o'y =0
7. If y= 500e + 600 e- then prove that
calculate AC, BC and (A+ B).C. Also verify that
A-E) C = AC BC 49y
8. If e+1) =1. prove that =-

I-1 4/ hence prove that=()

2 0 -2 prove thatA(BC) =(AB)C
3
3 MARK QUESTIONS
9) #A=-1 2&8- then verily
tA- B) =A -8& 4 +B)=A'+B J RELATION AND FUNCTION
1. Deterraine whether the relation R in
2 set A = , 2, 3, 13, 14)
calçulate AC. BC and (A + B).C Also verity that define -O) is
(A + B).C= AC + BC. relexive, symmetric and transitive
2. Show that the relation R in the set Z
of iategers given by
DETERMINANTS
R= ((x,y):2 divides (x-y)) is an
Solve the following system of linear equations using cquivalence relation,
matri method
3. PrOve that the relation R in the set Z
1, 2x+ 3y+ 3z=5x- 2 y 2 *4 and of all integers defincd as R= (x. y) x
- y is an integer) is an cquivaleL ce
2. 3x- 2y +3z 82x+y-z=1 and
4x-3y + 22 = 4. that the relatlon R in the set A
3. Xy +2z =7, 3x +4y- 5z = -5& (1,2, 3, 4, 5) given bvR = llo b)
2x-y+3z = 12 la - bl is eve), equivaeno
4. x-y+ z = 4,2x +y-3z =0and x + y+ rclation.
1=2 5. Show that the relation R In the
5. 2x 3y- 5z = 11.3r + 2y- 4z =
set A=(r:* EZ:0x12} given
-5 and x+y- 2z =-3. by R= ((a, b) : |a - b| is a multiple of
6. - y 27 = 1,2y - 3z = land3x -2y + 4}is an equivalence relation.
4z = 2. 6. Show that the relation R in R defined
7. x+y +1= b,y +3z = 11 and x- 2y+z= as R= ((a, b) : a s bj is reflexive and
0 transitive but not symmetric,
8. 4r + 3y + 22= 60,2r+ 4y + 6z = 7. Show that the relation R in the set R
9dnd Ery 32-70 of real numbers, defined azR
((a, bia sbi is ceither reflexive cor
VcONTINUITY AND DIFFERENTIABILITY syametric nor trausitive.

Most expected questlons for FIRST TEST

 I PU MATHEMATICS

* Check ehether the relation R in R

defined by R ((a b) as b'] is cONTINUITY AND DIFFERENTIABILITY
reflexive,syn mctric or transitive.
9 Check whcther thc relation R defined
in the set A = (1, 2,3.4,5.6] asR 1. If x=a (cost + log tan).y = a sint prove
o2a+ 1| 0s reflexiye or ttan
that t
symmetric 2. Findif a(cos8 + 9 sin 8) and
y a(sin @-cos 0)

INVERSE TRIGONOMETRIC 3. Ifx a a (0 sin 8) and y= a(1- cos )

FUNCTION Prove that tan;
4. Ifx = a(0-sin ) and y= a (1+
1. Show that cos () + cos (= cos0)then prove that - cot ()
5. Ifx a cos' @ and y =a sin' 8. prave that
Prove that 3 cos = cos '(4r
3a),r.1) r = asece,y =b tun Bthen tind
3 Prove that 3 sin = sin (3x 6.
7. Ifx sint,y = cos2t then find
4x'),r¬ [-: ty
8. Ifx = at andy = 2at then show thatF:
4 write tan).x 0in the simplest 9. Ifx = 2at and y = at then find
form
=Vatin andy = acort then prove
5 Express tan-().<rin the 10. ifx

simplest form that=

Solve 2 tan '(cos)= tan(2cosecx) 11. il a' then prove that
Prove that c o t i n H n !
VIint--HN 12. if y = (log x)cos then find y
rE (0.5) 13. Differentiate sin with respect to
V MATRICES 2 MARK QUESTIONS

1, ExpressA = a5 s sum of
RELATION AND FUNCTION
sym metrie and skew symmetric
matrix
1 If:R R defined by f(r) = (3-*)i then
2. Express A = a s sum of
Symmetric and skew symmetric prove that fof (x) =x.
matrix 2. Find gofand fog, if i0f()= |
3, If A and B are invertible matrices of and g(x) =5r- 2 (i i = 8 and
the same order, then (AB)= B
A Proye that for ith
any
rea number
sie
rix and A-A' is a skew INVERSE TRIGONOMETRIC FUNCTION
5.
vmmetric matrix
If 4 and B are svtnnetnc matrices of
sahe ordes then show that AB 1S P.Isin-(2xwI -x)= 2sin-'x sxs
Shetric if A and B commute that is
AF = KA
fcos -sin 01
O. show that : P.IS0n(2rv1-r)= 2cos ' r S
6. F ) sin Y cos 1

FIF)) a F( +y)
Most expocted questlons for FIRST TEST
 II PU MATHEMATICS

Write t a n , 0 <<nin simplest

form
6 Vktù=ViD, show that 0
1 Find, irx²+ xy y' 100
4 Wnte cot -where |x| > l in simplest
8 f sinx+ cos y= k then fnd
Wite Lan 0<<n in simplest 9 f sin x +cos' y Ithenfind
VioN
10 Find, If ax +by' = cos y.
form
11 fy +siny = cosx then find
Find the value of sin-(sin)
7 Find the value of sln-(sin) 12 If y= os).0<r<lthen find
3 I[y= sec(),0<x<
Evaluate sin;-sip() then find
Find the value of coscos
14 Ity = (logxjeos*, find
DETERMINANTS
Using determinants, find the area of the
triangle whose vertices are 16 ify = (stna)" ,tind
(3,8),(-42) and (5. 1)
Using determinants, find the area of the 17. Ify = log,(log x) then find
triangle whose vertices are
(-2,-3).(3,2) «nd (-1,-8)
Using determinants, find the area of the triangle
whose vertices are (1,0). (6.0) and (4,3)
Using determinants, find the area of the
( c e s are
(2 )
S Find values of k. ta
f trlangle is 4 sq-. units
and verices are (k. 0).(4.O). (0.2)usine
determinants.
Find values of k, if area of triangle is 4 sq. units
and vertices are (-2 0 (04),(0.k)using
determinants.
If the area of the triangle with vertices
2 -6).(5,4)and (k.4) is 35 square units,
find values of k using determinants.
Using determinants, find the equation of the
line joining (1,2) &(3,6).
4 Uslng determinants, find the equation of the
tine jolning (3, 1) & (9,3)
1 Using determ1nant show that points A(0,bt
c),B(b,c+ a) and C(c,a + brare collinear.
CONTINUITY AND DIFFERENTIABILITY

Find f2x+ 3y = siny

Fund, IH 2r 3y = sinx
I1yt siny = cos z then fmd
a My= x>0,find
ify = r find
Most expected questions for FIRST TEST