<br>
Consider f(x) = ax´ +
a +
a-3, g(x)
=
sinx 1
and
The least integral value of a, say a, for which f(x) = glx) has no real solution is equal to
(A) 4 (B) 5
(C) 0 (D) 7
=
+
a,x.t a, -3 = 0
has exactly ono real eoint of intersection wlh h(x) 0
Value of k for which aX
is equal to
(B) 2
(A) 1
(c) 23 4
((x) = 0 lie in (1, 2) then (a] (where [.]derolos the gro9st intoger
If atleast ono rool of equation
function) is (B) 1
1
(A)- •(D) none of those
(C) 2
y ((x) Is shown,
The graçlh of
then the number of soltlons of
I(l)) = 2 will be (B) (-3.2)
(A) 1 2
(C) 3 (D) 4
(-1,1)
(3. 0)
(12.0)
(6. -1)
(5, -1)
-6x11x-6 a
The vahue(s) of a, for whch =0 doesn have eal solun is aN
x- 10x -6 30
(8) 12
(A) -10 (D)-30
Match the equalion in List-l witlh(he
number of solutions of the cqualion
in List-1
List -1
(P) = 2x , List -1
sinx X> 0
19 1
6
(Q) sinx = .
19 X0 2
4
(R) (sin x) = 2x
,x>0
19 3. 5
(S) =
(sin|x]) (x},0<x<7
4 3
Codes:
R
(A) 1
2 3 4
(B) 4 3 2 1
(C) 1
4 3
(D) 4 3 1
<br>
O 100%
mucmonUs the greatost integcr function and ()
I the equation (where
denotes fraclional part of x) has three real and distinct roots say A,d, and A, (hen the value or
(A) 9 (B) 10
(C) 11 (D) 12
sin']+x-2tan-'x+Sin(cos x) &
he domain of 1
(where () and denotes fractional
+ In cos
integral part of x), is
(B) (0, 1)
(C) (-1, 1) (D) (0, 5/4)
The least integral value of K for the two curves y +k = 16 and xy= 1 to never meetl, is
(A) 218 (B)21 + 1
(C) 219 (D)2' +1
Consider f:R R SUch that f(3) = 1
for a e R
and f(%).f(y) +
then f(97) can be
(A) 1
(B)-1
(C) 2 (D) 97
= sin (k(x)) cos (k{x)) for k 0 and these exist at least
Let f(x) = sin (k{x})) + cos (k{x)) and g(x)
s range of values of k is (wrere (.)
xe
one c E +
such that f(X) g(«) ((c) g(c)
R
Rthen t
represents fracional part function).
ofo )
-o none of these
(C) k= orke() (D)
= COSX
)= sinx and g:(0.n][-11]:9(*) also
then
r'()+o'(x) a,f'(*)-g'(*)= pvxe(-1.1]
=
(B) max (a+2))
(A) max(2a + p) = 3n =2
+ 2p) =
(C) min
(2a +p) = 3n
(D) min (a
A function f: R
1
- R satisfies sinx cosy ((2x + 2y) - f(2x-2y)) = cosx.siny(((2x + 2y) +
(2x - 2y).
then the value of 4r"(«) + 1(x) is
2
M199--. x1
Ihen the souton of
theequaion f() -r')=0 is
(A)1 (B) x 0
(C) x= 1
(D) x = 2
<br>
Lett a)
(-3x-4)
-(x
-
2)-
I11(- , a] -Rsinjectve (one-one) funct1on, then the largest value
of 3 s
(8, S-3
2
3
(C)
(D) None of these
Paragraph for Quostion i
For x 0, 1, (i(x) E X, (,(x) r
fundions is closod under conpositions i.e. composition
.s(x) =
(-x,{x) =.
of any tO of thoso functions is again ono
1s(x) =
x) =
Thus farnily of
theso. of
Let F be a function such that
(A) f,
foF . Then F is oqual to
(B) (,
(C) (5
(D) f4
Let G be a function such that
Gof, = fo. Thon &is equal to
(C) 1, (8)4
(D) (2
Let
(A) fe
J
bc a function such that soJol =. Thon J is(B)oqual
, to
(C) 4 (D) (5
Let f(x) =
u cos x,
x+ x+ equation
. being an integer and u a real same
number. The number ordered
real roots
pair
is
of
(ls) (.. ) for which the
f(x) = 0 and f(((x)) =.0 have the
(B) 6
(non-empty) set of
(A) 4 (D) infinite
(C) 8
f(x +t
y) = f(xy) x4 and
Let f: NN where N is set of natural numbers be a function such that
y4, then (B) f(8)= f(9)
= f(5)
(A) (D) ((5)= f(6)
f(8)
a+x b, c e R, í(-1) = 0 and y = 1 is an asymptote of y = y=
h2G, f(). r')is inverse
of f(x); 1hen
(A) Area bounded between asymptotes of curve f() and asymptotes of f') is 9 sq. units
(B) (r')) is one-onc and into
(C) b+ c=1
() r'()=2X-1
X-1
f(7) is
periodic and f(3)=5, r(5)=11, then
non-periodic, g is periodic and g((%) is
I
Iis
(A) 23
(B) 17
(D) none of these
(C) 22
<br>
Match tho following Colunn- with Colurnn-l
K
Columh CAGRlA2 ColtúmnIHE
[X - 2m
(A) I:R»R, 1(*) - mx
XS-1 for all values of
one-one funcdlon
-4 X>-1 (p)
Ime (--2), thon ((«) Is
(8)
(*-a)(x-p)
(x-T)\X-6) (9) many•one fundion p(Pst)
5, then (() Is
(C) t:R-(o, 0), ,thon f(%) ls Into function
(D) Iff:R - R, (0) =
*° +2x' +6x + sinx +1, then f() (3) onto function
is
(1) ínvertible funcdion
Paragraph for Question Nos.
bnprehension 1:
If a, an expression, ((a, b) - (2 -b) answer the
beR' then, consklor 3a + b
Now,
following problerns.
Minimum valuo of (o, b) is for b
5+ 2
(A)
7 7
(C) (D)
7
The least value of f(a, b) is
(A) (2/2 -1) (e) 2(2-1)
(0) 2(/2 +1)
(C) 2-5
in greatest integer funcion
((x) =
In (|sin2x
+ |cos2x|), Where [.]
Letf:R -1,) and
fundamental period
then (B) f(:) is periodic vith
(A) 1(x) has range z
zl4 function
(D) ((%) is into
(C) f(-) is invertible in 0,|
- 5) range of ihe i:ncliony= X- 1
| The number of nteg•al values of a (a for which the
a- x* 1
containing any válues telonging to the interval-1, -is
1
The graph of the function f(%) =
(13 Is symmetrical about
1+x14ytt
2x
1+ xm
where m = 20-1
and xe (-1, 1)
the polnt
(A) (1. 1)
(C) (0, 1) (B) (1, 0)
(D) (1, -1)
<br>
If Range of valucs of x such that the is prime, is (x,,x,)u(x,, X4). Then
(i+x+**+ x)-2 =
((.] denotes the greatest integer function and x, <
x,)
If Range of values of 'x' salisfying the equality where a,b, c cN and D is in its
C
simplest form and a +b+c+abC = 5k, then k=
(.] denotes greatest integer function)
If afunction satisfies (x-y)f(x+y)-(x+y)f(x-y)=2(*'y-y°)xyeR =
andf (1) 2,then
(A)f(x) must be polynomial function (B) f(3) = 12
(C) f(0) =0 (D) f(x) may not be differentiable
The number of
integers in range of f(x) = xe", xe[-10] is.
If ee[-2017a,2017n]. then number of solutions of equation [sec9 -1]+ tan@tan=
sece-1+ tan 0 tan where (. )denotes fracional part functionand [.] denotes G..R)
(B)2016 (C)2017 (D)2018
(AJ0
f(x) =
(where J denotes the greatest integer fundion). now seled
0Ted oroperes for fx) ie
(A) doman of f(x) is [0, 3)
(3) domain of f(x) is [0, 2)
(C) range of f(x) is {0, cos (log 2)) (D) range of
f) iscos(1og2)
