03 maths questions
TRANSCRIPT
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Assertion Reason Type Questions
M HEM ICS
1.1. FUNCTIONS FUNCTIONS
Each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason). Each question
has 4 choices (A), (B), (C) and () out o! "hich #N$% #NE is correct.
(A) State&ent ' 1 is True, State&ent ' 2 is True State&ent ' 2 is a correct e*+anation !or
State&ent ' 1.
(B) State&ent ' 1 is True, State&ent ' 2 is True State&ent ' 2 is N#T a correct e*+anation !or
State&ent ' 1.
(C) State&ent ' 1 is True, State&ent ' 2 is a+se.
() State&ent ' 1 is a+se, State&ent ' 2 is True.
1. Let f(x) = cos3x + sin 3 x .
Statement 1 : f(x) is not a periodic function.Statement 2 : L.C.M. of rational and irrational does not exist
Ans. (A)
2. Statement 1: If f(x) = ax + b and the equation f(x) = f (x) is satisfied b! e"er! real "alue of
x# then a$ and b = .Statement 2: If f(x) = ax + b and the equation f(x) = f (x) is satisfied b! e"er! real "alue of
x# then a = and b$.Ans. (%)
3. Statements-1: If f(x) = x and &(x) ='x
x# then &(x) = f(x) ala!s
Statements-2: At x = # &(x) is not defined.Ans. (A)
4. Statement1 : If f(x) =
# x
x # # then the *raph of the function ! = f (f(f(x))# x is a
strai*ht line
Statement2 :f(f(x)))) = xAns. (C)
,ol. f(f(x)) =
x
f (x) x
x
= =
f(f(f(x))) =
xx f (f (x))
x
= =
5. Let f( + x) = f( x) and f(- + x) = f(- x)
Statement1 :f(x) is periodic ith period
Statement2 : is not necessaril! funda/ental period of f(x)Ans. (A)
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,ol. f( + x) = f( x) ... ()
f(- + x) = f(- x) ... (')
x x in () f( x) = f(x) ... (3)x - x in (') f(' x) f(0 x) = f(x) ... (-)() and (-) f(' x) = f(0 x) .... (1)2se x x x in (1)# e *etf(x) = f( + x)
f(x) is periodic ith period b"iousl! is not necessar! the funda/ental period.
6. Statement1 : 4eriod of the function f(x) = 5x6 sin 'x e+ + does not existStatement2 : LCM of rational and irrational does not exist
Ans. (A)
,ol. L.C.M. of 5# 6 does not exist(A) is the correct option.
7. Statement1 : %o/ain of f(x) =
7 x 7 xis (# )
Statement2 : 7 x 7 x for x $Ans. (A),ol. Clearl! both are true and state/ent II is correct explantion of ,tate/ent I .
8. Statement1 : $an*e of f(x) = '- x is 8# '9Statement2 : f(x) is increasin* for x ' and decreasin* for ' x .
Ans. (C)
,ol.'
xf (x)
- x
=
f(x) is increasin* for ' x and decreasin* for x '.
9. Let a# b $# a b and let f(x) =a x
b x
++
.
Statement1 : f is a oneone function.
Statement2 : $an*e of f is $ 56
Ans. (;)
,ol. ,uppose a b. ,tate/ent II is true as ( ) '
b af (x)
b x
= + # hich is ala!s ne*ati"e and hence
/onotonic in its continuous part. Alsox bli/ f (x)
+= and
x bli/ f (x)
= . Moreo"er
x xli/ f (x) and li/ f (x)
= + = .
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10. Statement1 : sin x + cos (x) is a nonperiodic function.Statement2 : Least co//on /ultiple of the periods of sin x and cos ( x) is an irrationalnu/ber.
Ans. (C)
,ol. ,tate/ent I is true# as period of sin x and cos x are 'and ' respecti"el! hose L.C.M doesnot exist.
b"iousl! state/ent II is false
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,ol. ,ince cos n is also e"en function. herefore solution of cosx = f(x) is ala!s s!/. also out !axis.
15. If f is a pol!no/ial function satisf!in* ' + f(x).f(!) = f(x) + f(!) + f(x!) x# !$Statement-1:f(') = 1 hich i/plies f(1) = '
Statement-2: If f(x) is a pol!no/ial of de*ree n satisf!in* f(x) + f(Dx) = f(x). f(Dx)# then
f(x) = xn+ Ans. (A)
16. Statement-1:he ran*e of the function sin>+ cos>x + tan>x is 8D-# 3D-9Statement-2: sin>x# cos>x are defined for 7x7 and tan>x is defined for all x.
Ans. (A)
17. A function f(x) is defined as f(x) = here x is rational
here x is irrational
Statement-1 : f(x) is discontinuous at xll x$Statement-2 : In the nei*hbourhood of an! rational nu/ber there are irrational nu/bers and inthe "incit! of an! irrational nu/ber there are rational nu/bers.
Ans. (A)
18. Let f(x) = sin ( ) ( )' 3 x cos 3 3 x + Statement-1 :f(x) is a periodic function
Statement-2: LCM of to irrational nu/bers of to si/ilar Eind exists.Ans. (A)
19. Statements-1: he do/ain of the function f(x) = cos
>
x + tan
>
x + sin
>
x is 8># 9Statements-2: sin>x# cos>x are defined for 7x7 and tan>x is defined for all x.Ans. (A)
,ol. ;oth A and $ are ob"iousl! correct.
20. Statement-1 : he period of f(x) = = sin'x cos 8'x9 cos'x sin 8'x9 is D'
Statements-2: he period of x 8x9 is # here 89 denotes *reatest inte*er function.Ans. (A)
,ol. f(x) = x 8x9
f(x + ) = x + (8x9 + ) = x 8x9,o# period of x 8x9 is .
Let f(x) = sin ('x 8'x9)
f x sin ' x ' x' ' '
+ = + +
= sin ('x + 8'x9 )
= sin ('x 8'x9)
,o# period is D'
21. Statements-1: If the function f : $ $ be such that f(x) = x 8x9# here 8 9 denotes the*reatest inte*er less than or equal to x# then f>(x) is equals to 8x9 + x
Statements-2: &unction Ff G is in"ertible iff is one>one and onto.
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Assertion Reason Type Questions
Ans. (%),ol. f() = =
f() =
f is not one>one
f>(x) is not defined
22. Statements-1 : 4eriod of f(x) = sin -5x6 + tan 8x9 ere# 89 H 56 denote e ?.I.&. Hfractional part respecti"el! is .
Statements-2: A function f(x) is said to be periodic if there exist a positi"e nu/ber
independent of x such that f( + x) = f(x). he s/allest such positi"e "alue of is called the
period or funda/ental period.Ans. (A)
,ol. Clearl! tan 8x9 = x$ and period of sin - 5x6 = .
23. Statements-1: f(x) =x
x
+
is one>one function
Statements-2:x
x
+
is /onotonicall! decreasin* function and e"er! decreasin* function is
one>one.
Ans. (A)
,ol. f(x) =x
x
+
f(x) = ' '(x ) (x ) '
(x ) (x )
+ = 7cos (D' + x)7)= sin (+ 'x) (7cosx7 > 7sinx7)= >sin'x (7cosx7 > 7sinx7)
= sin'x (7sinx7 > 7cosx7)
,o/eti/es f(x + r) = f(x) here r is less than the L.C.M. of periods of all the function# butaccordin* to definition of periodicit!# period /ust be least and positi"e# so FrG is the
funda/ental period.
,o FfG is correct.
25. Statements-1: ex= lnx has one solution.
Statements-2: If f(x) = x f(x) = f(x) ha"e a solution on ! = x.Ans. (%)
26. Statements-1: &(x) = x + sinx. ?(x) = >x
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periodic function H *(x) is a non>periodic function then h(x) =
f(x) *(x) ill be a periodic function.Ans. (C)
27. Statements-1:x # x
f(x)x # x
+ = /onotonic function.
Statements-2: "er! one to one function is /onotonic.Ans. (C)
,ol. &or one to one function if xx'f(x) f(x') for all x# x'%f 3 >
but f ( 3) f ()to>one
but non>/onotonic
29. Statement1 :Let f : 8# '9 81# 9 8# '9 81# 9 defined asx -# x 8# '9
f(x)
x N# x 81# 9
+ =
+
then
the equation f(x) = f(x) has to solutions.
Statements-2: f(x) = f(x) has solutions onl! on ! = x line.
Ans. (C)
,ol.3 3
# and #' ' ' '
both lie on ! = f(x) then the! ill also lie on ! = f(x) there are to
solutions and the! do not lie on ! = x.
30. Statements-1: he functionpx q
rx s
++
(ps qr ) cannot attain the "alue pDr.
Statements-2: he do/ain of the function *(!) = q s!r! p is all real except aDc.
Ans. (A).
,ol. If e taEe ! =px q
rx s
++
then x =q sx
rx p
x does not exist if ! = pDrhus state/ent> is correct and follos fro/ state/ent>'
31. Statements-1: he period of f(x) = sin 8'9 xcos 8'x9 cos'x sin 8'x9 is D'
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Statements-2: he period of x 8x9 is .Ans. (A)
,ol. f(x) = sin('x 8'x9
f(x + D') =
sin 'x ' x '
+ +
= sin ('x + 8'x9 9= sin ('x 8'x9.) i.e.# period is D'.
f(x) = x 8x9
f(x + ) = x + (8x9 + ) = x 8x9
i.e.# period is .
32. Statements-1: If f is e"en function# * is odd function thenb
*(* ) is an odd function.
Statements-2: If f(x) = f(x) for e"er! x of its do/ain# then f(x) is called an odd function and
if f(x) = f(x) for e"er! x of its do/ain# then f(x) is called an e"en function.Ans. (A)
,ol. Let h(x) =f(x)
*(x)
h(x) =f ( x) f (x) f (x)
h(x)*( x) *( x) *(x)
= = =
h(x) =f
*is an odd function.
33. Statements-1: f : A ; and * : ; C are to function then (*of)= fo*.Statements-2: f : A ; and * : ; C are biOections then fH *are also biOections.
Ans. (%)
,ol. Assertion : f : A ;# * : ; C are to functions then (*of)fo*(since functions neednot posses in"erses.$eason : ;iOecti"e functions are in"ertibles.
34. Statements-1: he do/ain of the function 'f (x) lo* sin x= is (-n + )'
# n .
Statements-2: xpression under e"en root should be Ans. (A)
,ol. for f(x) to be real lo*'(sin x) sin x 'P sin x =
x = (-n + )'
# n .
35. Statements-1: he function f : $ $ *i"en 'af (x) lo* (x x )= + + a # a is in"ertible.
Statements-2: f is /an! one into.
Ans. (C)
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,ol. f is inOecti"e since x ! (x# ! $)
{ } { }' 'a alo* x x lo* ! ! + + + + f(x) f(!)
f is onto because ( )'alo* x x !+ + =
! !a a
x'
= .
36. Statements-1: (x) = sin (cos x) x #'
is a one>one function.
Statements-2: (x) x #'
Ans. (A)
37. Statements-1: &or the equation Ex'+ (' E)x + = E $ 56 exactl! one root lie in(# ).
Statements-2: If f(E) f(E') Q (f(x) is a pol!no/ial) then exactl! one root of f(x) = lie in
(E# E').
Ans. (C)
38. Statements-1: %o/ain of
' xf (x) sin is 5 # 6
'x
+=
Statements-2:
x 'x+ hen x and
x 'x+ hen x Q .Ans. (A)
39. Statements-1: $an*e of f(x) = 7x7(7x7 + ') + 3 is 83# )Statements-2: If a function f(x) is defined x $ and for x if a f(x) b and f(x) ise"en function than ran*e of f(x) f(x) is 8a# b9.
Ans. (A)
40. Statements-1: 4eriod of 5x6 = .
Statements-2: 4eriod of 8x9 = Ans. (A)
,ol. ,ince 5x6 = x 8x95x + 6 = x + 8x + 9= x + 8x9 = x 8x9 = 8x9
4eriod of 8x9 =
41. Statements-1: %o/ain of f = . If f(x) =
8x9 x
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Statements-2: 8x9 x x$Ans. (A)
,ol. f(x) =
8x9 x 8x9 x
8x9 x 8x9 x It is i/posible or 8x9 x,o the do/ain of f is because reason 8x9 x
42. Statements-1: he do/ain of the function sinx + cosx + tanx is 8# 9
Statements-2: sinx# cosx are defined for 7x7 and tanx is defined for all FxGAns. (A)
2.2. LIMITS, CONTINUITY & DIFFERENTIABILITYLIMITS, CONTINUITY & DIFFERENTIABILITY
43. Statements-1: he set of all points here the function f(x) =D x
# x
x
# x e
=
+
is differentiable
is (# ).
Statements-2: Lf() = # $f() = and f(x) =D x D x
'
D x '
e x(e
x
( e )
+
+# hich exists x .
Ans. (%)
,ol. ,tate/ent> is ron* ,tate/ent>' is true.
44. Statements-1: f(x) =
'
3
3 x # x '
x # x '
>
+
then f(x) is differentiable at x =
Statements-2: A function ! = f(x) is said to ha"e a deri"ati"e if
h h
f (x h) f (x) f (x h) f (x)li/ li/
h h+
+ + =
Ans. (%)
45. Consider the function f(x) = (7x7 7x 7)'
Statement 1: f(x) is continuous e"er!here but not differentiable at x = and .
Statement 2: f () = # f (+) = -# f () = -# f (+) = .
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Ans. (A)
46. Statement 1:D x
D xx
e li/
e
+
does not exist
Statement 2: L.
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Ans. (A),ol. Clearl! state/ent I is true and state/ent II is the correct explnation of state/ent I.
52. Statement1 :
x
xli/ sec
x
+
does not exist.
Statement2 : sect is defined for those t# hose /odulus "alue is /ore than or equal to .
Ans. (A)
,ol. ,tate/ent II is true and correct reasonin* for state/ent I# becausex
xli/
x =
+.
( x) = x = cos )
x li/ '
'sin'
+ =
.
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Statement2 : he function h(x) = /ax 5> x# # x'6 b x $# is not differnetiable at to"alues of x.
Ans. (;)
,ol. f(x) =
x
t sin dtt
f (x) x sinx
=
clearl!# f (x) is a finite nu/ber at all x (# ). f(x) is differentiable at all x (# ).
h(x) =
'
'
x K x
K x
x K x
fro/ *raph it is clear that h(x) is continuous at all x and it is not differentiable at x = > # .
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Assertion Reason Type Questions
,ol. f (c) =f (b) f (a)
b a
lo*c =b( lo*b) a( lo*a)
b a
(a b) lo*c = b( lo*b) a ( lo*a)
59. Statement 1 : Let 5x6 denotes the fractional part of x. henx
tan5x6li/
5x6=
Statement 2 :x
tan xli/
x=
Ans. (%)
60. Statement 1 :
t
sin x dx = > costStatement 2 : sinx is continuous in an! closed inter"al 8# t9
Ans. (A)
61. Statement 1 :x
sin xli/
x =
here 89 ?.I.&.
Statement 2 :x
sin xli/
x
=
Ans. (%)
62. Statement 1 :he function f(x) =
x - is continuous at a point x = a -.Statement 2 : &or x = a# f(x) has a definite "alue and as x a# f(x) has a li/it hich is alsoequal to its definite "alue of x = a -.
Ans. (A)
63. Statements-1:x li/
+x sin
x=
Statements-2: !li/
! sin
!
=
Ans. (%)
,ol. he Statements-1: is false sin as x +# the function xsin x
= a qt!t. apron. Sero) T (finite
nu/ber beteen H ). husx
li/ xsin
x+ .
he ,tate/ent>' is true since it is equi"alent to standard li/itx
sin x li/
x
=
option (d) is correct.
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Assertion Reason Type Questions
64. Statements-1: f(x) =nli/
(sinx)'n# then the set of points of discontinuities of f is 5('n + ) D'#
nI6Statements-2: ,ince > Q sinx Q # as n # (sinx)'n# sinx = ()'n# n .
Ans. (A),ol. ption (a) is correct.
Statements-1: is the solution of ,tate/ent>'.
65. Statements-1: f(x) =nli/
(cosx)'n# then f is continuous e"er!here in (># )
Statements-2: f(x) = cosx is continuous e"er!here i.e.# in (># )Ans. (%)
,ol.nli/
x'n=
7 x 7
7 x 7
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,osin x
x
= for x because
sin x
xis odd function so it is correct for x Q .
,o# FdG is correct.
68. Statements-1: f(x) = /ax (# x'# x3) is differentiable x$ except x = ># Statements-2: "er! continuous function is differentiable
Ans. (C)
,ol. he *raph of /ax (# x'# x3) is as under clearl! function is NOTdifferentiable at x = # ."er! continuous function is not necessaril! differentiable.,o# FcG is correct.
69. Statements-1:x
sin('x ')li/ '
x
+=
Statements-2: ,ince sinx has a ran*e of 8># 9 x$ x
sin xli/
x=
Ans. (%)
70. Statements-1: f(x) =
7s inx7# x
x
# x
7 s inx 7# x
x
>
= ' both are true and ,tate/ent>' is the correct explanation ofStatements-1: .
84. Statements-1: : f(x) = xnsin
x
is differentiable for all real "alues of x (n ')
Statements-2: for n ' ri*ht hand deri"ati"e = Left hand deri"ati"e (for all real "alues of x).Ans. (A)
,ol. f () =n
h h
h sin
f ( h) f () hli/ li/
h h
+ =
=h li/
hnsin
h
(n ')
= finite nu/ber =
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Assertion Reason Type Questions
' both are true and ,tate/ent>' is the correct explanationof Statements-1: .
85. Statements-1: he function
D x
D x
e
# hen x f(x) e
# hen x
= + =
is discontinuous at x = .
Statements-2: f() = .Ans. (;)
,ol.x li/ f(x)
=
x li/ f (x)
+=
L.
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Assertion Reason Type Questions
,ol.D x '
x x
...
e x '@ xli/ li/x x+
+ + ++ =
= ' 3x
li/ ...x x 'x @+ + + + = (infinits)
D x
x
eli/
x+does not exist
Ans.A)
90. Statements-1:D x 3
x li/( 3x) e
+ =
Statements-2: sincex li/
( + x)Dx= e
Ans. (A)
,ol.
D x
x li/( 3x)
+ ( )
3D3x
x li/ 3x + = e3
because( )
D x
x li/ x e
+ =
91. Statements-1: sinx = has atleast one roots beteen ( D'# D')Statements-2: ,ince sinx is continuous in 8>D'# D'9 and sin (>D') = ># sin (D' = i.e. sinxhas opposite si*n is at x = >D'# x = D'# b! inter/ediate theore/
Ans. (A)
,ol. f(x) = sinx continuous in 8>D'# D'9b! inter/ediate "alue theore/
f(>D') = sin (>D') = >
f sin ' '
= =
f and f ' '
are of opposite si*n is
b! inter/ediate "alue theore/#a pointc8>D'# D'9 such that f(x) = s a point x8>D'# D'9 such that f(x) =
i.e.# sinx =
thus sinx = has at least one root beteen #' '
92. Statements-1: Let f(x) =D x D x
D x D x
e e# x
e e
+= # x = then f(x) has a Ou/p discontinuit! at
x = .
Statements-2: ,incex li/
f(x) =
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Assertion Reason Type Questions
andx li/
+f(x) =
Ans. (A)
,ol.
D x D x 'D x
D x D x ' D xx x x e e eli/f (x) li/ li/e e e
= + +
x li/ f (x)
+=
x li/ f (x)
=
x = # f() =
# ) 56
Statements-2: Lf() = # $f() = is
f(x) =D x D x
D x '
e e
( e )
++
. hich exists x
Ans. (A)
,ol. Lf () = f ( x) f () D xx x x
x
eli/ li/x
+=
$f() =
f ( x) f ()
x D x
x x
xli/ li/
e
x
+ +
=
+
=D x
x
li/
e+=
+
L f() $ f() so it is differentiable in (># ) 56
f(x) =D x D x
D x '
e e
( e )
+ ++
x
94. Statements-1: f(x) =8x9
# x x
# here 89 denotes *reatest inte*er function# then f(x) is
differentiable at x =
Statements-2: L f() x x f (x) f () 8x9
li/ li/ 7 x 7x
x
=
= x x
li/ li/
7 x 7 x
x
= =
f() does not exist.Ans. (A)
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Assertion Reason Type Questions
,ol. $f () = x f (x) f () 8x9
li/ x 7 x 7x
x
+
+
=
=
x x
x 7 x 7
li/ li/ x x x(x ) x+ +
+
= = =
Lf() = then f() does not exist.
3.3. APPLICATION OF DERIVATIVESAPPLICATION OF DERIVATIVES
95. Statements-1: &or the circle (x )'
+ (! )'
= # the tan*ent at the point (# ) is the x>axis.Statements-2: the deri"ati"e of a sin*le "alued function ! = f(x) at x = a is the slope of the
tan*ent dran to the cur"e at x = a.
Ans. (;),ol.
96. Statements-1: ;oth sin x# and cos x are decreasin* functions in #'
[ Good ]
Statements-2: If a differentiable function decreases is an inter"al (a# b) then its deri"ati"e also
decreases in (a# b).
Ans. (C)
97. Statements-1: ee > [ Good ]
Statements-2: he function
xx ( x )> has a local /axi/u/ at x = e
Ans. (A)
98. Statements-1: Conditions of LMU fail in f(x) = 7x 7 (x )
Statements-2: 7x 7 is not differentiable at x =
Ans. (%)
99. Let f(x) =n
'
i
i
(x x )=
Statement1 : Mini/u/ "alue of f(x) occurs at x = ix
n
Statement2 :Mini/u/ of f(x) = ax'+ bx + c (a ) occurs at x = bD'a.Ans. (A)
,ol. f(x) =n
'
i
i
(x x )=
represents an upard parabola hose xcoordinate of "ertex is xiDn
100. Statement1 : # for '.R Q Q
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Assertion Reason Type Questions
Statement2 :f(x) = elo* x
xis a decreasin* function for x e.
Ans. (A)
,ol. f(x) =e
lo* (x)
x
f(x) = x'
lo* x
x
Q for x f(x) function
Also Q f() f() e elo* lo*
>
lo*e lo*e .
101. Statement1 : otal nu/ber of critical points of f(x) = /ax. 5D'# sinx# cox6 x are 1Statement2 : otal nu/ber of critical points of f(x) = /ax 5D'# x# cosx6 x are '
Ans. (A)
,ol. Clearl! critical points are
D3# # D-# D'# 1D.
102. Let f(x) = 1p'+ -(x ) x'# x$ and p is a real constantStatement1 : If the /axi/u/ "alues of f(x) is '# then p = '.
Statement2 : If the /axi/u/ "alue of f(x) is '# then p = '.
Ans. (A),ol. f(x) = x'+ -x + (1p' -)
Uertex !coordinate ='% -(1p -)
-a -
+ =
?i"en that' 'p
'-
+ = p'= - p = '.
103. Let f(x) = sinx + cosx + tanx and x 8 # 9
Statements-1: $an*e of f(x) is3
#-
4
.
Statements-2: f(x) is an increasin* function.
Ans. (A)
,ol. f(x) =tan x
'
+
f (x) ='
x
>+
/ini/u/ "alue of f(x) is-
and /axi/u/ "alue of f(x) is
3
-
.
104. Let f(x) = x3
Statements-1: x = # in the point of inflexion for f(x)
Statements-2: f (x) Q for x Q and f (x) for x .Ans. (A)
,ol. f (x) x = f (x) < for x Q and f (x) > for x
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Assertion Reason Type Questions
105. ,uppose f(x) ='
x
'+ n x 'cos x+l
Statements-1: f is an increasin* function.
Statements-2: deri"ati"e of f(x) ith respect to x is ala!s *reater than Sero.
Ans. (A)
,ol. ,tate/ent II is true as f (x) = x +
x ' sin x#
f (x) # x# > as x +
'# x x
> # and 7' sin x7 '. (do/ain of f is (# ))
cos x).
Ans. (%)
,ol. f(x) = x' x sin x cos x
( )f (x) 'x x cos x x ' cos x = =
Indeed# f(x) = has onl! to solutions,ince f(x) is increasin* in (# ) and decreasin* in (> # ).
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Assertion Reason Type Questions
,ol. /=
'
x
d!
dx
=
=
/'= x
d!
dx '= = //'=
hence an*le is D'
109. Statement 1 :he cur"e ! = xD3 has a point of inflection at x =
Statement 2 : A point here !fails to exist can be a point of inflectionAns. (A)
110. Let f(x) and *(x) are to positi"e and increasin* function
Statement 1 :If (f(x)) *(x) is decreasin* then f(x) Q
Statement 2 : If f(x) is decreasin* then f(x) Q and increasin* then f(x) for all x.Ans. (A)
111. Statement 1 : If f() = # f(x) = ln (x + ' x+ )# then f(x) is positi"e for all x$Statements-2: f(x) is increasin* for x and decreasin* for x Q .
Ans. (A),ol. ption (a) is correct.
f(x) = ln (x + ' x+ ) = >ln '( x x+ )f(x) f(x) hen x Q f(x) is increasin* hen x .
f(x) f() f(x) .
A*ain f(x) is decreasin* in (># )f(x) f() f(x) .f(x) is positi"e for all x$hus ,tate/ent> is true and follos fro/ ,tate/ent>'.
112. Statements-1: he to cur"es !'= -x and x'+ !' x + = at the point (# ') intersect
ortho*onall!.
Statements-2: o cur"es ! = f(x) H !=*(x) intersect ortho*onall! at (x!) if (f (x).*((x))= .
Ans. (%)
,ol. !'
= -x#
d!
dx
at (# ') =
-
'.' =
and 'x + '!d!
dx
=
d! 'x
dx '!
=
d!
dx
at (# ') = '
' '
=
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Assertion Reason Type Questions
/= /'he to cur"e touch each other
113. Statements-1: If 'Na + Rb + 3c + d = # then the equation -ax 3+ 3bx'+ 'cx + d = has atleast
one real root l!in* beteen (# 3)Statements-2: If f(x) is continuous in 8a# b9# deri"able in (a# b)# then at least one point c(a#b) such that f(c) = .
Ans. (A)
,ol. Let f(x) = ax-+ bx3+ cx'+ dx in 8# 39f() =
f(3) = 3('Na + Rb + 3c + d) =
f() = f(3),ince f(x) is pol!no/ial
it is continuous in 8# 39 and deri"able in (# 3) also f() = f(3)f(x) = in x(# 3)-ax3+ 3bx'+ 'cx + d = in x# 3)
114. Statements-1: f(x) = 5x6 has local /ini/a at x = .
Statements-2: x = a ill be local /ini/a for ! = f(x) pro"idedx ali/ f (x) f (a)
> also
x ali/
+f(x) f(a).
Ans. (A)
,ol. he *raph of f(x) = 5x6 is as under clearl! x = is local /ini/a.
Alsox li/
f(x) f() #
x li/
+f(x) f()
,o FaG is correct.
115. Statements-1: f(x) =
x K'
x'
xQ x' xStatements-2: If f(x) f(x) x' xfunction is ala!s increasin*
Ans. (A)
117. Statements-1: he *raph of a continuous function ! = f(x) has a cusp at point x = c if f (x) hassa/e si*n on both sides of c.
Statements-2: he conca"it! at an! point x = c depends upon f (x). If f (x) Q or f (x) the function is either conca"e up or conca"e don.
Ans. (A)
118. Statements-1: If f be a function defined for all x such that 7f(x) f(!)7 Q (x !)'then f isconstant
Statements-2: If (x) Q (x) Q (x) for all x andx a x a x ali/ (x) li/ (x) L li/ (x) L
= = =Ans. (A)
,ol.f (x h) f (x)
7 h 7h
+
8 7cosx7 Q and 3
' 0 0
x3 3 '
+ +
9
f(x) is strictl! increasin*.
120. Statements-1: If H are an! to roots of equation excosx = # then the equationexsinx = has at least one root in (# )
Statements-2: f is continuous in 8# 9. f is deri"able in (# ). f() = f() then these existsx ( , )such that f(x) =
Ans. (A)
,ol. ?i"en excos= ... () and ecos= .. (')Let f(x) = e>x cosx# then f(x) is continuous and differentiable.
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Assertion Reason Type Questions
Also# f() = f() = (fro/ () H (')herefore b! $olleGs MU# f(x) = has at least one root in (# )>e>x+ sinx = for at least one x(# ) exsinx = has at least one root in (# ).
121. Statements-1: he /ini/u/ "alue of the expression x'+ 'bx + c is c b'.
Statements-2: he first order deri"ati"e of the expression at x = b is Sero and secondderi"ati"e is ala!s positi"e.
Ans. (A)
,ol. Mini/u/ "alue ='-ac b
-a
=
'- c -b
-a
Also f(x) = 'x + 'b = x = b.
122. Statements-1: Let (x) = sin (cosx) in #'
then (x) is decreasin* in #'
Statements-2: (x) x #'
Ans. (A)
,ol. ,tate/ent> is rue,tate/ent>' is rue
,tate/ent>' is the correct explanation of ,tate/ent>.
123. Statements-1: he function f(x) = x-0x3+ ''x''-x + ' is decreasin* for e"er!x ('# 3) (# )
Statements-2: f (x) for the *i"en "alues of x.Ans. (C)
,ol. # and -D3f(x) = (x ') ex+ ex(3x' 'x + ') + (3x' 'x + ') ex+ (x3 x'+ 'x 0) ex= x3 x + - = x = 'f(x) = (3x' ) ex+ ex(x3 x + -)= ex(x3+ 3x' x ')
f(') = e'(0 + ' ' ') = e'
126. Statements-1: Consider the function f(x) ' 'f (x x ) f (x ) f (x )
' '
+ + In>'= In# n.
Ans. (C),ol. ption (c) is correct.
In= tanxx dx =n ' ' n
(tan x.sec x tan x)dx In=
n
n '
tan xI
n
4ut n = # 1(I+ I-) = tan1x.
155. Statement-1: If a and b' -ac Q # then the "alue of the inte*ral'
dx
ax bx c+ + ill be of
the t!pe tan>x A
c;
+ + # here A# ;# C# are constants.
Statement-2: If a # b' -ac Q then ax'+ bx + c can be ritten as su/ of to squares.
Ans. (A),ol. If a H b' -ac Q# then
ax'+ bx + c =
' 'b -ac ba x
'a -a
+ +
'' '
dx dx
ax bx c ba x E
'a
=+ + + +
# here E'=
'-ac b
-a
>
hich ill ha"e an anser of the t!pe
bx
'a. tan Ca E D a E D a
+ +
or tan>x A
C;
+ +
.
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Assertion Reason Type Questions
156. Statements-1:
' xx
' 3D ' '
x x ee dx c
(x ) x
+= +
+ +
Statements-2:x
e (f (x) f (x)dx+ = ex
f(x) + cAns. (C)
,ol.'
x
' 3D '
x xe dx
(x )
+ +
=
'x
' 3D ' ' 3D '
x xe dx
(x ) (x )
+ + +
=
x
'
ec
x +
+
157. Statements-1:
'
'- '
x '
dxx '(x 1x -) tan
x
++ +
= lo* 7tan>(x + 'Dx)7 + c
Statements-2:
' '
dx xtan c
a x a a
= ++
Ans. (A)
,ol.
'
'- '
(x ')dx
x '(x 1x -) tan
x
+
+ +
4ut x + 'Dx = S#
( 'Dx') dx = dS
'
dS
(S ) tan S+= lo* 7tan>(x + 'Dx)7 + c
158. Statements-1:'
xln
xe c(ln x) ln x
= +
Statements-2: ex
(f(x) + f(x)) dx = ex
f(x) + c.Ans. (A)
,ol.' '
xln
ln x e dx dx(ln x) (ln x)
=
4ut lnx = t
x = et
dx dt
x=
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Assertion Reason Type Questions
= 't
t
e
tdt
=t
'
e dt
t t
..
=te
ct
+ =ln xe x
c cln x ln x
+ = + . ,o FaG is correct
159. Statements-1: -3 -
dx c
' xx x= + +
+
Statements-2: &or inte*ration b! parts e ha"e to follo ILA rule.
Ans. (;)
,ol. 3 -1
-
dx dx#
x x x
x
=
+ +
o +
-
t
x=
1
-dx dt
x =
= dt
- t = ' t c
- + =
-
c
' x + +
160. Statements-1: A function &(x) is an antideri"ati"e of a function f(x) if & (x) = f(x)Statements-2: he functions x'+ # x'# x'+ ' are all antideri"ati"es of the function 'x.
Ans. (;)
161. Statements-1: b
a x
xdx = b a # a Q b
Statements-2: If f(x) is a function continuous e"er! here in the inter"al (a# b) except x = c
then
b c b
a a c
f (x)dx f (x)dx f (x)dx= + Ans. (A)
162. Statements-1:
3
3
- 3 x dx ' 3 +
Statements-2: / and M be the least and the /axi/u/ "alue of a continuous function! = f(x) in 8a# b9 then
b
a
/(b a) f (x)dx M(b a) Ans. (A)
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Assertion Reason Type Questions
163. Statements-1:'
x
e dx e< is true
,tate/ent>' is false
ex(f(x) + f(x))dx = exf(x) + c
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168. Statements-1:x ' x x x e (x ) cos (x.e )dx x.e sin '(x.e ) C
' -+ = + +
Statements-2: ( ) { }f (x) (x)dx# (x) t = equals f(t)dt .Ans. (A),ol. ,ubstitutin* x.ex= t.
x ' xe (x ) cos (xe )dx+ reduces to 'cos tdt sin't
t C' '
1 = + +
169. Statements-1: lo* xdx x lo* x x c= +Statements-2:
duu"dx u "dx "dx dx
dx
= +
Ans. (C)
,ol.III
lo*xdx
lo* x 7 dx x dxx
=
= x lo* x x + c.
170. Statements-1:
' xx
' '
x -x ' ee dx C
x -x - (x ')
+ += + + + +
Statements-2: ( )
x xe f (x) f (x) dx e f (x) C+ = +Ans. (A)
171. Statements-1:
' '
sin x x x'
3 7 x 7 3 7 x 7
=
+ +
Statements-2:
a a a
a
f (x) dx f (x)dx f ( x)dx
= = + Ans. (A)
172. Statements-1: he "alue of
3
( x)( x )dx+ + can not exceed 10
Statements-2: If /
f(x)
M
x
8a# b9 then
b
a
/(b a) f (x)dx (b a)M
Ans. (A)
173. Statements-1:
D ' 1D '
1D ' 1D '
(sin x)dx
(sin x) (cos x) -
=
+
Statements-2: Area bounded b! ! = 3x and ! = x'isR
'= sq. units
Ans. (;)
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Assertion Reason Type Questions
174. Statements-1:R x
e
x
x lo*
x
++ dx = lo*7 x + x
7 + c
Statements-2:f (x)
dx lo*7 f (x) 7 c
f(x)
= +
Ans. (A)
,ol. I =R x
e
x
x lo* dx
x
+ ++
Let t = x+ x
dt = (xlo*e + xR) dx
b! substition /ethod
dt
t = lo* 7t7 + C= lo* 7x+ x7 + c
175. Statements-1:
x
' x
e ( x)dx
cos (xe )
+ = tan (xex) + c
Statements-2:'sec xdx tan x c= +
Ans. (A)
,ol. I =x
' x
e ( x)dx
cos (xe )
+
4ut t = xex
dt = ( + x)exdx
I ='
'
dtsec dt
cos t=
= tant + c
= tan (x ex) + c
176. Statement-1 : f(x) =
x
'
lntdt(x )#
t t>
+ + then f(x) = >
fx
Statements-2: f(x) =
x
lntdt
t + # then f(x) +
fx '
=
(ln x)'
Ans. (%)
,ol. ption (d) is correct.
f(Dx) =
D x
'
ln t dt
t t+ +
4ut t = DS# dt ='
dS
S #
x
'
'
ln (D S) dSf
x S
S S
= + +
=x x
' '
ln S dS ln t dtf(x)
S S t t= =
+ + + +
Assertion A is false
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Assertion Reason Type Questions
he $eason $ is true hich can be pro"ed in the sa/e a! in hich Assertion a has beendispro"ed.
177. Statement-1 :
' '
sin x x 'x
dx dx3 7 x 7 3 7 x 7
= .
Statements-2: ,incesin x
3 7 x 7is an odd function. ,o# that
sin x
3 7 x 7=
.Ans. (A),ol. ,tate/ent>' is a solution for ,tate/ent>
178. Statements-1 :
n t
7 s inx7dx+
= ('n + ) C,t ( t )
Statements-2:
b c b
a a cf (x)dx f (x)dx f (x)dx= +
and
na a
f (x)dx n f (x)dx= if f(a + x) = f(x)Ans. (A)
,ol.
n t
7 s inx7dx+
=
t n t
t
7 sin x 7dx 7 sin x 7dx+
+ = ('n + ) cost
179. Statements-1: he "alue of the inte*ral'
x
e dx belon*s to 8# 9Statements-2: If / H M are the loer bound and the upper bounds of f(x) o"er 8a# b9 and f is
inte*rable# then / (b a) b
a
f(x)dx M(b a).Ans. (%)
,ol. &or x e ha"e e 'xe e
e( ) '
x
e dx e( )
'
x
e dx e
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Assertion Reason Type Questions
180. Statements-1:
8cot x9dx
= cot# here 89 denotes *reatest inte*er function.
Statements-2:
b
a
f(x)dx
is defined onl! if f(x) is continuous in (a# b) 89 function isdiscontinuous at all inte*ers
Ans. (A)
,ol.
cot
cot
8cot x9dx 8cot x9dx 8cot x9dx
= +
=
cot
.dx + = cot.FaG is correct.
181. Statements-1: ( )-
' '
-
x x x x dx
+ + + =
Statements-2:
a
a
f (x)dx
= if f(x) is an odd function.Ans. (A)
,ol. f(x) = ' ' x x x x+ + +
f(>x) = ' ' x x x x + + + = ( )' ' x x x x + + + = >f(x)
,o# f(x) is odd. Also
a a a
a
f (x)dx f (x)dx f ( x)dx
= +
,o# FaG is correct.
182. Statements-1: All continuous functions are inte*rable
Statements-2: If a function ! = f(x) is continuous on an inter"al 8a#b9 then its definite inte*ral
o"er 8a# b9 exists.
Ans. (;)
183. Statements-1: If f(x) is continuous on 8a# b9# a b and ifb
a
f (x) dx = # then f(x) = at leastonce in 8a# b9
Statements-2: If f is continuous on 8a# b9# then at so/e point c in 8a# b9
f(c) =
b
a
f(x)dx
b a Ans. (A)
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Assertion Reason Type Questions
184. Statements-1:
-
-
7 x ' 7dx 1
+ =
Statements-2:
b c b
a c
f (x)dx f (x)dx f (x)dx= +
here C
(A# ;)
Ans. (A)
,ol.
- ' -
- - '
7 x ' 7 dx (x ')dx (x ')dx
+ = + + + = '.
185. Statements-1:
'
'
xlo* dx
x
+ =
Statements-2: If f is an odd function
a
a
f (x)dx
=Ans. (A)
,ol. f(x) = lo* x
x+
f(>x)= lo*(f(x)) = lo* x x
lo* x x
+ + =
= f(x) is odd function.
186. Statement-1 Ifax
e dx
a
= then / ax /
/@x e dx
a
+=
Statement-2 :n
Ex
n
d(e )
dx= EneEx and
n n
n n
d ( ) n@
dx x x + =
Ans. (A)
,ol.
ax
e dx a
=
%ifferential both sides .r.to FaG / ti/es
a
/ ax / /
/
/@x e ( ) dx ( )
a
+ =
/
/ ax
/ / /
( ) /@ /@x e dx
( ) a a
+ +
= =
is true
;ut ,tate/ent>' is falseperiod is not *i"en
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Assertion Reason Type Questions
188. Statements-1:
7 cos x 7 dx '
=
Statements-2:
b c b
a a c
f (x)dx f (x)dx f (x)dx= +
here a Q c Q b.Ans. (A)
,ol.
7 cosx7dx
D '
D '
7 cos x 7 dx 7 cos x 7 dx
= + D '
D '
cos x dx cos x dx
= = (
)
(
) = '.
189. Statements-1:
cosx
cosx cos x
edx
e e
= +
Statements-2:
b b
a a
f (x)dx f (a b x)dx= + Ans. (%)
,ol. cosx
cosx cosx
eI dx
e e
= +cosx
cosx cosx
eI dxe e
= +
'I dx
= = I'
= .
190. Statements-1:
x 8x9
e dx (e ) =
Statements-2:
n
x 8x9 x 8x9
e dx n e dx = Ans. (A)
,ol.
x 8x9
e dx x 8x9 is periodic ith period b! reasonn
x 8x9 x 8x9
e dx e dx =
=
x
e dx = (e )
191. Statements-1: tanx
dx
' '
=
+
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Assertion Reason Type Questions
Statements-2:
b b
a a
f (x)dx f (a b x)dx= + Ans. (A)
,ol. I = tanx
dx
'
+ ... ()
b! reason
b b
a a
f (x) dx f (a b x)dx= +
I =
a
tan( x) tan x
dx dx
' '
=+ + ... (')
() H (') 'I = tan x tan x
dx
' '
+ + +
'I =
tan x tan x
tan x tan x
' ' 'dx dx
' ' '
+ + = = + +
I = D'
05.05. STRAIGHT LINESSTRAIGHT LINES
192. Let the equation of the line ax + b! + c =
Statement-1:a# b# c are in A.4.hich force ax + b! + c = to pass throu*h a fixed point (# >')
Statement-2: !n" fa/il! of lines ala!s pass throu*h a fixed point
Ans. (C)
193. Statement-1:he area of the trian*le for/ed b! the points A(# ')# ;(# -)
C('# 3) is sa/e as the area for/ed b! A(# )# ;(# ')# C('# )Statement-2: he area of the trian*le is constant ith respect to translation of coordinate axes.
Ans. (A)
194. Statement-1: he lines (a + b)x + (a 'b) ! = a are concurrent at the point '#3 3
.
Statement-2: : he lines x + ! = and x '! = intersect at the point '
#3 3
.
Ans. (A)
,ol. he ,tate/ent> is true and follos fro/ reason $. ,ince the fa/il! of lines can be ritten as
a(x + ! ) + b(x '!) = .
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Assertion Reason Type Questions
195. Statement-1: ach point on the line ! x + ' = is equidistant fro/ the lines-! + 3x ' = # 3! + -x '- = .
Statement-2: : he locus of a point hich is equidistant fro/ to *i"en lines is the an*ular
bisector of the to lines.
Ans. (A)
,ol. e can sho that ! x + ' = is one of the bisectors of the lines -! + 3x ' = # 3! + -x '- =
A is true and follos fro/ $.
196. Statement-1: If A('a# -a) and ;('a# a) are to "ertices of a equilateral trian*le A;C and the
"ertex C is *i"en b! a1#3aa' + .
Statement-2: : An equilateral trian*le all the coordinates of three "ertices can be rational
Ans. (C)
,ol. Let A(x# !)# ;(x'# !') H C(x3# !3) are all rational coordinates
ar(A;C) =
' '
3 3
x ! 3
x ! ' -
x !
= 8(x x')'+ (! !')'9
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199. Statement-1: If the "ertices of a trian*le are ha"in* rational co>ordinate then its centroid#
circu/center H orthocenter are rational
Statement-2: : In an! trian*le# orthocenter# centroid and circu/ center are collinear and
centroid di"ides the line Ooinin* orthocenter and circu/center in the ratio ' : .Ans. (;)
,ol. Centroid = ' 3 ' 3x x x ! ! !#
3 3
+ + + +
is a rational point orthocenter is intersection point of
to altitudes hich ill bear rational coefficients hen expressed as a strai*ht line. ,o#
orthocenter is also rational
Clearl! circu/center is also rational.FbG is correct.
200. Statement-1: If line ! =
x -
3
+ # /aEes an an*le ith positi"e direction of x>axis# then
tan= >D3# cos=3
# sin
=
Statement-2: : he para/etric equation of line passin* throu*h (x# !) is *i"en b!
x x ! !
rcos sin
= =
here r is para/eter H 8# )
Ans. (%)
201. Statement-1: In A;C# A(# ') is "ertex H line x ! 1 = is equation of bisector of A;C#then (N # -) is a point l!in* on base ;C.
Statement-2: : ;isector beteen to lines is locus of points equi>distant fro/ both the lines.
Ans. (A)
202. Statement-1: Area of the trian*le for/ed b! -x + ! + = ith the co>ordinate axes is
' 7 - 7 0=
sq. units.
Statement-2: : Area of the trian*le /ade b! the line ax + b! + c = ith the co>ordinate axes
is
'c
' 7 ab 7.
Ans. (A)
,ol. 4ut in for/ula'c
' 7 ab 7 ' 7 - 7 0= =
sq. units.
203. Statement-1: If (ax + b! + c) + (a'x + b'! + c') + (a3x + b3! + c3) = then lines ax + b!+c= # a'x + b'! + c'= and a3x + b3! + c3= cannot be parallel
Statement-2: : If su/ of three strai*ht lines equations is identicall! Sero then the! are either
concurrent or parallel.
Ans. (%)
,ol. he state/ent> is false since (x ') + ('x 3) + (1 3x) = but the lines x ' = # 'x 3 =
and 1 3x = are parallel. he Statement-2: is a standard true result hose /ore *eneral
fro/ is. If L= L'= # L3= be three lines. If e could find # # " (not all Sero) such that
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Assertion Reason Type Questions
L+ L'+ UL3= then the three lines L= # L'= L3= are either concurrent or areparallel.
204. Statement-1: he three non>parallel lines ax + b! + c= # a'x + b'! + c'= # a3x + b3! + c3=
are concurrent if
' ' '
3 3 3
a b c
a b c
a b c
=
Statement-2: : he area of the trian*le for/ed b! three concurrent lines /ust be Sero.
Ans. (A),ol. ;oth ,tate/ent> and Statement-2: are true and Statement-2: is the correct explanation of
,tate/ent>.
205. Statement-1: he point (# ') lies inside the for/ed b! the lines 'x + 3! = #
x + '! 3 = # and 1x ! = for e"er!
3
# # ' '
Statement-2: : o points (x# !) and (x'# !') lie on the sa/e side of strai*ht line ax + b! + c= if ax+ b!+ c H ax'+ b!'+ c are of opposite si*n.
Ans. (C)
,ol. A# 4 lie on sa/e side of ;C 3
#'
on sa/e side of CA
# #3 '
on sa/e side of A; if ( )
# #3
taEin* intersection e
*et result.
206. Statement-1: he equation of the strai*ht line hich passes throu*h the point ('# 3) and thepoint of the intersection of the lines x + ! + - = and 3x ! 0 = is 'x ! N = Statement-2: : 4roduct of slopes of to perpendicular strai*ht lines is .
Ans. (;)
,ol. An! line throu*h the intersection of x + ! + - = H 3x ! 0 = is (x + ! + -) + (3x ! 0) = since it passes throu*h ('# 3) so = 3 hence required equation is 'x ! N = .
207. Statement-1: he incentre of a trian*le for/ed b! the lines
a. x cos !sinR R + = 0 0x cos !sin
R R + = K 3 3x cos !sin
R R + =
is (# ).
Statement-2: : he point (# ) is equidistant fro/ the lines
0 0x cos !sin # x cos !cos
R R R R
+ = + = and
3 3x cos !sin
R R
+ =
Ans. (;)
208. Statement-1: he co/bined equation of lines LH L'is 'x'+ x! + !'= and that of L3H L-
is -x'+ 0x! + !'= . If the an*le beteen LH L-is then an*le beteen L'H L3is also .
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Assertion Reason Type Questions
Statement-2: : If the pair of lines LL'= H L3L-= are equall! inclined lines then an*lebeteen LH L'= an*le beteen L'and L3.
Ans. (A)
06.06. AREA UNDER THE CURVESAREA UNDER THE CURVES
209. Let 7A7 be the area bounded beteen the cur"es ! = 7x7 and ! = 7x7 K 7A'7 be the area
bounded beteen the cur"es ! = 7x7 and ! = 7x7 .
Statement-1: 7A7 = 7A'7
Statement-2: Area of to si/ilar parallelo*ra/s are equal.
Ans. (A)
,ol. Clearl! 7A7 = 7A'7
210. Statement-1: Area bounded beteen the cur"es ! = 7x 37 and ! = cos(cosx) is 'D'Statement-2: 7x 37 = 3 x for 1D' x 3 cos(cosx) = x '# 'x 3
Ans. (A)
,ol. =' ( ) ( )
3 3
1 D ' 1 D 'x ' 3 x dx ' ('x 1 )dx
= =
'D'.
211. Statement-1: Area of the ellipse' 'x !
-
+ = in the first quadrant is equal to
Statement-2: Area of the ellipse' '
'
' '
x !a
a b+ = is ab.
Ans. (%)
,ol. Area of ellipse' '
x !
- + = in the first quadrant =
'
- '
= .
212. Statement-1: Area enclosed b! the cur"e 7 x 7 + 7 ! 7 = ' is 0 units
Statement-2: 7 x 7 7 ! 7 '+ = represents an square of side len*th 0 unit.Ans. (A)
,ol. Clearl! 7 x 7 + 7 ! 7 = ' represents a square of 0 units and area of square is equal to square of
the side len*th.
213. Statement-1: he area bounded b! ! = x(x )'# the !axis and the line ! = ' is
'
)
(x (x ')' ')dx is equal to3
).
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Assertion Reason Type Questions
Statement-2: he cur"e ! = x(x )'is intersected b! ! = ' at x = ' onl! and for Q x Q '# thecur"e ! = x(x )'lies belo the line ! = '.
Ans. (A)
,ol. ,ol"in* ! = x(x )'and ! = '# e *et x = '.
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Statement-2: he len*th of the se/i>/aOor axis of ellipse 'x '+ 3!'= is /ore than the radiusof the circle x'+ !' 'x + -! + - = .
Ans. (;)
,ol. he ellipse' 'x !
3 '+ = H the circles is (x )'+ (! + ')'= .
Area of ellipse = 3 ' = and area of circle = . ()'= he ,tate/ent>' is true in this particular exa/ple. In *eneral# this /a! not be true.
218. Statement-1: Area included beteen the parabolas ! = x'D-a and the cur"e
! =' '
0ab
x -a+is
'a( -)
3 sq. units.
Statement-2: ;oth the cur"es are s!//etrical about !>axis and required area is
'
x
'
x
(! ! )dxAns. (A)
,ol. $equired area =
'a 'a3 '
' '
0a x' dx dx
x -a -a
+
='a
3(> -)
219. Statement-1: he area of the re*ion bounded b! !'= -x # ! = 'x is D3 sq. units.
Statement-2: he area of the re*ion bounded b! !'= -ax# ! = /x is'
3
0a
3/sq. units.
Ans. (A)
,ol. $eq. area = ( )'-a D/
-ax /x dx
='
3
0a
3/sq. units
220. Statement-1: Area under the cur"e ! = sinx# abo"e FxG axis beteen to ordinates x = H x =
'is - units.
Statement-2:
'
sin x dx -
=Ans. (C)
,ol. [ ]'
'
sin x dx cos x
= = 8>cos'> (>cos())9= 8 ()9 = ,o# c is correct.
221. Statement-1: Area under the cur"e ! = 87sinx7 + 7cosx79# here 89 denotes the *reatest inte*er
function. abo"e FxG axis and beteen the ordinates = H x = is units.Statement-2: f(x) = 7sinx7 + 7cosx7 is periodic ith funda/ental period D'.
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Assertion Reason Type Questions
Ans. (;)
222. Statement-1: Area beteen ! = ' x'H ! = x is equal to'
'
(' x x )dx
+
Statement-2: hen a re*ion is deter/ined b! cur"es that intersect# the intersection points *i"ethe units of inte*ration.
Ans. (;)
,ol. 7sinx7 + 7cosx7 ' ,o 87sinx7 + 7cosx79 =
,o
.dx
= 223. Statement-1: Area of the re*ion bounded b! the lines '! = >x + 0# x>axis and the lines x = 3
and x = 1 is - sq. units.
Statement-2: Area of the re*ion bounded b! the lines x = a# x = b# x>axis and the cur"e ! =
f(x) is
b
a
f(x)dx .Ans. (A)
,ol. Area =
11 '
3 3
0 x xdx 0x
' ' '
=
= - sq. units.
224. Statement-1: he area of the re*ion included beteen the parabola'3x
!-
= and the line
3x '! + ' = is 'N sq. units.
Statement-2: he area bounded b! the cur"e ! = f(x) the x>axis and x = a# x = b is
b
a
f(x)dx#here f is a continuous function defined on 8a# b9.
Ans. (A),ol. $equired area
-
'
'
3x ' 3x dx
' -
+ = 'N sq. units.
225. Statement-1: he area of the re*ion
'(x# !) : ! x # '3
3 ! x # x '
+=
+ sq. units.
Statement-2: he area bounded b! the cur"es ! = f(x)# x>axis ordinates x = a# x = b is'
a
f(x)dxAns. (%),ol. $equired area is
'
'
'3(x )dx (x )dx
+ + + = sq. units.
226. Statement-1: Area bounded b! !'= -x and its latus rectu/ = 0D3
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Assertion Reason Type Questions
Statement-2: Area of the re*ion bounded b! !'= -ax and it is latus rectu/ 0a'D3
!ns. (A)
,ol. area = ar (A,)
=
' x dx
= '
3D '
' - -.x
3 3 3
= =
hose area =- 0
'3 3
= that is latus rectu/ b! reason ha"e latus rectu/ ='0a
3
07.07. DIFFERENTIAL EQUATIONDIFFERENTIAL EQUATION
227. Statement-1: he order of the differential equation hose *eneral solution is ! = c cos'x +
cos'sin'x + c3cos
'x + c-e'x+ c1
'x ce
+ is 3
Statement-2: otal nu/ber of arbitrar! para/eters in the *i"en *eneral solution in the
state/ent () is .Ans. (A)
,ol. ! = ccos'x + c'sin'x + c3cos
'x + c-e'x+ c1
'x ce
+
= ccos'x + c'c'x 'x
3 - 1
cos'x cos 'x c c e c e .e
' '
+ + +
=
'x3 3' '
- 1
c cc c
c cos 'x (c c )e' ' ' '
+ + + + = cos'x + 'e'x
+ 3otal nu/ber of independent para/eters in the *i"en *eneral solution is 3.
228. Statement-1: %e*ree of differential equation of parabolas ha"in* their axis alon* xaxis and"ertex at ('# ) is '.
Statement-2: %e*ree of differential equation of parabola ha"in* their axis alon* xaxis and
"ertex at (# ) is .Ans. (%)
,ol. quation of parabola ill be !'= ap (x )
'!d!
p
dx
= %.. is ! = 'd!
(x )
dx
de*ree of this %.. is .
229. Statement1 : ,olution of the differential equationd! !
xdx x
+ = is x! =3x
c3
+ .
Statement2 : ,olution of the differential equationd!
4W Ldx
+ = is
( )pdx pdxWe .e dx c = + here 4 and are function of x alone.Ans. (A)
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Assertion Reason Type Questions
,ol.dx
4dx xe e = = x
,ol. is x! = 'x dx c+ x! =3
xc
3+ .
230. Let the *eneral solution of a differential equation be ! = aebx + c.
Statement1 : rder of the differential equation is 3.
Statement2 : rder of the differential equation is equal to the nu/ber of actual constant of
the solution
Ans. (%),ol. ! = aebx + c= aec. ebx= Aebx
order is to.231. Let & be the fa/il! of ellipses on the Cartesian plane# hose directrices are x = '.
Statement1 : he order of the differential equation of the fa/il! & is '.
Statement2 : & is a to para/eter fa/il!.
Ans. (A)
,ol. ,tate/ent II is true as an! /e/ber of the fa/il! ill ha"e equation( )
( )
''
' ' '
!x
a a e
+ =
#
here Q e Q # a # b $ and ae = '.
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Assertion Reason Type Questions
Statement2 : A solution of the differential equation
'd! d!
x ! dx dx
+ =
is ! = '.
Ans. (C)
,ol. he *i"en equation can be rearran*ed as#d! ! !e
lo*dx x x
=
put ! = "x d! d"
" xdx dx
= +
d" "lo* " d" dx
dx x " lo* " x= = ! = xecx
for II# put'd!
p p xp ! dx
= + =
!= px p'
p = p + xdp dp
'pdx dx dp
dx = or x 'p = ! = 'x + c
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Assertion Reason Type Questions
let circle is (x h)'+ (! h)' = h'
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Statement-2: he order of the differential equation for/ed b! an! fa/il! of cur"e is equal tothe nu/ber of arbitrar! constants present in it.
Ans. (C)
,ol. ! = cex+ (c'+ c3) e
xT -ce = ex(c+ (c'+ c3) -c
e )
! = cex X () { }-c
' 3here c c (c c )e= + +
xd! cedx
=
x
d!
dxce
= 4ut in ()
! = xx
d!
dx ee
,od!
!dx
= and order is .
FcG is correct.
242. Statement-1: he de*ree of differential equation
' '
'
d! d !3 lo*
dx dx
+ =
is not defined.
Statement-2: he de*ree of differential equation is the poer of hi*hest order deri"ati"e hen
differential equation has been expressed as pol!no/ial of deri"ati"es.Ans. (A)
,ol.
' '
3'
d! d ! lo*
dx dx
+ =
3' '
'
d! d ! lo*
dx dx
+ =
de*ree is not defined as it is not a pol!no/ial of deri"ati"es.
FaG is correct.
243. Statement-1: he order of differential equation of fa/il! of circles passin* then ori*in is '.
Statement-2: he order of differential equation of a fa/il! of cur"e is the nu/ber ofindependent para/eters present in the equation of fa/il! of cur"es
Ans. (A)
244. Statement-1: Inte*ratin* factor ofxd!
3! xdx
+ = is x3
Statement-2: Inte*ratin* factor ofd!
p(x)! (x)dx
+ = is epdx
Ans. (A)
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Assertion Reason Type Questions
,ol. I.&. epdx=
3 dxxe
d! 3!
dx x+ = = x3.
245. Statement-1: he differentiable equation !3d! + (x + !') dx = beco/es ho/o*eneous if eput !'= t.
Statement-2: All differential equation of first order and first de*ree beco/es ho/o*eneous if
e put ! = tx.
Ans. (C)
,ol. $ is false sinced!
dx=
'
'
x !
! x
++
cannot be /ade ho/o*enous b! puttin* ! = tx.
;ut if e put !'= t in the differential equation in assertion A then '!d! dt
dx dx=
And differential equation beco/es t .
'dt + (x + t) dx =
or dxDdt t
'(x t)
+
hich is ho/o*eneous.
246. Statement-1: he *eneral solution ofd!
dx+ 4(x) ! = (x) is p( x) dxe c +
Statement-2: Inte*ratin* factor ofd!
dx+ 4(x) ! = (x) is p( x) dxe
Ans. (%)
,ol. ,tate/ent> is false
,tate/ent>' is true.
247. Statement-1: he *eneral solution of d! ! dx
+ = is !ex= ex+ c
Statement-2: he nu/ber of arbitrar! constants in the *eneral solution of the differential
equation is equal to the order of differential equation.Ans. (;)
,ol.d!
dx+ ! =
d!dx
!=
d!
!=
dx lo* ( !) = x ! = ex# !ex= ex+ c
order of differential equation is the nu/ber of arbitrar! constants.;oth one true# but ,tate/ent>' is not the correct explanation.
248. Statement-1: %e*ree of the differential equation
'd! d!
! x dx dx
= + +
is '.
Statement-2: In the *i"en equation the poer of hi*hest order deri"ati"e hen expressed as a
pol!no/ials in deri"ati"es is '.
Ans. (A)
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Assertion Reason Type Questions
,ol.
'd! d!
! x dx dx
= + +
beco/es
'
' 'd! d!
(x ) 'x! (! ) dx dx
+ = # hen expressed as a pol!no/ial in deri"ati"es.
249. Statement-1: he differential equation of the fa/il! of cur"es represented b! ! = A.e xis *i"en
b!d!
!dx
= .
Statement-2:d!
!dx
= is "alid for e"er! /e/ber of the *i"en fa/il!.
Ans. (A)
,ol. ! = A.ex
on differentiation e *et
xd! A.edx =
250. Statement-1: he differential equation ' 'd! 'x!
dx x !=
+can be sol"ed b! puttin* ! = "x
Statement-2: ,ince the *i"en differentiable equation is ho/o*enous
Ans. (A)
,ol. ' 'd! 'x!
dx x != + ... ()
his is ho/o*enous differential equation put ! = "x
fro/ ()d! d"
" xdx dx
= +
" +
'
' '
xd" 'x "
dx x ( " )=
+3 '
' ' '
d" '" '" " " "( " )x "
dx " " "
= = =
+ + +
'
'
( " ) dxd""( " ) x
+=
251. Statement-1: A differential equation'd! ! x
dx x+ = can be sol"ed b! findin*. If = 4dxe
=Dxdx lo*xe e x = = then solution !.x = x3dx + c
Statement-2: ,ince the *i"en differential equation in of the for/ d!Ddx + p! = herep# arefunction of x
Ans. (A)
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Assertion Reason Type Questions
,ol. d!Ddx + !Dx = x'... ()
his is ter/ of linear differential equation d!Ddx + p! = ... (')fro/ () and (') p = >Dx # = x'
I.f.
4dx D xdx x
e e
=
= e!.I.f = xTI.fd + c!x = x3dx + c.Ans. (A)
252. Statement-1: he differential equation of all circles in a plane /ust be of order 3.
Statement-2: here is onl! on circle passin* throu*h three non collinear points.
Ans. (A),ol. he equation of circle contains three independent constants if it passes throu*h three non>
collinear points therefore A is true and follos fro/ state/ent>'
08.08. CIRCLESCIRCLES
253. an*ents are dran fro/ the ori*in to the circle x'+ !'> 'hx > 'h! + h' = (h )Statement 1: An*le beteen the tan*ents is D'Statement 2: he *i"en circle is touchin* the co>ordinate axes.
Ans. (A)
,ol. he centre of circle is (h# h) and radius = h
he circle is touchin* the co>ordinate axes.
254. Consider to circles x'+ !' -x ! 0 = and x'+ !' 'x 3 =
Statement 1: ;oth circles intersect each other at to distinct points
Statement 2: ,u/ of radii of to circles in *reater than distance beteen the centres of tocircles
Ans. (;)
255. Cis a circle of radius ' touchin* xaxis and !axis. C'is another circle of radius *reater than '
and touchin* the axes as ell as the circle c.
Statement1 : $adius of circle c'= ' ( ' ) ( ' ')+ +
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Assertion Reason Type Questions
Statement2 : Centres of both circles ala!s lie on the line ! = x.Ans. (%)
,ol.
255. &ro/ the dia*ra/
,in-1Y =r
' ' ' r + + r = ' ( ' )( ' ')+ +
b"iousl! centres of both the circles /a! also lie on ! = x
(%) is the correct option.
256. &ro/ the point 4( '# )# tan*ents 4A and 4; are dran to the circle x'+ !'= -.
Statement1 :Area of the quadrilateral A4; (obe!in* ori*in) is -.
Statement2 : an*ents 4A and 4; are perpendicular to each other and therefore quadrilateral
A4; is a square.Ans. (A)
,ol. Clearl! ( '# ) lies on x'+ !'= 0# hich is the director circle of x'+ !'= -
an*ents 4A and 4; are perpendicular to each other.A4; is a squarearea of A4; = -.
257. Statement1 : an*ents dran fro/ ends points of the chord x + a! = of the parabola
!'= '-x /eet on the line x + =
Statement2 :4air of tan*ents dran at the end points of the parabola /eets on the directrix of
the parabolaAns. (A)
,ol. &or !'= '-x# focus is (# )
Clearl! x + a! = passes throu*h the point (# )
,ince e Eno pair of tan*ents dran at the end points of the focal chord of the parabola /eetson the directrix of the parabola.
258. Statement1 :u/ber of focal chords of len*th units that can be dran on the parabola ! ''! 0x + N = is Sero
Statement2 : Lotus rectu/ is the shortest focal chord of the parabola
Ans. (A),ol. ?i"en parabola is (! )'= 0 (x ')
Len*th of L$ = 0o focal chords of less than 0 is possible(A) is the correct option.
259. Statement1 : Centre of the circle ha"in* x + ! = 3 and x ! = as its nor/al is (# ').
Statement2 : or/als to the circle ala!s passes throu*h its centre.
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Assertion Reason Type Questions
Ans. (%),ol. 4oint of intersection of x + N = 3 and x ! = is ('# ).
260. Statement1 : he nu/ber of co//on tan*ents to the circle x'+ !'= - and x'+ !' x 0!
'- = # is oneStatement2 : If CC' = 'r r # then nu/ber of co//on tan*ents is three. hereCC'= %istance beteen the centres at both the circle and r# r'are the radius of the circle
respecti"el!Ans. (C)
,ol.
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Assertion Reason Type Questions
= for/ an
equilateral trian*le
Statement-2: he *i"en circles touch each other externall!.
Ans. (A)
269. Statement-1:he circle described on the se*/ent Ooinin* the points (>'# >)# (# >3) as dia/eter
cuts the circle x'+ !'+ 1x + ! + - = ortho*onall!Statement-2: o circles x' + !' + '*x + 'f! + c = x
' + !' + '*'x + 'f'! + c' = ortho*onall! if '**'+ 'ff'= c+ c'
Ans. (%)
270. Statement-1 : he equation of chord of the circle x'+ !' x + ! R = # hich is bisectedat (>'# -) /ust be x + ! ' = .
Statement-2: In notations# the equation of the chord of the circle , = bisected at (x # !) /ust
be = ,.Ans. (%)
,ol. he ,tate/ent>' is ell Enon.
$esult but applied to the data *i"en in assertion A ill !ield 1x R! + - =
is false# ' is rue.
271. Statement-1 : If to circles x'+ !'+ '*x + 'f! = and x'+ !'+ '*x + 'f! = touch eachother# then f* = f*Statement-2 : o circles touch other# if line Ooinin* their centres is perpendicular to all
possible co//on tan*ents.
Ans. (C)
,ol. he ,tate/ent ' is false bcoS line Ooinin* centres /a! not be parallel to co//on tan*ents.he state/ent> can be pro"ed easil! b! usin* distance beteen centres = su/ of radii.
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Assertion Reason Type Questions
272. Statement-1 : u/ber of circles passin* throu*h (# ')# (-# N) and (3# ) is one.
Statement-2 : ne and onl! circle can be /ade to pass throu*h three non>collinear points.Ans. (%)
,ol. ,lope of line Ooinin* its (# ') H (>-# N) = N ' - =
,lope of line Ooinin* points (# ') H (3# )
= '
3
=
points are collinearnu/ber circle can be dran
273. Statement-1 : he chord of contact of tan*ent fro/ three points A# ;# C to the circle x '+ !'=
a'are concurrent# then A# ;# C ill be collinear.
Statement-2 : A# ;# C ala!s lies on the nor/al to the circle x '+ !'= a'
Ans. (C)
,ol. quation of chord of contact fro/ A(x# !) is xx+ !! a'=
xx'+ !!' a'=
xx3+ !!3 a'=
i.e.#
' '
3 3
x !
x !
x !
=
A# ;# C are collinear.
274. Statement-1 :Circles x'+ !'= -- and x'+ !' x 0! = do not ha"e an! co//on tan*ent.
Statement-2 : If one circle lies co/pletel! inside the other circle then both ha"e no co//on
tan*ent.
Ans. (A),ol. Clearl! no co//on tan*ent
FaG is correct.
275. Statement-1 : he equation x'+ !' 'x 'a! 0 = represents for different "alues of FaG a
s!ste/ of circles passin* throu*h to fixed points l!in* on the x>axis.
Statement-2 : , = is a circle H L = is a strai*ht line# then , + L = represents the fa/il!of circles passin* throu*h the points of intersection of circle and strai*ht line. (here isarbitrar! para/eter).
Ans. (A)
,ol. x'+ !' 'x 'a! 0 =
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Assertion Reason Type Questions
x'+ !' 'x 0) 'a(!) = , + L = x'+ !' 'x 0 = sol"in* the to equation
! =
x' -x + 'x 0 = x' -x + 'x 0 =
x(x -) + ' (x -) =
x = -# x = >'
,o# (-# )# (>'# ) are the points of intersection hich lie on x>axis.FaG is correct.
276. Statement-1 : Len*ths of tan*ent dran fro/ an! point on the line x + '! = to the circles
x'+ !' = H x'+ !' -x 0! ' = are equal
Statement-2 : %irector circle is locus of point of intersection of perpendicular tan*ents.
Ans. (;)
277. Statement-1 : ne H onl! one circle can be dran throu*h three *i"en pointsStatement-2 : "er! trian*le has a circu/circle.
Ans. (A)
278. Statement-1 : he circles x'+ !'+ 'px + r = # x'+ !'+ 'q! + r = touch if ' '
p q r+ =
Statement-2 : o circles ith centre C# C'and radii r# r'touch each other if rr'= cc'Ans. (A)
,ol. o circles touch each other CC' = rr'
' ' ' 'p q p r q r+ = + = p'+ q'= ' ' ' 'p r q r ' (p r)(q r) + + ' '
r p q= +
279. Statement-1 : he equation of chord of the circle x'+ !' x + ! R = hich is bisected
at (>'# -) /ust be x + ! ' =
Statement-2 : In notations the equation of the chord of the circle s = bisected at (x# !) /ustbe = ,.
Ans. (%)
,ol. he state/ent>' x is ell Enon result but if applied to the data *i"en in state/ent> ill !ield
1x R! + - =
state/ent> is false# state/ent>' is true.
280. Statement-1 : he equation x'+ !' -x + 0! 1 = represent a circle.
Statement-2 : he *eneral equation of de*ree to ax
'
+ 'hx! + b!
'
'*x + 'f! + c = represents a circle# if a = b H h = . circle ill be real if *'+ f' c .Ans. (A)
,ol. ,tate/ent> is rue and ,tate/ent>' is rue. Also ,tate/ent>' is the correct explanation of,tate/ent>.
281. Statement-1 : he least and *reatest distances of the point 4(# N) fro/ the circle
x'+ !'-x '! ' = are 1 and 1 units respecti"el!.
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Assertion Reason Type Questions
Statement-2 : A point (x# !) lies outside a circle s = x'+ !'+ '*x + 'f! + c = if s
here s= x'+ !
'+ '*x+ 'f!+ c.
Ans. (;)
,ol. Centre of the circle ('# )r '1 1= =distance of (# N) fro/ ('# ) is units hence required distances are 1# 1 respecti"el!.
282. Statement-1 : he point (a# a) lies inside the circle x'+ !'-x + '! 0 = hen e"era (# -)Statement-2 : 4oint (x# !) lies inside the circle x
' + !' + '*x + 'f! + c = # i f ' '
x ! '*x 'f! c + + + + < .
Ans. (A)
,ol. ,ince point lies inside the circle
a'
+ a'
-a 'a 0 Q a'3a - Q Q a Q -
283. Statement-1 : If n 3 then the "alue of n for hich n circles ha"e equal nu/ber of radicalaxes as ell as radical centre is 1.
Statement-2 : If no to of n circles are concentric and no three of the centres are collinear
then nu/ber of possible radical centre = nC3.
Ans. (A)
284. Statement-1 : o circles x' + !'+ 'ax + c = and x' + !'+ 'b! + c = touches if
' '
a b c+ =Statement-2 : o circles centres c# c'and radii r# r'touches each other if rZ r'= cc'.
Ans. (A)
285. Statement-1 : u/ber of point (a # 3a)+ a I# l!in* inside the re*ion bounded b! thecircles x'+ !''x 3 = and x'+ !''x 1 = is .Statement-2 : ,u/ of squares of the len*ths of chords intercepted b! the lines x + ! = n# n on the circle x'+ !'= - is 0.
Ans. (;)
09.09. PARABOLA PARABOLA
286. Statement-1 :,lope of tan*ents dran fro/ (-# ) to parabola !'= Rx are R
#- -
.
Statement-2 : "er! parabola is s!//etric about its directrix.
Ans. (C)
,ol. ! = /x +a
/
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Assertion Reason Type Questions
= -/ .R D -
/
/' -/ + R =
/= ' R
#/- -=
"er! /= ' R
#/- -
=
"er! parabola is s!//etric about its axis.
287. Statement-1 : hou*h (# + ) there canGt be /ore than one nor/al to the parabola ! '= -x#if Q '.Statement-2 : he point (# + ) lies outside the parabola for all .
Ans. (;)
,ol. ption (;) is correct
An! nor/al to !'= -x isW + tx = 't + t3
If this passes throu*h (# + )# e *et+ + = 't + t3
t3+ t(' > ) > > = = f(t) (sa!)If Q '# then f(t) = 3t'+ (' > ) f(t) = ill ha"e onl! one real root. ,o A is true.,tate/ent ' is also true bcoS (+ )' -is true . he state/ent is true but does notfollo true state/ent>'.
288. Statement-1 : If x + ! = E is a nor/al to the parabola !'
= 'x# then E is R.Statement-2 : quation of nor/al to the parabola !'= -ax is ! /x + 'a/ + a/3=
Ans. (A),ol. &or the parabola !'= 'x# equation of a nor/al ith slope > is ! = >x >'. 3(>) >3 (>) 3
x + ! = R# E = R
289. Statement-1 : If b# E are the se*/ents of a focal chord of the parabola ! '= -ax# then E is
equal to abDb>a.
Statement-2 : Latus rectu/ of the parabola !'= -ax is
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Assertion Reason Type Questions
a =b a
ab
290. Statement-1 : o parabolas !'= -ax and x'= -a! ha"e co//on tan*ent x + ! + a =
Statement-2 : x + ! + a = is co//on tan*ent to the parabolas ! '= -ax and x'= -a! andpoint of contacts lie on their respecti"e end points of latus rectu/.
Ans. (;)
,ol. !'= -axequation of tan*ent of slope F/G
! = /x +a
/
If it touches x'= -a! then x'= -a (/x + aD/)
x' -a/x >'
-a /
= ill ha"e equal roots
% =
a'/'+'a
/
=
/3= > / = >,o ! = >x a x + ! + a = (a# >'a) H (>'a# a) lies on it
F;G is correct.
291. Statement-1 : In parabola !'= -ax# the circle dran taEin* focal radii as dia/eter touches
!>axis.
Statement-2 : he portion of the tan*ent intercepted beteen point of contact and directix
subtends RY an*le at focus.
Ans. (;),ol.
291. (x a) (x at') + ! (! 'at) =
,ol"e ith x = a't'+ ! (! 'at) =
!' 'at! + a't'=
If it touches !>axis then abo"e quadratic /ust ha"e equal roots.
,# % = -a't' -a't'= hich is correct.
F;G is correct.
292. Statement-1 : he Ooinin* points (0# >0) H (D'# ')# hich are l!in* on parabola !'= -ax#pass throu*h focus of parabola.
Statement-2 : an*ents dran at (0# >0) H (D'# >') on the parabola !'= -ax are perpendicular.
Ans. (;)
293. Statement-1 : here are no co//on tan*ents beteen circle x'+ !' -x + 3 = and parabola
!'= 'x.
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Assertion Reason Type Questions
Statement-2 : quation of tan*ents to the parabola x'= -a! is x = /! + aD/ here / denotesslope of tan*ent.
Ans. (C)
294. Statement-1 : hree distinct nor/als of the parabola !'= 'x can pass throu*h a point (h #)
here h .Statement-2 : If h 'a then three distinct nroa/ls can pass throu*h the point (h# ) to the
parabola !'= -ax.
Ans. (A)
295. Statement-1 : he nor/als at the point (-# -) and
# -
of the parabola !' = -x are
perpendicular.
Statement-2 : he tan*ents to the parabola at the and of a focal chord are perpendicular.
Ans. (A)
296. Statement-1 : hrou*h (# + ) there cannot be /ore than one>nor/al to the parabola ! '=-x if Q '.Statement-2 : he point (# + ) lines out side the parabola for all .
Ans. (;),ol. An! nor/al to the parabola !'= -x is ! + tx = 't + t 3
It this passes throu*h (# + )t3+ t(' > ) > > = = f(t) sa!)Q ' than f(t) = 3t'+ (' > ) f(t) = ill ha"e onl! one real root A is truehe state/ent>' is also true since (+ )' -is true for all . he state/ent>' is true butdoes not follo true state/ent>'.
297. Statement-1 : ,lope of tan*ents dran fro/ (-# ) to parabola !'= Rx are D-# RD-
Statement-2 : "er! parabola is s!//etric about its axis.
Ans. (A)
,ol. ! = /x +a
/
= -/ +R D -
//'-/ + R =
/= D-# /'= RD-
"er! parabola is s!//etric about its axis.
298. Statement-1 : If a parabola is defined b! an equation of the for/ ! = ax'+ bx + c here a# b# c
$ and a # then the parabola /ust possess a /ini/u/.Statement-2 : A function defined b! an equation of the for/ ! = ax'+ bx + c here a# b# c$and a # /a! not ha"e an extre/u/.
Ans. (C)
,ol. ,tate/ent> is true but ,tate/ent>' is false.
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Assertion Reason Type Questions
299. Statement-1 : he point (sin # cos ) does not lie outside the parabola '!'+ x ' = hen1 3
# #' '
Statement-2 : he point (x# !) lies outside the parabola !'
= -ax if !'
-ax .Ans. (;),ol. If the point (sin # cos ) lies inside or on the parabola '!'+ x ' = then 'cos'+ sin '
sin (' sin )
sin # or
sin'
.
300. Statement-1 : he line ! = x + 'a touches the parabola !'= -a(x + a).
Statement-2 : he line ! = /x + c touches !'= -a(x + a) if c = a/ + aD/.
Ans. (A),ol. ! = (x + a) + a is of the for/
! = /(x + a) + aD/ here / = .
) f(t) = ill ha"e onl! one real root.state/entI is true. ,tate/entII is also true since (+ )' -is true for all $ [ 56.,tate/ent I is true but does ot follo true state/ent II.
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Assertion Reason Type Questions
305. Let !'= -a (x + a) and !'= -b (x + b) are to parabolas
Statement-1 :an*ents are dran fro/ the locus of the point are /utuall! perpendicular
Statement-2: he locus of the point fro/ hich /utuall! perpendicular tan*ents can be dran
to the *i"en co/b is x + ! + b = Ans. (A)
10.10.
ELLIPSEELLIPSE
306. an*ents are dran fro/ the point (>3# -) to the cur"e Rx'+ !' = --.
ST!T#$#NT-1: he tan*ents are /utuall! perpendicular.
ST!T#$#NT-2: he locus of the points fro/ hich /utuall! perpendicular tan*ents can bedran to the *i"en cur"e is x'+ !'= '1.
Ans. (A)
307. Statement1 :Circle x'+ !' = R# and the circle (x 1) ( 'x 3) + ! ( '! ') = toucheseach other internall!.
Statement2 :Circle described on the focal distance as dia/eter of the ellipse -x '+ R!'= 3
touch the auxiliar! circle x'
+ !'
= R internall!Ans. (A)
,ol. llipse is' 'x !
R -
+ =
focus ( 1#) # e = 13
# An! point an ellipse (3 '
#' '
equation of circle as the dia/eter# Ooinin* the points ( )3D '# ' D ' and focus ( 1#) is( x 1 ) ( ' x 3) !( '.! ') + = (A) is the correct option.
308. Statement1 : If the tan*ents fro/ the point (# 3) to the ellipse' '
x ! R -
+ = are at ri*ht
an*les then is equal to '.Statement2 : he locus of the point of the intersection of to perpendicular tan*ents to the
ellipse' '
' '
x !
a b+ = # is x'+ !'= a'+ b'.
Ans. (A)
,ol. (# 3) should satisf! the equation x'+ !'= 3
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Assertion Reason Type Questions
= '.
309. Statement1 : x ! 1 = is the equation of the tan*ent to the ellipse Rx'+ !'= --.
Statement2 : he equation of the tan*ent to the ellipse
' '
' '
x !
a b+ = is of the for/ ! = /x ' ' '
a / b+ .Ans. (A)
,ol.
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Assertion Reason Type Questions
Statement-2 : If xand x'be an! to positi"e nu/bers then '
'
x xx x
'
+ +
Ans. (;)
313. Statement-1 : In an ellipse the su/ of the distances beteen foci is ala!s less than the su/of focal distances of an! point on it.
Statement-2 : he eccentricit! of an! ellipse is less than .Ans. (A)
,ol. ption (A) is correct
,u/ of the distance beteen foci = 'ae
,u/ of the focal distances ='a
e
ae Qa
ebcoS e Q .
;oth are true and it is correct reason.
314. Statement-1 : An! chord of the conic x'+ !'+ x! = # throu*h (# ) is bisected at (# )
Statement-2 : he centre of a conic is a point throu*h hich e"er! chord is bisected.
Ans. (A)
,ol. Let ! = /x be an! chord throu*h (# ). his ill /eet conic at points hose x>coordinates are*i"en b! x'+ /'x'+ /x'=
( + / + /') x' =
x + x'= 'x x
'
+=
Also != /x# !'= /x'
!+ !'= / (x+ x') = '
! !
'
+= /id>point of chord is (# ) /.
315. Statement-1 : A tan*ent of the ellipse x'+ -!'= - /eets the ellipse x'+ '!'= at 4 H . he
an*le beteen the tan*ents at 4 and of the ellipse x '+ '!'= is D'Statement-2 : If the to tan*ents fro/ to the ellipse x 'Da'+ !'Db'= are at ri*ht an*le# then
locus of 4 is the circle x'+ !'= a'+ b'.Ans. (A)
,ol. quation of 4 (i.e.# chord of contact) to the ellipse x'+ '!'=
hx E!
3+ = ... ()An! tan*ent to the ellipse x'+ -!'= - is
i.e.# xD' cos+ !sin= ... (')() H (') represent the sa/e line h = 3cos# E = 3sinLocus of $ (h# E) is x'+ !'= R
316. Statement-1 : he equation of the tan*ents dran at the ends of the /aOor axis of the ellipseRx'+ 1!' 3! = is ! = # ! = N.
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Assertion Reason Type Questions
Statement-1 : he equation of the tan*ent dran at the ends of /aOor axis of the ellipsex'Da'+ !'Db'= ala!s parallel to !>axis
Ans. (C)
,ol. x'D1 + (!>3)'DR =
nds of the /aOor axis are (# ) and (# )quation of tan*ent at (# ) and (# ) is ! = # and ! =
317. Statement-1 : an*ents dran fro/ the point (3# -) on to the ellipse' 'x !
R
+ = ill be
/utuall! perpendicular
Statement-2 : he points (3# -) lies on the circle x '+ !'= '1 hich is director circle to the
ellipse' 'x !
R
+ = .
Ans. (A)
,ol.' '
x ! R
+ = ill ha"e director circle x'+ !'= + R
x'+ !'= '1and e Eno that the locus of the point of intersection of to /utuall! perpendicular tan*ents
dran to an! standard ellipse is its director circle.FaG is correct.
318. Statement-1 : &or ellipse' 'x !
1 3
+ = # the product of the perpendicular dran fro/ focii on
an! tan*ent is 3.
Statement-2 : &or ellipse'x !
1 3
2
+ = # the foot of the perpendiculars dran fro/ foci on an!
tan*ent lies on the circle x'+ !'= 1 hich is auxiliar! circle of the ellipse.
Ans. (;),ol. ;! for/ula pp'= b
'
= 3.
also foot of perpendicular lies on auxiliar! circle of the ellipse.
F;G is correct.
319. Statement-1 : If line x + ! = 3 is a tan*ent to an ellipse ith foci (-# 3) H (# !) at the point(# ')# then ! = N.
Statement-2 : an*ent and nor/al to the ellipse at an! point bisects the an*le subtended b!
foci at that point.Ans. (A)
320. Statement-1 : an*ents are dran to the ellipse' 'x !
- '
+ = at the points# here it is
intersected b! the line 'x + 3! = . 4oint of intersection of these tan*ents is (0# ).
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Assertion Reason Type Questions
Statement-2 : quation of chord of contact to the ellipse' '
' '
x !
a b+ = fro/ an external point is
*i"en b! ' '
xx !!
a b
+ =
Ans. (%)
321. Statement-1 : In an ellipse the su/ of the distances beteen foci is ala!s less than the su/
of focal distances of an! point on it.
Statement-2 : he eccentricit! of an! ellipse is less than .
Ans. (A)
,ol. ,u/ of distances beteen foci = 'ae su/ of the focal distances = 'aDeae Q aDe since e Q .
322. Statement-1 : he equation x'+ '!'+ x! + 'x + 3! + = can ne"er represent a h!perbolaStatement-2 : he *eneral equation of second de*ree represent a h!perbola it h' ab.
Ans. (A),ol. he state/ent> is false. ,ince this ill represent h!perbola if h' ab
'
'-
> 77 ' '
hus reason $ bein* a standard result is true.
323. Statement-1 : he equation of the director circle to the ellipse -x'+ Rx'= 3 is x'+ !'= 3.
Statement-2 : he locus of the point of intersection of perpendicular tan*ents to an ellipse iscalled the director circle.
Ans. (A)
,ol. ;oth ,tate/ent> and ,tate/ent>' are rue and ,tate/ent>' is the correct explanation of
,tate/ent>.
324. Statement-1 : he equation of tan*ent to the ellipse -x' + R!'= 3 at the point (3# ') isx !
3 '
= .
Statement-2 : an*ent at (x# !) to the ellipse' '
' '
x !
a b+ = is ' '
xx !!
a b =
Ans. (C)
,ol. $equired tan*ent is
3x '! x ! or
R - 3 '
= =
325. Statement-1 : he /axi/u/ area of 4,,'here ,# ,'are foci of the ellipse' '
' '
x !
a b+ =
and 4 is an! "ariable point on it# is abe# here e is eccentricit! of the ellipse.
Statement-2 : he coordinates of pare (a sec # b tan ).Ans. (C)
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Assertion Reason Type Questions
,ol.
area of 4,,'= abe sin clearl! its /axi/u/ "alue is abe.
326. Statement-1 : In an ellipse the su/ of the distances beteen foci is ala!s less than the su/
of focal distance of an! point on it.
Statement-2 : he eccentricit! of ellipse is less than .
Ans. (A)
11.11. HYPERBOLAHYPERBOLA
327. Let W = ''
x R3
x83# ) and W= ''
x R3
be x(># >39 to cur"es.
Statement 1: he nu/ber of tan*ents that can be dran fro/
1#3
to the cur"e
W= ''
x R3
is Sero
Statement 2: he point
1#3 %&es on t'e ()*+e W=
'' x R3 .
Ans. (A)
,ol. an*ents cannot be dran fro/ one branch of h!perbola to the other branch.
328. Statement1 : If (3# -) is a point of a h!perbola ha"in* focus (3# ) and (# ) and len*th ofthe trans"erse axis bein* unit then can taEe the "alue or 3.Statement2 : , 4 ,4 'a = # here , and ,are the to focus 'a = len*th of the trans"erseaxis and 4 be an! point on the h!perbola.
Ans. (%)
,ol. ( ) '3 - + = = or .
329. Statement1 : he eccentricit! of the h!perbola Rx' !' N'x + R! -- = is1
-.
Statement2 : he eccentricit! of the h!perbola' '
' '
x !
a b = is equal to
'
'
b
a+ .
Ans. (A)
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Assertion Reason Type Questions
,ol.
' 'x - ! 3
R
=
e =
R 1
-+ =.
330. Let a# b# $ 56# here a# b are constants and is a para/eter.
Statement1 : All the /e/bers of the fa/il! of h!perbolas' '
' ' '
x !
a b+ =
ha"e the sa/e
pair of as!/ptotes.
Statement2 : Chan*e in # does not chan*e the slopes of the as!/ptotes of a /e/ber of the
fa/il!' '
' ' '
x !
a b+ =
.
Ans. (A)
,ol. ;oth state/ents are true and state/ent II is the correct reasonin* for state/ent I# as for an!
/e/ber# se/i trans"erse and se/i conOu*ate axes area
and
b
respecti"el! and hence
as!/ptoters are ala!s ! =bx
a .
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Assertion Reason Type Questions
,ol. he state/ent> is false bcoS this ill represent h!perbola if h' ab
'
'-
> 77 ' '
he stat/enet>'# bein* a standard result# is true.
333. Statement1 If a point (x# !) lies in the re*ion II of' '
' '
x !#
a b = shon in the fi*ure# then
' '
' '
x !
a b <
Statement2 If (4(x# !) lies outside the a h!perbola' '
' '
x !
a b = # then
' '
' '
x !
a b is false bcoS points in re*ion II lie belo the line ! = bDa x ' '
' '
x !
a b >
he re*ion>' is true (standard result). Indeed for points in re*ion II
Q' '
' '
x !
a b < .
334. Statement1 quation of tan*ents to the h!perbola 'x'3!'= hich is parallel to the line! = 3x + - is ! = 3x 1 and ! = 3x + 1.Statement2 ! = /x + c is a tan*ent to x'Da'!'Db'= if c'= a'/'+ b'.
Ans. (C)
,ol. x'Da'!'Db'= if c'= a'/'b'
c'= 3.3'' = '1c = Z 1
real tan*ents are ! = 3x + 1
335. Statement1 : here can be infinite points fro/ here e can dra to /utuall!
perpendicular tan*ents on to the h!perbola' 'x !
R
=
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Assertion Reason Type Questions
Statement2 :he director circle in case of h!perbola' 'x !
R
= ill not exist because a'Q
b'and director circle is x'+ !'= a' b'.
Ans. (%),ol. he locus of poi