03 maths questions

7/26/2019 03 Maths Questions

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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)

1


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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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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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,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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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. (%)


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 =

17


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' 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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,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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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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,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

21


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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

22


23/147


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 (> # ).


24/147


,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 (# ') = '

' '

=

24


25/147


/= /'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.


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.

26


27/147


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;

+ +

.

35


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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=

36


37/147


= '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)

37


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163. Statements-1:'

x

e dx e< is true

,tate/ent>' is false

ex(f(x) + f(x))dx = exf(x) + c

38


39/147


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. (;)

39


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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

4


41/147


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

41


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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)

42


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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


44/147


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

' '

=

+

44


45/147


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 '!) = .

45


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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

L


47/147


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

47


48/147


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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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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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'.

51


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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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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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,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 = '.


55/147


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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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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,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)

58


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,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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,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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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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62/147


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.


63/147


= 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.

63


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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 =

64


65/147


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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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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= -/ .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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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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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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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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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

71


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= '.

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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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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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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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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,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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,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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,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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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

03 maths questions

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