TARGET : JEE (Advanced) 2014

TEST SYLLABUS
Class : XII & XIII
Course: VIJETA (JPA) &
VIJA!(JRA)
a"e : 2#$04$2014
PP %o& :0'
(AT)E(ATIC*
a"e : +0$04$2014 TE*T : Par" Tes" $+(PT$+)
*,lla-us : Conic Section & Solution of Triangle
%o"e : Dear Students, DPP # 11 in Revision plan for JEE(dvanced! "ill #e of
$atrices & Deter%inant and $iscellaneous
Rev.s.on PP / '
C0%IC *ECTI0% And *012TI0% 03 TRIA%G1E
To"al (ar4s : 152 (a6& T.7e : 150 7.n&
*.n8le c9o.ce 0-:ec".ve (;1 ne8a".ve 7ar4.n8) <&1 "o <&+= (+ 7ar4s + 7.n&) >11#? 11#@
(ul".Ale c9o.ce o-:ec".ve (no ne8a".ve 7ar4.n8) <&40 "o <&42 (5 7ar4s? 4 7.n&) >15? 12@
*u-:ec".ve <ues".ons (;1 ne8a".ve 7ar4.n8) <&4+ (4 7ar4s? 5 7.n&) >4? 5@
(a"c9 "9e 3olloB.n8 (no ne8a".ve 7ar4.n8) <&44? 45 (C 7ar4s? C 7.n&) >1'? 1'@
1& T"o sides of an e&uilateral triangle 'C touc( t(e para#ola )
*
+ ,-, t(en points , ' & C lie on t(e curve
(! )
*
+ (- . 1!
*
. ,- ('! )
*
+ /(- . 1!
*
. -
(C! )
*
+ /(- . 1!
*
. ,- (D! )
*
+ (- . 1!
*
. -
2& 0f t(e e&uation of fa%il) of ellipse is

*
*
*
*
sin
)
cos
-
+ 1, "(ere
,

1
*

, t(en t(e locus of e-tre%ities of t(e

latus rectu% is
(! *)
*
(1 . -
*
! + (1 2 -
*
!
*
('! *-
*
(1 . )
*
! + (1 2 )
*
!
*
(C! *)(1 2 -
*
! + 1 . -
*
(D! *-(1 2 )
*
! + 1 . )
*
+& 0f tangent and nor%al at a point P to t(e ellipse
13
)
*4
-
* *
+ + 1 intersect %a5or a-is at points T and 6 respectivel)
in suc( a "a) t(at ratio of areas of

PT6 and

PSS7 is
38
91
, t(en area of

PSS7 is (S and S7 are focii of ellipse!

(! 3
/
s&: units ('! 1*
/
s&: units
(C! ,
/
s&: units (D! /
/
s&: units
4& T(e line 5oining t(e ort(ocentre and t(e centroid of t(e triangle for%ed #) a focal c(ord of t(e para#ola "it(
t(e tangents at its e-tre%ities is
(! Parallel to t(e a-is of t(e para#ola
('! Perpendicular to t(e a-is of t(e para#ola
(C! 6eit(er parallel nor perpendicular to t(e a-is of t(e para#ola
(D! 6ot(ing can #e certainl) said
Pa8e / 1
5& ll t(e vertices of a trape;iu% lie on t(e para#ola )
*
+ ,-: 0ts diagonals pass t(roug( (1, 8! and (ave a lengt(
,
*4
units eac(: T(en t(e area of t(e trape;iu% is
(! 188 s&: units ('! <4=, s&: unit (C! *4=, s&: units (D! 34 s&: units
'& line is dra"n fro% (2*, 8! to intersect para#oala )
*
+ ,- at P & > in t(e first &uadrant: 0f
,
1
>
1
P
1
< + ,
t(en t(e slope of t(e line is al"a)s
(! ?
/
('! 1
/
1
(C! 1
/
(D!
/
1
#& Consider t(e para#ola )
*
+ ,-: (,, 2,! & '(9, 3! #e t"o fi-ed points, on t(e para#ola: T(en a point C on t(e
para#ola for "(ic( t(e area of

'C is %a-i%u% is
(!

1 ,
,
1
('! (,, ,! (C!
( )
/ * , / (D!
( )
/ * 2 , /
C& 0f t(e #isector of

P', "(ere P and P' are t(e tangents to t(e para#ola )

*
+ ,a-, is e&uall) inclined to t(e
co@ordinate a-es, t(en t(e locus of P is
(! tangent at t(e verte-
('! directri- of t(e para#ola
(C! a circle "it( centre at origin and radius AaB units
(D! t(e latus rectu% of t(e para#ola
=& Cro% a given point on t(e para#ola )
*
+ ,a-, a focal c(ord ' and a tangent is dra"n: 6o" t"o circles are
dra"n: Dne "it( ' as dia%eter and t(e ot(er taEing intercept of t(e tangent #et"een point and P ("(ere
t(e tangent %eets t(e directri-! as dia%eter: T(en t(e co%%on c(ord of t(e t"o circles is
(! line 5oining focus S and P
('! line 5oining focus S and
(C! tangent line to para#ola at
(D! 6one of t(ese
10& 6or%als D,
1
&
*
one dra"n to t(e para#ola )
*
+ F- fro% ((, 8!: 0f triangle D
1

*
is e&uilateral, t(en
A(B can #e e&ual to
(! *, ('! *3 (C! *F (D! /8
11& 0f t(e nor%al to t(e para#ola at P %eets it again at > and if P> and t(e nor%al at > %aEe angles

&

"it(
t(e -@a-is respectivel), t(en tan

( tan

. tan

! (as a value e&ual to

(! 8 ('! 21 (C! 2* (D! 2 1=*
12& 6or%als to t(e para#ola )
*
+ ,a- at (-
1
, )
1
! and '(-
*
, )
*
! %eet on t(e para#ola suc( t(at -
1
. -
*
+ ,: T(en
G)
1
. )
*
G +
(!
*
('!
* *
(C!
* ,
(D!
* 3
1+& 0f t"o different tangents of )
*
+ ,- are nor%als to -
*
+ ,#) t(en
(! G#G ?
* *
1
('! G#G 1
* *
1
(C! G#G ?
*
1
(D! G#G 1
*
1
Pa8e / 2
Pa8e / +
14& 0f t(e line seg%ent 5oining t(e foci of t(e ellipse
*
*
*
*
#
)
a
-
+ + 1 (a ? #! does not su#tend a rig(t angle at an)
point on t(e ellipse, t(en t(e range of eccentricit) of t(e ellipse is
(!

1 ,
*
1
('!

1 ,
*
1
(C!

*
1
, 8
(D!

*
1
, 8
15& T(e point P(

! l)ing on t(e ellipse ,-

*
. /)
*
+ 1* in t(e first &uadrant, suc( t(at t(e area enclosed #) t(e
lines ) + -, ) +

, - +

, and t(e -@a-is is %ini%u% is

(!

1 ,
*
/
('!

/
* *
, 1
(C!

/ ,
*
/
(D!

*
14
,
,
/
1'& Hit( a given point and a given line as focus and directri-, a series of ellipses are descri#ed: T(en locus of t(e
e-tre%ities of t(eir %inor a-is is
(! an ellipse ('! para#ola
(C! ()per#ola (D! straig(t line
1#& T(e au-illiar) circle of a fa%il) of ellipse passes t(roug( origin and %aEes intercepts of F and 3 units on t(e
-@a-is and )@a-is respectivel): 0f t(e eccentricit) of all suc( ellipses is
*
1
, t(en t(e locus of t(eir focus is
(!
9
)
13
-
* *
+ + *4 ('! ,-
*
. ,)
*
2 /*- 2 *,) . <4 + 8
(C!
13
)
9
-
* *
+ + *4 (D! *-
*
. )
*
+ *
1C& Iet S #e t(e focus, C t(e centre, and P #e an) point on t(e directri- of an ellipse of eccentricit) 7e7: T(e line
PC %eets t(e ellipse at : T(en t(e angle #et"een PS and t(e tangent at is
(! tan
21
(e! ('!

=* (C! tan
21
(1 2 e
*
! (D!

=,
1=& 0f t(e ellipse
*
*
*
*
#
)
a
-
+ + 1 is inscri#ed in a rectangle "(ose lengt( to #readt( ratio is * : 1, t(en t(e area of
t(e rectangle is
(!
<
,
(a
*
. #
*
! ('!
/
,
(a
*
. #
*
!
(C!
4
1*
(a
*
. #
*
! (D!
4
F
(a
*
. #
*
!
Pa8e / 4
20& *
1
: Diagonals of a parallelogra% inscri#ed in an ellipse al"a)s intersect at its centre:
*
2
: Centre of an ellipse #isects ever) c(ord passing t(roug( it:
(! STTE$E6T@1 is true, STTE$E6T@* is true and STTE$E6T@* is correct e-planation for
STTE$E6T@1
('! STTE$E6T@1 is true, STTE$E6T@* is true and STTE$E6T@* is not correct e-planation for
STTE$E6T@1
(C! STTE$E6T@1 is true, STTE$E6T@* is false
(D! STTE$E6T@1 is false, STTE$E6T@* is true
(E! 'ot( STTE$E6TS are false
21& T"o tangents to t(e ()per#ola
*
*
*
*
#
)
2
a
-
+ 1 (aving slopes %
1
& %
*
cut t(e coordinate a-es in four con@c)clic
points: T(en %
1
%
*
is e&ual to
(! 1 ('! 21 (C!
#
a
(D! 2
a
#
22& 0f a varia#le line (as intercepts of e
1
and e
*
on t(e co@ordinate a-es, "(ere
*
e
1
&
*
e
*
are t(e eccentricities of
a ()per#ola and its con5ugate, t(en t(e line al"a)s touc(es a fi-ed circle -
*
. )
*
+ r
*
, "(ere ArB is
(! 1 ('! * (C! / (D! ,
2+& 0f tangents P> and PR are dra"n fro% a varia#le point P to ()per#ola
*
*
*
*
#
)
2
a
-
+ 1 (a ? #!, suc( t(at t(e
fourt( verte- S of parallelogra% P>SR lies on t(e circu%circle of

P>R, t(en t(e locus of P is

(! -
*
. )
*
+ #
*
('! -
*
. )
*
+ a
*
(C! -
*
. )
*
+ a
*
2 #
*
(D! -
*
. )
*
+ a
*
. #
*
24& T(e c(ord of contact of a point P "it( respect to a ()per#ola and its au-iliar) circle are at rig(t angles, t(en
P lies on
(! con5ugate ()per#ola ('! directrices
(C! one of t(e as)%ptotes (D! 6one of t(ese
25& T(e polar of an) point on an as)%ptote of a ()per#ola "it( respect to t(e ()per#ola is al"a)s
(! parallel to t(at as)%ptote
('! perpendicular to t(at as)%ptote
(C! neit(er parallel nor perpendicular to t(e as%ptotes
(D! so%e ti%es parallel, so%e ti%es perpendicular to t(at as%ptote
2'& Iet

'C #e suc( t(at

'C +
/
*

and ' + -: Curt(er ('!(C! + 1: 0f - varies, t(en t(e longest possi#le

lengt( of t(e angle #isector D is
(! * ('! 1 (C! 1=* (D! *=/
Pa8e / 5
2#& 0f t(e %edian D of

'C %aEes an angle

,

"it( side 'C, t(en Gcot ' 2 cot CG +

(! 1 ('! * (C! 1=/ (D! *=4
2C& 'C is an e&uilateral triangle of side F c%: 0f R, r, & ( are t(e circu%radius, inradius and altitude of

'C,
t(en
(
r R
+
(as a value e&ual to
(! , ('! * (C! 1 (D! /
2=& 0f in a triangle

/
1
*
1
r
r
2 1
r
r
2 1 + *, t(en t(e triangle is
(! rig(t angled ('! isosceles (C! e&uilateral (D! 6one of t(ese
+0& 0f J is t(e ort(ocentre of an acute angle triangle "(ose circu%@circle is -
*
. )
*
+ 13, t(en circu%dia%eter of

J'C is
(! 1 ('! * (C! , (D! F
+1& *$1 : 0f a, #, c are t(e sides of a triangle t(en t(e %ini%u% value of
c 2 # a
c *
# 2 a c
# *
a 2 c #
a *
+
+
+
+
+
is 9:
*$2 : :$:

K:$:

J:$:
(! STTE$E6T@1 is true, STTE$E6T@* is true and STTE$E6T@* is correct e-planation for
STTE$E6T@1
('! STTE$E6T@1 is true, STTE$E6T@* is true and STTE$E6T@* is not correct e-planation for
STTE$E6T@1
(C! STTE$E6T@1 is true, STTE$E6T@* is false
(D! STTE$E6T@1 is false, STTE$E6T@* is true
(E! 'ot( STTE$E6TS are false
+2& 0n

'C,

'C + 1*8L, ' + /c% and 'C + ,c%: 0f perpendicular constructed to ' at and to 'C at C
%eet at D, t(en CD +
(! / ('!
/
/ F
(C! 4 (D!
/
/ 18
++& 0f D, 'E and CC are t(e %edians of a

'C, t(en (D
*
. 'E
*
. CC
*
! : ('C
*
. C
*
. '
*
! is e&ual to
(! , : / ('! / : * (C! / : , (D! * : /
+4& 0f 'CD is a c)clic &uadrilateral suc( t(at 1*tan 2 4 + 8 and 4cos' . / + 8, t(en t(e e&uation "(ose roots
are cosC and tanD is
(! /9-
*
2 13- 2 ,F + 8 ('! /9-
*
. FF- . ,F + 8
(C! /9-
*
2 FF- . ,F + 8 (D! /9-
*
2 13- . ,F + 8
+5& 0f in a triangle "(ose circu%centre is origin, a

sin, t(en for an) point (a, #! l)ing inside t(e circu%circle of

'C,
(! Ga#G 1 1=F ('! 1=F 1 Ga#G 1 1=*
(C! Ga#G ? 1=* (D! 6ot(ing can #e said
+'&

'C is isosceles "it( '+ C and

C' + 183L: Point $ is an interior point suc( t(at

$' + <L and

$' + */L: T(en

$C +
(!F<L ('! 3<L (C! <,L (D! F/L
+#& 0f S & S

are t(e foci of t(e ellipse

*4
-
*
.
13
)
*
+ 1 and P in an) point on it, t(en t(e range of values of SP:S :S

P is
(! M9, 13N ('! M9, *4N (C! M13, *4N (D! M1, 13N
+C& T(e sides of a triangle are a + -
*
. - . 1, # + *- . 1, c + -
*
2 1: T(en t(e greatest angle of t(e triangle is
(! 98L ('! 1*8L (C! 38L (D! /8L
+=& Iet a

c #e t(e sides of a triangle: 0f a

*
. #
*
1 c
*
, t(en t(e triangle is
(! acute angled ('! rig(t angled
(C! o#tuse angled (D! not(ing can #e said
40& Consider t(e ellipse
! 11 E ( f
)
! 4 E * E ( f
-
*
*
*
+
+
+ +
+ 1, "(ere f(-! is a decreasing function, t(en
(! t(e set of values of E for "(ic( t(e %a5or a-is of t(e ellipse is -@a-is is (2/, *!
('! t(e set of values of E for "(ic( t(e %a5or a-is of t(e ellipse is )@a-is is (2

, *!
(C! t(e set of values of E for "(ic( t(e %a5or a-is of t(e ellipse is )@a-is is (2

, 2/!

(*,

!
(D! t(e set of values of E for "(ic( t(e %a5or a-is of t(e ellipse is -@a-is is (2/,

!
41& T"o concentric ellipses are suc( t(at t(e foci of one lie on t(e ot(er and t(e lengt( of t(eir %a5or@a-es are
e&ual: 0f e
1
& e
*
#e t(eir eccentricities, t(en
(! t(e &uadrilateral for%ed #) 5oining t(eir foci is a parallelogra%
('! t(e angle #et"een t(eir a-es is given #) cos

+ *
*
*
1
*
*
*
1
e e
1
2
e
1
e
1
+
(C! t(eir a-es are perpendicular if e
1
+
*
*
e 2 1
(D! 6one of t(ese
42& 0f t(e tangent at t(e point P(

! to t(e ellipse
13-
*
. 11)
*
+ *43 is also a tangent to t(e circle -
*
. )
*
2 *- + 14 t(en

+
(!
/
*

('!
/
,

(C!
/
4

(D!
/

4+& tangent dra"n to t(e ()per#ola

*
*
a
-
2
*
*
#
)
+ 1 at P

3
for%s a triangle of area /a
*
s&: units: "it( co@ordinate
a-es: 0f t(e eccenticit) of t(e ()per#ola is 7e7, t(en t(e value of e
*
2 9 is:
Pa8e / '
Pa8e / #
44& Tangents dra"n fro% a point P to t(e ellispe
*
*
*
*
#
)
a
-
+ + 1 (a ? #! %aEe angles

and

"it( t(e %a5or a-is of

t(e ellipse: T(en find t(e locus of P if
Colu7n$ I Colu7n$ II
(!

.

+
*
n

(n

6! (p! circle
('! tan

tan

R! (&! ellipse
(C! tan

. tan

R! (r! ()per#ola
(D! cot

. cot

+ c (c

R! (s! pair of straig(t lines

45& 0f e
1
and e
*
are t(e roots of t(e e&uation -
*
2 a- . * + 8, t(en %atc( t(e follo"ing
Colu7n$I Colu7n$II
(! 0f e
1
& e
*
are t(e eccentricities of an ellipse and a ()per#ola, t(en a (p! 3
can taEe values
('! e
1
& e
*
are t(e eccentricities of t"o ()per#olas, t(en a can taEe values (&! 4<=*8
(C! e
1
& e
*
are t(e eccentricities of a ()per#ola and its con5ugate, t(en a (r!
* *
can taEe values
(D! e
1
is eccentricit) of a ()per#ola for "(ic( t(ere e-ist infinite nu%#er of
points fro% "(ic( perpendicular tangents can #e dra"n & e
*
t(e eccentricit) (s! 4
of a ()per#ola for "(ic( no suc( points e-ist, t(en a can taEe values