MATHEMATICS
E
S.No.
CONTENTS
Page
1.
Point & Straight Line
01
2.
Circle
07
3.
Conic Section
13
4.
Vector
19
5.
Three Dimensional Geometry
26
Point & Straight Line
ALLEN
POINT & STRAIGHT LINE
1.
If a, b, g are the real roots of the equation
x3 – 3px2 + 3qx – 1 = 0, then the centroid of the
triangle whose vertices are (a,
is:(1) p, –q
2.
1
1
1
), (b, ) and (g, )
b
g
a
(3) (p, q)
æp qö
è2 2ø
(4) ç , ÷
(2) 1
(3) 2
(4) 3
If P is a moving point in the xy-plane in such a way
that perimeter of triangle PQR is 16
{where Q º (3,
8.
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
E
æ
-3p ö
4 ÷ø
æ
(2) ç 2,
è
3p ö
4 ÷ø
(4) None of these
The equation of the perpendicular bisectors of the
y = x and y = –x, respectively. If the point
A is (1, 2), then the area of DABC is :-
(1) 6
(2) 12
(1) 6 sq. units
(2) 3 sq. units
(3) 18
(4) 9
(3) 9 sq. units
(4) 2 sq. units
In a triangle ABC, co-ordinates of A are (1, 2) and
the equations to the medians through B and C are
x + y = 5 and x = 4 respectively. Then the
co-ordinates of B and C will be:-
9.
Line AB passes through point (2, 3) and intersects
the positive x and y axes at A(a, 0) and B(0, b)
respectively. If the area of DAOB is 11, the
(1) (–2, 7), (4, 3)
numerical value of 4b2 + 9a2, is :-
(2) (7, –2), (4, 3)
(1) 220
(2) 240
(3) (2, 7), (–4, 3)
(3) 248
(4) 284
Consider the family of lines x(a + b) + y = 1, where
a, b and c are the roots of the equation
x3 – 3x2 + x + l= 0 such that c Î [1,2]. If the given
family of lines makes triangle of area 'A' with
coordinate axis, then maximum value of 'A'
(in sq. units) will be (1)
6.
pö
4
sides AB an d AC of a triangle ABC are
5 ), R º (7, 3 5 )} then maximum
(4) (2, –7), (3, –4)
5.
æ
(3) ç 2,
è
area of triangle PQR is :-
4.
An insect is resting on the graph paper at a point
A(3, 2). Now it starts moving towards west direction
and covers a distance of 4 units and then it turns
towards south and covered a distance of 3 units and
reaches at point B then the polar co-ordinates of
point B will be :(1) ç 6 2, ÷
è
ø
Number of straight lines from (1, 1) which make
area of 1 sq. units with the coordinate axes is equal
to (1) 0
3.
(2) (–p, q)
7.
EXERCISE-I
1
4
(2) 1
(3)
1
8
(4)
(2) 64x + 8y + 35 = 0
The locus of the mid-point of the portion intercepted
between the axes by the line x cos a + y sin a = p,
(where p is constant is) :
(1) x2 + y2 = 4p2
1
2
The equations of bisectors of two lines L1 & L2 are
2x – 16y – 5 = 0 and 64x + 8y + 35 = 0. If the
line L1 passes through (– 11, 4), the equation of
acute angle bisector of L1 & L2 is :
(1) 2x – 16y – 5 = 0
10.
(3) x2 – y2 =
11.
4
p2
(2)
1
1
4
+ 2 = 2
2
x
y
p
(4)
1
1
2
+ 2 = 2
2
x
y
p
The point (a2, a+1) is a point in the angle between
th e lines 3x – y + 1 = 0 and x + 2y – 5 = 0
containing the origin, if(1) a ³ 1 or a £ –3
(2) a Î (0, 1)
æ1
è3
ö
ø
(3) data insufficient
(3) a Î (–3, 0) È ç , 1 ÷
(4) None of these
(4) None of these
1
JEE ( Main) - Mathematics
13.
If area of the triangle formed by the centroid and
two vertices of a triangle is 6 sq. unit then the area
of the triangle will be :(1) 6 Sq. unit
(2) 9 Sq. unit
(3) 18 Sq. unit
(4) 9/2 Sq. unit
If (–2, 6) is the image of the point (4, 2) with respect
to the line L = 0, then L =
(1) 3x – 2y + 5
(3) 2x + 3y – 5
14.
15.
(2) 780
(3) 901
(4) 861
Number of lines that can be drawn through the
point(4,–5) so that its distance from (–2,3) will be
equal to 12 is equal to-
(1) –2.5
(3) –1.5
18.
then the co-ordinates of mid-point of side opposite
to A is(1) (1,–11/3)
(3) (1,–3)
22.
23.
The line x= c cuts the triangle with corners (0,0);
(1,1) and (9,1) into two region. For the area of the
two regions to be the same c must be equal to(2) 3
(4) 3 or 15
If m and b are real numbers and mb > 0, then the
line whose equation is y = mx + b cannot contain
the point(1) (0,2009)
(3) (0,–2009)
24.
A point P(x,y) moves so that the sum of the distance
from P to the coordinate axes is equal to the distance
from P to the point A(1,1). The equation of the
locus of P in the first quadrant is 25.
(2) (x + 1) (y + 1) = 2
(2) (1,5)
(4) (1,6)
(1) 5/2
(3) 7/2
(2) –2
(4) –1
(1) (x + 1) (y + 1) = 1
If in triangle ABC, A º (1,10),
æ 1 2ö
æ 11 4 ö
÷ and orthocenter º ç , ÷
è 3 3ø
è 3 3ø
(2) (–1/4,0)
(4) (–4,0)
A line passes through (2,2) and cuts a triangle of
area 9 square units from the first quadrant. The
sum of all possible values for the slope of such a
line, is-
(2) 3
(4) 1
circumcenter º ç - ,
(2) 1
(4) 3
If the x intercept of the line y = mx + 2 is greater
than 1/2 then the gradient of the line lies in the
interval(1) (–1,0)
(3) (–¥,–4)
17.
21.
(2) 3x – 2y + 10
(4) 6x – 4y – 7
(1) 820
Two mutually perpendicular straight lines through
the origin from an isosceles triangle with the line
2x + y = 5. Then the area of the triangle is :
(1) 5
(3) 5/2
The number of points, having both co-ordinates as
integers, that lie in the interior of the triangle with
vertices (0, 0), (0, 41) and (41, 0) is :
(1) 0
(3) 2
16.
ALLEN
20.
(2) (2009,0)
(4) (20,–100)
If a and b are real numbers between 0 and 1 such
that the points (a, 1) (1, b) and (0, 0) form an
equilateral triangle, then a, b are (1) 2 - 3,2 - 3
(2) 3 - 1, 3 - 1
(3) 2 - 1, 2 - 1
(4) None of these
For a variable line
the locus of the foot of perpendicular drawn from
origin to it is -
(3) (x – 1)(y – 1) = 1
(4) (x – 1)(y – 1) = 2
19.
2
x y
1 1
1
+ = 1 where 2 + 2 = 2
a b
a
b
c
If A and B are the points (–3,4) and (2,1), then the
co-ordinates of the point C on AB produced such
that AC = 2BC are :
(1) (2,4)
(2) (3,7)
(3) (7,–2)
æ 1 5ö
(4) ç - , ÷
è 2 2ø
(1) x2 + y2 =
c2
2
(2) x2 + y2 = c2
(3) x2 + y2 = 2c
(4) None of these
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
12.
E
Point & Straight Line
ALLEN
26.
Two sides of on isosceles triangle are given by the
equations 7x – y + 3 = 0 and x + y – 3 = 0. If its
third side passes through the point (1, –10), then
its equations are (1) x – 3y – 7 = 0
29.
its intercept between the axis is bisect at p. Its
equation is (1) 3x – 4y + 7 = 0
or 3x + y – 31 = 0
(2) 4x + 3y = 24
(2) x – 3y – 31 = 0 or 3x + y – 7 = 0
(3) 3x + 4y = 25
(3) x – 3y – 31 = 0 or 3x + y + 7 = 0
(4) x + y = 7
(4) None of these
27.
30.
The incentre of the triangle fo rmed by
x = 0, y = 0 and 3x + 4y = 12 is -
æ 1 1ö
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
28.
E
(1) ç , ÷
è 2 2ø
(2) (1, 1)
1
(3) (1, )
2
1
(4) ( , 1)
2
A straight line through the point p(3, 4) is such that
If the straight line drawn through the point
P( 3 , 2) and making an angle p/6 with x-axis
meets the line
3 x – 4y + 8 = 0 at Q then the
length PQ is (1) 4
(2) 5
(3) 6
(4) None of these
The point (–4, 5) is the vertex of a square and one
of its diagonal is 7x–y + 8 = 0. The equation of the
other diagonal is :(1) 7x–y = 23
(2) x + 7y = 31
(3) x – 7y = 31
(4) None of these
ANSW ER KEY
Que.
E xercis e-I
A ns .
1
3
2
3
3
2
4
2
5
4
6
1
7
3
8
2
9
1
10
2
Que.
11
12
13
14
15
16
17
18
19
20
A ns .
3
3
1
2
1
4
1
2
3
1
Que.
21
1
22
2
23
1
24
1
25
2
26
3
27
2
28
2
29
2
30
3
A ns .
3
JEE ( Main) - Mathematics
ALLEN
PREVIOUS YEARS' QUESTIONS
2.
The centroid of a triangle is (2,3) and two of its
vertices are (5,6) and (–1,4). The third vertex of
the triangle is[AIEEE-2002]
(1) (2,1)
(2) (2,–1)
(3) (1,2)
(4) (1,–2)
3.
æ2 1 ö
÷
(2) ç ,
è3 3ø
æ2 3ö
(3) çç 3 , 2 ÷÷
è
ø
æ 1 ö
÷
(4) ç 1,
3ø
è
(3y)2
a2
b2
1)2
5.
(3y)2
a2
4
Let P(–1,0) Q=(0,0) and R(3, 3 3 ) be three points.
[AIEEE 2007], [IIT Scr. 2002]
9.
10.
3
y=0
2
(1)
3x+ y= 0
(2) x +
(3)
3
x+y=0
2
(4) x + 3 y = 0
The perpendicular bisector of the line segment
joining P(1, 4) and Q(k, 3) has y-intercept –4. Then
a possible value of k is[AIEEE-2008]
(1) 1
(2) 2
(3) –2
b2
(4) –4
The lines p(p + 1) x – y + q = 0 and
2
(p2 + 1)2x + (p2 + 1)y + 2q = 0 are
[AIEEE 2009]
(3) (3x –
+
=
+
2
2
2
(4) (3x + 1) + (3y) = a + b2
Perpendicular to a common line for :
Let A(2,–3) and B(–2,1) be vertices of a triangle
ABC. If the centroid of this triangle moves on the
line 2x + 3y = 1, then the locus of the vertex C is
the line[AIEEE-2004, 2011]
(2) More than two values of p
(1) Exactly two values of p
(1) 2x + 3y = 9
(2) 2x – 3y = 7
(3) 3x + 2y = 5
(4) 3x –2y = 3
(3) No value of p
(4) Exactly one value of p
11.
(1) y2 – 4x + 2 = 0
(2) y2 + 4x + 2 = 0
(3) x2 + 4y + 2 = 0
(4) x2 – 4y + 2 = 0
If a vertex of a triangle is (1,1) and the mid points
of two sides through this vertex are (–1,2) and (3,2),
then the centroid of the triangle is- [AIEEE - 2005]
7ö
æ
(1) ç -1, ÷
3ø
è
æ -1 7 ö
(2) ç , ÷
è 3 3ø
æ 7ö
(3) ç 1, ÷
è 3ø
æ1 7ö
(4) ç , ÷
è3 3ø
The line L given by
x y
+ = 1 passes through the
5 b
point (13, 32). The line K is parallel to L and has
Let P be the point (1,0) and Q a point on the curve
y2 = 8x. The locus of mid point of PQ is[AIEEE-2005]
6.
(2) 4x + 3y = 24
(4) x + y = 7
The equation of the bisector of the angle PQR is-
Locus of centroid of the triangle whose vertices are
(a cos t, a sin t), (b sin t, – b cos t) and (1,0), where
t is a parameter, is[AIEEE 2003]
(1) (3x +
+
=
–
2
2
2
(2) (3x – 1) + (3y) = a – b2
4.
8.
[AIEEE-2002]
æ
3ö
(1) çç 1, 2 ÷÷
è
ø
1)2
A straight line passing through the point A(3,4) is
such that its intercept between the axes is bisected
at A. Then its equation is[AIEEE 2006]
(1) 3x – 4y + 7 = 0
(3) 3x + 4y = 25
The incentre of the triangle with vertices (1, 3) ,
(0,0) and (2,0) is-
7.
the equation
x y
+ = 1 . Then the distance between
c 3
L and K is :
[AIEEE-2010]
(1)
12.
23
15
(2)
17
(3)
17
15
(4)
23
17
A line is drawn through the point (1, 2) to meet the
coordinate axes at P and Q such that it forms a
triangle OPQ, where O is the origin. If the area of
the triangle OPQ is least, then the slope of the line
PQ is :
[AIEEE-2012]
(1) -
1
2
(2) -
1
4
(3) –4
(4) –2
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
1.
EXERCISE-II
E
Point & Straight Line
ALLEN
13.
14.
15.
16.
If the point (1, a) lies in between the straight lines
x + y = 1 and 2(x + y) = 3 then a lies in
interval :[AIEEE-2012 (Online)]
æ 3ö
(1) ç 1, ÷
è 2ø
æ 1ö
(2) ç 0, ÷
è 2ø
(3) (–¥, 0)
æ3 ö
(4) ç , ¥ ÷
è2 ø
If two vertices of a triangle are (5, –1) and(–2, 3)
and its orthocentre is at (0, 0), then the third vertex
is :[AIEEE-2012 (Online)]
(1) (4, –7)
(2) (–4, 7)
(3) (–4, –7)
(4) (4, 7)
The x-coordinate of the incentre of the triangle that
has the coordinates of mid points of its sides as
(0, 1)(1, 1) and (1, 0) is :
[JEE(Main)-2013]
(1) 2 + 2
(2) 2 - 2
(3) 1 + 2
(4) 1 - 2
19.
on the
[JEE (Main)-2014 (Online)]
20.
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
E
(1) x – 2y + 1 = 0
(2) 5x – 2y = 0
(3) 2x + y – 9 = 0
(4) 2x – 5y = 0
Ihe base of an equilateral triangle is along the line
given by 3x + 4y = 9. If a vertex of the triangle is
(1, 2), then the length of a side of the triangle is :
[JEE (Main)-2014 (Online)]
(1)
21.
4 3
15
(2)
4 3
5
(2)
(3) y = 3x - 3
(4)
3y = x - 3
(3) 3bc – 2ad = 0
(4) 3bc + 2ad = 0
Let PS be the median of the triangle with vertices
P (2, 2), Q (6, –1) and R (7, 3). The equation of the
line passing through (1, –1) and parallel to PS is :
[JEE (Main)-2014 (Online)]
22.
(1) circle
(2) straight line
(3) parabola
(4) hyperbola
The circumcentre of a triangle lies at the origin and
its centroid is the mid point of the line segment
joining the points (a2 + l, a2 + l) and (2a, –2a),
a ¹ 0. Then for any a, the orthocentre of this triangle
lies on the line :
[JEE (Main)-2014 (Online)]
(1) y – (a2 + 1) x = 0
(2) y + x = 0
(3) (a – 1)2x – (a + 1)2 y = 0
(4) y – 2ax = 0
23.
If a line L is perpendicular to the line 5x – y = l,
and the area of the triangle formed by the line L
and the coordinate axes is 5, then the distance of
line L from the line x + 5y = 0 is :[JEE (Main)-2014 (Online)]
(1) 4x – 7y – 11 = 0
(1)
(3) 4x + 7y + 3 = 0
(4) 2x – 9y – 11 = 0
2 3
5
(a, b) lies on a :-
[JEE(Main)-2014]
(2) 2x + 9y + 7 = 0
(4)
x + 4ay + a = 0 are concurrent, then the point
Let a, b, c and d be non-zero numbers. If the point
of intersection of the lines 4ax + 2ay + c = 0 and
5bx + 2by + d = 0 lies in the fourth quadrant and
is equidistant from the two axes then :
(2) 2bc + 3ad = 0
2 3
15
x + 2ay + a = 0, x + 3by + b = 0 and
3y = x - 1
(1) 2bc – 3ad = 0
(3)
If the three distinct lines
A ray of light along x + 3y = 3 gets reflected
(1) y = x + 3
RQ is
the centroid of DPQR lies on the line:
[JEE(Main)-2014]
18.
x-axis. If equation of
x – 2y = 2 and PQ is parallel to the x-axis, then
upon reaching x-axis, the equation of the reflected
ray is :
[JEE(Main)-2013]
17.
Given three points P, Q, R with P(5, 3) and R lies
(3)
7
5
(2)
5
13
7
13
(4)
5
7
5
JEE ( Main) - Mathematics
24.
ALLEN
Locus of the image of the point (2, 3) in the line
(2x – 3y + 4) + k (x – 2y + 3) = 0, k Î R, is a
(1) circle of radius
(2) circle of radius
2
[JEE(Main)-2015]
3
(3) straight line parallel to x-axis
29.
(4) straight line parallel to y-axis
25.
Two sides of a rhombus are along the lines,
x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals
intersect at (–1, –2), then which one of the following
is a vertex of this rhombus ?
[JEE(Main)-2016]
æ 10 7 ö
,- ÷
è 3 3ø
26.
28.
(1) ç -
(2) (–3, –9)
(3) (–3, –8)
(4) ç , - ÷
æ1
è3
8ö
3ø
30.
Let k be an integer such that triangle with vertices
(k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then
the orthocentre of this triangle is at the point:
Orthocentre of the triangle whose vertices are
A(0,0), B(3,4) & C(4,0) is[IIT-2003]
æ 3ö
(1) ç 3, ÷
è 4ø
æ 5ö
(2) ç 3, ÷
è 4ø
(3) (3,12)
(4) (2,0)
Let O(0,0), P(3,4), Q(6,0) be the vertices of the
triangle OPQ. The point R inside the triangle OPQ
is such that the triangles OPR, PQR, OQR are of
equal area. The coordinates of R are- [IIT-2007]
æ4 ö
(1) ç ,3 ÷
è3 ø
æ 2ö
(2) ç 3, ÷
è 3ø
æ 4ö
(3) ç 3, ÷
è 3ø
æ4 2ö
(4) ç , ÷
è3 3ø
A straight line L through the point (3, –2) is inclined
at an angle 60° to the line
3x + y = 1 . If L also
intersect the x-axis, then the equation of L is
(1) y + 3x + 2 - 3 3 = 0
[IIT-2011]
[JEE(Main)-2017]
(2) y - 3x + 2 + 3 3 = 0
æ
è
1ö
÷
2ø
(2) ç 2, -
æ
è
1ö
÷
2ø
æ
è
3ö
÷
4ø
(4) ç 1, -
æ
è
3ö
÷
4ø
(1) ç 2,
(3) ç 1,
(4)
3y + x - 3 + 2 3 = 0
If P(1,2), Q(4,6) R(5,7) and S(a,b) are the vertices
of a parallelogram PQRS, then[IIT-1998]
(1) a = b, b = 4
(2) a = 3, b = 4
(3) a = 2, b = 3
(4) a = 3, b = 5
ANSW ER KEY
PR E V IOUS Y E A R S QUE STIONS
Que.
3y - x + 3 + 2 3 = 0
E xercise-II
A ns .
1
2
2
4
3
3
4
1
5
1
6
3
7
2
8
1
9
4
10
4
Que.
11
12
13
14
15
16
17
18
19
20
A ns .
4
4
2
3
2
2
3
2
4
1
Que.
21
2
22
3
23
2
24
1
25
4
26
1
27
3
28
1
29
3
30
2
A ns .
6
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\01-Point & Straight Line.p65
27.
(3)
E
Circle
ALLEN
CIRCLE
1.
In the figure shown, radius of circle C1 be r and that
of C2 be
6.
r
1
, where r = PQ, then length of AB is
2
3
K > 0, then the value of [K] is:-
Q
(1) 2
C2
2.
7.
(3) 3 3 r
(4)
3 3r
2
15
21
(2)
17
32
(3)
17
35
(4)
(3)
( 2 - 1)
2 (2 - 3 )
(2)
2
(4)
(
3(
)
3 - 1)
If in the adjacent figure PT is tangent to semicircle,
then radius of circle is -
8.
9.
If p and q be the longest distance and the shortest
dis tance
respectively
of
th e
po int
(–7, 2) from any point (a, b) on the curve whose
equation is x2 + y2 – 10x – 14y – 51 = 0 then G.M.
of p and q is equal to :(1) 2 11
(2) 5 5
(3) 13
(4) None of these
In the figure, OABC is a square of side 6 cm, then
the equation of the smallest circle is :-
y
C
30°
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
E
4 3 unit
(1) 4 3 unit
5.
A
B
C
(2) 4 unit
(3) 6 unit
(4) 2 3 unit
The equation of the circle having the lines
x2 + 2xy + 3x + 6y = 0 as its normals and having
the size just sufficient to contain the circle
x(x – 4) + y(y – 3) = 0 is:(1) x2 + y2 + 6x – 3y + 45 = 0
B
•
•
•
•
O
A
(4) x2 + y2 + 18x – 2y – 32 = 0
x
æ 3 2 - 3ö
(1) (x – 3) + (y – 3) = ç
è 2 ÷ø
2
æ 3 2 + 3ö
(2) (x – 3)2 + (y – 3)2 = ç
è 2 ÷ø
2
2
2
(3) (x – 3)2 + (y – 3)2 = 1
(2) x2 + y2 + 6x – 3y – 45 = 0
(3) x2 + y2 + 18x + 2y + 32 = 0
(4) 5
Equation of the circle passing through origin whose
centre lie in the first quadrant and length of intercept
on x and y-axis is 6 and 4 respectively, is-
T
P
(3) 4
(3) x2 + y2 – 3x – 2y = 0 (4) None of these
15
32
3 2- 3
(2) 3
(1) x2 + y2 – 4x – 6y = 0 (2) x2 + y2 – 6x – 4y = 0
Inside the unit circle S = {(x,y) |x2 + y2 = 1} there
are three smaller circles of equal radius 'a' tangent
to each other externally and to S internally. Then
the value of a is equal to (1)
4.
3 3r
4
Three parallel chords of a circle have lengths 2,
3, 4 & subtend angles a, b, a + b at the centre
respectively (a + b < p), then cosa is equal to (1)
3.
(2)
p
2p
and
, wheree
K
K
(Where [K] denotes the greatest integer less than
or equal to K)
A
(1) 2 3 r
3 + 1 apart. If the chords
subtend at the center, angles of
B
P
Two parallel chords of a circle of radius 2 are at
a distance
(where P and Q being centres of C 1 & C 2
respectively)
C1
EXERCISE-I
(4) (x – 3)2 + (y – 3)2 =
1
4
7
JEE(Main)-Mathematics
(1) 2 3
(2)
2p
3
3 +
(2) x2 + y2 + 32x + 4y – 235 = 0
(3) x2 + y2 + 32x – 4y – 235 = 0
(4) None of these
a2
(2)
6
a2
(3)
4
a2
(4)
3
In an equilateral triangle 3 coins of radii 1 unit each
are kept so that they touch each other and also the
sides of the triangle. Area of the triangle is :-
16.
17.
18.
(3) 12 +
13.
(4) 3 +
the ratio
A
is :B
9
16
(2)
3
4
20.
(3)
27
32
(4)
(4) 4/15
p
6
(2)
p
3
(3)
p
2
(4)
(1) 4 sq. units
(2) 8 sq. units
(3) 6 sq. units
(4) none
p
4
Two circles whose radii are equal to 4 and 8 intersect
at right angles. The length of their common chord
is-
16
5
(2) 8
(3) 4 6
(4)
8 5
5
The angle at which the circle (x–1)2 + y2 = 10 and
x2 + (y – 2)2 = 5 intersect is (1)
3 6
8
(3) 3/15
The area of the quadrilateral formed by the tangents
from t he point (4, 5) to the circle
x2 + y2 – 4x – 2y – 11 = 0 with the pair of radii
through the points of contact of the tangents is :
(1)
21.
p
6
(2)
p
4
(3)
p
3
(4)
p
2
The equation of a circle which touches the line
x + y = 5 at N(–2,7) and cuts the circle
x2 + y2 + 4x – 6y + 9 = 0 orthogonally, is (1) x2 + y2 + 7x – 11y + 38 = 0
(2) x2 + y2 = 53
y – 1 = m1(x – 3) and y – 3 = m2(x – 1) are two
family of straight lines, at right angled to each other.
The locus of their point of intersection is
(3) x2 + y2 + x – y – 44 = 0
(4) x2 + y2 – x + y – 62 = 0
22.
(2) x2 + y2 – 4x – 4y + 6 = 0
Tangents PA and PB are drawn to the circle
x2 + y2 = 4, then the locus of the point P if the
triangle PAB is equilateral, is equal to-
(3) x2 + y2 – 2x – 6y + 6 = 0
(1) x2 + y2 = 16
(2) x2 + y2 = 8
(4) x2 + y2 – 4x – 4y – 6 = 0
(3) x2 + y2 = 64
(4) x2 + y2 = 32
(1) x2 + y2 – 2x – 6y + 10 = 0
8
7 3
4
A square and an equilateral triangle have the same
perimeter. Let A be the area of the circle
circumscribed about the square and B be the area
of the circle cirumscribed about the triangle then
(1)
14.
7 3
4
(2) 2/13
The angle between the two tangents from the origin
to the circle (x – 7)2 + (y + 1)2 = 25 equals
(1)
(2) 6 + 4 3
3
Th e smallest dista nce betw een the circle
(x – 5)2 + (y + 3)2 = 1 and the line 5x + 12y – 4 = 0,
is
(1) 1/13
19.
(1) 4 + 2
The equation of the image of the circle x2 + y2 + 16x
– 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0
is
(1) x2 + y2 + 32x – 4y + 235 = 0
A circle is inscribed in an equilateral triangle of side
a. The area of any square inscribed in this circle is:-
a2
(1)
2
12.
15.
(4) x2 + y2 + 32x + 4y + 235 = 0
p
(3) 2 3 +
3
11.
ALLEN
What is the length of shortest path by which one can
go from (–2, 0) to (2, 0) without entering the interior
of circle, x2 + y2 = 1
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
10.
E
Circle
ALLEN
23.
24.
B and C are fixed points having co-ordinates (3, 0)
and (–3, 0) respectively. If the vertical angle BAC is
90°, then the locus of the centroid of the DABC has
the equation :
(1) x2 + y2 = 1
(2) x2 + y2 = 2
(3) 9(x2 + y2) = 1
(4) 9(x2 + y2) = 4
(2) 9
(3) 10
Tangents are drawn to the circle x2 + y2 = 10 at
the point where it meet by the circle
x2 + y2 + 4x – 3y + 2 = 0. The point of intersection
of these tangent is-
æ5
è2
(1) ç , –
Suppose that the equation of the circle having
(–3, 5) and (5, –1) as end points of a diameter is
(x – a)2 + (y – b)2 = r2. Then a + b + r, (r > 0) is
(1) 8
25.
28.
(4) 11
10 ö
÷
3 ø
5ö
æ 10
,– ÷
2ø
è 3
(2) ç –
æ 10 5 ö
, ÷
è 3 2ø
(3) ç –
29.
(4) None of these
point (3, 1) and touching the line |x – 1| = |y – 1|
is :-
Two circles intersects at the point P(2, 3) and the
line joining the other extremity of the two diameter
through P makes an angle p/6 with x-axis, then the
equation of the common chord of the two circles is-
(1) x2 + y2 – 3x + 4y + 11 = 0
(1) x +
3 y – (2 + 3 3 ) = 0
(2) x +
3 y – (2 3 + 2) = 0
Equation of the circle of radius
2 containing the
(2) x + y – 6x + 2y + 8 = 0
2
2
(3) x2 + y2 – 6x – 2y + 8 = 0
(4) None
26.
The locus of the middle points of chords of the curve
x2 + y2 = 9 which has gradient 3 is :(1) 2x – y = 7
30.
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
E
3 x + y – (2 3 + 3) = 0
(4)
3 x + y – (2 + 3 3 ) = 0
Any circle through the points of intersection of the
(2) 2x + y = 0
lines x +
(3) x + 3y = 0
these lines at points P and Q, then the angle
subtended by the arc PQ at its centre is-
(4) 3x – 4y + 5 = 0
27.
(3)
If the radius of the circle (x – 1)2 + (y – 2)2 = 1 and
(x – 7)2 + (y – 10)2 = 4 are increasing uniformly
w.r.t. time as 0.3 and 0.4 unit/sec, then they will
touch each other at t equal to-
3 y = 1 and
3 x – y = 2 if intersects
(1) 180°
(2) 90°
(3) 120°
(4) Depends on centre and radius
(1) 45 sec
(2) 90 sec
(3) 11 sec
(4) None of these
ANSW ER KEY
Que.
E xercis e-I
A ns .
1
4
2
2
3
2
4
1
5
2
6
2
7
2
8
1
9
1
10
3
Que.
11
12
13
14
15
16
17
18
19
20
A ns .
2
2
3
2
4
2
3
2
1
2
Que.
21
1
22
1
23
1
24
1
25
3
26
3
27
2
28
3
29
3
30
1
A ns .
9
JEE(Main)-Mathematics
ALLEN
PREVIOUS YEARS' QUESTIONS
If a circle passes through the point (a, b) and cuts
the circle x2 + y2 = 4 orthogonally, then the locus
of its centre is[AIEEE-2004]
7.
(1) 2ax + 2by + (a2 + b2 + 4) = 0
(2) 2ax + 2by – (a2 + b2 + 4) = 0
(3) 2ax – 2by + (a2 + b2 + 4) = 0
8.
(4) 2ax – 2by – (a2 + b2 + 4) = 0
2.
A variable circle passes through the fixed point
A(p, q) and touches x-axis. The locus of the other
end of the diameter through A is-
[AIEEE-2005]
[AIEEE-2004]
(1) (x – p) = 4qy
(2) (x – q) = 4py
(3) (y – p)2 = 4qx
(4) (y – q)2 = 4px
2
3.
2
If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie
along diameters of a circle of circumference 10p,
then the equation of the circle is[AIEEE-2004]
9.
(1) x2 + y2 – 2x + 2y – 23 = 0
(1) 3a2 – 10ab + 3b2 = 0
(2) 3a2 – 2ab + 3b2 = 0
(3) 3a2 + 10ab + 3b2 = 0
(4) 3a2 + 2ab + 3b2 = 0
Let C be the circle with centre (0, 0) and radius
3 units. The equation of the locus of the mid points
of the chords of the circle C that subtend an angle
of
(2) x + y – 2x – 2y – 23 = 0
2
If a circle passes through the point (a, b) and cuts
the circle x2 + y2 = p2 orthogonally, then the equation
of the locus of its centre is[AIEEE-2005]
2
2
2
2
(1) x + y – 3ax – 4by + (a + b – p2) = 0
(2) 2ax + 2by – (a2 – b2 + p2) = 0
(3) x2 + y2 – 2ax – 3by + (a2 – b2 – p2) = 0
(4) 2ax + 2by – (a2 + b2 + p2) = 0
If the pair of lines ax2 + 2(a + b)xy + by2 = 0 lie
along diameters of a circle and divide the circle into
four sectors such that the area of one of the sectors
is thrice the area of another sector then-
2
2p
at its centre is3
[AIEEE-2006, IIT-1996]
(3) x2 + y2 + 2x + 2y – 23 = 0
4.
(4) x2 + y2 + 2x – 2y – 23 = 0
(1) x2 + y2 = 1
(2) x2 + y2 =
27
4
The intercept on the line y = x by the circle
x2 + y2 – 2x = 0 is AB. Equation of the circle on AB
as a diameter is[AIEEE-2004]
(3) x2 + y2 =
9
4
(4) x2 + y2 =
3
2
10.
(1) x2 + y2 – x – y = 0
(2) x2 + y2 – x + y = 0
(3) x2 + y2 + x + y = 0
(4) x2 + y2 + x – y = 0
5.
If the circles x2 + y2 + 2ax + cy + a = 0 and
x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct
point P and Q then the line 5x + by – a = 0 passes
through P and Q for[AIEEE-2005]
(1) exactly one value of a
(3) infinitely many values of a
(4) exactly two values of a
A circle touches the x-axis and also touches the
circle with centre at (0, 3) and radius 2. The locus
of the centre of the circle is[AIEEE-2005]
(1) an ellipse
(3) a hyperbola
10
[AIEEE-2008]
12.
(2) no value of a
6.
11.
(2) a circle
(4) a parabola
Consider a family of circles which are passing
through the point (–1, 1) and are tangent to x-axis.
If (h, k) are the co-ordinates of the centre of the
circles, then the set of values of k is given by the
interval[AIEEE-2007]
(1) 0 < k < 1/2
(2) k ³ 1/2
(3) –1/2 £ k £ 1/2
(4) k £ 1/2
The point diametrically opposite to the point (1, 0)
on the circle x2 + y2 + 2x + 4y – 3 = 0 is(1) (3, –4)
(2) (–3, 4)
(3) (–3, –4) (4) (3, 4)
Three distinct points A, B and C are given in the
2–dimensional coordinate plane such that the ratio
of the distance of any one of them from the point
(1, 0) to the distance from the point (–1, 0) is equal
to
1
. Then the circumcentre of the triangle ABC
3
is at the point :-
æ5
è2
ö
æ5 ö
(2) ç , 0 ÷
ø
è3 ø
(1) ç , 0 ÷
[AIEEE-2009]
(3) (0, 0)
æ5
è4
ö
ø
(4) ç , 0 ÷
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
1.
EXERCISE-II
E
Circle
ALLEN
13.
If P and Q are the points of intersection of the circles
x 2 + y 2 + 3x + 7y + 2p – 5 = 0 and
x2 + y2 + 2x + 2y – p2 = 0, then there is a circle
passing through P, Q and (1, 1) for :[AIEEE-2009]
(1) All except two values of p
19.
The circle passing through (1, – 2) and touching the
axis of x at (3, 0) also passes through the point :
[JEE (Main)-2013]
(1) (–5, 2)
20.
(2) Exactly one value of p
(2) (2, –5)
[JEE-Main (on line)-2013]
(4) All except one value of p
For a regular polygon, let r and R be the radii of
the inscribed and the circumscribed circles.
A false statement among the following is :-
(1)
21.
[AIEEE-2010]
r 1
=
(1) There is a regular polygon with
R 2
15.
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
E
(4) 5
If the circle x2 + y2 – 6x – 8y + (25 – a2) = 0 touches
the axis of x, then a equals :(1) ±4
22.
(2) ±3
(3) 0
(4) ±2
The radius of a circle, having minimum area, which
touches the curve y = 4 – x2 and the lines, y = |x|
is :[JEE(Main)-2017]
(3) There is a regular polygon with
r 2
=
R 3
(1) 4
(
2 + 1)
(2) 2
(
2 + 1)
r
3
=
R
2
(3) 2
(
2 –1)
(4) 4
(4) There is a regular polygon with
(
2 –1)
23.
The circle x + y = 4x + 8y + 5 intersects the line
3x – 4y = m at two distinct points if :2
(2) – 35 < m < 15
(4) 35 < m < 85
24.
The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0)
touch each other if :[AIEEE-2011]
(1) a = 2c
(2) |a| = 2c
(3) 2|a| = c
(4) |a| = c
(3)
(4)
x2
+
y2
–x–y=0
+
y2
+ 2x + 2y – 7 = 0
The length of the diameter of the circle which
touches the x-axis at the point (1, 0) and passes
through the point (2, 3) is :
[AIEEE-2012]
(1) 5/3
(2) 10/3
(3) 3/5
(4) 6/5
(3) 3
(4)
3
Line 2x + 3y + 1 = 0 is a tangent to a circle at
(1, –1). This circle is orthogonal to a circle which is
drawn having diameter as a line segment with end
points (0, –1) and (– 2, 3). Find equation of circle.
(1) 2x2 + 2y2 – 10x – 5y + 1 = 0
(2) x2 + y2 – 10x – 5y + 1 = 0
(3) x2 + y2 – 5x – 5y + 1 = 0
(1) x2 + y2 + x + y – 2 = 0
x2
(2) 2
[IIT 2004]
The equation of the circle passing through the points
(1, 0) and (0, 1) and having the smallest radius is-
(2) x2 + y2 – 2x – 2y + 1 = 0
The radius of the circle, having centre at (2, 1),
whose one of the chord is a diameter of the circle
x2 + y2 – 2x – 6y + 6 = 0
[IIT 2004 (Scr)]
(1) 1
[AIEEE-2011]
18.
(3) 4
r
1
=
R
2
(1) – 85 < m < – 35
(3) 15 < m < 65
17.
(2) 2 5
[JEE-Main (on line)-2013]
[AIEEE-2010]
16.
57
(2) There is a regular polygon with
2
(4) (–2, 5)
If a circle C passing through (4, 0) touches the circle
x2 + y2 + 4x – 6y – 12 = 0 externally at a point
(1, –1), then the radius of the circle C is :-
(3) All values of p
14.
(3) (5, –2)
(4) 2x2 + 2y2 – 5x – 5y + 1 = 0
25.
A circle is given by x2 + (y – 1)2 = 1, another
circle C touches it externally and also the x-axis,
then the locus of its centre is
[IIT 2005 (Scr)]
(1) {(x, y) : x2 = 4y} È {(x, y) : y £ 0}
(2) {(x, y) : x2 + (y – 1)2 = 4} È {x, y) : y £ 0}
(3) {(x, y) : x2 = y} È {(0, y) : y £ 0}
(4) {(x, y) : x2 = 4y} È {(0, y) : y £ 0}
11
JEE(Main)-Mathematics
ALLEN
29.
Let ABCD be a quadrilateral with area 18, with
The straight line 2x – 3y = 1 divides the circular
region x2 + y2 £ 6 into two parts. If
side AB parallel t o t he s ide CD and
AB = 2CD. Let AD be perpendicular to AB and
ìæ 3 ö æ 5 3 ö æ 1 1 ö æ 1 1 ö ü
S = íç 2, ÷ , ç , ÷ , ç , - ÷, ç , ÷ ý ,
îè 4 ø è 2 4 ø è 4 4 ø è 8 4 ø þ
CD. If a circle is drawn inside the quadrilateral ABCD
touching all the sides, then its radius is
(1) 3
27.
(2) 2
[IIT 2007]
then the number of point(s) in S lying inside the
(4) 1
smaller part is :-
(3) 3/2
[IIT-2011]
Tangents drawn from the point P(l, 8) to the circle
(1) 1
(2) 2
x2
(3) 3
(4) 4
+
y2
– 6x – 4y – 11 = 0 touch the circle
at the points A and B. The equation of the
circumcircle of the triangle PAB is
(1)
x2
+
y2
30.
[IIT 2009]
The locus of the mid-point of the chord of contact
of tangents drawn from points lying on the straight
+ 4x – 6y + 19 = 0
2
(2) x2 + y2 – 4x – 10y + 19 = 0
28.
[IIT 2012]
(3) x2 + y2 – 2x + 6y – 29 = 0
(1) 20(x2 + y2) – 36x + 45y = 0
(4) x2 + y2 – 6x – 4y + 19 = 0
(2) 20(x2 + y2) + 36x – 45y = 0
The circle passing through the point (–1,0) and
(3) 36(x2 + y2) – 20x + 45y = 0
touching the y-axis at (0,2) also passes through the
point (2) ç -
æ 3 5ö
, ÷
è 2 2ø
(4) (–4,0)
(3) ç -
ANSWER KEY
PREVIOUS YEARS QUESTIONS
Ans.
Que.
Ans.
Que.
Ans.
12
2
æ 5 ö
,2÷
è 2 ø
(1) ç -
Que.
2
(4) 36(x + y ) + 20x – 45y = 0
[IIT 2011]
æ 3 ö
,0÷
è 2 ø
2
line 4x – 5y = 20 to the circle x + y = 9 is-
Exercise-II
1
2
2
1
3
1
4
1
5
2
6
4
7
4
8
4
9
3
10
2
11
3
12
4
13
4
14
3
15
2
16
4
17
3
18
2
19
3
20
4
21
1
22
4
23
3
24
1
25
4
26
2
27
2
28
4
29
2
30
1
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\02-Circle.p65
26.
E
Conic Section
ALLEN
CONIC SECTION
1.
2.
3.
Length of the latus rectum of the parabola
25[(x – 2)2 + (y – 3)2] = (3x – 4y + 7)2 is(1) 4
(2) 2
(3) 1/5
(4) 2/5
7.
(2) 4
(3) 6
(4) 8
If a focal chord of y 2 = 4x makes an angle
æ pù
a, a Î ç 0, ú with the positive direction of x-axis,
è 4û
then minimum length of this focal chord is -
Maximum number of common chords of a parabola
and a circle can be equal to
(1) 2
EXERCISE-I
(1)
(2)
2 2
(3) 8
8.
A variable circle is drawn to touch the line 3x – 4y = 10
and also the circle x2 + y2 = 1 externally then the
locus of its centre is (1) straight line
9.
(2) circle
4 2
(4) 16
If (2,–8) is one end of a focal chord of the parabola
y2 = 32x, then the other end of the focal chord, is(1) (32,32)
(2) (32,–32)
(3) (–2,8)
(4) (2,8)
Minimum distance between the curves y2 = x – 1
and x2 = y – 1 is equal to
(3) pair of real, distinct straight lines
(4) parabola
4.
5.
(1)
The straight line y = m(x – a) will meet the parabola
y2 = 4ax in two distinct real points if
10.
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
E
(2)
5 2
4
(3)
7 2
4
(4)
2
4
(1) m Î R
(2) m Î [–1, 1]
The length of a focal chord of the parabola
y2 = 4ax at a distance b from the vertex is c, then
(3) m Î (– ¥, 1] È [1, ¥)
(4) m Î R – {0}
(1) 2a2 = bc
(2) a3 = b2c
(3) ac = b2
(4) b2c = 4a3
The equation of the circle drawn with the focus of
the parabola (x - 1)2 - 8 y = 0 as its centre and
touching the parabola at its vertex is :
11.
(1) x2 + y2 - 4 y = 0
6.
3 2
4
(2)
x2
(3)
x2
(4)
x2
+
y2
- 4y + 1 = 0
+
y2
- 2x - 4y = 0
+
y2
- 2x - 4y + 1 = 0
12.
(2)
(3)
- 2 = - 2 cos t ; y = 4
x = tan t ;
t
2
y = sec t
(4) x = 1 - sin t ; y = sin
t
t
+ cos
2
2
(2) – 12
(3) – 9
(4) – 6
The points of contact Q and R of tangent from the
point P (2, 3) on the parabola y2 = 4x are
(2) (1, 2) and (4, 4)
(3) (4, 4) and (9, 6)
(1) x = 3 cos t ; y = 4 sin t
cos2
(1) – 18
(1) (9, 6) and (1, 2)
Which one of the following equations represented
parametrically, represents equation to a parabolic
profile?
x2
y-intercept of the common tangent to the parabola
y2 = 32x and x2 = 108y is
(4) (9, 6) and (
13.
1
, 1)
4
The equation of a straight line passing through the
point (3,6) and cutting the curve y =
orthogonally is(1) 4x + y – 18 =0
(3) 4x – y – 6 = 0
x
(2) x + y – 9 = 0
(4) none
13
JEE ( Main) - Mathematics
ALLEN
The equation of the common tangent touching the
circle (x – 3)2 + y2 = 9 and the parabola y2 = 4x
above the x-axis is (1)
3y = 3x + 1
(2)
3y = -(x + 3)
(3)
3y = x + 3
(4)
3y = -(3x + 1)
19.
(1)
20.
(x - h) 2 (y - k) 2
+
= 1 has major
15. If the ellipse
M
N
axis on the line y = 2, minor axis on the line
x =–1, major axis has length 10 and minor axis has
length 4. The number h,k,M,N (in this order only) are-
16.
(1) –1,2,5,2
(2) –1,2,10,4
(3) 1,–2,25,4
(4) –1,2,25,4
Let S(5,12) and S'(–12,5) are the foci of an ellipse
passing through the origin. The eccentricity of
ellipse equals (2)
1
3
The y-axis is the directrix of the ellipse with
eccentricity e = 1/2 and the corresponding focus
is at (3, 0), equation to its auxiliary circle is
(1)
(4)
2
3
2 2
3
(2)
5
3
(3)
8
9
(4)
2
3
(x - 2)2 (y + 1) 2
+
=1
25
16
(x + 2)2 (y - 1) 2
+
=1
(2)
25
9
(3) x2 + y2 – 8x + 9 = 0
(4) x2 + y2 = 4
Imagine that you have two thumbtacks placed at two
points, A and B. If the ends of a fixed length of string
are fastened to the thumtacks and the string is drawn
taut with a pencil, the path traced by the pencil will
be an ellipse. The best way to maximise the area
surrounded by the ellipse with a fixed length of string
occurs when
*(b)
(3)
(x - 2)2 (y + 1) 2
+
=1
9
25
(4)
(x + 2)2 (y - 1) 2
+
=1
9
25
Which of the following statement(s) is/are correct
for the ellipse of 21(a) ?
(1) auxiliary circle is (x + 2)2 + (y – 1)2 = 25
I
the two points A and B have the maximum
distance between them.
(2) director circle is (x + 2)2 + (y – 1)2 = 34
II
two points A and B coincide.
(3) Latus rectum =
III A and B are placed vertically.
IV The area is always same regardless of the
location of A and B .
(1) I
(2) II
(3) III
(4) IV
The latus rectum of a conic section is the width of
the function through the focus. The positive
difference between the length of the latus rectum of
3y = x2 + 4x – 9 and x2 + 4y2 – 6x + 16y = 24 is-
(1)
14
1
2
21.(a) Which of the following is an equation of the ellipse
with centre (–2,1), major axis running from (–2,6)
to (–2,–4) and focus at (–2,5) ?
(2) x2 + y2 – 8x – 12 = 0
18.
(3)
An ellipse is inscribed in a circle and a point within
the circle is chosen at random. If the probability that
this point lies outside the ellipse is 2/3 then the
eccentricity of the ellipse is :
(1)
(1) x2 + y2 – 8x + 12 = 0
17.
1
2
1
2
(2) 2
(3)
3
2
(4)
5
2
(4) eccentricity =
22.
18
5
4
5
The foci of a hyperbola coincide with the foci of the
x 2 y2
+
= 1 . Then the equation of the
ellipse
25 9
hyperbola with eccentricity 2 is
(1)
x 2 y2
=1
12 4
(3) 3x2 – y2 + 12 = 0
(2)
x 2 y2
=1
4 12
(4) 9x2 – 25y2 – 225 = 0
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
14.
E
Conic Section
ALLEN
23.
(1) line and an ellipse
(3) line and hyperbola
24.
(2) 6
2
The equation
the hyperbola
hyperbola
(3) 8
Locus of the point of intersection of the tangents at
the points with eccentric angles f and
(2) line and a parabola
(4) line and a point
The focal length of the
x2 – 3y2 – 4x – 6y – 11 = 0, is(1) 4
25.
27.
The graph of the equation x + y = x3 + y3 is the
union of -
(1) x = a
(4) 10
28.
2
x
y
+
= 1 (p ¹ 4, 29)
29 - p 4 - p
If
p
- f on
2
x2 y2
= 1 is :
a 2 b2
(2) y = b
(3) x = ab (4) y = ab
x2
y2
= 1 represents family of
cos2 a sin 2 a
hyperbolas where 'a' varies then -
represents -
(1) distance between the foci is constant
(1) an ellipse if p is any constant greater than 4
(2) distance between the two directrices is constant
(3) distance between the vertices is constant
(2) a hyperbola if p is any constant between 4 and
29.
(4) distances between focus and the corresponding
directrix is constant
(3) a rectangular hyperbola if p is any constant
greater than 29.
(4) no real curve is p is less than 29.
26.
A tangent to the ellipse
x 2 y2
+
= 1 with centre
9
4
C meets its director circle at P and Q. Then the
product of the slopes of CP and CQ, is -
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
(1)
E
9
4
(2)
-4
9
(3)
2
9
* Marked Question is multiple
(4) –
1
4
answer
ANSWER KEY
Que.
Ans.
Que.
Ans.
Que.
Ans.
Exercise-I
1
4
2
3
3
4
4
4
5
4
6
2
7
3
8
1
9
1
10
4
11
2
12
2
13
1
14
3
15
4
16
1
17
2
18
2
19
1
20
1
21(a)
4
21(b)
1,2,3,4
22
2
23
1
24
3
25
2
26
2
27
2
28
1
15
JEE ( Main) - Mathematics
ALLEN
PREVIOUS YEARS' QUESTIONS
The normal at the point (bt12, 2bt1) on a parabola
meets the parabola again in the point (bt22, 2bt2),
then[AIEEE-2003]
2
t1
2.
y=
(1) xy =
a x
– 2a is2
+
3
4
35
16
For the hyperbola
(3) (2, 4)
2
(2) y2 = 14x
(3) y2 = –104x
(4) y2 = –14x
If P 1 and P 2 are two points on the ellipse
[AIEEE-2012 (Online)]
(1)
105
(4) xy =
64
(2) (0, 2)
(1) y2 = 26x
the chord joining the points (0, 1) and (2, 0), then
the distance between P1 and P2 is :-
The equation of a tangent to the parabola y2 = 8x
is y = x + 2. The point on this line from which the
other tangent to the parabola is perpendicular to
the given tangents is[AIEEE-2007]
(1) (–1, 1)
10.
(4) (–2, 0)
The foci of the ellipse
(1) 9
11.
(3)
5
x2
y2
+ 2 = 1 and the
16
b
[AIEEE-2003] ; [AIEEE-2012 (Online)]
(2) 1
(3) 5
If the eccentricity of a hyperbola
[AIEEE-2007]
5.
(2) Abscissae of foci
(4) Directrix
passes through (k, 2) is
If two tangents drawn from a point P to the parabola
y2 = 4x are at right angles then the locus of P is :-
6.
(1) x = 1
(2) 2x + 1 = 0
(3) x = –1
(4) 2x – 1 = 0
The equation of the hyperbola whose foci are
(–2,0) and (2, 0) and eccentricity is 2 is given by :
(3) (x2 + y2)2 = 6x2 + 2y2
(3) 3x2 – y2 = 3
(4) –x2 + 3y2 = 3
(4) (x2 + y2)2 = 6x2 – 2y2
(1) 4
(2) 2
(3) 1
(4) 8
(4) 1
(2) (x2 – y2)2 = 6x2 – 2y2
(2) x2 – 3y2 = 3
The area of the triangle formed by the lines joining
the vertex of the parabola, x 2 = 8y, to the
extremities of its latus rectum is :-
(3) 18
The locus of the foot of perpendicular drawn from
the centre of the ellipse x2 + 3y2 = 6 on any tangent
to it is :
[JEE(Main)-2014]
(1) –3x2 + y2 = 3
[AIEEE-2012 (Online)]
16
(2) 8
(1) (x2 – y2)2 = 6x2 + 2y2
[AIEEE-2011]
7.
x2 y2
- = 1 , which
9 b2
[AIEEE-2012 (Online)]
(1) 2
12.
(4) 7
13
, then the value of k2
3
is :-
[AIEEE-2010]
(4) 2 3
1
y2
x2
–
=
coincide. Then the
25
81
144
value of b2 is-
2
following remains constant when a varies ?
(2) 2 2
10
hyperbola
x
y
- 2 = 1 , which of the
2
cos a sin a
(1) Abscissae of vertices
(3) Eccentricity
x2 y2
+
=1
16
3
x2
+ y 2 = 1 at which the tangents are parallel to
4
[AIEEE-2006]
(2) xy =
64
(3) xy =
105
4.
9.
2
a x
3
3ö
2
[AIEEE-2012 (Online)]
The locus of the vertices of the family of parabolas
3 2
3.
2
t1
(4) t2 = t1 –
æ
The normal at ç 2, ÷ to the ellipse,
è
ø
touches a parabola, whose equation is :
2
(2) t2 = –t1 –
t1
2
(1) t2 = t1 +
t1
(3) t2 = –t1 +
8.
13.
The slope of the line touching both, the parabolas
y2 = 4x and x2 = – 32 y is :
[JEE(Main)-2014]
(1)
1
2
(2)
3
2
(3)
1
8
(4)
2
3
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
1.
EXERCISE-II
E
Conic Section
ALLEN
14.
The area (in sq. units) of the quadrilateral formed
by the tangents at the end points of the latera recta
x2
y2
+
= 1 is :
9
5
to the ellipse
(1)
15.
16.
27
2
(3)
27
4
(2) x2 = 2y
(3) x2 = y
(4) y2 = x
21.
4
3
4
3
(3)
1
1
=1
2 +
2x
4y2
(2)
1
1
+
=1
2
2y 2
4x
(3)
x2
y2
+
=1
2
4
(4)
x2
y2
+
=1
4
2
A tangent is drawn at some point P of the ellipse
x2
y2
+
= 1 is intersecting to the coordinate
a2
b2
axes at points A and B the minimum area of the
DOAB is[IIT-2005]
The eccentricity of the hyperbola whose length of
the latus rectum is equal to 8 and the length of its
conjugate axis is equal to half of the distance
between its foci, is :
[JEE(Main)2016]
(2)
(1)
(4) 18
(1) y2 = 2x
3
Locus of middle point of segment of tangent to
ellipse x2 + 2y2 = 2. Which is intercepted between
the coordinate axes is[IIT-2004]
[JEE(Main)-2015]
Let O be the vertex and Q be any point on the
parabola, x2 = 8y. If the point P divides the line
segment OQ internally in the ratio 1 : 3, then the
locus of P is :[JEE(Main)-2015]
(1)
17.
(2) 27
20.
(4)
A hyperbola passes through the point P
(
2
3
(1) ab
(3)
22.
a 2 + b2
4
Con sider
a
branch
(2)
a 2 + b2
2
(4)
a2 + b2 - ab
3
of
th e
hyperbola
x2 – 2y2 – 2 2 x – 4 2 y – 6 = 0 with vertex at
2, 3 )
the point A. Let B be one of the end points of its
latus rectum. If C is the focus of the hyperbola nearest
to the point A. then the area of the triangle ABC
is[IIT-2008]
and has foci at (± 2, 0). Then the tangent to this
hyperbola at P also passes through the point :
[JEE(Main)2017]
(
)
(
)
(1) - 2, - 3
(3) 2 2, 3 3
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
18.
E
(
(2) 3 2, 2 3
(4)
(
)
3, 2 )
Consider a circle with centre lying on the focus of
the parabola y2 = 2px such that it touches the
directrix of the parabola. Then a point of
intersection of the circle and the parabola is-
23.
[IIT-1995]
19.
(1) (p/2, p)
(2) (–p/2, p)
(3) (–p/2, –p)
(4) None of these
The locus of the mid-point of the line segment joining
the focus to a moving point on the parabola
y2 = 4ax is another parabola with directrix[IIT-2002]
(1) x = –a
(2) x = –a/2
(3) x = 0
(4) x =a/2
(1) 1 –
2
3
(2)
3
–1
2
(3) 1 +
2
3
(4)
3
+1
2
The line passing through the extremity A of the
major axis and extremity B of the minor axis of the
ellipse x2 + 9y2 = 9 meets its auxiliary circle at the
point M. Then the area of the triangle with vertices
at A, M and the origin O is :[IIT-2009]
(1)
*24.
31
10
(2)
29
10
(3)
21
10
(4)
27
10
Let A and B be two distinct points on the parabola
y2 = 4x. If the axis of the parabola touches a circle
of radius r having AB as its diameter, then the slope
of the line joining A and B can be[IIT-2010]
(1) –
1
r
(2)
1
r
(3)
2
r
(4) –
2
r
17
JEE ( Main) - Mathematics
25.
ALLEN
Let P(6, 3) be a point on t he hyperbola
28.
be the common chord of the circle x2 + y2 – 2x – 4y = 0
x 2 y2
- = 1 . If the normal at the point P intersects
a 2 b2
and the given parabola. The area of the triangle
PQS is .
the x-axis at (9,0), then the eccentricity of the
hyperbola is [IIT-2011]
(1)
26.
5
2
(2)
3
2
(3)
(4)
2
3
29.
and D2 be the area of the triangle formed by
drawing tangents at P and at the end points of
(2) 6
(3) 2
(4) None of these
(2) 4
(3) 8
(4) 2
If the normals of the parabola y2 = 4x drawn at
of r2 is
æ1 ö
, 2 ÷ on the parabola,
è2 ø
(1) 4
(1) 16
the circle (x – 3)2 + (y + 2)2 = r2, then the value
rectum and the point P ç
D1
D 2 is
[IIT-2012]
the end points of its latus rectum are tangents to
Consider the parabola y2 = 8x. Let D1 be the area
of the triangle formed by the end points of its latus
the latus rectum. Then
Let S be the focus of the parabola y2 = 8x and let PQ
30.
[JEE (Advanced) 2015]
(1) 4
(2) 3
(3) 2
(4) 1
Let the curve C be the mirror image of the parabola
y2 = 4x with respect to the line x + y + 4 = 0. If
A and B are the points of intersection of C with
[IIT-2011]
the line y = –5, then the distance between A and
B is
x 2 x2
+
= 1 is inscribed in a
27. The ellipse E1 :
9 4
[JEE (Advanced) 2015]
(1) 10
(2) 6
(3) 8
(4) 4
rectangle R whose sides are parallel to the
coordinate axes. Another ellipse E2 passing through
the point (0,4) circumscribes the rectangle R. The
eccentricity of the ellipse E2 is [IIT-2012]
2
2
(2)
3
2
1
2
(3)
(4)
* Marked Question is multiple
answer
ANSWER KEY
PREVIOUS YEARS QUESTIONS
Que.
Ans.
Que.
Ans.
Que.
Ans.
18
3
4
Exercise-II
1
2
2
4
3
4
4
2
5
3
6
3
7
4
8
3
9
1
10
4
11
3
12
3
13
1
14
2
15
2
16
4
17
3
18
1
19
3
20
1
21
1
22
2
23
4
24
3,4
25
2
26
3
27
3
28
2
29
3
30
4
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\03-Conic Section.p65
(1)
E
Vector
ALLEN
VECTOR
1.
uuur
If ABCD is a parallelogram AB = 2iˆ + 4 ˆj - 5kˆ and
uuur
AD = ˆi + 2 ˆj + 3kˆ , then the unit vector in the
7.
2.
(1)
1 ˆ ˆ ˆ
(i + 2 j - 8k)
(2)
69
(3)
1
ˆ
( - ˆi - 2 ˆj + 8k)
69
1 ˆ ˆ ˆ
(- i - 2 j + 8k)
(4)
69
If a, b and c are perpendicular to b + c, c + a and
a + b respectively and if |a + b| = 6, |b + c| = 8
and |c + a| = 10 then |a + b + c| =
(1) 5 2
3.
4.
5.
(2) 50
(3) 10 2
(4) 10
The position vector of coplanar points A, B, C, D
are a, b, c and d respectively, in such away that
(a – d).(b–c)=(b–d).(c–a)= 0, then the point D of
the triangle ABC is :(1) Incentre
(2) Circumcentre
(3) Orthocentre
(4) None of these
r
8.
(4) None of these
9
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
E
(3) 3
(4) None
If a, b and c are three unit vectors then minimum
(2) 2
(3) 1
(4) 4
If four vector a, b, c and d are coplanar then
(a ´ b) ´ (c ´ d) :(1) 3
10.
11.
(2) 1
If a, b, c are the pth, qth, rth term of an A.P.
(4) None
(1) Linearly dependent
(2) Linearly Independent
(3) Parallel vector
(4) None
r
r
If p and q are two unit vectors inclined at an angle
r r
a to each other then | P + q | < 1 If :-
2p
4p
<a<
3
3
(3) a >
12.
(3) 2
Vectors ˆi + ˆj + kˆ , 2iˆ + 6 ˆj - kˆ and 9iˆ - ˆj + 3kˆ aree
(1)
2p
3
(2) a <
p
3
(4) a =
p
2
r r r
r
If three vectors a,b,c are such that a ¹ 0 and
r
r r
r r r r
a ´ b = 2a ´ c , | a | = | c |= 1, | b | = 4 and the angle
r
and x = (q - r)iˆ + (r - p) ˆj + (p - q)kˆ &
r
r
y = aiˆ + bjˆ + ckˆ , then -
r
between b and c is cos–1
r r
r
1
r
r
then b - 2c = la
4
where l is equal to :-
(1) x, y are parallel vectors
r r
(2) x ´ y = ˆi + ˆj + kˆ
(1) ±2
r r
(3) x.y = 1
13.
r r
(4) x, y are orthogonal vectors
6.
(2) 2
(1) 3
r r r r
r r r
u ×v + u = w and w ×u = v , then the value of
r r r
[ u v w ] is(3) 0
(1) 1
value of | a + b |2 + | b + c |2 + | c + a |2 is :-
r
(2) –1
r r r
rr
r r r
Let u an d v are un it vectors su ch t hat
(1) 1
rr
rr
Value of a.a ' + b.b' + c.c ' , (where a ', b', c ' form a
reciprocal system of vectors with the vectors a, b, c )
direction of BD is :-
1 ˆ ˆ ˆ
(i + 2 j - 8k)
69
EXERCISE-I
(2) ±4
(3)
1
2
(4)
1
4
ABCDEF is a regular hexagon where centre O is
the origin. If the position vector of A is ˆi - ˆj + 2kˆ
r
uuur
A straight line is given by r =(1+ t) î +3t ĵ +(1–t) k̂
then BC is equal to :-
whe re t Î R. If this line lies in the plane
x + y + cz = d then the value of (c + d) is
(1) ˆi - ˆj + 2kˆ
(2) -ˆi + ˆj - 2kˆ
(1) 9
(3) 3iˆ + 3jˆ - 4kˆ
(4) None of these
(2) 1
(3) –1
(4) 7
19
JEE ( Main) - Mathematics
14.
ALLEN
A point I is the centre of a circle inscribed
in a triangle ABC, then the vector sum
21.
uuur uur uuur uur uuur uur
BC IA + CA IB + AB IC is :-
uur uur uur
IA + IB + IC
(2)
3
(1) Zero
(3) 3
15.
r
r
(4) None
r
If a , b , c are coplanar then the value of the
rr rr
r
a.a b.a c.a
rr rr rr
determinant b.a b.b b.c is
rr rr rr
c.a c.b c.c
(1) 0
16.
(2) 3
22.
r
r r
If vectors c , a = xiˆ + yjˆ + zkˆ and b = ˆj are such
r
r r r
that a , c , b form a right handed system then c is:r
(1) ziˆ - xkˆ
(2) 0
(3) yjˆ
(4) - ziˆ + xkˆ
r
r
r
The vector a lies in the plane of vectors b and c
which of the following is correct :-
r r
(2) a . b ×c = 1
r r r
(4) a . b ×c = 3
r
r r r
(1) a . ( b ×c ) = 0
(3) 1
(4) None
r r
r
r r
r r r
The value of (a + 2b - c). (a - b) ´ (a - b - c) is
{
r r r
(3) a . b ×c = –1
}
23.
Area of parllologram whose adjacent sides are
ˆi + 2ˆj + 3kˆ and 3iˆ - 2ˆj + kˆ is :-
equal to :-
rrr
(3) 3 [a b c]
17.
rrr
(1) 5 2
(2) 8 3
rrr
(3) 6
(4) None
(2) 2 [a b c]
(4) 4 [a b c]
24.
r
For any vector P the value of
r
r
3 r ˆ2
| P ´ i | + | P ´ ˆj |2 + | P ´ kˆ |2 is
2
{
}
(1) tan–1 (5
r2
r
where P 2 = P :-
18.
r r
(2) a + b
r r
(4) b ´ a
(3) a - b
19.
r
20
r
r
r
r
25.
(4) cot–1 (3
2)
The volume of the tetrahedron formed by the
r r r
r r r
r
(3) 0
5)
the parallelepiped formed by the coterminus
r
r
(4) 25
r r r
If a , b , c are coplanar then r r r
r r
r r
( a + b + c ). (( a + b ) × ( a + c )) equals –
r r r
(1) 0
(2) [ a , b , c ]
r r r
r r r
(4) –[ a , b , c ]
(3) 2[ a , b , c ]
r r r
edges a + b, b + c, c + a is
r
r
r
r
If | u | = 3; | v | = 4 and | w | = 5 then
r r r r r r
u . v + v . w + w . u is :(2) –25
2)
(2) cos–1 (2
coterminus edges a , b, c is 3. Then the volume of
Let u , v , w be vectors such that u + v + w = 0
(1) 47
20.
r
r
2)
(3) cosec–1 (5
r
r
r
r
(1) P 2
(2) 2P 2
(3) 3P 2
(4) 4P 2
rr
rr
rr ˆ ˆ
is equal to :[a b ˆi]iˆ + [a b ˆj]jˆ + [a b k]k
(1) a ´ b
Position vectors of the four angular points of a
tetrahedron ABCD are A(3, – 2, 1); B(3, 1, 5);
C(4, 0, 3) and D(1, 0, 0). Acute angle between the
plane faces ADC and ABC is
26.
(1) 6
(2) 18
(3) 36
(4) 9
r r
r
a, b and c be three vectors having magnitudes
r r r r
1,1 and 2 respectively. If a ´ ( a ´ c ) + b = 0 , then
the acute angle between
r r
a & c is :
(1) p/6
(2) p/4
(3) p/3
(4) 5p/12
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
rrr
(1) [a b c]
E
Vector
ALLEN
27.
A vector of magnitude 5 5 coplanar with vectors
30.
vectors parallel and perpendicular to the vector
î+ 2 ĵ & ĵ+ 2k̂ and the perpendicular vector
$i + $j + k$ then the vectors are :
2î + ĵ+ 2k̂ is
(
(1) ± 5 5î + 6ˆj - 8k̂
(
(1) - $i + $j + k$
)
the n
the
(2) 3
(
)
& 8 $i - $j - 4 k$
(
)
& 4 $i - 5 $j - 8 k$
(4) none
maximum
r r r r r r
é a ´ b b´ g g ´ a ù is equal to
ë
û
(1) 2
& 7 $i - 2 $j - 5 k$
(3) + 2 $i + $j + k$
r
r
r
28. Let a = 2iˆ + 3jˆ - kˆ and b = ˆi + ˆj . If g is a unit
vector,
)
$ $ $
(2) - 2 i + j + k
(
)
(3) ± 5 5 (5î + 6ˆj - 8k̂ )
(4) ± (5î +6ˆj-8k̂ )
5 5î +6ˆj-8k̂
(2) ±
If the vector 6 $i - 3 $j - 6 k$ is decomposed into
(3) 4
value
31.
r r
r
Given three vectors a , b & c each two of which
r
(r )
are non collinear. Further if a + b is collinear
of
r
(
r
r
r
)
with c , b + c is collinear with a &
r
(4) 9
r
r
29. If the vectors a = 3 î + ˆj- 2 k̂ , b = - $i + 3 $j + 4 k$
r
& c = 4 $i - 2 $j - 6 k$ constitute the sides of a D
r
r
½ a ½=½ b ½ = ½ c ½ =
2 . Then the value of
(1) is 3
(2) is - 3
(3) is 0
(4) cannot be evaluated
r r
r r r r
a .b +b .c + c.a :
ABC, then the length of the median bisecting the
r
vector c is
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
(1)
E
(2)
2
14
(3)
74
(4)
6
ANSWER KEY
Que.
Exercise-I
1
3
2
4
3
3
4
1
5
4
6
1
7
3
8
1
9
4
10
2
Ans.
Que.
Ans.
11
1
12
2
13
2
14
1
15
1
16
3
17
3
18
1
19
2
20
1
21
1
22
1
23
2
24
1
25
3
26
1
27
4
28
2
29
4
30
1
Que.
Ans.
31
2
Ans.
Que.
21
JEE ( Main) - Mathematics
ALLEN
PREVIOUS YEARS' QUESTIONS
r
r
r
Let a = ˆj - kˆ and c = iˆ - ˆj - kˆ . Then the vector b
r
r
r
r
r r
5.
(p¹q¹r¹1) are coplanar, then the value of
pqr – (p + q + r) is :
[AIEEE-2011]
satisfying a ´ b + c = 0 and a . b = 3 is :
[AIEEE-2010]
(1) -ˆi + ˆj - 2kˆ
(2) 2iˆ - ˆj + 2kˆ
2.
6.
(1) –2
(2) 2
(3) 0
(4) –1
r r r
Let a, b,c be three non-zero vectors which are
(4) ˆi + ˆj - 2kˆ
and b + 2cr is colliner with a , then a + 3b + 6c is
r
(l, m) =
(1) (–3, 2)
(2) (2, –3)
(3) (–2, 3)
(4) (3, –2)
r
If a =
(1) a + c
r r
(2) a
7.
r 1
1 ( ˆ ˆ)
3i + k and b = 2iˆ + 3ˆj - 6kˆ , then
10
7
(1) 5
(2) 3
(3) – 5
(4) – 3
)
r
(4) 0
r
r
r
r r r r
r
c and d are two vectors satisfying : b ´ c = b ´ d
r
rr
and a.d
= 0 . Then the vector d is equal to :-
r
r æ b.cr ö r
(1) b + ç r r ÷ c
è a.b ø
rr
r æ a.c ö r
(2) c - ç r r ÷ b
è a.b ø
r
r æ b.cr ö r
(3) b - ç r r ÷ c
è a.b ø
rr
r æ a.c ö r
(4) c + ç r r ÷ b
è a.b ø
[AIEEE-2011]
r
Let â and b̂ be two unit vectors. If the vectors
r
r
c = aˆ + 2bˆ and d = 5aˆ - 4bˆ are perpendicular to
each other, then the angle between â and b̂ is :
[AIEEE-2012]
(1)
8.
The vectors a and b are not perpendicular and
r
r
(3) c
[AIEEE-2010]
(
r
r
r
[AIEEE-2011]
r
r
If the vectors a = ˆi - ˆj + 2kˆ , b = 2iˆ + 4ˆj + kˆ and
[AIEEE-2011]
22
r
pairwise non-collinear. If a + 3b is collinear with c
r
r r
r r
r
the value of ( 2a - b) . éë( a ´ b ) ´ ( a + 2b ) ùû is :-
4.
r
r
(3) ˆi - ˆj - 2kˆ
r
c = lˆi + ˆj + mkˆ are mutually orthogonal, then
3.
ˆ ˆ + qjˆ + kˆ and ˆi + ˆj + rkˆ
If the vectors piˆ + ˆj + k,i
p
4
(2)
p
6
(3)
p
2
(4)
p
3
Let ABCD be a parallelogram suc h th at
uuur r uuur r
AB = q, AD = p and ÐBAD be an acute angle.
r
If r is the vector that coincides with the altitude
r
directed from the vertex B to the side AD, then r
is given by :
[AIEEE-2012]
r 3 ( pr . qr ) r
r
r
=
3q
+ r r p
(1)
(p . p)
r
r
(2) r = 3q -
r r
3( p . q ) r
r r p
(p . p)
r æ pr . qr ö r
r
(3) r = - q + ç r r ÷ p
èp.pø
r r
r r æ p . q ör
(4) r = q - ç r r ÷ p
èp. pø
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
1.
EXERCISE-II
E
Vector
ALLEN
9.
ABCD is a parallelogram. The position vectors of
A and C are respectively, 3iˆ + 3ˆj + 5kˆ and
14.
are vectors in 3-dimensional space, then the
r r r
maximum possible value of | u ´ v.w | is :-
ˆi - 5ˆj - 5kˆ . If M is the mid-point of the diagonal DB,
uuuur
then the magnitude of the projection of OM on
uuur
OC , where O is the origin is :-
[AIEEE-2012 (Online)]
(1)
[AIEEE-2012 (Online)]
7
50
(1)
10.
(2) 7 50
7
51
(3)
(4) 7 51
r
r
ˆ r = ˆi - 3k,
ˆ and w
If u = ˆj + 4k,v
= cos q ˆi + sin q ˆj
(2) 5
14
(3) 7
15.
(4)
13
uuur
if the vectors AB = 3iˆ + 4kˆ and
uuur
AC = 5iˆ - 2 ˆj + 4kˆ are the sides of a triangle ABC,
r
r
ˆ b = 2iˆ + 3ˆj - kˆ and
If a = ˆi - 2ˆj + 3k,
then the length of the median through A is :
r
c = lˆi + ˆj + (2l – 1) k̂ are coplanar vectors, then
l is equal to :-
11.
[AIEEE-2012 (Online)]
(1) 1
(2) 2
(3) –1
(4) 0
r r
r
r r
r
r
If a + b + c = 0,| a |= 3,| b |= 5 and | c |= 7, then the
r
r
angle between a and b is :(1)
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
12.
E
13.
p
3
(2)
p
2
16.
(1)
18
(2)
72
(3)
33
(4)
45
r rr rr r
rrr
2
If éë a ´ b b ´ c c ´ a ùû = l éëa b c ùû then l is equal to:
[AIEEE-2012 (Online)]
(3)
p
6
(4)
[JEE (Main)-2014]
p
4
r
r
[JEE (Main)-2013]
ˆ b = 2iˆ + 3ˆj - kˆ and
If a = ˆi - 2ˆj + 3k,
17.
(1) 2
(2) 3
(3) 0
(4) 1
Let a , b and c be non-zero vectors such that
1
| b || c | a . If q is the acute
3
r
r
c = riˆ + ˆj + (2r - 1)kˆ are three vectors such that c
r
r
is parallel to the plane of a and b , then r is equal
angle between the vectors b and c , then sinq
to :-
equals-
(a× b) × c =
[AIEEE-2012 (Online)]
(1) 0
(2) 2
(3) –1
(4) 1
(1)
A unit vector which is perpendicular to the vector
r
r r
2i - j + 2k and is coplanar with the vectors
r r r
r r r
i + j - k and 2i + j - k is :-
18.
1
3
(1)
(3)
17
r
r
r
3i + 2 j + 2k
17
2
3
(2)
® ®
(3)
®
æ®
è
®
ö
ø
®
®
c , then the angle between a and b is :-
5
r
r r
2i + 2 j - k
(4)
3
2 2
3
3 æ ® ®ö ®
ç b + c ÷ . If b is not parallel to
2 è
ø
®
(2)
(4)
®
that a ´ ç b ´ c ÷ =
2jˆ + kˆ
2
3
Let a, b and c be three unit vectors such
[AIEEE-2012 (Online)]
r
r
r
3i + 2 j - 2k
[JEE (Main)-2015]
[JEE (Main)-2016]
(1)
5p
6
(2)
3p
4
(3)
p
2
(4)
2p
3
23
JEE ( Main) - Mathematics
ALLEN
*23.
ˆi + ˆj + 2kˆ and ˆi + 2ˆj + kˆ , and perpendicular to
r r
r
and the angle between c and a ´ b be 30º. Then
r r
a·c is equal to :
20.
the vector ˆi + ˆj + kˆ is/are
25
(2)
8
(3) 2
(4) 5
r
Let u be a vector coplanar with the vectors
r
b = ˆj + kˆ . If
rr
r
perpendicular to a and u.b = 24, then
an d
r
u
24.
is
(3) ˆi - ˆj
(4) -ˆj + kˆ
r
r
r
ˆ b = -ˆi + ˆj and c = ˆi + 2 ˆj + 3kˆ be
Let a = -ˆi - k,
r
is
(1) 315
(2) 256
(3) 84
(4) 336
Two adjacent sides of a parallelogram ABCD are
uuur
uuuur
25.
given by AB = 2iˆ + 10jˆ + 11kˆ and AD = -ˆi + 2ˆj + 2kˆ
The side AD is rotated by an acute angle a in the
plane of the parallelogram so that AD becomes
AD'. If AD' makes a right angle with the side AB,
then the cosine of the angle a is given by -
(1)
8
9
(2)
17
9
(3)
1
9
(4)
4 5
9
[IIT-2011]
(1) 8
(2) 9
(3) 6
(4) None of these
r
r r
r
a and b are vectors such that | a + b | = 29
r
r
ˆ = (2iˆ + 3jˆ + 4k)
ˆ ´ b , then
and a ´ (2iˆ + 3jˆ + 4k)
r r
ˆ is
a possible value of (a + b).( -7iˆ + 2ˆj + 3k)
If
[IIT-2012]
[IIT-2010]
26.
(1) 0
(2) 3
(3) 4
(4) 8
r r
r
c are unit vectors satisfying
r r
r r
r r
| a - b |2 + | b - c |2 + | c - a |2 = 9 , then
If a, b and
r r r
| 2a + 5b + 5c | is
r
r
ˆ b = ˆi - ˆj + kˆ and rc = ˆi - ˆj - kˆ
Let a = ˆi + ˆj + k,
r
r
be three vectors. A vector v in the plane of a
1
r
r
and b , whose projection on c is
, is given
3
by
24
(2) -ˆi + ˆj
rr
r r r r and r r
r.a = 0 , then the value of r.b
r ´b = c´ b
r2
u is
[JEE (Main)-2018]
22.
(1) ˆj - kˆ
three given vectors. If r is a vector such that
equal to-
21.
[IIT-2011]
[JEE (Main)-2017]
1
(1)
8
r
a = 2iˆ + 3ˆj - kˆ
The vector(s) which is/are coplanar with vectors
[IIT-2011]
(1) ˆi - 3ˆj + 3kˆ
(2) -3iˆ - 3 ˆj - kˆ
(3) 3iˆ - ˆj + 3kˆ
(4) ˆi + 3ˆj - 3kˆ
27.
Let
uuur
PR = 3iˆ + ˆj - 2kˆ
[IIT-2012]
and
uuur
SQ = ˆi - 3jˆ - 4kˆ
determine diagonals of a parallelogram PQRS and
uuur
PT = ˆi + 2jˆ + 3kˆ be another vector. Then the
volume of the parallelepiped determined by the
uuur
uuur uuur
vectors PT, PQ and PS is [JEE-Advanced 2013]
(1) 5
(2) 20
(3) 10
(4) 30
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
19.
r
r
r
Let a = 2iˆ + ˆj - 2kˆ and b = ˆi + ˆj . Let c be a
r r r
r r
vector such that | c - a | = 3, (a ´ b) ´ c = 3
E
Vector
ALLEN
28.
P.
Match List-I with List-II and select the correct answer
using the code given below the lists.
List-I
List-II
Volume of parallelepiped
1. 100
r r
determined by vectors a, b and
r
c is 2. Then the volume of the
parallelepiped determined by
r r
r r
vectors 2 a ´ b ,3 b ´ c and
(
r r
(c ´ a)
) (
*29.
2 and the angle between each pair of them is
p
r
r
. If a is a nonzero vector perpendicular to x
3
r
r r
and y ´ z and b is nonzero vector perpendicular
r r
r
to y and z ´ x , then
[JEE(Advanced)-2014]
)
is
Q. Volume of parallelepiped
r r
r
determined by vectors a, b and c
is 5. Then the volume of the
parallelepiped determined by
r r r r
vectors 3 a + b , b + c and
(
)(
r
r r r r
(1) b = (b. z) (z - x)
2. 30
r
r r r r
(2) a = (a . y) (y - z)
)
r r
r r r r
(3) a . b = -(a . y) (b . z)
r r
2 ( c + a ) is
R
r r
and a - b
(
S
r
r r r r
(4) a = (a . y) (z - y)
Area of a triangle with adjacent
3. 24
sides determined by vectors
r
r
a and b is 20. Then the area of
the triangle with adjacent sides
r
r
determined by vectors 2a + 3b
(
)
30.
r
r r
Let a, b , and c be three non-coplanar unit vectors
such that the angle between every pair of them
)
is
is S.
Area of a parallelogram with
adjacent sides determined by
r
r
vectors a and b is 30.
Then the area of the
parallelogram with adjacent
sides determined by vectors
r r
r
a + b and a is
(
r r
r
Let x, y and z be three vectors each of magnitude
4. 60
r r
p
r r r r
r
. If a ´ b + b ´ c = pa + qb + rc , where p,q and
3
r are scalars, then the value of
p2 + 2q 2 + r 2
is
q2
[JEE(Advanced)-2014]
)
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\04-Vector.p65
[JEE-Advanced 2013]
E
Codes :
P
(1)
4
(2)
2
(3)
3
(4)
1
Q
2
3
4
4
R
3
1
1
3
* Marked Questions
S
1
4
2
2
are multiple answer
ANSWER KEY
PREVIOUS YEARS QUESTIONS
Que.
Ans.
Que.
Ans.
Que.
Ans.
Exercise-II
1
1
2
1
3
3
4
2
5
1
6
4
7
4
8
3
9
3
10
4
11
1
12
1
13
1
14
2
15
3
16
4
17
4
18
1
19
3
20
4
21
2
22
3
23
1,4
24
2
25
3
26
3
27
3
28
3
29
1,2,3
30
4
25
JEE ( Main) - Mathematics
ALLEN
THREE DIMENSIONAL GEOMETRY
P is a fixed point (a, a, a) on a line through the
7.
The equation of the plane through the point
origin equally inclined to the axes, then any plane
through P perpeneicular to OP, makes intercepts
(–1, 2, 0) and parallel to the lines
on the axes, then sum of whose reciprocals is equal
to(1) a
1
(3)
a
2.
(1)
(2)
(3)
(4)
(4) None of these
The distance between two points P and Q is
d and the length of their projections of PQ
8.
d12 + d 22 + d 23 = kd2 where 'k' is-
OP = r, then centroid of the triangle ABC is-
(2) 5
(3) 3
(4) 2
The position vectors of two points P and Q are
3i + j + 2k and i – 2j – 4k respectively. The
to PQ is(1) r.(2i + 3j + 6k) = 28
9.
æ r2 r2 r 2 ö
÷
(3) ç , ,
è 3f 3g 3h ø
(4) None of these
In a three dimentional co-ordinate system P, Q and
R are the images of a point A(a, b, c) in xy, yz and
zx planes respectively. If G is the centroid of triangle
PQR, then area of triangle AOG is (O is the origin)
Th e sh ortest d ista nce bet ween the lines
(1) 0
s being parameters) is(1)
21
(3) 4
(3)
102
(2)
(4) 3
10.
(2) a2 + b2 + c2
2 2
(a + b2 + c2)
3
(4) None of these
The projections of a line on the axes are 9, 12,8
The four lines drawn from the vertices of any
the length of the line is
tetrahedron to the centroid of the opposite faces
(1) 7
(2) 17
meet in a point whose distance from each vertex
(3) 21
(4) 25
is k times the distance from each vertex to the
opposite face, where k is (1) 1/3
(2) 1/2
(3) 3/4
(4) 5/4
11.
æ 26 15 17 ö
, ,
7 7 7 ÷ø
æ 26 15 17 ö
,- , ÷
7
7 7ø
(2) ç
è
æ 15 26 -17 ö
(3) ç , ,
è 7 7 7 ÷ø
æ 26 17 -15 ö
(4) ç , ,
è 7 7 7 ÷ø
3/2
(3) cosq = 1/3
12.
(1) ç
è
The angle between any two diagonals of a cube is
(1) cos q =
The reflection of the point (2, –1, 3) in the plane
3x – 2y – z = 9 is -
26
æ r2 r2 r2 ö
(2) ç 2 , 2 , 2 ÷
è 3f 3g 3h ø
(3) r.(2i + 3j + 6k) + 28 = 0
r = (3i – 2j – 2k) + it and r = i – j + 2k + js (t and
6.
æ f g hö
(1) ç , , ÷
è 3r 3r 3r ø
(2) r.(2i + 3j + 6k) = 32
(4) None of these
5.
Through a point P(f, g, h) a plane is drawn at right
angles to OP, to meet the axes in A, B, C. If
equation of the plane through Q and perpendicular
4.
2x + 3y + 6z – 4 = 0
x– 2y + 3z + 5 = 0
x + y – 3z + 1 = 0
x + y + 3z – 1 = 0
on the co-ordinate planes are d1, d2, d3. Then
(1) 1
3.
x - 1 2y + 1 2z + 1
=
=
1
2
-1
and
3
(2)
2a
x y +1 z - 2
=
=
3
0
-1
(2) cos q = 1/ 2
(4) cos q = 1/ 6
r
A straight line is given by r =(1+ t) î +3t ĵ +(1–t) k̂
whe re t Î R. If this line lies in the plane
x + y + cz = d then the value of (c + d) is
(1) 9
(2) 1
(3) –1
(4) 7
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
1.
EXERCISE-I
E
Three Dimensional Geometry
ALLEN
13.
If a line makes angles a, b, g, d with the diagonals
of a cube then which of the following statement is
19.
Find the value of K if line
incorrect ?
3x + 2y + kz + 5 = 0 are parallel :-
4
(1) cos2a + cos2b + cos2g + cos2d =
3
(2) sin2a + sin2b + sin2g + sin2d =
8
3
(3) cos2a + cos2b + cos2g + cos2d =
20.
-4
3
The value o f m for which straight line
3x – 2y + z + 3 = 0 = 4x – 3y + 4z + 1 is parallel
21.
15.
(2) 8
(3) –18
(4) 11
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
If the line
The lines
(2) 3
(3)
31
10
22.
(2) k = 2
(3) k = 1
(4) k = 5
Projection of the line segment joining the points
7
22
(4)
(1) k = 0 or –1
(2) k = 1 or –1
(3) k = 0 or –3
(4) k = 3 or –3
The equation of right bisector plane of the segment
(1) x + y + z + 15 = 0
(3) x – y + z – 15 = 0
(4) None of these
(1) 1
23.
(3) Data not sufficient
(4) None
11
7
(4)
7
11
(2) 2
(3)
1
2
(4) 4
The angle between lines 3x + 2y + z = 0 = x + y – 2z
and 2x – y – z = 0 = 7x + 10y – 8z is :(1)
24.
p
6
(2)
p
2
(3)
p
3
(4) 0
The equation of line x + y + z – 1 = 0 = 4x + y – 2z + 2
written in the symmetrical form is : where
(A) º
x +1
y-2
z-0
=
=
1
–2
1
(B) º
x
y
z -1
=
=
1
–2
1
If sum of two unit vectors is a unit vector then find
(2) 0
(3)
The distance of the point (1, –2, 3) from the plane
(2) x + y + z – 15 = 0
(1) 1
22
7
x
y
z
=
= – is :2
3
6
31
30
x- 2
y-3
z- 4
=
=
and
1
1
–k
the magnitude of their difference :-
(2)
x – y + z = 5 measured parallel to the line
joining (2, 3, 4) and (6, 7, 8) is
18.
x -3 y -4 z -5
=
=
lies in the plane
2
3
4
(1) k = 3
(1)
x -1
y-4
z -5
=
=
are coplanar if
k
2
1
E
(4) None
Sum of the length intercepts on axes of the plane
(1) 0
17.
(3) –8
2, 1 :-
r ˆ ˆ
ˆ + µ(iˆ - 2ˆj + 3k)
ˆ is
r = i + j + l (iˆ + ˆj + k)
16.
(2) –9
(–1, 0, 3) and (2, 5, 1) on the line whose Dr's are 6,
to the plane 2x – y + mz – 2 = 0 is
(1) –2
(1) 17
4x + 4y – kz – d = 0 then :-
(4) None of these
14.
x - 2 y -1
=
= z plane
3
4
(C)
x + 1/ 2
y -1
z - 1/ 2
=
=
1
–2
1
(1) (A) and (B)
(2) (B) and (C)
(3) (A) and (C)
(4) (A), (B) and (C)
27
JEE ( Main) - Mathematics
25.
ALLEN
x - x1 y - y1 z - z1
=
=
29. The line
is
0
1
2
A plane passes through the point P(4, 0, 0) and
Q(0, 0, 4) and is parallel to the y-axis. The distance
of the plane from the origin is
(1) 2
(3)
26.
(2) 4
(4)
2
2 2
30.
If the plane 2x – 3y + 6z – 11 = 0 makes an angle
sin–1(k) with x-axis, then k is equal to
27.
(1)
3 2
(2) 2/7
(3)
2 3
(4) 1
(1) parallel to x-axis
(2) perpendicular to x-axis
(3) perpendicular to YOZ plane
(4) parallel to y-axis
The distance of the point (–1, –5, – 10) from the
point o f in tersection o f th e line
x - 2 y +1 z - 2
=
=
and the plane x – y + z = 5
2
4
12
is
The value of 'a' for which the lines
(1) 2 11
(2)
(3) 13
(4) 14
126
x -a y-7 z+2
x - 2 y - 9 z - 13
=
=
=
=
and
-1
2
-3
2
3
1
intersect, is
(1) – 5
(3) 5
28.
For the line
(2) – 2
(4) – 3
x -1 y - 2 z - 3
=
=
, which one of
1
2
3
the following is incorrect?
(1) it lies in the plane x – 2y + z = 0
(2) it is same as line
x y z
= =
1 2 3
ANSWER KEY
Que.
Ans.
Que.
Ans.
Que.
Ans.
28
Exercise-I
1
3
2
4
3
1
4
3
5
3
6
2
7
4
8
3
9
1
10
2
11
3
12
4
13
4
14
1
15
3
16
3
17
2
18
4
19
4
20
4
21
2
22
1
23
2
24
4
25
4
26
2
27
4
28
3
29
2
30
3
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
(3) it passes through (2, 3, 5)
(4) it is parallel to the plane x – 2y + z – 6 = 0
E
Three Dimensional Geometry
ALLEN
PREVIOUS YEARS' QUESTIONS
1.
2.
EXERCISE-II
The distance of the point (1, –5, 9) from the plane
x – y + z = 5 measured along a straight line
x = y = z is :
[AIEEE-2011]
(1) 3 5
(2) 10 3
(3) 5 3
(4) 3 10
6.
The distance of the point - ˆi + 2ˆj + 6kˆ from the
straight line that passes through the point
2iˆ + 3ˆj - 4kˆ an d is parallel to the vector
6iˆ + 3ˆj - 4kˆ is :
An equation of a plane parallel to the plane
x – 2y + 2z – 5 = 0 and at a unit distance from
the origin is :
[AIEEE-2012]
7.
(1) x – 2y + 2z + 5 = 0
(2) x – 2y + 2z – 3 = 0
(3) x – 2y + 2z + 1 = 0
[AIEEE-2012 (Online)]
(1) 8
(2) 7
(3) 10
(4) 9
A line with positive direction cosines passes through
the point P(2, –1, 2) and makes equal angles with
the coordinate axes. If the line meets the plane
2x + y + z = 9 at point Q, then the length PQ equals
[AIEEE-2012 (Online)]
(4) x – 2y + 2z – 1 = 0
3.
If
the
lines
x -1 y +1 z -1
=
=
2
3
4
and
x-3 y-k z
intersect, then k is equal to:
=
=
1
2
1
8.
(1) 2
(2)
3
(3) 1
(4)
2
The values of a for which the two points
(1, a, 1) and (–3, 0, a) lie on the opposite sides of
the plane 3x + 4y – 12z + 13 = 0, satisfy :-
[AIEEE-2012]
(1) 0
4.
(2) – 1
(3)
2
9
(4)
[AIEEE-2012 (Online)]
9
2
(1) 0 < a < 1/3
(2) a = 0
The equation of a plane containing the line
(3) –1 < a < 0
x +1 y - 3 z + 2
and the point (0, 7, –7) is:
=
=
-3
2
1
(4) a < –1 or a > 1/3
9.
[AIEEE-2012 (Online)]
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
(1) x + 2y – z = 21
E
(2) x + y + z = 0
If the three planes x = 5, 2x – 5ay + 3z – 2 = 0 and
3bx + y – 3z = 0 contain a common line, then
(a, b) is equal to :[AIEEE-2012 (Online)]
æ 1 8ö
÷
è 5 15 ø
(2) ç -
æ1
è5
(4) ç 15 , - 5 ÷
è
ø
(1) ç - ,
(3) 3x – 2y + 3z + 35 = 0
æ 8 1ö
, ÷
è 15 5 ø
(4) 3x + 2y + 5z + 21 = 0
5.
Consider the following planes :
(3) ç , -
P : x + y – 2z + 7 = 0
Q : x + y + 2z + 2 = 0
R : 3x + 3y – 6z – 11 = 0
10.
8ö
15 ÷ø
æ8
1ö
The coordinates of the foot of perpendicular from
[AIEEE-2012 (Online)]
(1) P and R are perpendicular
the point (1, 0, 0) to the line
x - 1 y + 1 z + 10
=
=
2
-3
8
(2) P and Q are parallel
are :-
[AIEEE-2012 (Online)]
(3) P and R are parallel
(1) (5, –8, –4)
(2) (2, –3, 8)
(3) (3, –4, –2)
(4) (1, –1, –10)
(4) Q and R are perpendicular
29
JEE ( Main) - Mathematics
11.
ALLEN
Distan ce between t wo parallel planes
2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is :-
15.
[JEE (Main)-2013]
12.
(1)
3
2
(2)
(3)
7
2
(4)
If the lines
The equation of the plane containing the line
2x – 5y + z = 3 ; x + y + 4z = 5, and parallel
to the plane, x + 3y + 6z = 1, is
(1) x + 3y + 6z = 7
5
2
(2) 2x + 6y + 12z = – 13
9
2
(3) 2x + 6y + 12z = 13
(4) x + 3y + 6z = – 7
x -2 y -3 z -4
=
=
and
1
1
-k
16.
The distance of the point (1, 0, 2) from the point
of intersection of the line
x -1 y - 4 z - 5
=
=
are coplanar, then k can
k
2
1
have :
[JEE (Main)-2015]
x - 2 y +1 z - 2
=
=
3
4
12
and the plane x – y + z = 16, is :
[JEE (Main)-2015]
[JEE (Main)-2013]
(1) any value
(1) 3 21
(2) 13
(3) 2 14
(4) 8
(2) exactly one value
(3) exactly two values
17.
The image of the line
The distance of the point (1, –5, 9) from the plane
x – y + z = 5 measured along the line x = y = z
is :
[JEE (Main)-2016]
x -1 y - 3 z - 4
=
=
in the plane
3
1
-5
(1)
(4) exactly three values.
2x – y + z + 3 = 0 is the line :
(1)
(3) 10 3
x +3 y -5 z-2
=
=
3
1
-5
18.
x -3 y+5 z-2
=
=
3
1
-5
(3)
30
10
3
x -3 y + 2 z + 4
=
=
lies in the plane,
2
–1
3
[JEE (Main)-2016]
19.
The angle between the lines whose direction cosines
satisfy the equations l + m + n = 0 and
l2 = m2 + n2 is :
[JEE (Main)-2014]
(1)
If the line,
(4)
lx + my – z = 9, then l2 + m2 is equal to :-
x -3 y+5 z-2
=
=
(4)
-3
-1
5
14.
(2) 3 10
[JEE (Main)-2014]
x +3 y -5 z+2
=
=
(2)
5
-3
-1
(3)
20
3
p
3
(2)
p
6
(4)
p
4
p
2
(1) 2
(2) 26
(3) 18
(4) 5
If the image of the point P(1, –2, 3) in the plane,
2x + 3y – 4z + 22 = 0 measured parallel to line,
x y z
= = is Q, then PQ is equal to :1 4 5
[JEE (Main)-2017]
(1) 6 5
(2) 3 5
(3) 2 42
(4)
42
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
13.
E
Three Dimensional Geometry
ALLEN
20.
The distantce of the point (1, 3, –7) from the plane
passing through the point (1, –1, –1), having normal
perpendicular to both the lines
and
(1)
21.
20
74
(3)
2
3
(2)
1
3
(3)
1
3 2
(2)
1
2 2
(3)
10
83
(4)
5
83
2
3
(4)
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
E
(2) y + z = –1
(3) y – z = –1
(4) y – 2z = –1
26.
1
2
(4)
Perpendiculars are drawn from points on the line
x + 2 y +1 z
=
= to the plane x + y + z = 3. The
2
-1
3
2
3
feet of perpendiculars lie on the line
[JEE-Advanced 2013]
(1)
x y -1 z -2
=
=
-13
5
8
(2)
x y -1 z - 2
=
=
2
3
-5
(3)
x y -1 z - 2
=
=
4
3
-7
(4)
x y -1 z - 2
=
=
-7
2
5
1
4 2
[IIT-2012]
24.
[IIT-2012]
(1) y + 2z = –1
The point P is the intersection of the straight line
joining the points Q(2,3,5) and R(1,–1,4) with the
plane 5x – 4y – z = 1. If S is the foot of the
perpendicular drawn from the point T(2,1,4) to
QR, then the length of the line segment PS is -
1
(1)
2
x -1 y +1 z
=
=
and
2
k
2
containing these two lines is(are)
If L1 is the line of intersection of the planes
2x – 2y + 3z – 2 = 0, x – y + z + 1 = 0 and L2
is the line of intersection of the planes
x + 2y – z – 3 =0, 3x – y + 2z – 1 = 0, then the
distance of the origin from the plane, containing the
lines L1 and L2 is :
[JEE (Main)-2018]
(1)
23.
(2)
If the straight lines
x +1 y +1 z
=
= are coplanar, then the plane(s)
5
2
k
[JEE (Main)-2017]
The length of the projection of the line segment
joining the points (5, –1, 4) and (4, –1, 3) on the
plane, x + y + z = 7 is :
[JEE (Main)-2018]
(1)
22.
x –1 y + 2 z - 4
=
=
1
3
-2
x – 2 y +1 z + 7
=
=
, is :2
-1
–1
10
74
*25.
*27.
A line l passing through the origin is perpendicular
to the lines
ˆ -¥ < t < ¥
l1 : ( 3 + t ) ˆi + ( -1 + 2t ) ˆj + ( 4 + 2t ) k,
(2)
2
(3) 2
(4)
2 2
The equation of a plane passing through the line of
intersection of the planes x + 2y + 3z = 2 and
2
x – y + z = 3 and at a distance
from the point
3
(3, 1, –1) is
[IIT-2012]
2x + y = 3 2 - 1
(1) 5x – 11y + z = 17
(2)
(3) x + y + z =
(4) x -
3
2y = 1 - 2
ˆ -¥ < s < ¥
l 2 : ( 3 + 2s ) ˆi + ( 3 + 2s ) ˆj + (2 + s ) k,
Then , the coordinate(s) of the point(s) on l2 at
a distance of
17 from the point of intersection
of l and l1 is(are) -
[JEE-Advanced 2013]
æ7 7 5ö
(1) ç , , ÷
è3 3 3ø
(2) (–1,–1,0)
(3) (1,1,1)
æ7 7 8ö
(4) ç , , ÷
è9 9 9ø
31
JEE ( Main) - Mathematics
*28.
Two lines L 1 : x = 5,
L 2 : x = a,
ALLEN
y
z
and
=
3 - a -2
y
z
are coplanar. Then a can
=
-1 2 - a
take value(s)
(1) 1
29.
30.
[JEE(Advanced)-2014]
[JEE-Advanced 2013]
(2) 2
(3) 3
From a point P(l,l,l), perpendiculars PQ and PR
are drawn respectively on the lines y = x,
z = 1 and y = –x, z = –1. If P is such that ÐQPR
is a right angle, then the possible value(s) of l is(are)
(1)
2
(2) 1
(3) –1
(4) – 2
(4) 4
Consider the lines
L1 :
y
x -1
z+3
x-4 y+3 z+3
=
=
,L 2 :
=
=
-1
2
1
1
1
2
and the planes P1:7x+y + 2z = 3, P2 :
3x + 5y – 6z = 4. Let ax + by + cz = d be the
equation of the plane passing through the point
of intersection of lines L1 and L2 and perpendicular
to planes P1 and P2.
Match List-I with List-II and select the correct answer
using the code given below the lists.
[JEE-Advanced 2013]
List-I
List-II
P.
a =
1.
13
Q.
b =
2.
–3
R.
c =
3.
1
S.
d =
4.
–2
P
Q
R
S
(1)
3
2
4
1
(2)
1
3
4
2
(3)
3
2
1
4
(4)
2
4
1
3
* Marked Questions
are multiple answer
ANSWER KEY
PREVIOUS YEARS QUESTIONS
Que.
Ans.
Que.
Ans.
Que.
Ans.
32
Exercise-II
1
2
2
2
3
4
4
2
5
3
6
2
7
2
8
4
9
3
10
3
11
3
12
3
13
1
14
1
15
1
16
2
17
3
18
1
19
3
20
3
21
3
22
1
23
1
24
1
25
2,3
26
4
27
2,4
28
1,4
29
1
30
3
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
Codes :
E
node06\B0AI-BO\Kota\JEE(MAIN)\Booster Course Sheet\Maths\Eng\2D, 3D & Vector\05-Three Dimensional Geometry.p65
ALLEN
E
Three Dimensional Geometry
Important Notes
33