Question 1: Palette 1/48

JEE Advanced 2025 MCQ

Let S denote the locus of the point of intersection of the pair of lines

$4x - 3y = 12\alpha$,

$4\alpha x + 3\alpha y = 12$,

where $\alpha$ varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points $(p, 0)$ and $(0, q)$, $q > 0$, and parallel to the line $4x - \frac{3}{\sqrt{2}} y = 0$.

Then the value of $pq$ is :

A

$-6\sqrt{2}$

B

$-3\sqrt{2}$

C

$-9\sqrt{2}$

D

$-12\sqrt{2}$

Question 2: Palette 2/48

JEE Advanced 2013 MCQ

For $a > b > c > 0,$ the distance between $(1, 1)$ and the point of intersection of the lines $ax + by + c = 0$ and $bx + ay + c = 0$ is less than $\left( {2\sqrt 2 } \right)$. Then

A

$a + b - c > 0$

B

$a - b + c < 0$

C

$a - b + c = > 0$

D

$a + b - c < 0$

Question 3: Palette 3/48

JEE Advanced 2011 MCQ

A straight line $L$ through the point $(3, -2)$ is inclined at an angle ${60^ \circ }$ to the line $\sqrt {3x} + y = 1.$ If $L$ also intersects the x-axis, then the equation of $L$ is

A

$y + \sqrt {3x} + 2 - 3\sqrt 3 = 0$

B

$y - \sqrt {3x} + 2 + 3\sqrt 3 = 0$

C

$\sqrt {3y} - x + 3 + 2\sqrt 3 = 0$

D

$\sqrt {3y} + x - 3 + 2\sqrt 3 = 0$

Question 4: Palette 4/48

JEE Advanced 2008 MCQ

Consider three points $P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )$ and $R = (\cos (\beta - \alpha + \theta ),\sin (\beta - \theta ))$ where $0 < \alpha ,\beta ,\theta < {\pi \over 4}$. Then :

A

P lies on the line segment RQ

B

Q lies on the line segment PR

C

R lies on the line segment QP

D

P, Q, R are non-collinear

Question 5: Palette 5/48

JEE Advanced 2008 MCQ

Consider the lines given by:

${L_1}:x + 3y - 5 = 0$

${L_2}:3x - ky - 1 = 0$

${L_3}:5x + 2y - 12 = 0$

Match the Statement/Expressions in Column I with the Statements/Expressions in Column II.

	Column I		Column II
(A)	L$_1$, L$_2$, L$_3$ are concurrent, if	(P)	$K = - 9$
(B)	One of L$_1$, L$_2$, L$_3$ is parallel to atleast one of the other two, if	(Q)	$K = - {6 \over 5}$
(C)	L$_1$, L$_2$, L$_3$ form a triangle, if	(R)	$K = {5 \over 6}$
(D)	L$_1$, L$_2$, L$_3$ do not form a triangle, if	(S)	$K = 5$

A

A - iv; B - ii; C - iii; D - i, ii

B

A - iv; B - i, ii; C - iii; D - i, ii, iv

C

A - iv; B - i; C - iii; D - i, ii

D

A - ii; B - i, iii; C - iii; D - i, ii, iv

Question 6: Palette 6/48

JEE Advanced 2008 MCQ

Let a and b be non-zero real numbers. Then, the equation

$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$ represents :

A

four straight lines, when c = 0 and a, b are of the same sign

B

two straight lines and a circle, when a = b, and c is of sign opposite to that of a

C

two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a

D

a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Question 7: Palette 7/48

JEE Advanced 2008 MSQ

A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then :

A

${1 \over {PS}} + {1 \over {ST}} < {2 \over {\sqrt {QS \times SR} }}$

B

${1 \over {PS}} + {1 \over {ST}} > {2 \over {\sqrt {QS \times SR} }}$

C

${1 \over {PS}} + {1 \over {ST}} < {4 \over {QR}}$

D

${1 \over {PS}} + {1 \over {ST}} > {4 \over {QR}}$

Question 8: Palette 8/48

JEE Advanced 2007 MCQ

The lines ${L_1}:y - x = 0$ and ${L_2}:2x + y = 0$ intersect the line ${L_3}:y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between ${L_1}$ and ${L_2}$ intersects ${L_3}$ at $R$.

Statement-1: The ratio $PR$ : $RQ$ equals $2\sqrt 2 :\sqrt 5 $. because
Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

A

Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement- 1

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

C

Statement-1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True.

Question 9: Palette 9/48

JEE Advanced 2007 MCQ

Let $O\left( {0,0} \right),P\left( {3,4} \right),Q\left( {6,0} \right)$ be the vertices of the triangles $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR$, $PQR$, $OQR$ are of equal area. The coordinates of $R$ are

A

$\left( {{4 \over 3},3} \right)$

B

$\left( {3,{2 \over 3}} \right)$

C

$\left( {3,{4 \over 3}} \right)$

D

$\left( {{4 \over 3},{2 \over 3}} \right)$

Question 10: Palette 10/48

JEE Advanced 2007 MCQ

Let $\mathrm{O(0,0), P(3,4), Q(6,0)}$ 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

A

$\left(\frac{4}{3}, 3\right)$

B

$\left(3, \frac{2}{3}\right)$

C

$\left(3, \frac{4}{3}\right)$

D

$\left(\frac{4}{3}, \frac{2}{3}\right)$

Question 11: Palette 11/48

JEE Advanced 2007 MCQ

Lines $\mathrm{L}_{1}: y-x=0$ and $\mathrm{L}_{2}: 2 x+y=0$ intersect the line $\mathrm{L}_{3}: y+2=0$ at $\mathrm{P}$ and $\mathrm{Q}$, respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$.

STATEMENT - 1 : The ratio PR : RQ equals $2 \sqrt{2}: \sqrt{5}$.

STATEMENT - 2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

A

Statement-1 is True, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True

Question 12: Palette 12/48

JEE Advanced 2007 MCQ

Consider the following linear equations

$ax + by + cz = 0$

$bx + cy + az = 0$

$cx + ay + bz = 0$

Match the conditions/expressions in Column I with statements in Column II.

	Column I		Column II
(A)	$a + b + c \ne 0$ and ${a^2} + {b^2} + {c^2} = ab + bc + ca$	(P)	the equations represent planes meeting only at a single point.
(B)	$a + b + c = 0$ and ${a^2} + {b^2} + {c^2} \ne ab + bc + ca$	(Q)	the equations represent the line $x=y=z$.
(C)	$a + b + c \ne 0$ and ${a^2} + {b^2} + {c^2} \ne ab + bc + ca$	(R)	the equations represent identical planes.
(D)	$a + b + c = 0$ and ${a^2} + {b^2} + {c^2} = ab + bc + ca$	(S)	the equations represent the whole of the three dimensional space.

A

A - (q), B - (r), C - (p), D - (s)

B

A - (r), B - (q), C - (s), D - (p)

C

A - (r), B - (p), C - (q), D - (s)

D

A - (r), B - (q), C - (p), D - (s)

Question 13: Palette 13/48

JEE Advanced 2005 MCQ

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through $(h, k)$ with the lines $y=x$ and $x+y=2$ is $4 h^{2}$. Find the locus of point $P$.

A

$3x=\pm~(y-1)$

B

$x=\pm~3(y-1)$

C

$2x=\pm~(y-1)$

D

$x=\pm~5(y-1)$

Question 14: Palette 14/48

JEE Advanced 2004 MCQ

Area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight line ${x^2} - {y^2} + 2y = 1$ is

A

2 sq. units

B

4 sq. units

C

6 sq. units

D

8 sq. units

Question 15: Palette 15/48

JEE Advanced 2003 MCQ

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $\left( {0,0} \right),\left( {0,21} \right)$ and $\left( {21,0} \right)$, is

A

133

B

190

C

233

D

105

Question 16: Palette 16/48

JEE Advanced 2003 MCQ

Orthocentre of triangle with vertices $\left( {0,0} \right),\left( {3,4} \right)$ and $\left( {4,0} \right)$ is

A

$\,\,\left( {3,{5 \over 4}} \right)$

B

$\left( {3,12} \right)$

C

$\left( {3,{3 \over 4}} \right)$

D

$\left( {3,9} \right)$

Question 17: Palette 17/48

JEE Advanced 2002 MCQ

Let $0 < \alpha < {\pi \over 2}$ be fixed angle. If $P = \left( {\cos \theta ,\,\sin \theta } \right)$ and $Q = \left( {\cos \left( {\alpha - \theta } \right),\,\sin \left( {\alpha - \theta } \right)} \right),$ then $Q$ is obtained from $P$ by

A

clockwise rotation around origin through an angle $\alpha $

B

anticlockwise rotation around origin through an angle $\alpha $

C

reflection in the line through origin with slope tan $\alpha $

D

reflection in the line through origin with slope tan $\left( {\alpha /2} \right)$

Question 18: Palette 18/48

JEE Advanced 2002 MCQ

A straight line through the origin $O$ meets the parallel lines $4x+2y=9$ and $2x+y+6=0$ at points $P$ and $Q$ respectively. Then the point $O$ divides the segemnt $PQ$ in the ratio

A

$1 : 2$

B

$3 : 4$

C

$2 : 1$

D

$4 : 3$

Question 19: Palette 19/48

JEE Advanced 2002 MCQ

Let $P = \left( { - 1,\,0} \right),\,Q = \left( {0,\,0} \right)$ and $R = \left( {3,\,3\sqrt 3 } \right)$ be three points.
Then the equation of the bisector of the angle $PQR$ is

A

${{\sqrt 3 } \over 2}x + y = 0$

B

$x + \sqrt 3 y = 0$

C

$\sqrt 3 x + y = 0$

D

$x + {{\sqrt 3 } \over 2}y = 0$

Question 20: Palette 20/48

JEE Advanced 2002 MCQ

If the pair of lines $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ intersect on the $y$ axis then

A

$2fgh = b{g^2} + c{h^2}$

B

$b{g^2} \ne c{h^2}$

C

$\,abc = 2fgh$

D

none of these

Question 21: Palette 21/48

JEE Advanced 2002 MCQ

A triangle with vertices $(4, 0), (-1, -1), (3, 5)$is

A

isosceles and right angled

B

isosceles but not right angled

C

right angled but not isosceles

D

neither right angled nor isosceles

Question 22: Palette 22/48

JEE Advanced 2002 MCQ

Locus of mid point of the portion between the axes of $x$ $\cos \alpha + y\sin \alpha = p$ where $p$ is constant is

A

${x^2} + {y^2} = {4 \over {{p^2}}}\,\,\,$

B

${x^2} + {y^2} = 4{p^2}$

C

${1 \over {{x^2}}} + {1 \over {{y^2}}} = {2 \over {{p^2}}}$

D

${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$

Question 23: Palette 23/48

JEE Advanced 2002 MCQ

The pair of lines represented by
$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for

A

two values of $a$

B

$\forall \,a$

C

for one values of $a$

D

for no values of $a$

Question 24: Palette 24/48

JEE Advanced 2001 MCQ

The number of integer values of $m$, for which the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer, is

A

2

B

0

C

4

D

1

Question 25: Palette 25/48

JEE Advanced 2001 MCQ

Area of the parallelogram formed by the lines $y = mx$, $y = mx + 1$, $y = nx$ and $y = nx + 1$ equals

A

$\left| {m + n} \right|/{\left( {m - n} \right)^2}$

B

$2/\left| {m + n} \right|$

C

$1/\left( {\left| {m + n} \right|} \right)$

D

$1/\left( {\left| {m - n} \right|} \right)$

Question 26: Palette 26/48

JEE Advanced 2000 MCQ

The incentre of the triangle with vertices $\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)$ and $\left( {2,\,0} \right)$ is

A

$\left( {1,\,{{\sqrt 3 } \over 2}} \right)$

B

$\left( {{2 \over 3},\,{1 \over {\sqrt 3 }}} \right)$

C

$\left( {{2 \over 3},\,{{\sqrt 3 } \over 2}} \right)$

D

$\left( {1,\,{1 \over {\sqrt 3 }}} \right)$

Question 27: Palette 27/48

JEE Advanced 2000 MCQ

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

A

$2x - 9y - 7 = 0$

B

$2x - 9y - 11 = 0$

C

$2x + 9y - 11 = 0$

D

$2x + 9y + 7 = 0$

Question 28: Palette 28/48

JEE Advanced 1999 MCQ

If ${x_1},\,{x_2},\,{x_3}$ as well as ${y_1},\,{y_2},\,{y_3}$, are in G.P. with the same common ratio, then the points $\left( {{x_1},\,{y_1}} \right),\left( {{x_2},\,{y_2}} \right)$ and $\left( {{x_3},\,{y_3}} \right).$

A

lie on a straight line

B

lie on an ellipse

C

lie on a circle

D

are vertices of a triangle

Question 29: Palette 29/48

JEE Advanced 1999 MCQ

Lt $PQR$ be a right angled isosceles triangle, right angled at $P(2, 1)$. If the equation of the line $QR$ is $2x + y = 3,$ then the equation representing the pair of lines $PQ$ and $PR$ is

A

$3{x^2} - 3{y^2} + 8xy + 20x + 10y + 25 = 0$

B

$3{x^2} - 3{y^2} + 8xy - 20x - 10y + 25 = 0$

C

$3{x^2} - 3{y^2} + 8xy + 10x + 15y + 20 = 0$

D

$3{x^2} - 3{y^2} - 8xy - 10x - 15y - 20 = 0$

Question 30: Palette 30/48

JEE Advanced 1999 MSQ

Let ${L_1}$ be a straight line passing through the origin and ${L_2}$ be the straight line $x + y = 1$. If the intercepts made by the circle ${x^2} + {y^2} - x + 3y = 0$ on ${L_1}$ and ${L_2}$ are equal, then which of the following equations can represent ${L_1}$?

A

$x + y = 0$

B

$x -y = 0$

C

$x + 7y = 0$

D

$x - 7y = 0$

Question 31: Palette 31/48

JEE Advanced 1998 MCQ

The diagonals of a parralleogram $PQRS$ are along the lines $x + 3y = 4$ and $6x - 2y = 7$. Then $PQRS$ must be a.

A

rectangle

B

square

C

cyclic quadrilateral

D

rhombus.

Question 32: Palette 32/48

JEE Advanced 1998 MCQ

If $\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)$ and $S\left( {a,b} \right)$ are the vertices of a parrallelogram $PQRS,$ then

A

$a = 2,\,b = 4$

B

$a = 3,\,b = 4$

C

$a = 2,\,b = 3$

D

$a = 3,\,b = 5$

Question 33: Palette 33/48

JEE Advanced 1998 MSQ

If the vertices $P, Q, R$ of a triangle $PQR$ are rational points, which of the following points of the triangle $PQR$ is (are) always rational point(s)?

A

centroid ( A rational point is a point both of whose co-ordinates are rational numbers.)

B

incentre. ( A rational point is a point both of whose co-ordinates are rational numbers.)

C

circumcentre ( A rational point is a point both of whose co-ordinates are rational numbers.)

D

orthocentre ( A rational point is a point both of whose co-ordinates are rational numbers.)

Question 34: Palette 34/48

JEE Advanced 1995 MCQ

The orthocentre of the triangle formed by the lines $xy=0$ and $x+y=1$ is

A

$\left( {{1 \over 2},\,{1 \over 2}} \right)$

B

$\left( {{1 \over 3},\,{1 \over 3}} \right)$

C

$\left( {0,\,0} \right)$

D

$\left( {{1 \over 4},\,{1 \over 4}} \right)$

Question 35: Palette 35/48

JEE Advanced 1994 MCQ

The locus of a variable point whose distance from $\left( { - 2,\,0} \right)$ is $2/3$ times its distance from the line $x = - {9 \over 2}$ is

A

ellipse -

B

parabola

C

hyperbola

D

none of these

Question 36: Palette 36/48

JEE Advanced 1994 MCQ

The equations to a pair of opposites sides of parallelogram are ${x^2} - 5x + 6 = 0$ and ${y^2} - 6y + 5 = 0,$ the equations to its diagonals are

A

$x + 4y = 13,\,y = 4x - 7$

B

$4x + y = 13,\,4y = x - 7$

C

$4x + y = 13,\,y = 4x - 7$

D

$y - 4x = 13,\,y + 4x = 7$

Question 37: Palette 37/48

JEE Advanced 1992 MCQ

If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is

A

square

B

circle

C

straight line

D

two intersecting lines

Question 38: Palette 38/48

JEE Advanced 1990 MCQ

Line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$, then

A

${a^2} + {b^2} = {p^2} + {q^2}$

B

${1 \over {{a^2}}} + {1 \over {{b^2}}} = {1 \over {{p^2}}} + {1 \over {{q^2}}}$

C

${a^2} + {p^2} = {b^2} + {q^2}$

D

${1 \over {{a^2}}} + {1 \over {{p^2}}} = {1 \over {{b^2}}} + {1 \over {{q^2}}}$

Question 39: Palette 39/48

JEE Advanced 1988 MCQ

If $P=(1, 0),$ $Q=(-1, 0)$ and $R=(2, 0)$ are three given points, then locus of the point $S$ satisfying the relation $S{Q^2} + S{R^2} = 2S{P^2},$ is

A

a straight line parallel to x-axis

B

a circle passing through the origin

C

a circle with the centre at the origin

D

a straight line parallel to y-axis.

Question 40: Palette 40/48

JEE Advanced 1988 MCQ

The lines $2x + 3y + 19 = 0$ and $9x + 6y - 17 = 0$ cut the coordinates axes in concyclic points.

A

TRUE

B

FALSE

Question 41: Palette 41/48

JEE Advanced 1986 MCQ

The points $\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)$ and $\left( {82,\,30} \right)$ are vertices of

A

an obtuse angled triangle

B

an acute angled triangle

C

a right angled triangle

D

none of these

Question 42: Palette 42/48

JEE Advanced 1986 MCQ

A vector $\overline a $ has components $2p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $\overline a $ has components $p + 1$ and $1$, then

A

$p = 0$

B

$p = 1$ or $p = - {1 \over 3}$

C

$\,p = - 1$ or $p = {1 \over 3}$

D

$p = 1$ or $p = -1$

Question 43: Palette 43/48

JEE Advanced 1986 MSQ

All points lying inside the triangle formed by the points $\left( {1,\,3} \right),\,\left( {5,\,0} \right)$ and $\left( { - 1,\,2} \right)$ satisfy

A

$3x + 2y \ge 0$

B

$2x + y - 13 \ge 0$

C

$2x - 3y - 12 \le 0$

D

$ - 2x + y \ge 0$

Question 44: Palette 44/48

JEE Advanced 1985 MSQ

Three lines $px + qy + r = 0$, $qx + ry + p = 0$ and $rx + py + q = 0$ are concurrent if

A

$p + q + r = 0$

B

${p^2} + {q^2} + {r^2} = qr + rp + pq$

C

${p^3} + {q^3} + {r^3} = 3pqr$

D

none of these.

Question 45: Palette 45/48

JEE Advanced 1983 MCQ

The straight lines $x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0$ form a triangle which is

A

isosceles

B

equilateral

C

right angled

D

none of these

Question 46: Palette 46/48

JEE Advanced 1983 MCQ

The straight line $5x + 4y = 0$ passes through the point of intersection of the straight lines $x + 2y - 10 = 0$ and $2x + y + 5 = 0.$

A

TRUE

B

FALSE

Question 47: Palette 47/48

JEE Advanced 1980 MCQ

The point $\,\left( {4,\,1} \right)$ undergoes the following three transformations successively.
Reflection about the line $y=x$.
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle $p/4$ about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.

A

$\left( {{1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$

B

$\left( { - \sqrt 2 ,\,7\sqrt 2 } \right)$

C

$\left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$

D

$\left( { \sqrt 2 ,\,7\sqrt 2 } \right)$

Question 48: Palette 48/48

JEE Advanced 1979 MCQ

The points $\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)$ and $\left( {{a^2},\,ab} \right)$ are :

A

Collinear

B

Vertices of a parallelogram

C

Vertices of a rectangle

D

None of these