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Straight Lines and Pair of Straight Lines
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Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ ...
JEE Main 2022 (Online) 27th July Morning Shift
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A point $$P$$ moves so that the sum of squares of its distances from the points $$(1,2)$$ and $$(-2,1)$$ is 14. Let $$f(...
JEE Main 2022 (Online) 26th July Morning Shift
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Let the point $$P(\alpha, \beta)$$ be at a unit distance from each of the two lines $$L_{1}: 3 x-4 y+12=0$$, and $$L_{2}...
JEE Main 2022 (Online) 25th July Evening Shift
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A line, with the slope greater than one, passes through the point $$A(4,3)$$ and intersects the line $$x-y-2=0$$ at the ...
JEE Main 2022 (Online) 25th July Morning Shift
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Let $$\alpha$$1, $$\alpha$$2 ($$\alpha$$1 2) be the values of $$\alpha$$ fo the points ($$\alpha$$, $$-$$3), (2, 0) and ...
JEE Main 2022 (Online) 30th June Morning Shift
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Let the area of the triangle with vertices A(1, $$\alpha$$), B($$\alpha$$, 0) and C(0, $$\alpha$$) be 4 sq. units. If th...
JEE Main 2022 (Online) 24th June Evening Shift
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Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. ...
JEE Main 2022 (Online) 26th June Morning Shift
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If two straight lines whose direction cosines are given by the relations $$l + m - n = 0$$, $$3{l^2} + {m^2} + cnl = 0$$...
JEE Main 2022 (Online) 27th June Morning Shift
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In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie ...
JEE Main 2022 (Online) 27th June Morning Shift
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Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : $$-$$4x + 3y = 12 and the line L3, which passes through t...
JEE Main 2022 (Online) 28th June Evening Shift
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The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $...
JEE Main 2022 (Online) 29th June Evening Shift
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The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is...
JEE Main 2022 (Online) 29th June Morning Shift
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Let A be the set of all points ($$\alpha$$, $$\beta$$) such that the area of triangle formed by the points (5, 6), (3, 2...
JEE Main 2021 (Online) 31st August Evening Shift
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If p and q are the lengths of the perpendiculars from the origin on the lines,x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ ...
JEE Main 2021 (Online) 31st August Morning Shift
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The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl...
JEE Main 2021 (Online) 27th August Evening Shift
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Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisect...
JEE Main 2021 (Online) 27th August Morning Shift
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Let ABC be a triangle with A($$-$$3, 1) and $$\angle$$ACB = $$\theta$$, 0 < $$\theta$$ < $${\pi \over 2}$$. If th...
JEE Main 2021 (Online) 26th August Morning Shift
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Let the equation of the pair of lines, y = px and y = qx, can be written as (y $$-$$ px) (y $$-$$ qx) = 0. Then the equa...
JEE Main 2021 (Online) 25th July Evening Shift
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Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of...
JEE Main 2021 (Online) 27th July Evening Shift
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The point P (a, b) undergoes the following three transformations successively :(a) reflection about the line y = x.(b) t...
JEE Main 2021 (Online) 27th July Evening Shift
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Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be al...
JEE Main 2021 (Online) 18th March Evening Shift
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The equation of one of the straight lines which passes through the point (1, 3) and makes an angles $${\tan ^{ - 1}}\lef...
JEE Main 2021 (Online) 18th March Morning Shift
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The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is ...
JEE Main 2021 (Online) 18th March Morning Shift
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In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation ...
JEE Main 2021 (Online) 17th March Morning Shift
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Let A($$-$$1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at poin...
JEE Main 2021 (Online) 16th March Evening Shift
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The intersection of three lines x $$-$$ y = 0, x + 2y = 3 and 2x + y = 6 is a :
JEE Main 2021 (Online) 26th February Morning Shift
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The image of the point (3, 5) in the line x $$-$$ y + 1 = 0, lies on :
JEE Main 2021 (Online) 25th February Morning Slot
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A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts $${1 \over 4}$$ of this li...
JEE Main 2021 (Online) 24th February Morning Slot
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A ray of light coming from the point (2, $$2\sqrt 3 $$) is incident at an angle 30o on the line x = 1 at the point A. Th...
JEE Main 2020 (Online) 6th September Morning Slot
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If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, th...
JEE Main 2020 (Online) 4th September Evening Slot
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Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. ...
JEE Main 2020 (Online) 4th September Morning Slot
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If a $$\Delta $$ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates :
JEE Main 2020 (Online) 3rd September Evening Slot
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The set of all possible values of $$\theta $$ in the interval (0, $$\pi $$) for which the points (1, 2) and (sin $$\the...
JEE Main 2020 (Online) 2nd September Evening Slot
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Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of th...
JEE Main 2020 (Online) 9th January Morning Slot
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Let two points be A(1, –1) and B(0, 2). If a point P(x', y') be such that the area of $$\Delta $$PAB = 5 sq. units and i...
JEE Main 2020 (Online) 8th January Morning Slot
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The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is:
JEE Main 2020 (Online) 7th January Evening Slot
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A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perp...
JEE Main 2019 (Online) 12th April Evening Slot
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Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance $${3 \over 5}$$ from the origin. Then which one of ...
JEE Main 2019 (Online) 10th April Evening Slot
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The region represented by| x – y | $$ \le $$ 2 and | x + y| $$ \le $$ 2 is bounded by a :
JEE Main 2019 (Online) 10th April Morning Slot
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If the two lines x + (a – 1) y = 1 and 2x + a2y = 1 (a$$ \in $$R – {0, 1}) are perpendicular, then the distance of their...
JEE Main 2019 (Online) 9th April Evening Slot
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Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is
JEE Main 2019 (Online) 9th April Morning Slot
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Suppose that the points (h,k), (1,2) and (–3,4) lie on the line L1 . If a line L2 passing through the points (h,k) and ...
JEE Main 2019 (Online) 8th April Evening Slot
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If the system of linear equations x – 2y + kz = 1 2x + y + z = 2 3x – y – kz = 3 has a solution (x,y,z), z $$ \ne $$ 0, ...
JEE Main 2019 (Online) 8th April Evening Slot
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Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of $$\Delta $$AOP is 4,...
JEE Main 2019 (Online) 8th April Morning Slot
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A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in :
JEE Main 2019 (Online) 8th April Morning Slot
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If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes i...
JEE Main 2019 (Online) 12th January Evening Slot
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If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, $$\beta ...
JEE Main 2019 (Online) 12th January Morning Slot
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If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation ...
JEE Main 2019 (Online) 11th January Evening Slot
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Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in whi...
JEE Main 2019 (Online) 10th January Evening Slot
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Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), t...
JEE Main 2019 (Online) 10th January Evening Slot
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If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of th...
JEE Main 2019 (Online) 10th January Morning Slot
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A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid ...
JEE Main 2019 (Online) 10th January Morning Slot
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If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to -
JEE Main 2019 (Online) 10th January Morning Slot
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Let the equations of two sides of a triangle be 3x $$-$$ 2y + 6 = 0 and 4x + 5y $$-$$ 20 = 0. If the orthocentre of this...
JEE Main 2019 (Online) 9th January Evening Slot
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Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is tru...
JEE Main 2019 (Online) 9th January Morning Slot
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The foot of the perpendicular drawn from the origin, on the line, 3x + y = $$\lambda $$ ($$\lambda $$ $$ \ne $$ 0) is P....
JEE Main 2018 (Online) 15th April Evening Slot
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The sides of a rhombus ABCD are parallel to the lines, x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0. If the diagonals of the...
JEE Main 2018 (Online) 15th April Evening Slot
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In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x + y ...
JEE Main 2018 (Online) 15th April Morning Slot
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A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the orig...
JEE Main 2018 (Offline)
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A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through th...
JEE Main 2017 (Online) 9th April Morning Slot
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Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the o...
JEE Main 2017 (Offline)
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A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O...
JEE Main 2016 (Online) 10th April Morning Slot
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A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then r...
JEE Main 2016 (Online) 10th April Morning Slot
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The point (2, 1) is translated parallel to the line L : x− y = 4 by $$2\sqrt 3 $$ units. If the newpoint Q lies in the t...
JEE Main 2016 (Online) 9th April Morning Slot
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If a variable line drawn through the intersection of the lines $${x \over 3} + {y \over 4} = 1$$ and $${x \over 4} + {y...
JEE Main 2016 (Online) 9th April Morning Slot
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Two sides of a rhombus are along the lines, $$x - y + 1 = 0$$ and $$7x - y - 5 = 0$$. If its diagonals intersect at $$(...
JEE Main 2016 (Offline)
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The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $$(0,...
JEE Main 2015 (Offline)
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Let $$a, b, c$$ and $$d$$ be non-zero numbers. If the point of intersection of the lines $$4ax + 2ay + c = 0$$ and $$5bx...
JEE Main 2014 (Offline)
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Let $$PS$$ be the median of the triangle with vertices $$P(2, 2)$$, $$Q(6, -1)$$ and $$R(7, 3)$$. The equation of the li...
JEE Main 2014 (Offline)
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The $$x$$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $$(0, 1) (1,...
JEE Main 2013 (Offline)
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A ray of light along $$x + \sqrt 3 y = \sqrt 3 $$ gets reflected upon reaching $$X$$-axis, the equation of the reflected...
JEE Main 2013 (Offline)
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If the line $$2x + y = k$$ passes through the point which divides the line segment joining the points $$(1, 1)$$ and $$(...
AIEEE 2012
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The lines $${L_1}:y - x = 0$$ and $${L_2}:2x + y = 0$$ intersect the line $${L_3}:y + 2 = 0$$ at $$P$$ and $$Q$$ respect...
AIEEE 2011
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The line $$L$$ given by $${x \over 5} + {y \over b} = 1$$ passes through the point $$\left( {13,32} \right)$$. The line ...
AIEEE 2010
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Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance o...
AIEEE 2009
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The lines $$p\left( {{p^2} + 1} \right)x - y + q = 0$$ and $$\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right...
AIEEE 2009
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The shortest distance between the line $$y - x = 1$$ and the curve $$x = {y^2}$$ is :
AIEEE 2009
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The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of ...
AIEEE 2008
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If one of the lines of $$m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$$ is a bisector of angle between the lines $...
AIEEE 2007
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Let A $$\left( {h,k} \right)$$, B$$\left( {1,1} \right)$$ and C $$(2, 1)$$ be the vertices of a right angled triangle wi...
AIEEE 2007
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Let $$P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)$$ and $$R = \left( {3,3\sqrt 3 } \right)$$ be three point. ...
AIEEE 2007
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If $$\left( {a,{a^2}} \right)$$ falls inside the angle made by the lines $$y = {x \over 2},$$ $$x > 0$$ and $$y = 3x,...
AIEEE 2006
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A straight line through the point $$A (3, 4)$$ is such that its intercept between the axes is bisected at $$A$$. Its equ...
AIEEE 2006
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If a vertex of a triangle is $$(1, 1)$$ and the mid points of two sides through this vertex are $$(-1, 2)$$ and $$(3, 2)...
AIEEE 2005
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The line parallel to the $$x$$ - axis and passing through the intersection of the lines $$ax + 2by + 3b = 0$$ and $$bx -...
AIEEE 2005
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If one of the lines given by $$6{x^2} - xy + 4c{y^2} = 0$$ is $$3x + 4y = 0,$$ then $$c$$ equals
AIEEE 2004
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If the sum of the slopes of the lines given by $${x^2} - 2cxy - 7{y^2} = 0$$ is four times their product $$c$$ has the v...
AIEEE 2004
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Let $$A\left( {2, - 3} \right)$$ and $$B\left( {-2, 1} \right)$$ be vertices of a triangle $$ABC$$. If the centroid of ...
AIEEE 2004
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The equation of the straight line passing through the point $$(4, 3)$$ and making intercepts on the co-ordinate axes who...
AIEEE 2004
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If the equation of the locus of a point equidistant from the point $$\left( {{a_{1,}}{b_1}} \right)$$ and $$\left( {{a_{...
AIEEE 2003
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Locus of centroid of the triangle whose vertices are $$\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \rig...
AIEEE 2003
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If $${x_1},{x_2},{x_3}$$ and $${y_1},{y_2},{y_3}$$ are both in G.P. with the same common ratio, then the points $$\left(...
AIEEE 2003
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If the pair of straight lines $${x^2} - 2pxy - {y^2} = 0$$ and $${x^2} - 2qxy - {y^2} = 0$$ be such that each pair bisec...
AIEEE 2003
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A square of side a lies above the $$x$$-axis and has one vertex at the origin. The side passing through the origin makes...
AIEEE 2003
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If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$-axis then
AIEEE 2002
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The pair of lines represented by $$$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$$ are perpendicular to each ot...
AIEEE 2002
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Locus of mid point of the portion between the axes of $$x$$ $$cos$$ $$\alpha + y\,\sin \alpha = p$$ where $$p$$ is c...
AIEEE 2002
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A triangle with vertices $$\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$$ is
AIEEE 2002
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Numerical
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The equations of the sides $$\mathrm{AB}, \mathrm{BC}$$ and $$\mathrm{CA}$$ of a triangle $$\mathrm{ABC}$$ are $$2 x+y=0...
JEE Main 2022 (Online) 26th July Morning Shift
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Let $$A\left( {{3 \over {\sqrt a }},\sqrt a } \right),\,a > 0$$, be a fixed point in the xy-plane. The image of A in y-a...
JEE Main 2022 (Online) 24th June Morning Shift
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A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through ...
JEE Main 2022 (Online) 28th June Morning Shift
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A man starts walking from the point P($$-$$3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2)....
JEE Main 2021 (Online) 1st September Evening Shift
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Let the points of intersections of the lines x $$-$$ y + 1 = 0, x $$-$$ 2y + 3 = 0 and 2x $$-$$ 5y + 11 = 0 are the mid ...
JEE Main 2021 (Online) 1st September Evening Shift
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Consider a triangle having vertices A($$-$$2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of ...
JEE Main 2021 (Online) 20th July Evening Shift
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A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, t...
JEE Main 2021 (Online) 18th March Morning Shift
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Let tan$$\alpha$$, tan$$\beta$$ and tan$$\gamma$$; $$\alpha$$, $$\beta$$, $$\gamma$$ $$\ne$$ $${{(2n - 1)\pi } \over 2}$...
JEE Main 2021 (Online) 17th March Evening Shift
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The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y $$ \le $$ 100 and 4x + 3y $$ \le $$ 75 for ...
JEE Main 2021 (Online) 17th March Morning Shift
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Let $$\lambda$$ be an integer. If the shortest distance between the lines x $$-$$ $$\lambda$$ = 2y $$-$$ 1 = $$-$$2z and...
JEE Main 2021 (Online) 24th February Evening Slot
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If the line, 2x - y + 3 = 0 is at a distance $${1 \over {\sqrt 5 }}$$ and $${2 \over {\sqrt 5 }}$$ from the lines 4x - 2...
JEE Main 2020 (Online) 5th September Morning Slot
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If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through ($$\alpha $$, 7, 1) is $$\left...
JEE Main 2020 (Online) 7th January Evening Slot
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Let A(1, 0), B(6, 2) and C $$\left( {{3 \over 2},6} \right)$$ be the vertices of a triangle ABC. If P is a Point inside ...
JEE Main 2020 (Online) 7th January Morning Slot
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