JEE Mains Previous Year Questions (PYQ)

JEE Main 2021 27th August Evening Shift

Let $ A(a,0) $, $ B(b, 2b + 1) $ and $ C(0,b) $, $ b \neq 0 $, $ |b| \neq 1 $, be points such that the area of triangle $ ABC $ is $ 1 \, \text{sq. unit} $, then the sum of all possible values of $ a $ is :

7.

A variable line $ L $ passes through the point $ (3,5) $ and intersects the positive coordinate axes at the points $ A $ and $ B $. The minimum area of the triangle $ OAB $, where $ O $ is the origin, is:

8.

The equations of the sides $ AB $, $ BC $ and $ CA $ of a triangle $ ABC $ are $ 2x+y=0 $, $ x+py=15a $ and $ x-y=3 $ respectively. If its orthocentre is $ (2,a) $, $ -\dfrac{1}{2}<a<2$, then $p$ is equal to________.

9.

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 bisector of $ AB $ meets the $ y $-axis at $ C $. The locus of the mid-point $ P $ of $ MC $ is:

10.

A line passing through the point $ A(9, 0) $, makes an angle of $ 30^{\circ} $ with the positive direction of $ x $-axis. If this line is rotated about $ A $ through an angle of $ 15^{\circ} $ in the clockwise direction, then its equation in the new position is:

11.

Let $ C(\alpha, \beta) $ be the circumcenter of the triangle formed by the lines $ 4x + 3y = 69 $, $ 4y - 3x = 17 $ and $ x + 7y = 61 $. Then $ (\alpha - \beta)^2 + \alpha + \beta $ is equal to:

12.

Let $ A\!\left(\dfrac{3}{\sqrt{a}}, \sqrt{a}\right) $, $ a>0 $, be a fixed point in the $ xy $-plane. The image of $ A $ in $ y $-axis be $ B $ and the image of $ B $ in $ x $-axis be $ C $. If $ D(3\cos \theta, a \sin \theta) $ is a point in the fourth quadrant such that the maximum area of $ \triangle ACD $ is $ 12 $ square units, then $ a $ is equal to _________.

13.

If the line $ \ell_{1}: 3y-2x=3 $ is the angular bisector of the lines $ \ell_{2}: x-y+1=0 $ and $ \ell_{3}: \alpha x+\beta y+17=0 $, then $ \alpha^{2}+\beta^{2}-\alpha-\beta $ is equal to ______.

14.

The vertices of a triangle are $ A(-1,3) $, $ B(-2,2) $ and $ C(3,-1) $. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

15.

In an equilateral triangle $ PQR $, let the vertex $ P $ be at $ (3, 5) $ and the side $ QR $ be along the line $ x + y = 4 $. If the orthocentre of the triangle $ PQR $ is $ (\alpha, \beta) $, then $ 9(\alpha + \beta) $ is equal to:

16.

A light ray emits from the origin making an angle $ 30^\circ $ with the positive $ x $-axis. After getting reflected by the line $ x+y=1 $, if this ray intersects $ x $-axis at $ Q $, then the abscissa of $ Q $ is:

17.

Let $ \triangle ABC $ be a triangle formed by the lines $ 7x-6y+3=0 $, $ x+2y-31=0 $ and $ 9x-2y-19=0 $. Let the point $ (h,k) $ be the image of the centroid of $ \triangle ABC $ in the line $ 3x+6y-53=0 $. Then $ h^2+k^2+hk $ is equal to:

18.

Two sides of a parallelogram are along the line $ 4x+5y=0 $ and $ 7x+2y=0 $. If the equation of one of the diagonals of the parallelogram is $ 11x+7y=9 $, then other diagonal passes through the point:

19.

Let $ PQR $ be a triangle. The points $ A, B $ and $ C $ are on the sides $ QR, RP $ and $ PQ $ respectively such that $ \frac{QA}{AR}=\frac{RB}{BP}=\frac{PC}{CQ}=\frac{1}{2} $. Then $ \frac{\text{Area}(\triangle PQR)}{\text{Area}(\triangle ABC)} $ is equal to:

20.

Among the statements:
(S1): If $ A(5,-1) $ and $ B(-2,3) $ are two vertices of a triangle, whose orthocentre is $ (0,0) $, then its third vertex is $ (-4,-7) $
and
(S2): If positive numbers $ 2a,b,c $ are three consecutive terms of an A.P., then the lines $ ax+by+c=0 $ are concurrent at $ (2,-2) $,

21.

Let the lines $ 3x - 4y - \alpha = 0 $, $ 8x - 11y - 33 = 0 $ and $ 2x - 3y + \lambda = 0 $ be concurrent. If the image of the point $ (1, 2) $ in the line $ 2x - 3y + \lambda = 0 $ is $ \left( \frac{57}{13}, \frac{-40}{13} \right) $, then $ |\alpha \lambda| $ is equal to:

22.

Let the area of a $ \triangle PQR $ with vertices $ P(5, 4) $, $ Q(-2, 4) $ and $ R(a, b) $ be $ 35 $ square units. If its orthocenter and centroid are $ O\!\left(2, \dfrac{14}{5}\right) $ and $ C(c, d) $ respectively, then $ c + 2d $ is equal to:

23.

If $ (\alpha,\beta) $ is the orthocenter of the triangle $ ABC $ with vertices $ A(3,-7) $, $ B(-1,2) $ and $ C(4,5) $, then $ 9\alpha-6\beta+60 $ is equal to:

24.

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 also an integer, is:

25.

Let $ A(\alpha,-2), B(\alpha,6) $ and $ C\!\left(\dfrac{\alpha}{4},-2\right) $ be vertices of a $ \triangle ABC $. If $ \left(5,\dfrac{\alpha}{4}\right) $ is the circumcentre of $ \triangle ABC $, then which of the following is NOT correct about $ \triangle ABC $?

26.

If $ p $ and $ q $ are the lengths of the perpendiculars from the origin on the lines, $ x cosec \alpha - y \,\text{sec}\, \alpha = k \cot 2\alpha $ and $ x \sin \alpha + y \cos \alpha = k \sin 2\alpha $ respectively, then $ k^{2} $ is equal to:

27.

Let the vertex $ A $ of a triangle $ ABC $ be $ (1, 2) $, and the mid-point of the side $ AB $ be $ (5, -1) $. If the centroid of this triangle is $ (3, 4) $ and its circumcenter is $ (\alpha, \beta) $, then $ 21(\alpha + \beta) $ is equal to:

28.

Let $ m_1,m_2 $ be the slopes of two adjacent sides of a square of side $ a $ such that $ a^2+11a+3\!\left(m_1^2+m_2^2\right)=220 $. If one vertex of the square is $ \left(10(\cos\alpha-\sin\alpha),\,10(\sin\alpha+\cos\alpha)\right) $, where $ \alpha\in\!\left(0,\dfrac{\pi}{2}\right) $ and the equation of one diagonal is $ (\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10 $, then $ 72(\sin^4\alpha+\cos^4\alpha)+a^2-3a+13 $ is equal to:

29.

Let the angles made with the positive $ x $-axis by two straight lines drawn from the point $ P(2,3) $ and meeting the line $ x+y=6 $ at a distance $ \sqrt{\frac{2}{3}} $ from the point $ P $ be $ \theta_1 $ and $ \theta_2 $. Then the value of $ (\theta_1+\theta_2) $ is:

30.

Let two straight lines drawn from the origin $ O $ intersect the line $ 3x + 4y = 12 $ at the point $ P $ and $ Q $ such that $ \triangle OPQ $ is an isosceles triangle and $ \angle POQ = 90^\circ $. If $ l = OP^2 + PQ^2 + OQ^2 $, then the greatest integer less than or equal to $ l $ is:

31.

A rod of length $ \text {eight} \, \text{units} $ moves such that its ends $ A $ and $ B $ always lie on the line $ x - y + 2 = 0 $ and $ y + 2 = 0 $, respectively. If the locus of the point $ P $, that divides the rod $ AB $ internally in the ratio $ 2:1 $ is $ 9( x^{2} + \alpha y^{2} + \beta xy + \gamma x + 28y) - 76 = 0 $, then $ \alpha - \beta - \gamma $ is equal to:

32.

Let $ A(1,0) $, $ B(2,-1) $ and $ C\!\left(\frac{7}{3}, \frac{4}{3}\right) $ be three points. If the equation of the bisector of the angle $ ABC $ is $ \alpha x + \beta y = 5 $, then the value of $ \alpha^2 + \beta^2 $ is:

33.

A ray of light coming from the point $ P(1, 2) $ gets reflected from the point $ Q $ on the $ x $-axis and then passes through the point $ R(4, 3) $. If the point $ S(h, k) $ is such that $ PQRS $ is a parallelogram, then $ hk^2 $ is equal to:

34.

Let a point $ A $ lie between the parallel lines $ L_1 $ and $ L_2 $ such that its distances from $ L_1 $ and $ L_2 $ are $ 6 $ and $ 3 $ units, respectively. Then the area (in sq. units) of the equilateral triangle $ ABC $, where the points $ B $ and $ C $ lie on the lines $ L_1 $ and $ L_2 $, respectively, is:

35.

Let the equation of the pair of lines, $ y = px $ and $ y = qx $, can be written as $ (y - px)(y - qx) = 0 $. Then the equation of the pair of the angle bisectors of the lines $ x^2 - 4xy - 5y^2 = 0 $ is:

36.

Let $ \alpha, \beta, \gamma, \delta \in \mathbb{Z} $ and let $ A(\alpha,\beta) $, $ B(1,0) $, $ C(\gamma,\delta) $ and $ D(1,2) $ be the vertices of a parallelogram $ ABCD $. If $ AB=\sqrt{10} $ and the points $ A $ and $ C $ lie on the line $ 3y=2x+1 $, then $ 2(\alpha+\beta+\gamma+\delta) $ is equal to:

37.

The portion of the line $ 4x + 5y = 20 $ in the first quadrant is trisected by the lines $ L_1 $ and $ L_2 $ passing through the origin. The tangent of an angle between the lines $ L_1 $ and $ L_2 $ is:

38.

Let $ \triangle ABC $ be an isosceles triangle in which $ A $ is at $ (-1, 0) $, $ \angle A = \dfrac{2\pi}{3} $, $ AB = AC $ and $ B $ is on the positive $ x $-axis. If $ BC = 4\sqrt{3} $ and the line $ BC $ intersects the line $ y = x + 3 $ at $ (\alpha, \beta) $ then $ \dfrac{\beta^{4}}{\alpha^{2}} $ is __________.

39.

Let a circle of radius $ 4 $ pass through the origin $ O $, the point $ A(-\sqrt{3}\,a, 0) $ and $ B(0, -\sqrt{2}\,b) $, where $ a $ and $ b $ are real parameters and $ ab \neq 0 $. Then the locus of the centroid of $ \triangle OAB $ is a circle of radius:

40.

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 point $ P $ and $ Q $ respectively. Let $ A_{1} $ and $ A_{2} $ be the area of $ \triangle ABC $ and $ \triangle PQC $ respectively, such that $ A_{1}=3A_{2} $, then the value of $ m $ is equal to:

41.

A line passing through the point $ P(a, 0) $ makes an acute angle $ \alpha $ with the positive $ x $-axis. Let this line be rotated about the point $ P $ through an angle $ \dfrac{\alpha}{2} $ in the clock-wise direction. If in the new position, the slope of the line is $ 2 - \sqrt{3} $ and its distance from the origin is $ \dfrac{1}{\sqrt{2}} $, then the value of $ 3a^2 \tan^2 \alpha - 2\sqrt{3} $ is:

42.

Let the line $ L_1 : x + 3 = 0 $ intersect the lines $ L_2 : x - y = 0 $ and $ L_3 : 3x + y = 0 $ at the points $ A $ and $ B $, respectively. Let the bisector of the obtuse angle between the lines $ L_2 $ and $ L_3 $ intersect the line $ L_1 $ at the point $ C $. Then $ BC^2 : AC^2 $ is equal to:

43.

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 point $ B $. If the length of the line segment $ AB $ is $ \dfrac{\sqrt{29}}{3} $, then $ B $ also lies on the line:

44.

Let the point $ P(\alpha,\beta) $ be at a unit distance from each of the two lines $ L_1: 3x-4y+12=0 $, and $ L_2: 8x+6y+11=0 $. If $ P $ lies below $ L_1 $ and above $ L_2 $, then $ 100(\alpha+\beta) $ is equal to:

45.

Let $ B $ and $ C $ be the two points on the line $ y + x = 0 $ such that $ B $ and $ C $ are symmetric with respect to the origin. Suppose $ A $ is a point on $ y - 2x = 2 $ such that $ \triangle ABC $ is an equilateral triangle. Then, the area of the $ \triangle ABC $ is:

46.

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 the point $ Q(5, 4) $. Let $ R $ be the point that divides the line segment $ AQ $ internally into the ratio $ 2\!:\!1 $. Let the co-ordinates of the foot of the perpendicular $ M $ from $ R $ on the bisector of the angle $ PAQ $ be $ (\alpha, \beta) $. Then, the value of $ 7\alpha + 3\beta $ is equal to ________.

47.

Let $ A(1,2) $ and $ C(-3,-6) $ be two diagonally opposite vertices of a rhombus, whose sides $ AD $ and $ BC $ are parallel to the line $ 7x - y = 14 $. If $ B(\alpha,\beta) $ and $ D(\gamma,\delta) $ are the other two vertices, then $ |\alpha + \beta + \gamma + \delta| $ is equal to:

48.

Let $ A(-1,1) $ and $ B(2,3) $ be two points and $ P $ be a variable point above the line $ AB $ such that the area of $ \triangle PAB $ is $ 10 $. If the locus of $ P $ is $ ax+by=15 $, then $ 5a+2b $ is:

49.

If the image of the point $ (-4,5) $ in the line $ x+2y=2 $ lies on the circle $ (x+4)^{2}+(y-3)^{2}=r^{2} $, then $ r $ is equal to:

50.

Let $ a $ be the length of a side of a square $ OABC $ with $ O $ being the origin. Its side $ OA $ makes an acute angle $ \alpha $ with the positive $ x $-axis and the equations of its diagonals are $ (\sqrt{3}+1)x + (\sqrt{3}-1)y = 0 $ and $ (\sqrt{3}-1)x - (\sqrt{3}+1)y + 8\sqrt{3} = 0 $. Then $ a^{2} $ is equal to:

51.

Let $ A $ be the set of all points $ (\alpha,\,\beta) $ such that the area of triangle formed by the points $ (5,\,6) $, $ (3,\,2) $ and $ (\alpha,\,\beta) $ is $ 12 $ square units. Then the least possible length of a line segment joining the origin to a point in $ A $, is:

52.

Let a triangle be bounded by the lines $ L_1: 2x + 5y = 10 $; $ L_2: -4x + 3y = 12 $ and the line $ L_3 $, which passes through the point $ P(2,3) $, intersects $ L_2 $ at $ A $ and $ L_1 $ at $ B $. If the point $ P $ divides the line-segment $ AB $, internally in the ratio $ 1:3 $, then the area of the triangle is equal to:

53.

Let the mid points of the sides of a triangle $ ABC $ be $ \left(\dfrac{5}{2},7\right) $, $ \left(\dfrac{5}{2},3\right) $ and $ (4,5) $. If its incentre is $ (h,k) $, then $ 3h+k $ is equal to:

54.

Let the area of the triangle formed by a straight line $ L: x + by + c = 0 $ with co-ordinate axes be $ 48 $ square units. If the perpendicular drawn from the origin to the line $ L $ makes an angle of $ 45^\circ $ with the positive $ x $-axis, then the value of $ b^2 + c^2 $ is:

55.

The point $ P(a, b) $ undergoes the following three transformations successively:
(a) Reflection about the line $ y = x $
(b) Translation through $ 2 $ units along the positive direction of x-axis
(c) Rotation through angle $ \dfrac{\pi}{4} $ about the origin in the anti-clockwise direction
If the co-ordinates of the final position of the point $ P $ are $ \left( -\dfrac{1}{\sqrt{2}}, \dfrac{7}{\sqrt{2}} \right) $, then the value of $ 2a + b $ is equal to:

56.

Let $ ABC $ be an equilateral triangle with orthocenter at the origin and the side $ BC $ on the line $ x + 2\sqrt{2}\,y = 4 $. If the co-ordinates of the vertex $ A $ are $ (\alpha, \beta) $ then the greatest integer less than or equal to $ \lvert \alpha + \sqrt{2}\beta \rvert $ is:

57.

The equations of the sides $ AB $ and $ AC $ of a triangle $ ABC $ are $ (\lambda + 1)x + \lambda y = 4 $ and $ \lambda x + (1-\lambda)y + \lambda = 0 $ respectively. Its vertex $ A $ is on the y-axis and its orthocentre is $ (1, 2) $. The length of the tangent from the point $ C $ to the part of the parabola $ y^{2} = 6x $ in the first quadrant is:

58.

Let $ \tan\alpha, \tan\beta $ and $ \tan\gamma \,;\ \alpha,\beta,\gamma \neq \dfrac{(2n-1)\pi}{2},\ n\in \mathbb{N} $ be the slopes of three line segments $ OA $, $ OB $ and $ OC $, respectively, where $ O $ is origin. If circumcentre of $ \triangle ABC $ coincides with origin and its orthocentre lies on $ y $-axis, then the value of $ \left(\dfrac{\cos 3\alpha+\cos 3\beta+\cos 3\gamma}{\cos\alpha\cos\beta\cos\gamma}\right)^{2} $ is equal to ________.

59.

The equations of two sides $ AB $ and $ AC $ of a triangle $ ABC $ are $ 4x + y = 14 $ and $ 3x - 2y = 5 $, respectively. The point $ \left(2, \frac{4}{3}\right) $ divides the third side $ BC $ internally in the ratio $ 2:1 $, the equation of the side $ BC $ is:

60.

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $ L_1: 2x + y + 6 = 0 $ and $ L_2: 4x + 2y - p = 0, \, p > 0 $, at the points $ A $ and $ B $, respectively. If $ AB = \frac{9}{\sqrt{2}} $ and the foot of the perpendicular from the point $ A $ on the line $ L_2 $ is $ M $, then $ \frac{AM}{BM} $ is equal to:

61.

Let $ A(a,b) $, $ B(3,4) $ and $ C(-6,-8) $ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $ P(2a+3,7b+5) $ from the line $ 2x+3y-4=0 $ measured parallel to the line $ x-2y-1=0 $ is:

62.

Let the line $ x + y = 1 $ meet the axes of $ x $ and $ y $ at $ A $ and $ B $, respectively. A right angled triangle $ AMN $ is inscribed in the triangle $ OAB $, where $ O $ is the origin and the points $ M $ and $ N $ lie on the lines $ OB $ and $ AB $, respectively. If the area of the triangle $ AMN $ is $\displaystyle \frac{4}{9} $ of the area of the triangle $ OAB $ and $ AN : NB = \lambda : 1 $, then the sum of all possible value(s) of $ \lambda $ is:

63.

Let $ R $ be the interior region between the lines $ 3x-y+1=0 $ and $ x+2y-5=0 $ containing the origin. The set of all values of $ a $, for which the points $ (a^2,a+1) $ lie in $ R $, is:

64.

Let the equation of two adjacent sides of a parallelogram $ ABCD $ be $ 2x - 3y = -23 $ and $ 5x + 4y = 23 $. If the equation of its one diagonal $ AC $ is $ 3x + 7y = 23 $ and the distance of $ A $ from the other diagonal is $ d $, then $ 50\,d^2 $ is equal to $ \_\_\_\_\_ $.

65.

The distance of the point $ (2,3) $ from the line $ 2x-3y+28=0 $, measured parallel to the line $ \sqrt{3}x-y+1=0 $, is equal to:

66.

Consider a triangle having vertices $ A(-2,3) $, $ B(1,9) $ and $ C(3,8) $. If a line $ L $ passing through the circum-center of triangle $ ABC $, bisects line $ BC $, and intersects y-axis at point $ \left(0,\dfrac{\alpha}{2}\right) $, then the value of real number $ \alpha $ is ________.

67.

If the orthocenter of the triangle formed by the lines $ y=x+1 $, $ y=4x-8 $ and $ y=mx+c $ is at $ (3,-1) $, then $ m-c $ is:

68.

Let the points of intersections of the lines $ x - y + 1 = 0 $, $ x - 2y + 3 = 0 $ and $ 2x - 5y + 11 = 0 $ are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is:

69.

Let the centroid of an equilateral triangle $ ABC $ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $ x + y = 3 $. If $ R $ and $ r $ be the radius of circumcircle and incircle respectively of $ \triangle ABC $, then $ (R + r) $ is equal to:

70.

Let the points $ \left(\frac{11}{2}, \alpha\right) $ lie on or inside the triangle with sides $ x + y = 11 $, $ x + 2y = 16 $ and $ 2x + 3y = 29 $. Then the product of the smallest and the largest values of $ \alpha $ is equal to:

71.

Let $ A(-2,-1), B(1,0), C(\alpha ,\beta ) $ and $ D(\gamma ,\delta ) $ be the vertices of a parallelogram $ ABCD $. If the point $ C $ lies on $ 2x-y=5 $ and the point $ D $ lies on $ 3x-2y=6 $, then the value of $ \left|\alpha +\beta +\gamma +\delta \right| $ is equal of ________.

72.

Let $ \left(5, \frac{a}{4}\right) $ be the circumcenter of a triangle with vertices $ A(a, -2) $, $ B(a, 6) $ and $ C\!\left(\frac{a}{4}, -2\right) $. Let $ \alpha $ denote the circumradius, $ \beta $ denote the area and $ \gamma $ denote the perimeter of the triangle. Then $ \alpha + \beta + \gamma $ is:

73.

Let $ C $ be the circle $ x^{2} + (y - 1)^{2} = 2 $, $ E_{1} $ and $ E_{2} $ be two ellipse whose centres lie at the origin and major axes lie on $ x $-axis and $ y $-axis respectively. Let the straight line $ x + y = 3 $ touch the curves $ C, E_{1} $ and $ E_{2} $ at $ P(x_{1},y_{1}), Q(x_{2},y_{2}) $ and $ R(x_{3},y_{3}) $ respectively. Given that $ P $ is the mid-point of the line segment $ QR $ and $ PQ = \dfrac{2\sqrt{2}}{3} $, the value of $ 9(x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}) $ is equal to ______.

74.

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 on the line $ x + 3y = 7 $. If $ (\alpha, \beta) $ is the centroid of $ \triangle ABC $, then $ 15(\alpha + \beta) $ is equal to:

75.

Two vertices of a triangle $ ABC $ are $ A(3,-1) $ and $ B(-2,3) $, and its orthocenter is $ P(1,1) $. If the coordinates of the point $ C $ are $ (\alpha,\beta) $ and the centre of the circle circumscribing the triangle $ PAB $ is $ (h,k) $, then the value of $ (\alpha+\beta)+2(h+k) $ equals

76.

Let $ ABC $ be a triangle with $ A(-3,1) $ and $ \angle ACB=\theta $, $ 0<\theta<\frac{\pi}{2} $. If the equation of the median through $ B $ is $ 2x+y-3=0 $ and the equation of angle bisector of $ C $ is $ 7x-4y-1=0 $, then $ \tan\theta $ is equal to:

77.

If a straight line drawn through the point of intersection of the lines $ 4x+3y-1=0 $ and $ 3x+4y-1=0 $, meets the co-ordinate axes at the points $ P $ and $ Q $, then the locus of the mid-point of $ PQ $ is:

78.

A rectangle is formed by the lines $ x = 0 $, $ y = 0 $, $ x = 3 $ and $ y = 4 $. Let the line $ L $ be perpendicular to $ 3x + y + 6 = 0 $ and divide the area of the rectangle into two equal parts. Then the distance of the point $ \left(\frac{1}{2}, -5\right) $ from the line $ L $ is equal to:

79.

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum, of all possible distinct values of $ m $, is:

80.

If the line segment joining the point $ (5,2) $ and $ (2,a) $ subtends an angle $ \dfrac{\pi}{4} $ at the origin, then the absolute value of the product of all possible values of $ a $ is:

81.

Consider the lines $ x(3\lambda+1)+y(7\lambda+2)=17\lambda+5 $, $ \lambda $ being a parameter, all passing through a point $ P $. One of these lines (say $ L $) is farthest from the origin. If the distance of $ L $ from the point $ (3, 6) $ is $ d $, then the value of $ d^2 $ is:

82.

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) $. The man is walking at a constant speed. If the man reaches the point $ Q $ in the minimum time, then $ 50\big((PR)^2 + (RQ)^2\big) $ is equal to:

83.

In a $ \triangle ABC $, suppose $ y = x $ is the equation of the bisector of the angle $ B $ and the equation of the side $ AC $ is $ 2x - y = 2 $. If $ 2AB = BC $ and the points $ A $ and $ B $ are respectively $ (4, 6) $ and $ (\alpha, \beta) $, then $ \alpha + 2\beta $ is equal to:

84.

The distance of the origin from the centroid of the triangle whose two sides have the equations $ x - 2y + 1 = 0 $ and $ 2x - y - 1 = 0 $ and whose orthocenter is $ \left(\frac{7}{3}, \frac{7}{3}\right) $ is:

85.

Let a variable line of slope $ m > 0 $ passing through the point $ (4,-9) $ intersect the coordinate axes at the points $ A $ and $ B $. The minimum value of the sum of the distance of $ A $ and $ B $ from the origin is:

86.

Let A be the point of intersection of the lines $ 3x + 2y = 14 $, $ 5x - y = 6 $ and B be the point of intersection of the lines $ 4x + 3y = 8 $, $ 6x + y = 5 $. The distance of the point $ P(5,-2) $ from the line AB is:

87.

The straight lines $ l_1 $ and $ l_2 $ pass through the origin and trisect the line segment of the line $ L: 9x + 5y = 45 $ between the axes. If $ m_1 $ and $ m_2 $ are the slopes of the lines $ l_1 $ and $ l_2 $, then the point of intersection of the line $ y = (m_1 + m_2)x $ with $ L $ lies on:

88.

Let $ A(a,0) $, $ B(b, 2b + 1) $ and $ C(0,b) $, $ b \neq 0 $, $ |b| \neq 1 $, be points such that the area of triangle $ ABC $ is $ 1 \, \text{sq. unit} $, then the sum of all possible values of $ a $ is :

89.

Let the triangle $ PQR $ be the image of the triangle with vertices $ (1, 3) $, $ (3, 1) $ and $ (2, 4) $ in the line $ x + 2y = 2 $. If the centroid of $ \triangle PQR $ is the point $ (\alpha, \beta) $, then $ 15(\alpha - \beta) $ is equal to:

90.

Let $ \triangle ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ be $ 3y - x = 2 $ and $ x + y = 2 $, respectively, and the points $ B $ and $ C $ lie on $ x $-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

91.

Let the area of the triangle with vertices $ A(1,\alpha) $, $ B(\alpha,0) $ and $ C(0,\alpha) $ be $ 4 \, \text{sq. units} $. If the points $ (\alpha,-\alpha) $, $ (-\alpha,\alpha) $ and $ (\alpha^{2},\beta) $ are collinear, then $ \beta $ is equal to:

92.

If the sum of squares of all real values of $ \alpha $, for which the lines $ 2x - y + 3 = 0 $, $ 6x + 3y + 1 = 0 $ and $ a x + 2y - 2 = 0 $ do not form a triangle is $ p $, then the greatest integer less than or equal to $ p $ is __________.

93.

If the orthocentre of the triangle, whose VERTICES are $ (1,2),(2,3) $ and $ (3,1) $ is $ (\alpha,\beta) $, then the quadratic equation whose roots are $ \alpha+4\beta $ and $ 4\alpha+\beta $, is:

94.

If $ P(6,1) $ be the orthocentre of the triangle whose vertices are $ A(5,-2), B(8,3) $ and $ C(h,k) $, then the point $ C $ lies on the circle:

95.

Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$, $x + y = 5$ and $7x + 4y = 15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x = 0$, $y = 0$ and $x + y = 1$ is:

96.

The area of the quadrilateral $ ABCD $ with vertices $ A(2,1,1) $, $ B(1,2,5) $, $ C(-2,-3,5) $ and $ D(1,-6,-7) $ is equal to:

97.

Let the circumcentre of a triangle with vertices $ A(a,3) $, $ B(b,5) $ and $ C(a,b) $, $ ab > 0 $ be $ P(1,1) $. If the line $ AP $ intersects the line $ BC $ at the point $ Q(k_1,k_2) $, then $ k_1 + k_2 $ is equal to:

98.

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. Then the area of $ \triangle PQR $ :

99.

Let $ A(6,8) $, $ B(10\cos\alpha,-10\sin\alpha) $ and $ C(-10\sin\alpha,10\cos\alpha) $, be the vertices of a triangle. If $ L(a,9) $ and $ G(h,k) $ be its orthocenter and centroid respectively, then $ (5a-3h+6k+100\sin 2\alpha )$ is equal to ____________.

100.

Let $ A(1, 1), \; B(-4, 3), \; C(-2, -5) $ be vertices of a triangle $ ABC $. $ P $ be a point on side $ BC $, $ \Delta_{1} $ and $ \Delta_{2} $ be the areas of triangles $ APB $ and $ ABC $, respectively. If $ \Delta_{1} : \Delta_{2} = 4 : 7 $, then the area enclosed by the lines $ AP $, $ AC $ and the $ x $-axis is:

101.

Let a ray of light passing through the point $ (3,10) $ reflects on the line $ 2x+y=6 $ and the reflected ray passes through the point $ (7,2) $. If the equation of the incident ray is $ ax+by+1=0 $, then $ a^2+b^2+3ab $ is equal to ________.

102.

The equation of one of the straight lines which passes through the point $ (1, 3) $ and makes an angle $ \tan^{-1}(\sqrt{2}) $ with the straight line, $ y + 1 = 3\sqrt{2}\,x $ is:

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JEE Mains | PYQ