Practice Questions

Q68.If the two lines x + (a −1)y = 1 and 2x + a2y = 1, (a ∈R −{0,1}) are perpendicular, then the distance of their point of intersection from the origin is (1) 2 (2) √2 √5 5 (3) 2 (4) 5 √25

Q68.The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, −3), then its radius is (1) 3√2 (2) 3 (3) 2 (4) 2√2

Q68.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 (1) √7−1 (2) 1–√7 √7+1 1+√7 (3) √5−1 (4) 1–√5 √5+1 1+√5

Q68.If the line 3x + 4y −24 = 0 intersects the x-axis is at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is: (1) (4, 4) (2) (3, 4) (3) (4, 3) (4) (2, 2)

Q68.The maximum value of 3 cos θ + 5 sin(θ −π6 ) for any real value of θ is : (1) √19 (2) √31 (3) √79 (4) √34 2

Q68.The number of solutions of the equation 1 + sin4𝑥= cos23𝑥, 𝑥∈- , is: 2 2 (1) 5 (2) 7 (3) 3 (4) 4

Q68.A point on the straight line, 3𝑥+ 5𝑦= 15 which is equidistant from the coordinate axes will lie only in: (1) 1𝑠𝑡 and 2𝑛𝑑 quadrants (2) 1𝑠𝑡, 2𝑛𝑑 and 4th (3) 1𝑠𝑡 quadrant (4) 4𝑡ℎ quadrant quadrants

Q68.If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4 (x −a2) = 0 and the other two vertices are the points of intersection of the parabola and y -axis, is 250 sq. units, then a value of 'a' is : (1) 5√5 (2) 5 (21/3) (3) (10)33 (4) 5

Q68.If the area of an equilateral triangle inscribed in the circle x2 + y2 + 10x + 12y + c = 0 is 27√3 sq. units, then c is equal to: (1) 25 (2) 13 (3) −25 (4) 20

Q68.A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (−1, 1) and (2, 3) . Then the centroid of this triangle is: (1) ( 31 , 1) (2) (1, 73 ) (3) ( 31 , 2) (4) ( 13 , 35 )

Q68.The tangent and the normal lines at the point √3, 1 to the circle 𝑥2 + 𝑦2 = 4 and the 𝑥 -axis form a triangle. The area of this triangle (in square units) is: 1 2 (1) (2) 3 √3 4 1 (3) (4) √3 √3

Q69.A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an equation of the line L is: Note: In actual JEE Main paper, two options were correct for this question. Hence, we have changed one option. + + = 8√2 (2) x + √3y = 8 1)x (√3 −1)y (1) (√3 + √3y = 8√2 (3) √3x + y = 8 (4) (√3 −1)x

Q69.Let the length of the latus rectum of an ellipse with its major axis along x -axis and centre at the origin, be 8 . If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it? (1) (4√2, 2√2) (2) (4√3, 2√2) (3) (4√3, 2√3) (4) (4√2, 2√3)

Q69.If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is: (1) 120 (2) 60 13 13 13 13 (3) (4) 5 2

Q69.If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is : (1) (x2 + y2)(x + y) = R2xy (2) (x2 + y2)3 = 4R2x2y2 (3) (x2 + y2) 2 = 4R2x2y2 (4) (x2 + y2) 2 = 4Rx2y2

Q69.The locus of the centres of the circles, which touch the circle, 𝑥2 + 𝑦2 = 1 externally, also touch the 𝑦-axis and lie in the first quadrant, is: (1) 𝑦= √1 + 2𝑥, 𝑥≥0 (2) 𝑦= √1 + 4𝑥, 𝑥≥0 (3) 𝑥= √1 + 2𝑦, 𝑦≥0 (4) 𝑥= √1 + 4𝑦, 𝑦≥0

Q69.The tangent to the parabola 𝑦2 = 4𝑥 at the point where it intersects the circle 𝑥2 + 𝑦2 = 5 in the first quadrant, passes through the point: (1) 1 3 (2) -1 4 4, 4 3, 3 1 1 3 7 (3) - 4, 2 (4) 4, 4

Q69.Let S be the set of all triangles in the xy -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is: (1) 36 (2) 32 (3) 9 (4) 18

Q69.A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (−8, 5) and (6, 5), then the area of the rectangle (in sq. units ) is: (1) 72 (2) 98 (3) 56 (4) 84

Q69.A square is inscribed in the circle x2 + y2 −6x + 8y −103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is: (1) 6 (2) √137 (3) √41 (4) 13

Q69.Three circles of radii 𝑎, 𝑏, 𝑐, 𝑎< 𝑏< 𝑐 touch each other externally. If they have 𝑥- axis as a common tangent, then: (1) 1 1 1 (2) 𝑎, 𝑏, 𝑐 are in A.P. √𝑎= √𝑏+ √𝑐 √𝑎, √𝑏, √𝑐 are in A.P. (3) √𝑏=1 √𝑎+1 √𝑐1 (4)

Q69.If the straight line 2x −3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β) , then β equals : (1) −5 (2) 353 (3) 5 (4) −353

Q69.If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y −1 = 0, (K ∈R), intersect at the points P and Q, then the line 4x + 5y −K = 0 , passes through P and Q, for: (1) exactly two values of K (2) no value of K (3) exactly one value of K (4) infinitely many values of K y2

Q69.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 of ΔPQR is a line: (1) with slope 2 (2) with slope 3 3 2 (3) parallel to y-axis (4) parallel to x-axis

Q69.The length of the chord of the parabola x2 = 4y having equation x −√2y + 4√2 = 0 is JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper (1) 6√3 units (2) 8√2 units (3) 2√11 units (4) 3√2 units y2 x2 = r ≠±1. Then S represents: y) ∈R2 : 1+r − 1−r