Practice Questions

Q67.A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of 10 1 1 3 + 1 is (2 2(3) 3 ) (1) 1 : 4(16) 1 1 3 (2) 4(36) 3 : 1 3 (3) 2(36) 1 1 3 : 1 (4) 1 : 2(6)

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.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.Lines are drawn parallel to the line 4𝑥- 3𝑦+ 2 = 0, at a distance units from the origin. Then which one of 5 the following points lies on any of these lines? JEE Main 2019 (10 Apr Shift 2) JEE Main Previous Year Paper 1 1 1 2 (1) 4, - 3 (2) - 4, 3 (3) -1 - 2 (4) 1 1 4, 3 4, 3

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

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 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

Q68.If 0 ≤x < π2 , then the number of values of x for which sin x −sin 2x + sin 3x = 0, is: (1) 4 (2) 3 (3) 2 (4) 1

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.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.In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. if x2 −c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is (1) 3 y (2) c 2 √3 (3) 3c (4) √3y

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

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.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.The sum of the squares of the lengths of the chords intercepted on the circle, 𝑥2 + 𝑦2 = 16, by the lines, 𝑥+ 𝑦= 𝑛, 𝑛∈𝑁, where 𝑁 is the set of all natural numbers is: (1) 210 (2) 105 (3) 320 (4) 160

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 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.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.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 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.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.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.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

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