Practice Questions

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.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.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.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

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.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.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.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.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 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.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.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 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.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.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 a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is: (1) x2 + y2–16x2y2 = 0 (2) x2 + y2–4x2y2 = 0 (3) x2 + y2–2xy = 0 (4) x2 + y2–2x2y2 = 0

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

Q70.If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 , then the eccentricity of the hyperbola is: (1) 13 (2) 2 12 (3) 13 (4) 13 6 8

Q70.If the normal to the ellipse 3𝑥2 + 4𝑦2 = 12 at a point 𝑃 on it is parallel to the line, 2𝑥+ 𝑦= 4 and the tangent to the ellipse at 𝑃 passes through 𝑄( 4,4 ) then 𝑃𝑄 is equal to: (1) √61 (2) 5√5 2 2 (3) √157 (4) √221 2 2

Q70.The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y −24 = 0 also passes through the point: (1) (4, −2) (2) (−4, 6) (3) (6, −2) (4) (−6, 4) JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper

Q70.Let C1 and C2 be the centres of the circles x2 + y2 −2x −2y −2 = 0 and x2 + y2 −6x −6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1 QC2 is : JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) 6 (2) 4 (3) 8 (4) 9

Q70.A circle touching the x− axis at (3, 0) and making an intercept of length 8 on the y− axis passes through the point: (1) (3, 10) (2) (2, 3) (3) (3, 5) (4) (1, 5)