Practice Questions

Q66.A hyperbola passes through the foci of the ellipse x2 = 1 and its transverse and conjugate axes coincide 25 + 16 with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is: (1) x2 = 9 9 −y216 = 1 (2) x2 −y2 (3) x2 9 −y225 = 1 (4) x29 −y24 = 1

Q66.Consider the parabola with vertex 2, 4 and the directrix 𝑦= 2 . Let P be the point where the parabola meets the line 𝑥= - 12. If the normal to the parabola at P intersects the parabola again at the point Q . then ( PQ ) 2 is equal to : 25 75 (1) (2) 2 8 (3) 125 (4) 15 16 2

Q66.Consider the following three statements: (A) If 3 + 3 = 7 then 4 + 3 = 8 (B) If 5 + 3 = 8 then earth is flat. (C) If both (A) and (B) are true then 5 + 6 = 17. Then, which of the following statements is correct? (1) (A) is false, but (B) and (C) are true (2) (A) and (C) are true while (B) is false (3) (A) is true while (B) and (C) are false (4) (A) and (B) are false while (C) is true

Q66.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 (1) 3x2 + 2y −6 = 0 (2) 2x2 −3y + 9 = 0 (3) 3x2 −2y −6 = 0 (4) 2x2 + 3y −9 = 0

Q66.Let (1 + x + 2x2) 20 = a0 + a1x + a2x2 + … + a40x40, then a1 + a3 + a5 + … + a37 is equal to (1) 220(220 −21) (2) 219(220 −21) (3) 219(220 + 21) (4) 220(220 + 21) Q67. 1 + sin2 x sin2 x sin2 x The solutions of the equation cos2 x 1 + cos2 x cos2 x = 0, (0 < x < π), are 4 sin 2x 4 sin 2x 1 + 4 sin 2x (1) 12 π , π6 (2) π6 , 5π6 (3) 5π 12 , 7π12 (4) 7π12 , 11π12

Q66.The line 12x cos θ + 5y sin θ = 60 is tangent to which of the following curves ? (1) x2 + y2 = 30 (2) 144x2 + 25y2 = 3600 (3) x2 + y2 = 169 (4) 25x2 + 12y2 = 3600

Q66.Let ABC be a triangle with A(−3, 1) and ∠ACB = θ, 0 < θ < π2 . 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 θ is equal to: (1) 3 (2) 4 4 3 (3) 2 (4) 12

Q66.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 the parallelogram is 11x + 7y = 9, then other diagonal passes through the point: (1) (1, 2) (2) (2, 2) (3) (2, 1) (4) (1, 3)

Q66.The locus of the mid points of the chords of the hyperbola x2 −y2 = 4, which touch the parabola y2 = 8x, is : (1) y2(x −2) = x3 (2) x3(x −2) = y2 (3) x2(x −2) = y3 (4) y3(x −2) = x2 lim n=1 n(n+1)x2+2(2n+1)x+4x ) is equal to :

Q66.Let f(x) be a differentiable function at x = a with f ′(a) = 2 and f(a) = 4. Then lim x−a x→a (1) a + 4 (2) 2a −4 (3) 4 −2a (4) 2a + 4

Q66.The image of the point (3, 5) in the line x −y + 1 = 0, lies on : (1) (x −2)2 + (y −4)2 = 4 (2) (x −4)2 + (y −4)2 = 8 (3) (x −4)2 + (y + 2)2 = 16 (4) (x −2)2 + (y −2)2 = 12

Q66.Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x -axis and y-axis at point P and Q , respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r2 is equal to (1) 529 (2) 125 64 72 (3) 625 (4) 585 72 66

Q66.In the circle given below, let OA = 1 unit, OB = 13 unit and PQ ⊥OB. Then, the area of the triangle PQB (in square units) is : (1) 24√3 (2) 26√3 (3) 24√2 (4) 26√2 √3 sin( π6 +h)−cos( π6 +h) is :

Q66.The locus of the mid-point of the line segment joining the focus of the parabola 𝑦2 = 4𝑎𝑥 to a moving point of the parabola, is another parabola whose directrix is: (1) 𝑥= 𝑎 (2) 𝑥= 0 (3) 𝑥= - 𝑎 (4) 𝑥= 𝑎 2 2

Q66.Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 −y2 = 3. If L is also a tangent to the parabola y2 = αx, then α is equal to: (1) 12 (2) −12 (3) 24 (4) −24

Q66.The locus of mid-points of the line segments joining -3, - 5 and the points on the ellipse 𝑥2 + 𝑦2 = 1 is : 4 9 (1) 36𝑥2 + 16𝑦2 + 90𝑥+ 56𝑦+ 145 = 0 (2) 36𝑥2 + 16𝑦2 + 108𝑥+ 80𝑦+ 145 = 0 (3) 9𝑥2 + 4𝑦2 + 18𝑥+ 8𝑦+ 145 = 0 (4) 36𝑥2 + 16𝑦2 + 72𝑥+ 32𝑦+ 145 = 0

Q66.Let a tangent be drawn to the ellipse x2 cos θ, sin θ ∈(0, π2 ). Then the value of θ 27 + y2 = 1 at (3√3 θ) where such that the sum of intercepts on axes made by this tangent is minimum is equal to : (1) π (2) π 8 4 (3) π (4) π 6 3 x-axis at Q and latus

Q67.Let L be a tangent line to the parabola y2 = 4x −20 at (6, 2). If L is also a tangent to the ellipse x2 y2 2 + b = 1, then the value of b is equal to : (1) 11 (2) 14 (3) 16 (4) 20 JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper

Q67. x→2(∑9 (1) 5 (2) 7 24 36 (3) 1 (4) 9 5 44

Q67.The statement among the following that is a tautology is: JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper (1) 𝐴∨𝐴∧𝐵 (2) 𝐴∧𝐴∨𝐵 (3) 𝐵→𝐴∧𝐴→𝐵 (4) 𝐴∧𝐴→𝐵→𝐵

Q67.For the system of linear equations: x −2y = 1, x −y + kz = −2, ky + 4z = 6, k ∈R Consider the following statements: (A) The system has unique solution if k ≠2, k ≠−2. (B) The system has unique solution if k = −2. (C) The system has unique solution if k = 2. JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (D) The system has no-solution if k = 2. (E) The system has infinite number of solutions if k ≠−2. Which of the following statements are correct? (1) (A) and (E) only (2) (B) and (E) only (3) (A) and (D) only (4) (C) and (D) only

Q67.The value of lim cos h−sin h) } h→0{ √3h(√3 (1) 43 (2) √32 (3) 23 (4) 43

Q67.The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, x2 is : 9 −y216 = 1 (1) (x2 + y2)2 −16x2 + 9y2 = 0 (2) (x2 + y2)2 −9x2 + 144y2 = 0 2 2 (3) (x2 + y2) −9x2 −16y2 = 0 (4) (x2 + y2) −9x2 + 16y2 = 0

Q67.The mean of 6 distinct observations is 6. 5 and their variance is 10. 25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are: (1) 10, 11 (2) 3, 18 (3) 8, 13 (4) 1, 20

Q67.If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (−30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is: (1) 5 (2) 7 (3) 3√5 (4) 5√3 y2