Practice Questions

Q61.The number of pairs 𝑎, 𝑏 of real numbers, such that whenever 𝛼 is a root of the equation 𝑥2 + 𝑎𝑥+ 𝑏= 0, 𝛼2 - 2 is also a root of this equation, is : (1) 6 (2) 8 (3) 4 (4) 2

Q61.If the real part of the complex number (1 −cos θ + 2i sin θ)−1 is 15 for θ ∈(0, π), then the value of the x dx is equal to: integral ∫θ0 sin JEE Main 2021 (20 Jul Shift 2) JEE Main Previous Year Paper (1) 1 (2) 2 (3) −1 (4) 0

Q61.The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is: (1) 77 (2) 82 (3) 42 (4) 35

Q61.Let α = max{82 sin 3x ⋅44 cos 3x} and β = min sin 3x ⋅44 cos 3x}. If 8x2 + bx + c = 0 is a quadratic equation x∈R x∈R{82 whose roots are α1/5 and β1/5, then the value of c −b is equal to : (1) 42 (2) 47 (3) 43 (4) 50 JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper

Q62.Let the lines (2 −i)z = (2 + i)z and (2 + i)z + (i −2)z −4i = 0, (here i2 = −1) be normal to a circle C . If ¯the line iz + z + 1 + i = 0 is tangent to this circle C , then its radius is : (1) 3 (2) 3√2 √2 (3) 3 (4) 1 2√2 2√2

Q62.Let C be the set of all complex numbers. Let S1 = {z ∈C : |z −2| ≤1} and 2 ¯S2 = {z ∈C : z(1 + i) + z(1 −i) ≥4}. Then, the maximum value of z −52 for z ∈S1 ∩S2 is equal to : (1) 3+2√2 (2) 5+2√2 4 2 (3) 3+2√2 (4) 5+2√2 2 4

Q62.If 𝑏 is very small as compared to the value of 𝑎, so that the cube and other higher powers of 𝑏 can be neglected 𝑎 in the identity 1 1 1 1 … . + 𝛼𝑛+ 𝛽𝑛2 + 𝛾𝑛3 𝑎- 𝑏+ 𝑎- 2𝑏+ 𝑎- 3𝑏+ 𝑎- 𝑛𝑏= then the value of 𝛾 is : (1) 𝑎2 + 𝑏 (2) 𝑎+ 𝑏 3𝑎3 3𝑎2 (3) 𝑏2 (4) 𝑎+ 𝑏2 3𝑎3 3𝑎3

Q63.If the greatest value of the term independent of x in the expansion of (x sin α + a cosx α )10 is (5!)210! value of a is equal to: (1) −1 (2) 1 (3) −2 (4) 2 10100 1

Q63. Let 𝑆𝑛= 1 · ( 𝑛- 1 ) + 2 · ( 𝑛- 2 ) + 3 · ( 𝑛- 3 ) + … + ( 𝑛- 1 ) · 1, 𝑛⩾4 . ∞ 2 Sn 1 The sum ∑n = 4 n! - ( n - 2 ) ! is equal to : 𝑒- 2 e - 1 (1) (2) 6 3 (3) e (4) e 6 3 20 1 4 = . If the sum of this 𝐴. 𝑃. is 189, then a6a16

Q63.If 0 < a, b < 1 , and tan−1 a + tan−1 b = π4 , then the value of (a + b) −( a2+b22 ) ( a3+b33 ) −( a4+b44 ) is : (1) loge( 2e ) (2) e (3) e2 −1 (4) loge 2

Q64.A circle C touches the line x = 2y at the point (2, 1) and intersects the circle C1 : x2 + y2 + 2y −5 = 0 at two points P and Q such that PQ is a diameter of C1 . Then the diameter of C is : (1) 4√15 (2) √285 (3) 15 (4) 7√5 = 1 having eccentricity √52 . If the tangent and

Q64.Let [x] denote greatest integer less than or equal to x . If for n ∈N, (1 −x + x3) n = ∑3nj=0 ajxj , then [ 3n2 ] [ 3n−12 ] ∑ j=0 a2j + 4 ∑ j=0 a2j+1 is equal to : (1) 2 (2) 2n−1 (3) 1 (4) n

Q64.Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (−4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y −4 = 0. If r1 = a + b√2, then r2 a + b is equal to: (1) 3 (2) 11 (3) 5 (4) 7

Q64.Let the circle S : 36x2 + 36y2 −108x + 120y + C = 0 be such that it neither intersects nor touches the co- ordinate axes. If the point of intersection of the lines, x −2y = 4 and 2x −y = 5 lies inside the circle S, then: (1) 25 9 < C < 133 (2) 100 < C < 165 (3) 81 < C < 156 (4) 100 < C < 156 = 1, a > b. Let E2 be another ellipse such that it touches the end points of major axis of E1

Q64.If 0 < x, y < π and cos x + cos y −cos(x + y) = 23 , then sin x + cos y is equal to: (1) 1 (2) √3 2 2 (3) 1−√3 (4) 1+√3 2 2 JEE Main 2021 (25 Feb Shift 2) JEE Main Previous Year Paper

Q65.Let an ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎2 > 𝑏2, passes through 3 1 and has eccentricity 1 If a circle, centered at √ 2, √3. 𝑎2 𝑏2 2 focus 𝐹( 𝛼, 0 ) , 𝛼> 0, of 𝐸 and radius √3, intersects 𝐸 at two points 𝑃 and 𝑄, then 𝑃𝑄2 is equal to : (1) 8 (2) 4 3 3 16 (3) (4) 3 3

Q65.The sum of solutions of the equation 1+sin x = |tan 2x|, x ∈(−π2 , π2 ) −{−π4 , π4 } is: (1) 10 π (2) −7π30 (3) −π15 (4) −11π30

Q65.Two tangents are drawn from the point P(−1, 1) to the circle x2 + y2 −2x −6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to: + (1) 2 (2) (3√2 2) (3) 4 (4) 3(√2 −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.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.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 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.A tangent and a normal are drawn at the point P(2, −4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to (1) −12 (2) −20 (3) −16 (4) −18

Q67.Let A = {(x, y) ∈R × R ∣2x2 + 2y2 −2x −2y = 1} B = {(x, y) ∈R × R ∣4x2 + 4y2 −16y + 7 = 0} and C = {(x, y) ∈R × R ∣x2 + y2 −4x −2y + 5 ≤r2}. Then the minimum value of |r| such that A ∪B ⊆C is equal to (1) 3+√10 (2) 2+√10 2 2 (3) 3+2√5 (4) 1 + √5 2

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