Practice Questions

Q90.Let 𝑦= 𝑝𝑥 be the parabola passing through the points – 1, 0, 0, 1 and 1, 0. If the area of the region 𝑥, 𝑦: 𝑥+ 12 + 𝑦- 12 ≤1, 𝑦≤𝑝𝑥 is 𝐴, then 12𝜋- 4𝐴 is equal to ________ . JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the foot of perpendicular from the point 𝐴4, 3, 1 on the plane 𝑃: 𝑥- 𝑦+ 2𝑧+ 3 = 0 be 𝑁. If 𝐵( 5, 𝛼, 𝛽) , 𝛼, 𝛽∈ℤ is a point on plane 𝑃 such that the area of the triangle 𝐴𝐵𝑁 is 3√2, then 𝛼2 + 𝛽2 + 𝛼𝛽 is equal to _____________ JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper

Q90.Let a line 𝐿 pass through the origin and be perpendicular to the lines 𝐿1: →𝑟= ^𝑖- 11 ^𝑗- 7 ^𝑘+ 𝜆 ^𝑖+ 2 ^𝑗+ 3 ^𝑘, 𝜆∈ℝ and 𝐿2: →𝑟= - ^𝑖+ ^𝑘+ 𝜇2 ^𝑖+ 2 ^𝑗+ ^𝑘, 𝜇∈ℝ. If 𝑃 is the point of intersection of 𝐿 and 𝐿1, and ,𝑄𝛼, 𝛽, 𝛾 is the foot of perpendicular from 𝑃 on 𝐿2, then 9𝛼+ 𝛽+ 𝛾 is equal to ________. JEE Main 2023 (11 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let the plane P contain the line 2x + y −z −3 = 0 = 5x −3y + 4z + 9 and be parallel to the line x+2 2 = 3−y−4 = z−75 . Then the distance of the point A(8, −1, −19) from the plane P measured parallel to the line −3 x = y−54 = −122−z is equal to _________. JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let S = {w1, w2, … . } be the sample space associated to a random experiment. Let P(wn) = P(wn−1)2 , . Let A = {2k + 3l; k, l ∈N} and B = {wn; n ∈A} . Then P(B) is equal to (1) 3 (2) 3 32 64 (3) 1 (4) 1 16 32 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q90.If 𝜆1 < 𝜆2 are two values of 𝜆 such that the angle between the planes 𝑃1: →𝑟(3 ^i - 5 ^𝑗+ ^𝑘) = 7 and , then the square of the length of perpendicular from the point 𝑃2: →𝑟· (𝜆 ^𝑖+ ^𝑗- 3 ^𝑘) = 9 is sin-12√65 38𝜆1, 10𝜆2, 2 to the plane 𝑃1 is JEE Main 2023 (30 Jan Shift 1) JEE Main Previous Year Paper

Q90.If 𝑦= 𝑦( 𝑥) is the solution of the differential equation 𝑑𝑦 4𝑥 𝑥+ 25 , 𝑥> 1 such that 𝑑𝑥+ 𝑥2 - 1𝑦= 𝑥2 - 1 2 2 𝑦(2) = 9log𝑒2 + √3 and 𝑦√2 = 𝛼log𝑒√𝛼+ 𝛽+ 𝛽- √𝛾, 𝛼, 𝛽, 𝛾∈ℕ, then 𝛼𝛽𝛾 is equal to JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q61.Let a circle 𝐶 in complex plane pass through the points 𝑧1 = 3 + 4𝑖, 𝑧2 = 4 + 3𝑖 and 𝑧3 = 5𝑖. If 𝑧≠𝑧1 is a point on 𝐶 such that the line through 𝑧 and 𝑧1 is perpendicular to the line through 𝑧2 and 𝑧3, then arg𝑧 is equal to 2 (1) tan-124 - 𝜋 (2) tan-1 - 𝜋 7 √5 (3) tan-13 - 𝜋 (4) tan-13 - 𝜋 4 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper 1 1 1 𝐾

Q62.Consider two G.Ps. 2, 22, 23, … and 4, 42, 43, … of 60 and n terms respectively. If the geometric mean of all 225 the 60 + n terms is (2) 8 , then ∑nk=1 k(n −k) is equal to: (1) 560 (2) 1540 (3) 1330 (4) 2600 n(S) + ∑θ∈S(sec( π4 + 2θ) cosec ( π4 + 2θ)) is equal

Q62.Let for some real numbers α and β, a = α −iβ . If the system of equations 4ix + (1 + i)y = 0 and ¯8(cos 2π3 + i sin 2π3 )x + ay = 0 has more than one solution then αβ is equal to (1) 2 −√3 (2) 2 + √3 (3) −2 + √3 (4) −2 −√3

Q62.The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives atleast 4 and atmost 7 candies, C3 receives atleast 2 and atmost 6 candies, is equal to (1) 205 (2) 615 (3) 510 (4) 430 JEE Main 2022 (28 Jun Shift 2) JEE Main Previous Year Paper

Q62.Let 𝑆= 𝑧= 𝑥+ 𝑖𝑦: 𝑧- 1 + 𝑖≥𝑧, 𝑧< 2, 𝑧+ 𝑖= 𝑧- 1. Then the set of all values of 𝑥, for which 𝑤= 2𝑥+ 𝑖𝑦∈𝑆 for some 𝑦∈ℝ, is 1 1 1 (2) - (1) -√2, 4 2√2 √2, (3) -√2, 1 (4) - 1 1 2 √2, 2√2

Q62.Let (z) represent the principal argument of the complex number z. The, |z| = 3 and arg(z −1) −arg(z + 1) = π4 intersect: (1) Exactly at one point (2) Exactly at two points (3) Nowhere (4) At infinitely many points.

Q63.The value of cos( 2π7 ) + cos( 4π7 ) + cos( 6π7 ) is equal to (1) −1 (2) −12 (3) −13 (4) −14

Q63.Let 𝑎𝑛𝑛=∞ 0 be a sequence such that 𝑎0 = 𝑎1 = 0 and 𝑎𝑛+ 2 = 3𝑎𝑛+ 1 - 2𝑎𝑛+ 1, ∀𝑛≥0. Then 𝑎25𝑎23 - 2𝑎25𝑎22 - 2𝑎23𝑎24 + 4𝑎22𝑎24 is equal to (1) 483 (2) 528 (3) 575 (4) 624 Q64. ∑𝑟=20 1 𝑟2 + 1𝑟! is equal to (1) 22! - 21! (2) 22! - 221! (3) 21! - 220! (4) 21! - 20!

Q63.If the constant term in the expansion of (3x3 −2x2 + x5 ) is 2k. l, where l is an odd integer, then the value of k is equal to (1) 6 (2) 7 (3) 8 (4) 9

Q63.The sum of the infinite series 1 + 65 + 1262 + 2263 + 3564 + 5165 + 7066 + … is equal to: (1) 425 (2) 429 216 216 (3) 288 (4) 280 125 125

Q63.Consider the sequence 𝑎1, 𝑎2, 𝑎3, … … such that 𝑎1 = 1, 𝑎2 = 2 and 𝑎𝑛+ 2 = + 𝑎𝑛 for 𝑛= 1, 2, 3, … 𝑎𝑛+ 1 1 1 1 1 𝑎1 + 𝑎2 𝑎2 + 𝑎3 𝑎3 + 𝑎4 𝑎30 + 𝑎31 If · · … = 2𝛼61𝐶31 then 𝛼 is equal to 𝑎3 𝑎4 𝑎5 𝑎32 (1) -30 (2) -31 (3) -60 (4) -61

Q63.Let S = {θ ∈[0, 2π] : 82 sin2 θ + 82 cos2 θ = 16} . Then to: (1) 0 (2) −2 (3) −4 (4) 12

Q64.Let S = {θ ∈(0, π2 ) : ∑9m=1 sec(θ + (m −1) π6 ) sec(θ + mπ6 ) = −8√3 }. Then (1) S = { 12π } (2) S = { 2π3 } (3) ∑θ∈S θ = π2 (4) ∑θ∈S θ = 3π4

Q64.Let a line L pass through the point of intersection of the lines bx + 10y −8 = 0 and 2x −3y = 0, b ∈R −{ 34 }. If the line L also passes through the point (1, 1) and touches the circle 17(x2 + y2) = 16, then x2 y2 the eccentricity of the ellipse 5 + b2 = 1 is (1) 2 (2) √5 √35 (3) 1 (4) √5 √25

Q64.Let the hyperbola H : x2 −y2 = 1 pass through the point . A parabola is drawn whose focus is a2 b2 (2√2, −2√2) same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H . If the length of the latus rectum of the parabola is e times the length of the latus rectum of H , where e is the eccentricity of H , then which of the following points lies on the parabola? (1) (2√3, 3√2) (2) (3√3, −6√2) (3) (√3, −√6) (4) (3√6, 6√2)

Q65.Let the locus of the centre 𝛼, 𝛽, 𝛽> 0, of the circle which touches the circle 𝑥2 + 𝑦- 12 = 1 externally and also touches the 𝑥-axis be 𝐿. Then the area bounded by 𝐿 and the line 𝑦= 4 is (1) 32√2 (2) 40√2 3 3 64 32 (3) (4) 3 3

Q65.For 𝑡∈0, 2𝜋, if 𝐴𝐵𝐶 is an equilateral triangle with vertices 𝐴sin𝑡, - cos𝑡, 𝐵cos𝑡, sin𝑡 and 𝐶𝑎, 𝑏 such that its 1 orthocentre lies on a circle with centre 1, 3, then 𝑎2 - 𝑏2 is equal to (1) 8 (2) 8 3 77 80 (3) (4) 9 9 11

Q65.A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q . If the y-axis bisects the segment PQ , then C is a parabola with (1) length of latus rectum 3 (2) length of latus rectum 6 (3) focus ( 34 , 0) (4) focus (0, 33 ) y2