Practice Questions

Q86.If 1 1 / 1 𝑛 𝑥21 + 𝑥14 + 𝑥72𝑥14 + 3𝑥7 + 6 7𝑑𝑥= where 𝑙, 𝑚, 𝑛∈𝑁, 𝑚 and 𝑛 are co-prime then 𝑙+ 𝑚+ 𝑛 ∫0 𝑙11𝑚/ is equal to _____ .

Q86.The vector →a = −ˆi + 2ˆj + ˆk is rotated through a right angle, passing through the y-axis in its way and the → → resulting vector is b. Then the projection of 3→a+ √2 b on →c= 5ˆi + 4ˆj + 3ˆk is (1) 3√2 (2) 1 (3) √6 (4) 2√3

Q87.Let the line L pass through the point (0, 1, 2), intersect the line x−12 = y−23 = z−34 and be parallel to the plane 2x + y −3z = 4. Then the distance of the point P(1, –9, 2) from the line L is (1) √74 (2) √69 (3) √54 (4) 9 = −5. If P

Q87.Let the quadratic curve passing through the point -1, 0 and touching the line 𝑦= 𝑥 at 1, 1 be 𝑦= 𝑓𝑥. Then the 𝑥-intercept of the normal to the curve at the point 𝛼, 𝛼+ 1 in the first quadrant is

Q87.Let the lines L1 : x+53 = y+41 = z−α−2 and L2 : 3x + 2y + z −2 = 0 = x −3y + 2z −13 be coplanar. If the point P(a, b, c) on L1 is nearest to the point Q(−4, −3, 2), then |a| + |b| + |c| is equal to (1) 12 (2) 14 (3) 8 (4) 10

Q87.Let 𝐶 be the largest circle centred at 2, 0 and inscribed in the ellipse 𝑥2 + 𝑦2 = 1. If 1, 𝛼 lies on 𝐶, then 10𝛼2 is 36 16 equal to ______ 𝜋 2 cos𝑥2023

Q87.For a, b ∈Z and |a −b| ≤10 , let the angle between the plane P : a x + y −z = b and the line L : x −1 = a −y = z + 1 be cos−1( 13 ) If the distance of the point (6, −6, 4) from the plane P is 3√6 , then a4 + b2 is equal to (1) 32 (2) 85 (3) 25 (4) 48

Q87.Let f(x) = ∫ 2 . If f(0) = 0 and f(1) = αβ1 tan−1( αβ ), √3 (3+4x2)√4−3x2 equal to _______.

Q87.Let for 𝑥∈𝑅, 𝑓𝑥= 𝑥+ 𝑥 and 𝑔𝑥= 𝑥, 𝑥< 0 . Then area bounded by the curve 𝑦= 𝑓𝑜𝑔𝑥 and the lines 2 𝑥2, 𝑥≥0 𝑦= 0, 2𝑦- 𝑥= 15 is equal to _____ . 2

Q88.The plane 2x −y + z = 4 intersects the line segment joining the points A(a, −2, 4) and B(2, b, −3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is √5 . If ab < 0 and P is the point (a −b, b, 2 b −a) then CP2 is equal to: (1) 17 (2) 16 3 3 (3) 73 (4) 97 3 3

Q88.Let 𝑓: ℝ→ℝ be a differentiable function such that 𝑓'𝑥+ 𝑓𝑥= ∫0 𝑓𝑡𝑑𝑡. If 𝑓0 = 𝑒-2, then 2𝑓0 - 𝑓2 is equal to _____ .

Q88.Let the co-ordinates of one vertex of ΔABC be A(0, 2, α) and the other two vertices lie on the line x+α 5 = y−12 = z+43 . For α ∈Z , if the area of ΔABC is 21 sq. units and the line segment BC has length 2√21 units, then α2 is equal to _______.

Q88.If the area of the region 𝑆= ( 𝑥, 𝑦) : 2𝑦- 𝑦2 ≤𝑥2 ≤2𝑦, 𝑥≥𝑦 is equal to 𝑛+ 2 - 𝜋 then the natural number 𝑛+ 1 𝑛- 1, 𝑛 is equal to _______ JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper

Q88.Let the plane P : 4x −y + z = 10 be rotated by an angle π2 about its line of intersection with the plane x + y −z = 4 . If α is the distance of the point (2, 3, −4) from the new position of the plane P , then 35α is equal to (1) 85 (2) 105 (3) 126 (4) 90

Q88.Let the line passing through the points P(2, −1, 2) and Q(5, 3, 4) meet the plane x −y + z = 4 at the point R. Then the distance of the point R from the plane x + 2y + 3z + 2 = 0 measured parallel to the line x−7 2 = y+32 = z−21 is (1) √61 (2) √189 (3) √31 (4) 3

Q89.Let the equation of the plane passing through the line x −2y −z −5 = 0 = x + y + 3z −5 and parallel to the line x + y + 2z −7 = 0 = 2x + 3y + z −2 be ax + by + cz = 65. Then the distance of the point (a, b, c) from the plane 2x + 2y −z + 16 = 0 is _____ .

Q89.Let P1 be the plane 3x −y −7z = 11 and P2 be the plane passing through the points (2, −1, 0), (2, 0, −1), and (5, 1, 1). If the foot of the perpendicular drawn from the point (7, 4, −1) on the line of intersection of the JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper planes P1 and P2 is (α, β, γ), then α + β + γ is equal to

Q89.Let λ1, λ2 be the values of λ for which the points ( 25 , 1, λ) and (−2, 0, 1) are at equal distance from the plane 2x + 3y −6z + 7 If λ1 > λ2 then the distance of the point (λ1 −λ2, λ2, λ1) from the line x−51 = y−12 = z+72 is ______

Q89.If the line x = y = z intersects the line x sin A + y sin B + z sin C −18 = 0 = x sin 2A + y sin 2B + z sin 2C −9, where A, B, C are the angles of a triangle ABC , then 80(sin A2 sin B2 sin C2 ) is equal to _________.

Q89.Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that N −2, √3 N, N + 2 are in geometric progression be 48k . Then the value of k is (1) 2 (2) 4 (3) 16 (4) 8 Q90. 25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non- smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is k .Then the 10 JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper value of k is _____ . JEE Main 2023 (25 Jan Shift 2) JEE Main Previous Year Paper

Q89.Let →𝑣= 𝛼 ^𝑖+ 2 ^𝑗- 3 ^𝑘, →𝑤= 2𝛼 ^𝑖+ ^𝑗- ^𝑘, and →𝑢 be a vector such that →𝑢= 𝛼> 0. If the minimum value of the 2 𝑚 where 𝑚 and 𝑛 are coprime natural numbers, then scalar triple product →𝑢 →𝑣 →𝑤 is -𝛼√3401, and →𝑢. ^𝑖 = 𝑛 𝑚+ 𝑛 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper Q90.𝐴2, 6, 2, 𝐵-4, 0, 𝜆, 𝐶2, 3, - 1 and 𝐷4, 5, 0, 𝜆≤5 are the vertices of a quadrilateral 𝐴𝐵𝐶𝐷. If its area is 18 square units, then 5 - 6𝜆 is equal to _____ . JEE Main 2023 (01 Feb Shift 1) JEE Main Previous Year Paper

Q89.Let a line L pass through the point P(2, 3, 1) and be parallel to the line x + 3y −2z −2 = 0 = x −y + 2z. If the distance of L from the point (5, 3, 8) is α, then 3α2 is equal to ________

Q89.Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is (1) 5 (2) 2 24 15 (3) 1 (4) 5 6 36

Q89.Let 𝑓: -2, 2 →ℝ be defined by 𝑓𝑥= 𝑥𝑥 , -2 < 𝑥< 0 where 𝑥 denotes the greatest integer function. If 𝑥- 1𝑥 , 0 ≤𝑥< 2 𝑚 and 𝑛 respectively are the number of points in – 2, 2 at which 𝑦= 𝑓𝑥 is not continuous and not differentiable, then 𝑚+ 𝑛 is equal to ________.

Q89.The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the point A , B, C is (2, a, 4), a ∈N . If the volume of the tetrahedron OABC is 144 unit3 , then which of the following points is NOT on P ? (1) (0, 4, 4) (2) (3, 0, 4) (3) (0, 6, 3) (4) (2, 2, 4)