Practice Questions

Q83.The number of solutions of sin2 x + (2 + 2x −x2) sin x −3(x −1)2 = 0, where −π ≤x ≤π, is________

Q84.Let a conic C pass through the point (4, −2) and P(x, y), x ≥3, be any point on C . Let the slope of the line touching the conic C only at a single point P be half the slope of the line joining the points P and (3, −5). If the focal distance of the point (7, 1) on C is d, then 12d equals ______

Q84.Let A be a 2 × 2 symmetric matrix such that A [ 11] [ 37] where I is an identity matrix of order 2 × 2 , then α + β equals _______

Q84.Consider a circle 𝑥- 𝛼2 + 𝑦- 𝛽2 = 50, where 𝛼, 𝛽> 0. If the circle touches the line 𝑦+ 𝑥= 0 at the point P, whose distance from the origin is 4√2 , then ( 𝛼+ 𝛽) 2 is equal to _______.

Q84.In a triangle ABC, BC = 7, AC = 8, AB = α ∈N and cos A = 32 . If 49 cos(3C) + 42 = mn , where gcd(m, n) = 1, then m + n is equal to________ Q85. 2x + 7y + λz = 3 If the system of equations 3x + 2y + 5z = 4 has infinitely many solutions, then (λ −μ) is equal x + μy + 32z = −1 to________

Q84.If limx→1 (5x+1)1/3−(x+5)1/3 = m√5 , where gcd(m, n) = 1, then 8 m + 12n is equal to______ (2x+3)1/2−(x+4)1/2 n(2n)2/3

Q84.Let the foci and length of the latus rectum of an ellipse 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏 be ±5, 0 and √50, respectively. 𝑎2 𝑏2 𝑥2 𝑦2 Then, the square of the eccentricity of the hyperbola − = 1 equals 𝑏2 𝑎2𝑏2

Q84.If the orthocentre of the triangle formed by the lines 2x + 3y −1 = 0, x + 2y −1 = 0 and ax + by −1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3, 4) and (−6, −8), then the value of |a −b| is_______ is

Q84.Let a line perpendicular to the line 2x −y = 10 touch the parabola y2 = 4(x −9) at the point P . The distance of the point P from the centre of the circle x2 + y2 −14x −8y + 56 = 0 is __________ = α + β√17, where

Q84.Let S be the focus of the hyperbola x23 −y25 = 1 A(√6, √5) and passing through the point S . If O is the origin and SAB is a diameter of C , then the square of the area of the triangle OSB is equal to___________

Q84.Let P(α, β) be a point on the parabola y2 = 4x. If P also lies on the chord of the parabola x2 = 8y whose mid point is (1, 54 ), then (α −28)(β −8) is equal to _______.

Q84.If lim 𝑎𝑥2𝑒𝑥−𝑏log𝑒1 + 𝑥+ 𝑐𝑥𝑒−𝑥 = 1, then 16𝑎2 + 𝑏2 + 𝑐2 is equal to ______. 𝑥→0 𝑥2sin𝑥 JEE Main 2024 (31 Jan Shift 2) JEE Main Previous Year Paper

Q84.Let the line 𝐿: √2𝑥+ 𝑦= 𝛼 pass through the point of the intersection 𝑃(in the first quadrant)of the circle 𝑥2 + 𝑦2 = 3 and the parabola 𝑥2 = 2𝑦. Let the line 𝐿 touch two circles 𝐶1 and 𝐶2 of equal radius 2√3. If the centres 𝑄1 and 𝑄2 of the circles 𝐶1 and 𝐶2 lie on the 𝑦- axis, then the square of the area of the triangle 𝑃𝑄1𝑄2 is equal to _________.

Q84.Suppose AB is a focal chord of the parabola y2 = 12x of length l and slope m < √3 . If the distance of the chord AB from the origin is d , then l d2 is equal to _______ for

Q84.Let 𝛼= and 𝛽= 𝑛- 1 ∑𝑘= 0 𝑘+ 1 ∑𝑘= 0 𝑘+ 2 . If 5𝛼= 6𝛽, then 𝑛 equals

Q84.Let the latus rectum of the hyperbola x2 = 1 subtend an angle of π3 at the centre of the hyperbola. If b2 9 −y2b2 is equal to l (1 + √n), where l and m are co-prime numbers, then l2 + m2 + n2 is equal to __________. m

Q84.Consider the circle C : x2 + y2 = 4 and the parabola P : y2 = 8x. If the set of all values of α, for which three chords of the circle C on three distinct lines passing through the point (α, 0) are bisected by the parabola P is the interval (p, q), then (2q −p)2 is equal to ________ JEE Main 2024 (09 Apr Shift 2) JEE Main Previous Year Paper

Q84.Let n − 2n + n − 8n + … + n − 2n⋅n2 be πk , limn→∞( √n4+1 (n2+1)√n4+1 √n4+16 (n2+4)√n4+16 √n4+n4 (n2+n2)√n4+n4 ) using only the principal values of the inverse trigonometric functions. Then k2 is equal to ________

Q84.Equations of two diameters of a circle are 2x −3y = 5 and 3x −4y = 7. The line joining the points (−227 , −4) and (−17 , 3) intersects the circle at only one point P(α, β). Then 17β −α is equal to = 1 lie on the curve y2 = 3x2 ,

Q84.Let 𝐴= 𝐼2 −2𝑀𝑀𝑇, where 𝑀 is real matrix of order 2 × 1 such that the relation 𝑀𝑇𝑀= 𝐼1 holds. If 𝜆 is a real number such that the relation 𝐴𝑋= 𝜆𝑋 holds for some non-zero real matrix 𝑋 of order 2 × 1, then the sum of squares of all possible values of 𝜆 is equal to:

Q85.If 𝑦= √𝑥+ 1𝑥2 −√𝑥 1 then 96𝑦'𝜋 is equal to: 𝑥√𝑥+ 𝑥+ √𝑥+ 153cos2𝑥−5cos3𝑥, 6 𝑥

Q85.Let A = {2, 3, 6, 7} and B = {4, 5, 6, 8}. Let R be a relation defined on A × B by (a1, b1)R (a2, b2) if and only if a1 + a2 = b1 + b2 . Then the number of elements in R is _________

Q85.In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to ______ Q86. ⎡ 1⎤ ⎡1⎤ Let A be a 3 × 3 matrix of non-negative real elements such that A 1 = 3 1 . Then the maximum value of ⎣ 1⎦ ⎣1⎦ det(A) is ______ π a, b ∈N, then a + b is equal to_________

Q85.Let a > 0 be a root of the equation 2x2 + x −2 = 0. If limx→1a 16(1−cos(2+x−2x2))(1−ax)2 α, β ∈Z , then α + β is equal to_______

Q85.Consider the matrices : A = [ 23 −5m ], B = [ 20m ] and X = [ xy ] . Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0 ), be the interval (a, b). Then 8 ∫ba |A|dm is equal to_________