Practice Questions

Q85.Let 𝐴= 2 -2 and𝐵= -1 2 . Then the number of elements in the set {𝑛, 𝑚: 𝑛, 𝑚∈1, 2, … … . 10 and 1 -1 -1 2 𝑛𝐴𝑛+ 𝑚𝐵𝑚= 𝐼} is _____.

Q85.The number of functions f , from the set A = {x ∈N : x2 −10x + 9 ≤0} to the set B = {n2 : n ∈N} such that f(x) ≤(x −3)2 + 1 , for every x ∈A , is _______.

Q85.Let M = 0 −α , where α is a non-zero real number and N = ∑49k=1 M [α 0 ] positive integral value of α is ______.

Q85.Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = _____.

Q85.Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is

Q85.Let x = sin(2 tan−1 α) and y = sin( 12 tan−1 43 ). If S = {α ∈R : y2 = 1 −x}, then ∑α∈S 16α3 is equal to _______.

Q85.The number of values of 𝑥 in the interval 4, 4 for which 14 cosec2 𝑥- 2sin2𝑥= 21 - 4cos2𝑥 holds, is ______.

Q85.If 1 + (2 + 49C1 + 49C2 + … . +49C49)(50C2 + 50C4 + … . . +50C50) is equal to 2n. m, where m is odd, then n + m is equal to _____ .

Q85.Let H : x2 −y2 = 1, a > 0, b > 0 , be a hyperbola such that the sum of lengths of the transverse and the a2 b2 H is √11 + , then value of a2 + b2 is equal to ______. 2 conjugate axes is 4(2√2 √14). If the eccentricity ) + 2 Q86. 50 tan(3 tan−1( 21 cos−1( √51 ))+4√2 tan( 21 tan−1(2√2)) is equal to ______.

Q85.Let the common tangents to the curves 4(x2 + y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then l is equal to e2 ______.

Q85.The sum of diameters of the circles that touch (i) the parabola 75𝑥2 = 645𝑦- 3 at the point 5, 5 and (ii) the 𝑦- axis, is equal to _____ .

Q85.The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If 𝜎 is the standard deviation of the data after omitting the two wrong observations from the data, then 38𝜎2 is equal to _______. JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper

Q85.Let 𝑆= 𝑥, 𝑦∈ℕ× ℕ: 9𝑥- 32 + 16𝑦- 42 ≤144 and 𝑇= 𝑥, 𝑦∈ℝ× ℝ: 𝑥- 72 + y - 42 ≤36 The 𝑛𝑆∩𝑇 is equal to ______. Q86. 1 -1 2 3 Let 𝑥= 1 and 𝐴= 0 1 6 . For 𝑘∈ℕ, if 𝑋'𝐴𝑘𝑋= 33, then 𝑘 is equal to 1 0 0 -1

Q85.Let the lines y + 2x = √11 + 7√7 and 2y + x = 2√11 + 6√7 be normal to a circle C , then the value of C : (x −h)2 + (y −k)2 = r2 . If the line √11y −3x = 5√773 + 11 is tangent to the circle (5h −8k)2 + 5r2 is equal to ______.

Q85.The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x −y = 3 respectively. If its orthocentre is (2, a), −12 < a < 2 , then p is equal to

Q85.The maximum number of compound propositions, out of p ∨r ∨s, p ∨r ∨~s, p ∨~q ∨s, ~p ∨~r ∨s, ~p ∨~r ∨~s, ~p ∨q ∨~s, q ∨r ∨~s, q ∨~r ∨~s, ~p ∨~q ∨~s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to

Q85.A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the 2 parabola y = (x −14 ) + α, where α > 0 . Then (4α −8)2 is equal to ______. Q86. ⎡ 14 28 −14 ⎤ The positive value of the determinant of the matrix A , whose Adj(Adj(A)) = −14 14 28 , is ______. ⎣ 28 −14 14 ⎦

Q85.For the hyperbola 𝐻: 𝑥2 - 𝑦2 = 1 and the ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎> 𝑏> 0, let the 𝑎2 𝑏2 (1) eccentricity of 𝐸 be reciprocal of the eccentricity of 𝐻, and 𝐾 be a common tangent of 𝐸 and 𝐻. (2) the line 𝑦= √ 52𝑥+ Then 4𝑎2 + 𝑏2 is equal to 100 Q86. 𝑥 𝑥+ 2cos𝑥3 + 2𝑥+ 2cos𝑥2 + 3sin𝑥+ 2cos𝑥 lim is equal to 𝑥→0 𝑥+ 23 + 2𝑥+ 22 + 3sin𝑥+ 2 JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q86.For the curve C : (x2 + y2 −3) + (x2 −y2 −1) 5 = 0 , the value of 3y′ −y3y′′ , at the point (α, α), α > 0 , on C , is equal to ________.

Q86.Let the equation of two diameters of a circle 𝑥2 + 𝑦2 - 2𝑥+ 2𝑓𝑦+ 1 = 0 be 2𝑝𝑥- 𝑦= 1 and 2𝑥+ 𝑝𝑦= 4𝑝. Then the slope 𝑚∈0, ∞ of the tangent to the hyperbola 3𝑥2 - 𝑦2 = 3 passing through the centre of the circle is equal to _____. Q87. 2 -1 -1 √3i - 1 Let 𝐴= 1 0 -1 and 𝐵= 𝐴- 𝐼. If 𝜔= , then the number of elements in the set 2 1 -1 0 𝑛∈1, 2, … , 100: 𝐴𝑛+ 𝜔𝐵𝑛= 𝐴+ 𝐵 is equal to _____ .

Q86.The number of distinct real roots of the equation x5(x3 −x2 −x + 1) + x(3x3 −4x2 −2x + 4) −1 = 0 is

Q86.Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S →S as f(n) = { 2n2n,−11 ifif nn == 1,6, 2,7, 3,8, 4,9, 510 + 1 , if n is odd Let g : S ≥S be a function such that fog(n) = , then {nn −1 , if n is even g(10)(g(1) + g(2) + g(3) + g(4) + g(5)) is equal to

Q86.Let the mirror image of a circle c1 : x2 + y2 −2x −6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx +10fy + 38 = 0. If r is the radius of circle c2 , then α + 6r2 is equal to ______

Q86.Let a line L1 be tangent to the hyperbola x216 −y24 = 1 perpendicular to L1 . If the locus of the point of intersection of L1 and L2 is (x2 + y2)2 = αx2 + βy2 , then α + β is equal to ______. Q87. ⎡ 0 1 0 ⎤ 2 Let X = 0 0 1 , Y = αl + βX + γX and Z = α2I −αβX + (β2 −αγ)X 2, α, β, γ ∈R. ⎣ 0 0 0 ⎦ 1 −2 1 5 5 5 ⎡ ⎤ If Y−1 = 0 51 −25 , then (α −β + γ)2 is equal to ______. 1 ⎣ 0 0 5 ⎦ is equal to _____.

Q86.Let S be the set containing all 3 × 3 matrices with entries from {−1, 0, 1} . The total number of matrices A ∈S such that the sum of all the diagonal elements of ATA is 6 is ______.