Practice Questions

Q84.Let [t] denote the greatest integer ≤t and {t} denote the fractional part of t . Then integral value of α for α2[x]+{x}+[x]−1 which the left hand limit of the function f(x) = [1 + x] + 2[x]+{x} at x = 0 is equal to α −43 is _____

Q84.The number of one-one functions f : {a, b, c, d} →{0, 1, 2, … , 10} such that 2f(a) −f(b) + 3f(c) + f(d) = 0 is _____ −3x −7 if x ⩽−1Q85. ⎧ 2x2 The number of points where the function f(x) = [4x2 −1] if −1 < x < 1 , where [t] denotes the ⎨ ⎩|x + 1| + |x −2| if x ⩾1 greatest integer ⩽t, is discontinuous is ______ π π

Q84.Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of , in the increasing powers of α , then α is 4√2 + 1 be 4√6 : 1. If the sixth term from the beginning is ( n 1 ) 4√3 4√3 4√3 equal to _______.

Q84.Let 𝐶𝑟 denote the binomial coefficient of 𝑥𝑟 in the expansion of 1 + 𝑥10. If for 𝛼, 𝛽∈𝑅, 𝛼× 211 𝐶1 𝐶2 𝐶1 + 3 · 2𝐶2 + 5 · 3𝐶3 + … upto 10 terms = (𝐶0 + 2 + 3 + … upto 10 terms) then the value of 2𝛽- 1 𝛼+ 𝛽 is equal to _____. 𝜋 7𝜋

Q84.If the sum of solutions of the system of equations 2sin2𝜃- cos2𝜃= 0 and 2cos2𝜃+ 3sin𝜃= 0 in the interval 0, 2𝜋 is 𝑘𝜋, then 𝑘 is equal to _______.

Q84.A ray of light passing through the point P(2, 3) reflects on the X -axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1 . Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of 7α + 3β is equal to _____.

Q84.If a1(> 0), a2, a3, a4, a5 are in a G.P. , a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4 , then a2 + a4 + 2a5 is equal to _____. JEE Main 2022 (26 Jun Shift 2) JEE Main Previous Year Paper

Q84.Let a circle C : (x −h)2 + (y −k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to _____.

Q85.The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation 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.The number of values of 𝑥 in the interval 4, 4 for which 14 cosec2 𝑥- 2sin2𝑥= 21 - 4cos2𝑥 holds, is ______.

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 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.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.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.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.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 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.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.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.Let S = {θ ∈(0, 2π) : 7 cos2 θ −3 sin2 θ −2 cos2 2θ = 2}. Then, the sum of roots of all the equations x2 −2(tan2 θ + cot2 θ)x + 6 sin2 θ = 0 θ ∈S, is _______.

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.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.If 40C0 + 41C1 + 42C2 + ⋯+ 60C20 = mn × 60C20 where m & n are co-prime, then m + n is equal to and let L2 be the line passing through the origin and

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 = _____.