Practice Questions

Q83.The remainder on dividing 1 + 3 + 32 + 33 + … + 32021 by 50 is _____.

Q83.A common tangent T to the curves C1 : x24 + y29 = 1 and C2 : x242 − 143y2 = 1 quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to _______. JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper Q84. ⎡ α β γ ⎤ Consider a matrix A = α2 β2 γ 2 , where α, β, γ are three distinct natural numbers. ⎣β + γ γ + α α + β⎦ If det(adj(adj(adj(adjA))) = 232 × 316 , then the number of such 3 - tuples (α, β, γ) is _______. (α−β)16(β−γ)16(γ−α)16

Q83.The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 5 19 65 … is equal to 3, 9, 27, 81, ______

Q84.If the coefficients of x and x2 in the expansion of (1 + x)p(1 −x)q, p, q ≤15 , are −3 and −5 respectively, then the coefficient of x3 is equal to ______.

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 [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.If sin2(10°) sin(20°) sin(40°) sin(50°) sin(70°) = α− 161 sin(10°), then 16 + α−1 is equal to _____.

Q84.The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is ______. 2k . If (I −M 2)N = −2I , then the

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.An ellipse E : x2a2 + y2b2 = 1 passes through the vertices of the hyperbola H : x249 −y264 = −1 and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H . Let the product of the eccentricities of E and H be 1 . If l is the length of the latus rectum of the ellipse E , then the 2 value of 113l is equal to _______.

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.If x→1(lim sin(3x2−4x+1)−x2+12x3−7x2+ax+b ) = −2, then the value of (a −b) 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.A rectangle R with end points of the one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x −y + 4 = 0, then the area of R is _____.

Q84.If the mean deviation about the mean of the numbers 1, 2, 3, … … , 𝑛, where 𝑛 is odd, is , then 𝑛 is equal 𝑛 to ______.

Q84.Let the coefficients of the middle terms in the expansion of 4 6 + , β > 0 , βx) , (1 −3βx)2 and (1 −β2 x) ( √61 respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50 −2d is β2 equal to _____ .

Q84.The number of solutions of the equation 2θ −cos2 θ + √2 = 0 in R is equal to ______.

Q84.Let 𝑥1, 𝑥2, 𝑥3, … . . , 𝑥20 be in geometric progression with 𝑥1 = 3 and the common ration 12. A new data is constructed replacing each 𝑥𝑖 by 𝑥𝑖- 𝑖2. If 𝑥 is the mean of new data, then the greatest integer less than or equal to 𝑥 is

Q84.Let 𝐴𝐵 be a chord of length 12 of the circle 169 𝑥- 22 + 𝑦+ 12 = 4 JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper If tangents drawn to the circle at points 𝐴 and 𝐵 intersect at the point 𝑃, then five times the distance of point 𝑃 from chord 𝐴𝐵 is equal to _____.

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.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.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 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 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 _______.