Practice Questions

Q81.If for some p, q, r ∈R, all have positive sign, one of the roots of the equation q2+r2 (p2 + q2)x2 −2q(p + r)x + q2 + r2 = 0 is also a root of the equation x2 + 2x −8 = 0 , then p2 is equal to-

Q81.The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______.

Q81.Let z = a + ib, b ≠0 be complex numbers satisfying z2 = ¯z ⋅21−|z| . Then the least value of n ∈N , such that zn = (z + 1)n , is equal to _____ .

Q81.In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, −2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is _____ JEE Main 2022 (24 Jun Shift 1) JEE Main Previous Year Paper

Q81.Let S = {4, 6, 9} and T = {9, 10, 11, … , 1000}. If A = {a1 + a2 + … + ak : k ∈N, a1, a2, a3, … , ak ∈S} then the sum of all the elements in the set T −A is equal to _______.

Q81.Let S ={ z ∈C : |z −3| ≤1 and z(4 + 3i) + z(4 −3i) ≤24}. If α + iβ is the point in S which is closest to 4i , then 25(α + β) is equal to ______.

Q81.Sum of squares of modulus of all the complex numbers z satisfying z = iz2 + z2 −z is equal to

Q82.Let f(x) = 2x2 −x −1 and S = {n ∈Z : |f(n)| ≤800} . Then, the value of ∑n∈S f(n) is equal to _______.

Q82.The number of 5 -digit natural numbers, such that the product of their digits is 36 , is

Q82.Let b1b2b3b4 be a 4-element permutation with bi ∈{1, 2, 3, … … … , 100} for 1 ≤i ≤4 and bi ≠bj for i ≠j , such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to ______.

Q82.The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _____.

Q82.Let A( √a3 , √a), a > 0 , be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C . If D(3 cos θ, a sin θ), is a point in the fourth quadrant such that the maximum area of ΔACD is 12 square units, then a is equal to _____

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.Let a circle C of radius 5 lie below the x-axis. The line L1 = 4x + 3y + 2 passes through the centre P of the circle C and intersects the line L2 : 3x −4y −11 = 0 at Q . The line L2 touches C at the point Q . Then the distance of P from the line 5x −12y + 51 = 0 is

Q83.Let A = ∑10i=1 ∑10j=1 min{i, j} and B = ∑10i=1 ∑10j=1 max{i, j}. Then A + B is equal to _____.

Q83.Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.

Q83.Let A = {1, 2, 3, 4, 5, 6, 7} . Define B ={ T ⊆A : either 1 ∉T or 2 ∈T } and C ={ T ⊆A : T the sum of all the elements of T is a prime number.} Then the number of elements in the set B ∪C is _______. Q84. 1 a a 1 48 2160 Let A = ⎡0 1 b ⎤ , a, b ∈R. If for some n ∈N, An = ⎡0 1 96 ⎤ then n + a + b is equal to _______. 0 0 1 0 0 1 ⎣ ⎦ ⎣ ⎦

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.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.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.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.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

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.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.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