Practice Questions

Q82.For real numbers α and β, consider the following system of linear equations: x + y −z = 2, x + 2y + αz = 1 and 2x −y + z = β. If the system has infinite solutions, then α + β is equal to ______.

Q82.Let z = 1−i√32 , i = √−1. Then the value of 21 + (z + 1z ) 3 + (z2 + z21 ) 3 + (z3 + z31 ) 3 + … + (z21 + z211 ) 3 is______.

Q82.If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is ___ .

Q82.Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.

Q82.If the real part of the complex number z = 1−3i3+2i coscos θθ , θ ∈(0, π2 ) is zero, then the value of sin2 3θ + cos2 θ is equal to ______.

Q82.The equation of a circle is Re (z2) + 2(Im(z))2 + 2 Re (z) = 0, where z = x + iy. A line which passes through the centre of the given circle and the vertex of the parabola, x2 −6x −y + 13 = 0, has y-intercept equal to _________.

Q83.The term independent of x in the expansion of 10 [ x2/3−x1/3+1x+1 − x−x1/2x−1 ] , x ≠1 , is equal to ___.

Q83.Let n ∈N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms of nC0, 3 ⋅nC1, 5 ⋅nC2, 7 ⋅nC3, … is equal to 2100 ⋅101, then 2[ n−12 ] is equal to n is equal to :

Q83.Let tan α, tan β and tan γ; α, β, γ ≠(2n−1)π2 , OC, respectively, where O is origin. If circumcentre of ΔABC coincides with origin and its orthocentre lies 2 on y-axis, then the value of ( coscos3α+cosα⋅cos3β+cosβ⋅cos γ 3γ ) is equal to :

Q83.Let m, n ∈N and gcd(2, n) = 1 . If 30(300 ) 30 30 30 29( 1 ) +2(28 ) 1(29 ) n equal to _______. (Here = nCk) (k )

Q83.The sum of all 3 -digit numbers less than or equal to 500, that are formed without using the digit 1 and they all are multiple of 11, is ______.

Q83.If the value of 1 + 1 1 1 + + + … . . upto ∞ is 𝑙, then 𝑙2 is equal to 3 32 33 .

Q83.The locus of the point of intersection of the lines (√3)kx + ky −4√3 = 0 and √3x −y −4(√3)k conic, whose eccentricity is a −b −tan( 2θ )

Q83.The total number of 4 -digit numbers whose greatest common divisor with 18 is 3 is _____.

Q83.Let 𝐴= {𝑛∈𝑁: 𝑛 is a 3 - digit number } 𝐵= 9𝑘+ 2: 𝑘∈𝑁 and 𝐶= 9𝑘+ 𝑙: 𝑘∈𝑁 for some 𝑙0 < 𝑙< 9. If the sum of all the elements of the set 𝐴∩𝐵∪𝐶 is 274 × 400, then 𝑙 is equal to Q84. 3 -1 -2 Let 𝑃= 2 0 𝛼 , where 𝛼∈𝑅. Suppose 𝑄= 𝑞𝑖𝑗 is a matrix satisfying 𝑃𝑄= 𝑘𝐼3 for 3 -5 0 𝑘 𝑘2 some non-zero 𝑘∈𝑅. If 𝑞23 = - 8 and 𝑄= 2 , then 𝛼2 + 𝑘2 is equal to_________.

Q83.Let y = mx + c, m > 0 be the focal chord of y2 = −64x, which is tangent to (x + 10)2 + y2 = 4 . Then, the m + value of 4√2( c) is equal to______ x2 ) is equal to ea , then a is equal to_____.

Q83.If the coefficient of 𝑎7𝑏8 in the expansion of ( 𝑎+ 2𝑏+ 4𝑎𝑏) 10 is 𝐾· 216, then 𝐾 is equal to

Q83.The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is______.

Q83.Let n be a positive integer. Let A = ∑nk=0 (−1)k × nCk[( 12 + ( 43 ) k + ( 87 ) k + ( 1615 ) k + ( 3231 ) k]. If 63A = 1 − 1 , then n is equal to ______ . 230

Q83.Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C2 meet the side AB at E . If the length of EB is α + √3β, where α, β are integers, then α + β is equal to ________. JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper aex−b cos x+ce−x

Q83.The students S1, S2, … , S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is __________.

Q83. sin2 x −2 + cos2 x cos 2x Let f(x) = 2 + sin2 x cos2 x cos 2x , x ∈[0, π]. Then the maximum value of f(x) is equal to sin2 x cos2 x 1 + cos 2x

Q83.Let the equation x2 + y2 + px + (1 −p)y + 5 = 0 represent circles of varying radius r ∈(0, 5]. Then the number of elements in the set S ={ q : q = p2 and q is an integer} is ___________ y2

Q83.If the sum of the coefficients in the expansion of ( 𝑥+ 𝑦) 𝑛 is 4096, then the greatest coefficient in the expansion is _____.

Q83.For k ∈N, let α(α+1)(α+2)…….(α+20) 2 1 = ∑20K=0 α+kAk , where α > 0. Then the value of 100( A14+A15A13 ) is equal to ____________.