Practice Questions

Q81.If 𝛼, 𝛽 are roots of the equation 𝑥2 + 5√2𝑥+ 10 = 0, 𝛼> 𝛽 and 𝑃𝑛= 𝛼𝑛- 𝛽𝑛 for each positive integer 𝑛, then the value of 𝑃17𝑃20 + 5√2𝑃17𝑃192 is equal to 𝑃18𝑃19 + 5√2𝑃18

Q81.The number of the real roots of the equation (x + 1)2 + x −5 = 274 is ________.

Q81.The number of solutions of the equation log(x+1)(2x2 + 7x + 5) + log(2x+5)(x 1)2

Q81.If for the complex numbers 𝑧 satisfying |𝑧- 2 - 2𝑖| ≤1, the maximum value of |3𝑖𝑧+ 6| is attained at 𝑎+ 𝑖𝑏, then 𝑎+ 𝑏 is equal to _____ .

Q82.The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

Q82.The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles' equations x2 + y2 −10x −10y + 41 = 0 x2 + y2 −24x −10y + 160 = 0 is ________ then the value of det (A4)+ det (A10 −(Adj (2 A))10) 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.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.Let A1, A2, A3, … . . be squares such that for each n ⩾1, the length of the side of An equals the length of diagonal of An+1 . If the length of A1 is 12 cm, then the smallest value of n for which area of An is less than one, is = 0 is a

Q82.The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is k is equal to is the term, independent of x, in the binomial expansion of ( x4 −12x2 )12, then

Q82.Let the coefficients of third, fourth and fifth terms in the expansion of (x + x2a )n, 12 : 8 : 3. Then the term independent of x in the expansion, is equal to _______. n ∈N be the slopes of the three line segments OA, OB and

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.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 _________.

Q82.The sum of 162th power of the roots of the equation x3 −2x2 + 2x −1 = 0 is ______. + + … . . + = n. 2m , then n + m is

Q82.The number of rational terms in the binomial expansion of 1 1 120 4 + 5 6 (4 ) is_______.

Q82.Let i = √−1. If (−1+i√3) 21 + (1+i√3) 21 = k, and n = [|k|] be the greatest integral part of |k|. (1−i)24 (1+i)24 Then ∑n+5j=0 (j + 5)2 −∑n+5j=0 (j + 5) is equal to ________.

Q82.If one of the diameters of the circle 𝑥2 + 𝑦2 - 2𝑥- 6𝑦+ 6 = 0 is a chord of another circle '𝐶', whose center is at 2, 1, then its radius is_____.

Q82.If ∑10r=1 r!(r3 + 6r2 + 2r + 5) = α(11!), then the value of α is equal to ___ .

Q82.The least positive integer n such that , i = √−1, is a positive integer, is ______. (1−i)n−2

Q83.If the coefficient of 𝑎7𝑏8 in the expansion of ( 𝑎+ 2𝑏+ 4𝑎𝑏) 10 is 𝐾· 216, then 𝐾 is 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 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.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 number of solutions of the equation cot x = cot x + sin1 x in the interval [0, 2π] is

Q83.If the constant term, in binomial expansion of (2xr + x21 ) 10 is 180, then r is equal to ____________.