Practice Questions

Q71.If ( 1−i1+i ) 2 = ( i−11+i ) 3 = 1, (m, n ∈N) then the greatest common divisor of the least values of m and n is 3 + 321 + 331 +….∞) is __________

Q71.If the mean and variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x ⋅y is equal to

Q71.For a positive integer n, (1 + x ) is expanded in increasing powers of x . If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then n is equal to

Q71.If the sum of the coefficients of all even powers of x in the product (1 + x + x2 + … + x2n)(1 −x + x2 −x3 + … + x2n) is 61, then n is equal to

Q71.The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word EXAMINATION is

Q71.The number of words (with or without meaning) that can be formed from all the letters of the word ′′LETTER′′ in which vowels never come together is.....

Q71.The number of distinct solutions of the equation, log 1 |sin x| = 2 −log 1 |cos x| in the interval [0, 2π], is 2 2 ________

Q72.Consider the data on x taking the values 0, 2, 4, 8, . . . . . , 2n with frequencies nC0, nC1, nC2, . . . . , nCn respectively. If the mean of this data is 728 , then n is equal to ....... . 2n

Q72.If the variance of the terms in an increasing A. P. b1b2, b3, … … . . , b11 is 90 then the common difference of this A. P. is

Q72.Let A(1, 0), B(6, 2) and C( 32 , 6) be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point (−76 , −13 ), is

Q72.The natural number m, for which the coefficient of x in the binomial expansion of (xm + x21 ) 22 is 1540, is

Q72.If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4th A.M. is equal to 2nd G.M., then m is equal to:

Q72.Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A →B ∣2 ∈f(A) and f is not one-one } is …

Q72.The numbers of integral values of k for which the line, 3x + 4y = k intersects the circle, x2 + y2 −2x −4y + 4 = 0 at two distinct points is.... x+x2+x3+...+xn−n

Q72.The value of 0. 16 log2.5( 1

Q72.Set A has melements and set B has nelements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m ⋅n is___.

Q72.The sum, ∑7n=1 n(n+1)(2n+1)4 , is equal to √2sinα

Q72.If Cr ≡25Cr and C0 + 5 ∙C1 + 9 ∙C2 + … + (101) ∙C25 = 225 ∙k, then k is equal to ____________.

Q72.Let PQ be a diameter of the circle x2 + y2 = 9. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of αβ is _______

Q72.An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then, the number of ways in which 4 marbles can be drawn so that at the most three of them are red is ___________.

Q72.If the system of equations x −2 y + 3z = 9 2x + y + z = b x −7y + az = 24, has infinitely many solutions, then a −b is equal to ______

Q72.The coefficient of x4 in the expansion of (1 + x + x2) 10 is ________

Q72.Let X = {n ∈N : 1 ≤n ≤50}. If A = {n ∈X : n is a multiple of 2} and B = {n ∈X : n is a multiple of 7}, then the number of elements in the smallest subset of X , containing both A and B , is.

Q73.The sum ∑20k=1(1 + 2 + 3 + … + k) is ___________.

Q73. lim 3x+33−x−12 is equal to x→2 3−x2 −31−x