Practice Questions

Q54.If the constant term in the binomial expansion of (√x − x2k ) 10 (1) 9 (2) 1 (3) 3 (4) 2

Q54.If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)n+5 are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is : (1) 462 (2) 330 (3) 792 (4) 252

Q54.Let an be the nth term of a G.P. of positive terms. If ∑100n=1 a2n+1 = 200 and ∑100n=1 a2n = 100, then ∑200n=1 an is equal to: (1) 300 (2) 225 (3) 175 (4) 150

Q54.If 32 sin 2α−1, 14 and 34−2 sin 2α are the first three terms of an A.P. for some α , then the sixth term of this A.P. is (1) 66 (2) 81 (3) 65 (4) 78

Q54.Five numbers are in A. P. , whose sum is 25 and product is 2520. If one of these five numbers is −12 , then the greatest number amongst them is (1) 27 (2) 7 (3) 21 (4) 16 2

Q54.If |x| < 1, |y| < 1 and x ≠1 , then the sum to infinity of the following series (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3)+. . . . . is (1) x+y−xy (2) x+y+xy (1+x)(1+y) (1+x)(1+y) (3) x+y−xy (4) x+y+xy (1−x)(1−y) (1−x)(1−y)

Q54.The value of ( 2 ⋅1 P0 −3 ⋅2 P1 + 4 ⋅3 P2−. . . . . . . . up to 51th term) +( 1! −2! + 3!−. . . . . . . up to 51th term) is equal to (1) 1 −51(51)! (2) 1 + (51)! (3) 1 + (52)! (4) 1 1 1 n 2 + 5 8 is exactly 33, then the least value of n is

Q54.If α and β, be the coefficients of x4 and x2 , respectively in the expansion of 6 6 + √x2 + −√x2 (x −1) (x −1) , then (1) α + β = 60 (2) α + β = −30 (3) α −β = 60 (4) α −β = −132

Q54.If 1 + (1 −22 ⋅1) + (1 −42 ⋅3) + (1 −62 ⋅5) + … … + (1 −202 ⋅19) = α −220 β , then an ordered pair (α, β) is equal to: (1) (10, 97) (2) (11, 103) (3) (10, 103) (4) (11, 97)

Q54.The product 2 41 ∙4 161 ∙8 481 ∙16 1 128 ∙. . . . to ∞ is equal to: (1) 2 21 (2) 2 41 (3) 1 (4) 2

Q54.Let two points be A(1, −1) and B(0, 2). If a point P(x', y') be such that the area of ΔPAB = 5 sq. units and it lies on the line 3x + y −4λ = 0, then a value of λ is (1) 4 (2) 3 (3) 1 (4) −3

Q54.If the sum of first 11 terms of an A.P. , a1, a2, a3 … … is 0(a1 ≠0) then the sum of the A.P a1, a3, a5, … . . a23 is ka1 where k is equal to (1) −12110 (2) 12110 (3) 725 (4) −725

Q55.If the number of integral terms in the expansion of (3 ) (1) 264 (2) 128 (3) 256 (4) 248

Q55.If the sum of the first 20 terms of the series log(71/2) x + log(71/3) x + log(71/4) x + … is 460 , then x is equal to: (1) 72 (2) 71/2 (3) e2 (4) 746/21

Q55.If the perpendicular bisector of the line segment joining the points P(1, 4) and Q(k, 3) has y-intercept equal to −4, then a value of k is; (1) −2 (2) −4 (3) √14 (4) √15

Q55.The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + … + 492 + 49 + 1, is (1) 32 (2) 63 (3) 60 (4) 35

Q55.If x = ∑∞n=0 (−1)ntan2θ and y = ∑∞n=0 cos2nθ, for 0 < θ < π4 , then: (1) x(1 + y) = 1 (2) y(1 −x) = 1 (3) y(1 + x) = 1 (4) x(1 −y) = 1 x when π

Q55.If a ΔABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates: (1) (– 3, 3) (2) (3, −3) (3) (−35 , 53 ) (4) ( 53 , −35 )

Q55.The locus of a point which divides the line segment joining the point (0, −1) and a point on the parabola x2 = 4y internally in the ratio 1 : 2 is: (1) 9x2 −12y = 8 (2) 9x2 −3y = 2 (3) x2 −3y = 2 (4) 4x2 −3y = 2

Q55.If {p} denotes the fractional part of the number p, then { 32008 } (1) 5 (2) 7 8 8 (3) 3 (4) 1 8 8 is incident at an angle 30° on the line x = 1 at the point A . The

Q55.If the common tangent to the parabolas, y2 = 4 x and x2 = 4 y also touches the circle, x2 + y2 = c2, then c is equal to : (1) 1 (2) 1 2√2 √2 (3) 41 (4) 12 P is any point on the

Q55.The coefficient of x7 in the expression (1 + x)10 + x(1 + x)9 + x2(1 + x)8+. . . . +x10 , is (1) 210 (2) 330 (3) 120 (4) 420

Q55.Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (−1, −4) in the line is : (1) ( 115 , 285 ) (2) ( 295 , 85 ) (3) ( 85 , 295 ) (4) ( 295 , 115 )

Q55.Let α > 0, β > 0 be such that α3 + β2 = 4 . If the maximum value of the term independent of x in the 1 10 10k, then k is equal to binomial expansion of (αx 9 + βx−16 ) is (1) 336 (2) 352 (3) 84 (4) 176

Q55.Let S be the sum of the first 9 term of the series : {x + ka} + {x2 + (k + 2)a} + {x3 + (k + 4)a} + {x4 + (k + 6)a} + … where a ≠0 and x ≠ 1 . If x10−x+45a(x−1) S = x−1 , then k is equal to (1) −5 (2) 1 (3) −3 (4) 3