Practice Questions

Q62.The number of complex numbers z such that |z −1| = |z + 1| = |z −i| equals (1) 1 (2) 2 (3) ∞ (4) 0

Q63.There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is (1) 36 (2) 66 (3) 108 (4) 3

Q64.A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = … … = a10 = 150 and a10, a11, … … are in A.P. with common difference −2, then the time taken by him to count all notes is JEE Main 2010 JEE Main Previous Year Paper (1) 34 minutes (2) 125 minutes (3) 135 minutes (4) 24 minutes

Q65.Let S1 = ∑10j=1 j(j −1)10Cj, S2 = ∑10j=1 j10Cj and S3 = ∑10j=1 j210Cj . Statement-1: S3 = 55 × 29 Statement-2: S1 = 90 × 28 and S2 = 10 × 28 . (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is false Statement-2 is not the correct explanation for Statement-1 (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Q66.Let cos(α + β) = 54 and let sin(α −β) = 135 , where 0 ≤α, β ≤π4 , then tan 2α = (1) 3356 (2) 1912 (3) 20 (4) 25 7 16 y

Q67.The line L given by x b = 1 passes through the point (13, 32). The line K is parallel to L and has the 5 + equation x c + 3y = 1. Then the distance between L and K is (1) √17 (2) 17 √15 (3) 23 (4) 23 √17 √15

Q68.The circle x2 + y2 = 4x + 8y + 5 intersects the line 3x −4y = m at two distinct points if (1) −35 < m < 15 (2) 15 < m < 65 (3) 35 < m < 85 (4) −85 < m < −35

Q69.If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is (1) 2x + 1 = 0 (2) x = −1 (3) 2x −1 = 0 (4) x = 1 =

Q70.Let f : R →R be a positive increasing function with limx→∞ f(3x)f(x) = 1. Then limx→∞ f(2x)f(x) (1) 2 (2) 3 3 2 (3) 3 (4) 1

Q71.For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is (1) 11 (2) 6 2 (3) 13 (4) 5 2 2

Q72.For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is (1) There is a regular polygon with 1 (2) There is a regular polygon with r = R R r = 32 √2 (3) There is a regular polygon with Rr = √32 (4) There is a regular polygon with Rr = 21

Q73.Let S be a non-empty subset of R. Consider the following statement: P : There is a rational number x ∈S such that x > 0. Which of the following statements is the negation of the statement P ? JEE Main 2010 JEE Main Previous Year Paper (1) There is no rational number x ∈S such that (2) Every rational number x ∈S satisfies x ≤0 x ≤0 (3) x ∈S and x ≤0 ⇒x is not rational (4) There is a rational number x ∈S such that x ≤0

Q74.Consider the following relations: R = {(x, y) ∣x, y are real numbers and x = wy for some rational number w ∣m, n, p and q are integers such that n, q ≠0 and qm = pn} . Then } ; S = {( mn , pq ) (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Q75.The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0 , is (1) 5 (2) 6 (3) at least 7 (4) less than 4

Q76.Let A be a 2 × 2 matrix with non-zero entries and let A2 = 1 , where 1 is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A . Statement-1: Tr(A) = 0 Statement-2: |A| = 1 (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is false Statement-2 is not the correct explanation for Statement-1 (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Q77.Consider the system of linear equations: x1 + 2x2 + x3 = 3 2x1 + 3x2 + x3 = 3 3x1 + 5x2 + 2x3 = 1 The system has (1) exactly 3 solutions (2) a unique solution (3) no solution (4) infinite number of solutions

Q78.Let f : R →R be a continuous function defined by f(x) = ex+2e−x1 . Statement-1: f(c) = 31 , for some c ∈R. Statement-2: 0 < f(x) ≤ 1 , for all x ∈R 2√2 (1) Statement-1 is true, Statement-2 is true; (2) Statement-1 is true, Statement-2 is false Statement-2 is not the correct explanation for Statement-1 (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1 JEE Main 2010 JEE Main Previous Year Paper

Q79.Let f : (−1, 1) →R be a differentiable function with f(0) = −1 and f ′(0) = 1 . Let g(x) = [f(2f(x) + 2)]2 . Then g′(0) = (1) −4 (2) 0 (3) −2 (4) 4

Q80.The equation of the tangent to the curve y = x + 4 , that is parallel to the x-axis, is x2 (1) y = 1 (2) y = 2 (3) y = 3 (4) y = 0 . If f has a local minimum at x = −1, then a

Q81.Let f : R →R be defined by f(x) = {k2x−2x,+ 3, ifif xx ≤−1> −1 possible value of k is (1) 0 (2) −12 (3) −1 (4) 1

Q82.Let p(x) be a function defined on R such that p′(x) = p′(1 −x), for all x ∈[0, 1], p(0) = 1 and p(1) = 41 . Then ∫10 p(x)dx equals (1) 21 (2) 41 (3) 42 (4) √41

Q83.The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = 3π2 is (1) 4√2 + 2 (2) 4√2 −1 (3) 4√2 + 1 (4) 4√2 −2

Q84.Solution of the differential equation cos xdy = y(sin x −y)dx, 0 < x < π2 is (1) y sec x = tan x + c (2) y tan x = sec x + c (3) tan x = (sec x + c)y (4) sec x = (tan x + c)y

Q85.Let →a = ^j −^k and →c = ^i −^j −^k. Then vector →b satisfying →a × →b + →c =→0 and →a ⋅→b = 3 is (1) 2^i −^j + 2^k (2) ^i −^j −2^k (3) ^i + ^j −2^k (4) −^i + ^j −2^k →

Q86.If the vectors →a = ^i −^j + 2^k, b = 2^i + 4^j + ^k and→c= λ^i +^j + μ^k are mutually orthogonal, then (λ, μ) = (1) (2, −3) (2) (−2, 3) (3) (3, −2) (4) (−3, 2)