Practice Questions

Q81.For x ∈(0, 5π2 ), define f(x) = ∫x0 √t sin (1) local minimum at π and 2π (2) local minimum at π and local maximum at 2π (3) local maximum at π and local minimum at 2π (4) local maximum at π and 2π

Q82.The area of the region enclosed by the curves y = x, x = e, y = x1 and the positive x-axis is JEE Main 2011 JEE Main Previous Year Paper (1) 1 square units (2) 3 square units 2 (3) 5 square units (4) 1 square units 2 2

Q83.If dy = y + 3 > 0 and y(0) = 2, then y(ln 2) is equal to dx (1) 5 (2) 13 (3) -2 (4) 7

Q84.Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equation dV(t)dt = −k(T −t), where k > 0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is (1) I −kT2 (2) 1 −k(T−t)22 (3) e−kT (4) T2 −1k → → → → 1 1 is

Q85.If →a = (3^i + ^k) and b = 7 (2^i + 3^j −6^k), then the value of (2→a− b) ⋅[(→a× b) × (→a+ 2 b)] √10 (1) −3 (2) 5 (3) 3 (4) −5

Q86.The vector →a and →b are not perpendicular and →c and →d are two vectors satisfying: →b × →c = →b × →d and →a ⋅→d = 0 . Then the vector →d is equal to →a⋅→c →b⋅→c (1) →c + (2) →b + ( →a⋅→b )→b ( →a⋅→b )→c →a⋅→c →b⋅→c (3) →c (4) →b −( →a⋅→b )→b −( →a⋅→b )→c z−3 , then λ equals and the plane x + 2y + 3z = 4 is cos−1 λ

Q87.If the angle between the line x = y−12 = 14 (√5 ) (1) 3 (2) 2 2 5 (3) 5 (4) 2 3 3

Q89.Consider 5 independent Bernoulli's trials each with probability of success p . If the probability of at least one failure is greater than or equal to 31 , then p lies in the interval 32 (1) ( 34 , 1112 ] (2) [0, 12 ] (3) ( 1112 , 1] (4) ( 12 , 34 ]

Q90.If C and D are two events such that C ⊂D and P(D) ≠0 , then the correct statement among the following is JEE Main 2011 JEE Main Previous Year Paper (1) P(C ∣D) ≥P(C) (2) P(C ∣D) < P(C) (3) P(C ∣D) = P(D)P(C) (4) P(C ∣D) = P(C) JEE Main 2011 JEE Main Previous Year Paper

Q61.If α and β are the roots of the equation x2 −x + 1 = 0, then α2009 + β2009 = (1) −1 (2) 1 (3) 2 (4) −2

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

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

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

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