Practice Questions

Q82.Let p be the statement " x is an irrational number", q be the statement " y is a transcendental number", and r be the statement " x is a rational number iff y is a transcendental number". Statement −1 : r is equivalent to either q or p Statement −2 : r is equivalent to ∼(p ↔∼q). (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1.

Q83.The statement p →(q →p) is equivalent to (1) p →(p →q) (2) p →(p ∨q) (3) p →(p ∧q) (4) p →(p ↔q)

Q84.The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b ? (1) a = 0, b = 7 (2) a = 5, b = 2 (3) a = 1, b = 6 (4) a = 3, b = 4 JEE Main 2008 JEE Main Previous Year Paper

Q85. AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60∘ . He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45∘ . Then the height of the pole is (1) 7√3 + 1)m 2 ⋅ √3−11 m (2) 7√32 ⋅(√3 (3) 7√3 2 ⋅(√3 −1)m (4) 7√32 ⋅ √3+11

Q86.Let R be the real line. Consider the following subsets of the plane R × R. S = {(x, y) : y = x + 1 and 0 < x < 2}, T = {(x, y) : x −y is an integer }. Which one of the following is true? (1) neither S nor T is an equivalence relation on R (2) both S and T are equivalence relations on R (3) S is an equivalence relation on R but T is not (4) T is an equivalence relation on R but S is not

Q87.Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A . Assume that A2 = 1. Statement -1: If A ≠1 and A ≠−1, then det A = −1. Statement −2 : If A ≠1 and A ≠−1, then tr(A) ≠0. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1

Q88.Let A be a square matrix all of whose entries are integers. Then which one of the following is true? (1) If det A = ±1, then A−1 exists but all its entries (2) If det A ≠±1, then A−1 exists and all its entries are not necessarily integers are non-integers (3) If det A = ±1, then A−1 exists and all its entries (4) If det A = ±1, then A−1 need not exist are integers

Q89.Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal to (1) 2 (2) −1 (3) 0 (4) 1

Q90.The value of cot (cosec−1 53 + tan−1 23 ) is (1) 6 (2) 3 17 17 (3) 4 (4) 5 17 17

Q91.Let f : N →Y be a function defined as f(x) = 4x + 3, where Y = {y ∈N : y = 4x + 3 for some x ∈N}. Show that f is invertible and its inverse is (1) g(y) = 3y+43 (2) g(y) = 4 + y+34 (3) g(y) = y+34 (4) g(y) = y−34 1 ), if x ≠1 x−1 . Then which one of the following is true?

Q92.Let f(x) = −1) sin ( {(x0, if x = 1 JEE Main 2008 JEE Main Previous Year Paper (1) f is neither differentiable at x = 0 nor at x = 1 (2) f is differentiable at x = 0 and at x = 1 (3) f is differentiable at x = 0 but not at x = 1 (4) f is differentiable at x = 1 but not at x = 0

Q93.Suppose the cube x3 −px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds? (1) The cubic has minima at √p3 and maxima at (2) The cubic has minima at −√p3 and maxima at −√p3 √p3 and The cubic has maxima at both and (3) The cubic has minima at both √p3 −√p3 (4) √p3 −√p3

Q94.How many real solutions does the equation x7 + 14x5 + 16x3 + 30x −560 = 0 have? (1) 7 (2) 1 (3) 3 (4) 5

Q95.The value of √2 ∫ sin xdx is sin(x−π4 ) (1) x + log cos (x −π4 ) + c (2) x −log sin (x −π4 ) + c (3) x + log sin (x −π4 ) + c (4) x −log cos (x −π4 ) + c dx. Then which one of the following is true?

Q96.Let I = ∫10 sin√xx dx and J = ∫10 cos√xx (1) I > 32 and J > 2 (2) I < 23 and J < 2 (3) I < 32 and J > 2 (4) I > 23 and J < 2

Q97.The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal to (1) 5 (2) 1 3 3 (3) 2 (4) 4 3 3

Q98.The solution of the differential equation dx dy = x+yx satisfying the condition y(1) = 1 is (1) y = ln x + x (2) y = x ln x + x2 (3) y = xe(x−1) (4) y = x ln x + x

Q99.The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is (1) (x −2)y′2 = 25 −(y −2)2 (2) (y −2)y′2 = 25 −(y −2)2 (3) (y −2)2y′2 = 25 −(y −2)2 (4) (x −2)2y′2 = 25 −(y −2)2 Q100.The non-zero verctors →a,→b and →c are related by →a = 8→b and →c = −7→b. Then the angle between →a and→cis (1) 0 (2) π/4 (3) π/2 (4) π Q101.The vector →a = α^i + 2^j + β^k lies in the plane of the vectors →b = ^i + ^j and →c = ^j + ^k and bisects the angle between →b and →c. Then which one of the following gives possible values of α and β ? (1) α = 2, β = 2 (2) α = 1, β = 2 (3) α = 2, β = 1 (4) α = 1, β = 1 Q102.The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz− plane at the point (0, 172 , −132 ). Then JEE Main 2008 JEE Main Previous Year Paper (1) a = 2, b = 8 (2) a = 4, b = 6 (3) a = 6, b = 4 (4) a = 8, b = 2 Q103.If the straight lines x−1 k = y−22 = z−33 and x−23 = y−3k = z−12 intersect at a point, then the integer k is equal to (1) −5 (2) 5 (3) 2 (4) −2 Q104.It is given that the events A and B are such that P(A) = 41 , P ( BA ) = 12 and P ( BA ) = 32 . Then P(B) is (1) 1 (2) 1 6 3 (3) 2 (4) 1 3 2 Q105.A die is thrown. Let A be the event that the number obtained is greater than 3 . Let B be the event that the number obtained is less than 5 . Then P(A ∪B) is (1) 3 (2) 0 5 (3) 1 (4) 2 5 JEE Main 2008 JEE Main Previous Year Paper

Q83.If the difference between the roots of the equation x2 + ax + 1 = 0 is less than √5, then the set of possible values of a is JEE Main 2007 JEE Main Previous Year Paper (1) (−3, 3) (2) (−3, ∞) (3) (3, ∞) (4) (−∞, −3)

Q84.If |z + 4| ≤3 , then the maximum value of |z + 1| is (1) 4 (2) 10 (3) 6 (4) 0

Q85.The set S = {1, 2, 3, … , 12) is to be partitioned into three sets A, B, C of equal size. Thus, A ∪B ∪C = S, A ∩B = B ∩C = A ∩C = ϕ . The number of ways to partition S is (1) 12! (2) 12! 3!(4!)3 3!(3!)4 (3) 12! (4) 12! (4!)3 (3!)4

Q86.In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals (1) 1 2 (1 −√5) (2) 21 √5 (3) √5 (4) 12 (√5 −1)

Q87.If p and q are positive real numbers such that p2 + q2 = 1 , then the maximum value of (p + q) is (1) 2 (2) 1/2 (3) 1 (4) √2 √2

Q88.The sum of the series 2! 1 −13! + 4!1 −… upto infinity is (1) e−2 (2) e−1 (3) e−1/2 (4) e1/2

Q89.In the binomial expansion of (a −b)n, n ≥5 , the sum of 5th and 6th terms is zero, then ab equals (1) 5 (2) 6 n−4 n−5 (3) n−5 (4) n−4 6 5