Practice Questions

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

Q90.The sum of the series 20C0 −20C1 + 20C2 −20C3 + … −… + 20C10 is (1) −20C10 (2) 12 20C10 (3) 0 (4) 20C10

Q91.Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' k ' can take is given by (1) {1, 3} (2) {0, 2} (3) {−1, 3} (4) {−3, −2}

Q92.Let P = (−1, 0), Q = (0, 0) and R = (3, 3√3) be three points. The equation of the bisector of the angle PQR (1) √3x + y = 0 (2) x + √32 y = 0 (3) √3 x + y = 0 (4) x + √3y = 0 2

Q93.If one of the lines of my2 + (1 −m2)xy −mx2 = 0 is a bisector of the angle between the lines xy = 0 , then m is JEE Main 2007 JEE Main Previous Year Paper (1) −1/2 (2) −2 (3) 1 (4) 2

Q94.Consider a family of circles which are passing through the point (−1, 1) and are tangent to x− axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval (1) 0 < k < 1/2 (2) k ≥1/2 (3) −1/2 ≤k ≤1/2 (4) k ≤1/2

Q95.The equation of a tangent to the parabola y2 = 8x is y = x + 2 . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is (1) (−1, 1) (2) (0, 2) (3) (2, 4) (4) (−2, 0) y2 x2

Q96.For the hyperbola = 1 , which of the following remains constant when α varies? cos2 α α − sin2 (1) eccentricity (2) directrix (3) abscissae of vertices (4) abscissae of foci

Q97.The function f : R ∼{0} →R given by f(x) = x1 − e2x−12 can be made continuous at x = 0 by defining f(0) as (1) 2 (2) −1 (3) 0 (4) 1

Q98.The average marks of boys in a class is 52 and that of girls is 42 . The average marks of boys and girls combined is 50 . The percentage of boys in the class is (1) 40 (2) 20 (3) 80 (4) 60

Q99.A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB(= a) subtends an angle of 60∘ at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30∘ . The height of the tower is (1) 2a (2) 2a√3 √3 (3) a (4) a√3 √3 Q100. 5 5α α Let A = ⎡ 0 α 5α ⎤. If A2 = 25 , then |α| equals 0 0 5 ⎣ ⎦ (1) 52 (2) 1 (3) 1/5 (4) 5 Q101. 1 1 1 If D = 1 1 + x 1 for x ≠0, y ≠0 then D is 1 1 1 + y (1) divisible by neither x nor y (2) divisible by both x and y (3) divisible by x but not y (4) divisible by y but not x Q102.If sin−1 ( x5 ) + cosec−1 ( 54 ) = π2 then a value of x is JEE Main 2007 JEE Main Previous Year Paper (1) 1 (2) 3 (3) 4 (4) 5 Q103.The largest interval lying in (−π2 , π2 ) for which the function [f(x) = 4−x2 + cos−1 ( x2 −1) + log(cos x)] is defined, is (1) [0, π] (2) (−π2 , π2 ) (3) [−π4 , π2 ) (4) [0, π2 ) Q104.Let f : R →R be a function defined by f(x) = Min{x + 1, |x| + 1}. Then which of the following is true? (1) f(x) ≥1 for all x ∈R (2) f(x) is not differentiable at x = 1 (3) f(x) is differentiable everywhere (4) f(x) is not differentiable at x = 0 Q105.The normal to a curve at P(x, y) meets the x-axis at G . If the distance of G from the origin is twice the abscissa of P , then the curve is a (1) ellipse (2) parabola (3) circle (4) pair of straight lines Q106.A value of C for which the conclusion of Mean Value Theorem holds for the function f(x) = loge x on the interval [1, 3] is (1) 2 log3 e (2) 21 loge 3 (3) log3 e (4) loge 3 Q107.The function f(x) = tan−1(sin x + cos x) is an increasing function in (1) ( π4 , π2 ) (2) (−π2 , π4 ) (3) (0, π2 ) (4) (−π2 , π2 ) Q108. ∫ dx equals cos x+√3 sin x (1) 1 2 log tan ( x2 + 12π ) + c (2) 21 log tan ( x2 − 12π ) + c (3) log tan ( x2 + 12π ) + c (4) log tan ( x2 − 12π ) + c dt. Then F(e) equalsQ109.Let F(x) = f(x) + f ( x1 ), where f(x) = ∫x1 log1+tt (1) 1 (2) 0 2 (3) 1 (4) 2 = π2 isQ110.The solution for x of the equation ∫x√2 t√t2−1dt (1) 2 (2) π (3) √3 (4) None of these 2 Q111.The area enclosed between the curves y2 = x and y = |x| is (1) 2/3 (2) 1 (3) 1/6 (4) 1/3 Q112.The differential equation of all circles passing through the origin and having their centres on the x-axis is (1) x2 = y2 + xy dxdy (2) x2 = y2 + 3xy dxdy (3) y2 = x2 + 2xy dxdy (4) y2 = x2 −2xy dxdy JEE Main 2007 JEE Main Previous Year Paper Q113.The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be √19 N . The value of P is (1) 5 N (2) 6 N (3) 3 N (4) 4 N Q114.If ^u and ^v are unit vectors and θ is the acute angle between them, then 2^u × 3^v is a unit vector for (1) exactly two values of θ (2) more than two values of θ (3) no value of θ (4) exactly one value of θ – Q115.Let –a = ^i +^j + ^k, b = ^i −^j + 2^k and –c = x^i + (x −2)^j −^k. If the vector –c lies in the plane of ¯a and ¯b, then x equals (1) 0 (2) 1 (3) −4 (4) −2 Q116.Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2 . If L makes an angles α with the positive x-axis, then cos α equals (1) 1 (2) 1 √3 2 (3) 1 (4) 1 √2 Q117.If a line makes an angle of π with the positive directions of each of x-axis and y-axis, then the angle that the 4 line makes with the positive direction of the z−axis is (1) π (2) π 6 3 (3) π (4) π 4 2 Q118.If (2, 3, 5) is one end of a diameter of the sphere x2 + y2 + z2 −6x −12y −2z + 20 = 0 , then the coordinates of the other end of the diameter are (1) (4, 9, −3) (2) (4, −3, 3) (3) (4, 3, 5) (4) (4, 3, −3) Q119.A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is (1) 1/729 (2) 8/9 (3) 8/729 (4) 8/243 Q120.Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2 , respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is (1) 0.06 (2) 0.14 (3) 0.2 (4) None of these JEE Main 2007 JEE Main Previous Year Paper