Practice Questions

Q70.The x−coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as (0, 1), (1, 1) and (1, 0) is (1) 1 + √2 (2) 1 −√2 (3) 2 + √2 (4) 2 −√2

Q71.If a circle C passing through (4, 0) touches the circle x2 + y2 + 4x −6y −12 = 0 externally at a point (1, −1) , then the radius of the circle C is : (1) 5 (2) 2√5 (3) 4 (4) √57

Q72.Given : A circle, 2x2 + 2y2 = 5 and a parabola, y2 = 4√5x. Statement - I : An equation of a common tangent to these curves is y = x + √5 . Statement - II : If the line, y = mx + √5m (m ≠0) is their common tangent, then m satisfies m4 −3m2 + 2 = 0 . JEE Main 2013 (07 Apr) JEE Main Previous Year Paper (1) Statement - I is true; Statement - II is false. (2) Statement - I is false; Statement - II is true. (3) Statement - I is true; Statement - II is true; (4) Statement - I is true; Statement - II is true; Statement - II is a correct explanation for Statement - II is not a correct explanation for statement - I. statement - I.

Q73.If a and c are positive real numbers and the ellipse x2 + y2 = 1 has four distinct points ir common with the 4c2 c2 circle x2 + y2 = 9a2 , then (1) 9ac −9a2 −2c2 < 0 (2) 6ac + 9a2 −2c2 < 0 (3) 9ac −9a2 −2c2 > 0 (4) 6ac + 9a2 −2c2 > 0

Q74.If the extremities of the base of an isosceles triangle are the points (2a, 0) and (0, a) and the equation of one of the sides is x = 2a, then the area of the triangle, in square units, is : (1) 5 a2 (2) 5 a2 4 2 (3) 25a2 (4) 5a2 4

Q76.A common tangent to the conics x2 = 6y and 2x2 −4y2 = 9 is: (1) x −y = 32 (2) x + y = 1 (3) x + y = 92 (4) x −y = 1 Then the number of non-singular matrices in the set S is : : aij ∈{0, 1, 2}, a11 = a22}

Q84.If x = ∫y0 √1+t2dt , then dx2d2y (1) y (2) √1 + y2 (3) x (4) y2 √1+y2

Q86.Let →a = 2^i + ^j −2^k,→b = ^i + ^j. If →c is a vector such that →a ∙→c = |→c|, |→c −→a| = 2√2 and the angle between →a × →b and →c is 30∘ , then |(→a × →b) × →c| equals: (1) 1 (2) 3√3 2 2 (3) 3 (4) 23

Q61.If a, b, c, d and p are distinct real numbers such that (a2 + b2 + c2)p2 −2p(ab + bc + cd) + (b2+ c2 + d2) ≤0, then (1) a, b, c, d are in A.P. (2) ab = cd (3) ac = bd (4) a, b, c, d are in G.P.

Q64.Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + … … + (361 + 380 + 400) is 8000 . Statement 2 : ∑nk=1 (k3 −(k −1)3) = n3 for any natural number n. (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; statement (4) Statement 1 is true, statement 2 is false 2 is not a correct explanation for statement 1

Q65.The sum of the series 1 + 34 + 109 + 2728 + … upto n terms is (1) 67 n + 16 − 3.2n−12 (2) 53 n −76 + 2.3n−11 (3) n + 21 − 2.3n1 (4) n −13 − 3.2n−11

Q71.Statement 1 : An equation of a common tangent to the parabola y2 = 16√3x and the ellipse 2x2 + y2 = 4 is y = 2x + 2√3 . Statement 2 : If the line y = mx + 4√3m , (m ≠0) is a common tangent to the parabola y2 = 16√3x and the ellipse 2x2 + y2 = 4 , then m satisfies m4 + 2m2 = 24 . (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; statement (4) Statement 1 is true, statement 2 is false 2 is not a correct explanation for statement 1

Q72. limx→0 ( x−sinx x ) sin ( x1 ) (1) equals 1 (2) equals 0 (3) does not exist (4) equals −1

Q75.Statement 1: The variance of first n odd natural numbers is n2−1 Statement 2: The sum of first n odd natural 3 n(4n2+1) number is n2 and the sum of square of first n odd natural numbers is . 3 (1) Statement 1 is true, Statement 2 is false. (2) Statement 1 is true, Statement 2 is true; Statement 2 is not a 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 a correct explanation for Statement 1. Q76. ⎡ 1 0 0⎤ ⎡ 1 0 0 ⎤ If A = 2 1 0 and B = −2 1 0 then AB equals ⎣−3 2 1⎦ ⎣ 7 −2 1 ⎦ (1) I (2) A (3) B (4) 0

Q75.In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to (1) 5π (2) π 6 6 (3) π (4) 3π 4 4 JEE Main 2012 (Offline) JEE Main Previous Year Paper Q76. ⎛1 0 0⎞ ⎛1⎞ ⎛0⎞ Let A = 2 1 0 . If u1 and u2 are column matrices such that Au1 = 0 and Au2 = 1 , then ⎝3 2 1⎠ ⎝0⎠ ⎝0⎠ u1 + u2 is equal to (1) ⎛−1⎞ (2) ⎛ −1⎞ 1 1 ⎝ 0 ⎠ ⎝ −1⎠ (3) ⎛−1⎞ (4) ⎛ 1 ⎞ −1 −1 ⎝ 0 ⎠ ⎝ −1⎠

Q77.Let P and Q be 3 × 3 matrices with P ≠Q. If P 3 = Q3 and P 2Q = Q2P , then determinant of (P 2 + Q2) is equal to (1) −2 (2) 1 (3) 0 (4) −1

Q80.Let f : [1, 3] →R be a function satisfying x ≤f(x) ≤√6 −x, for all x ≠2 and f(2) = 1, where R is the [x] set of all real numbers and [x] denotes the largest integer less than or equal to x. Statement 1: limx→2−f(x) exists. Statement 2: f is continuous at x = 2. (1) Statement 1 is true, Statement 2 is true, (2) Statement 1 is false, 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.

Q82.If dx d G(x) = etanx x , x ∈(0, π/2), then ∫1/21/4 x2 ⋅etan(πx2)dx is equal to (1) G(π/4) −G(π/16) (2) 2[G(π/4) −G(π/16)] (3) π[G(1/2) −G(1/4)] (4) G(1/√2) −G(1/2)

Q85.The parabola y2 = x divides the circle x2 + y2 = 2 into two parts whose areas are in the ratio (1) 9π + 2 : 3π −2 (2) 9π −2 : 3π + 2 (3) 7π −2 : 2π −3 (4) 7π + 2 : 3π + 2 x dy)

Q87.If the three planes x = 5, 2x −5ay + 3z −2 = 0 and 3bx + y −3z = 0 contain a common line, then (a, b) is equal to (1) ( 158 , −15 ) (2) ( 15 , −815 ) (3) (−815 , 51 ) (4) (−15 , 158 )

Q87.Let ABCD be a parallelogram such that AB→ =→q, AD→ = →p and ∠BAD be an acute angle. If →r is the vector that coincides with the altitude directed from the vertex B to the side AD, then →r is given by (1) →r = 3→q −3(→p⋅→q) →p (2) →r = −→q+ (→p⋅→p) ( →p⋅→p→p⋅→q )→p →p⋅→q 3(→p⋅→q) (3) →r = →q (4) →r = −3→q + →p −( →p⋅→p )→p (→p⋅→p)

Q88.A unit vector which is perpendicular to the vector 2^i −^j + 2^k and is coplanar with the vectors ^i + ^j −^k and 2^i + 2^j −^k is (1) 2^j+^k (2) 3^i+2^j−2^k √5 √17 (3) 3^i+2^j+2^k (4) 2^i+2^j−^k √17 3

Q67.The lines L1 : y −x = 0 and L2 : 2x + y = 0 intersect the line L3 : y + 2 = 0 at P and Q respectively. The bisector of the acute angle between L1 and L2 intersect L3 at R. This question has Statement −1 and Statement −2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1 : The ratio PR : RQ equals 2√2 : √5 . Statement-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles. (1) Statement −1 is true, Statement −2 is true; (2) Statement −1 is true, Statement- 2 is false. Statement −2 is not a 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 a correct explanation for Statement −1

Q77. x x < 0 ⎧ sin(p+1)x+sinx The value of p and q for which the function f(x) = is continuous for all x in R, is ⎨ q , x = 0 √x+x2−√x , x > 0 ⎩ x3/2 (1) p = 52 , q = 12 (2) p = −32 , q = 12 (3) p = 21 , q = 32 (4) p = 12 , q = −32

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