Practice Questions

Q67.Let L be a tangent line to the parabola y2 = 4x −20 at (6, 2). If L is also a tangent to the ellipse x2 y2 2 + b = 1, then the value of b is equal to : (1) 11 (2) 14 (3) 16 (4) 20 JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper

Q67.The value of lim cos h−sin h) } h→0{ √3h(√3 (1) 43 (2) √32 (3) 23 (4) 43

Q67.A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1 . Which of the following points does NOT lie on it? (1) (0, 3) (2) (4, 5) (3) (5, 4) (4) (−6, 0) y2

Q67. x→2(∑9 (1) 5 (2) 7 24 36 (3) 1 (4) 9 5 44

Q67.If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and 203 , respectively, then the value of |a −b| is equal to: (1) 9 (2) 11 (3) 7 (4) 1

Q67.Consider a circle C which touches the y− axis at (0, 6) and cuts off an intercept 6√5 on the x− axis. Then the radius of the circle C is equal to : (1) √53 (2) 9 (3) 8 (4) √82 x lim x ) is equal to : 8√1−sin x−8√1+sin

Q67. lim sin2(π cos4 x) is equal to : x4 x→0 (1) 2π2 (2) π2 (3) 4π2 (4) 4π

Q67.If 𝛼= lim tan3𝑥- tan𝑥𝜋 and 𝛽= lim are the roots of the equation, 𝑎𝑥2 + 𝑏𝑥- 4 = 0, then the ordered 𝑥→𝜋/ 4 cos𝑥+ 4 𝑥→0cos𝑥cot𝑥 pair 𝑎, 𝑏 is : (1) -1, 3 (2) 1, - 3 (3) 1, 3 (4) -1, - 3

Q67.Two poles AB of length a metres and CD of length a + b(b ≠a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan ∠ACB = 12 , then: (1) x2 + 2(a + 2b)x −b(a + b) = 0 (2) x2 + 2(a + 2b)x + a(a + b) = 0 (3) x2 −2ax + b(a + b) = 0 (4) x2 −2ax + a(a + b) = 0 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper

Q68.The value of lim cos−1(x−[x]2)⋅sin−1(x−[x]2) , where [x] denotes the greatest integer ≤x is: x→0+ x−x3 (1) π (2) 0 (3) π (4) π 4 2

Q68.If in a triangle ABC, AB = 5 units, ∠B = cos−1( 53 ) and radius of circumcircle of ΔABC is 5 units, then the area (in sq. units) of ΔABC is: (1) 10 + 6√2 (2) 8 + 2√2 (3) 6 + 8√3 (4) 4 + 2√3 a ∈R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix.

Q68.The value of x→0( (1) 0 (2) 4 (3) −4 (4) −1

Q68.Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is: (1) 25 (2) 30 (3) 20√3 (4) 25√3

Q68.If for the matrix, A = [ α1 −αβ ], (1) 3 (2) 1 (3) 2 (4) 4

Q68.Consider the following system of equations: x + 2y −3z = a 2x + 6y −11z = b x −2y + 7z = c where a, b and c are real constants. Then the system of equations : (1) has a unique solution when 5a = 2b + c (2) has no solution for all a, b and c (3) has infinite number of solutions when (4) has a unique solution for all a, b and c 5a = 2b + c

Q68.The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is: (1) 1 (2) 2 (3) 3 (4) 0

Q68.On the ellipse x2 8 + 4 = 1, let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S′ be the foci of the ellipse and e be its eccentricity. If A is the JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper area of the triangle SPS′ , then the value of (5 −e2) ⋅A is (1) 12 (2) 6 (3) 14 (4) 24

Q68.Let *, □∈{∧, ∨} be such that the Boolean expression (p*~q) ⇒(p □q) is a tautology. Then : (1) *= ∨, □= ∧ (2) *= ∨, □= ∨ (3) *= ∧, □= ∨ (4) *= ∧, □= ∧

Q68.If α, β are the distinct roots of x2 + bx + c = 0, then lim e2(x2+bx+c)−1−2(x2+bx+c) is equal to x→β (x−β)2 (1) 2(b2 + 4c) (2) b2 −4c (3) 2(b2 −4c) (4) b2 + 4c

Q68.Let in a right angled triangle, the smallest angle be θ. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin θ is equal to: (1) √5+1 (2) √5−1 4 2 (3) √2−1 (4) √5−1 2 4

Q68.Let the equation of the pair of lines, y = px and y = qx, can be written as (y −px)(y −qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 −4xy −5y2 = 0 is: (1) x2 −3xy + y2 = 0 (2) x2 + 4xy −y2 = 0 (3) x2 + 3xy −y2 = 0 (4) x2 −3xy −y2 = 0

Q68.The value of lim [r]+[2r]+...+[nr] , where r is non-zero real number and [r] denotes the greatest integer less than n→∞ n2 or equal to r, is equal to : (1) r (2) r 2 (3) 2r (4) 0

Q69.The values of λ and μ such that the system of equations x + y + z = 6, 3x + 5y + 5z = 26 and x + 2y + λz = μ has no solution, are: (1) λ = 3, μ = 5 (2) λ = 3, μ ≠10 (3) λ ≠2, μ = 10 (4) λ = 2, μ ≠10

Q69.Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy sin−1( 3x5 ) + sin−1( 4x5 ) = sin−1 x is equal to: (1) 2 (2) 1 (3) 3 (4) 0

Q69.The equation of one of the straight lines which passes through the point (1, 3) and makes an angles with the straight line, y + 1 = 3√2x is tan−1(√2) + + = 0 (1) 4√2x + 5y −(15 4√2) = 0 (2) 5√2x + 4y −(15 4√2) + = 0 (3) 4√2x + 5y −4√2 = 0 (4) 4√2x −5y −(5 4√2)