Practice Questions

Q73.If the tangent to the curve 𝑦= 𝑥3 at the point 𝑃𝑡, 𝑡3 meets the curve again at 𝑄, then the ordinate of the point which divides 𝑃𝑄 internally in the ratio 1: 2 is: (1) 0 (2) -2𝑡3 (3) -𝑡3 (4) 2𝑡3

Q73.If cot−1(α) = cot−1 2 + cot−1 8 + cot−1 18 + cot−1 32 + … . upto 100 terms, then α is: JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper (1) 1. 01 (2) 1. 00 (3) 1. 02 (4) 1. 03

Q73.A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is (1) 10 (2) 5 2+3√3 3+√3 (3) 10 (4) 5 3+2√3 2+√3 + … + n2

Q74.Let f : [0, ∞) →[0, 3] be a function defined by f(x) = {max{sin2 + cos x,t :x0>≤tπ ≤π}, x ∈[0, π] the following is true ? (1) f is continuous everywhere but not differentiable (2) f is differentiable everywhere in (0, ∞) exactly at one point in (0, ∞) (3) f is not continuous exactly at two points in (4) f is continuous everywhere but not differentiable (0, ∞) exactly at two points in (0, ∞)

Q74.The value of the integral ∫10 (1+x)(1+3x)(3+x)√xdx is: (1) π 4 (1 −√32 ) (2) π8 (1 −√36 ) (3) π 8 (1 −√32 ) (4) π4 (1 −√36 )

Q74.Let a be a real number such that the function f(x) = ax2 + 6x −15, x ∈R is increasing in (−∞, 43 ) and decreasing in ( 34 , ∞) . Then the function g(x) = ax2 −6x + 15, x ∈R has a (1) local maximum at x = −34 (2) local minimum at x = −34 (3) local maximum at x = 34 (4) local minimum at x = 34

Q74.If ∫100π0 sin2x xx dx = 1+4π2απ3 π −[ π ]) e ( α is: (1) 200(1 −e−1) (2) 100(1 −e) (3) 50(e −1) (4) 150(e−1 −1)

Q74.Let 1 / 2 𝑥𝑛 ∀𝑛> 𝑚 and 𝑛, 𝑚∈𝑁. Consider a matrix 𝐴= where 𝐽𝑛, 𝑚= ∫0 𝑥𝑚- 1𝑑𝑥, 𝑎𝑖𝑗3 × 3 J6 + 𝑖, 3 - J𝑖+ 3, 3 , 𝑖≤𝑗 a𝑖𝑗= Then adj A-1 is : 0 , 𝑖> 𝑗. (1) (15 ) 2 × 234 (2) (15 ) 2 × 242 (3) (105 ) 2 × 236 (4) (105 ) 2 × 238

Q74.Let α, β, γ be the real roots of the equation, x3 + ax2 + bx + c = 0, ( a, b, c ∈R and a, b ≠0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0, γu + αv + βw = 0 has non-trivial solution, then the value of a2 is b (1) 5 (2) 3 (3) 1 (4) 0

Q74.Let f(x) cos(2 sin(cot−1 √1−x )), (1) (1 −x)2f ′(x) + 2(f(x))2 = 0 (2) (1 + x)2f ′(x) + 2(f(x))2 = 0 (3) (1 −x)2f ′(x) −2(f(x))2 = 0 (4) (1 + x)2f ′(x) −2(f(x))2 = 0

Q74.The value of the integral ∫ sin θ⋅sin 2θ(sin6 θ+sin41−cosθ+sin22θθ)√2 sin4 θ+3 sin2 θ+6 (1) 1 32 (2) 1 32 18 [11 −18 sin2 θ + 9 sin4 θ −2 sin6 θ] + c 18 [9 −2 sin6 θ −3 sin4 θ −6 sin2 θ] + c (3) 1 32 (4) 1 −32 18 [11 −18 cos2 θ + 9 cos4 θ −2 cos6 θ] + c 18 [9 −2 cos6 θ −3 cos4 θ −6 cos2 θ] + c

Q74.Let 𝑓 be any continuous function on 0, 2 and twice differentiable on 0, 2 . If 𝑓0 = 0, 𝑓1 = 1 and 𝑓2 = 2, then : (1) 𝑓"𝑥> 0 for all 𝑥∈0, 2 (2) 𝑓'𝑥= 0 for some 𝑥∈0, 2 (3) 𝑓"𝑥= 0 for some 𝑥∈0, 2 (4) 𝑓"𝑥= 0 for all 𝑥∈0, 2 2 𝜋𝑥

Q74.Let f : R →R be defined as f(x) = e−x sin x. If F : [0, 1] →R is a differentiable function such that F(x) = ∫x0 f(t)dt, then the value of ∫10 (F ′(x) + f(x))exdx lies in the interval (1) [ 327360 , 360329 ] (2) [ 360330 , 360331 ] (3) [ 331360 , 360334 ] (4) [ 360335 , 360336 ] dx = αe−1 + βe−12 + γ, where α, β, γ are integers and [x] denotes the greatest

Q75.Let f be a twice differentiable function defined on R such that f(0) = 1, f ′(0) = 2 and f ′(x) ≠0 for all f(x) f ′(x) x ∈R. If = 0, for all x ∈R, then the value of f(1) lies in the interval f ′(x) f ′′(x) JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (1) (9, 12) (2) (3, 6) (3) (0, 3) (4) (6, 9)

Q75.If f(x) = { 5x + 1, xx >≤22 (1) f(x) is not continuous at x = 2 (2) f(x) is everywhere differentiable (3) f(x) is continuous but not differentiable at x = 2 (4) f(x) is not differentiable at x = 1

Q75.Let f : (a, b) →R be twice differentiable function such that f(x) = ∫xa g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g′(x) = 0 has at least : (1) twelve roots in (a, b) (2) five roots in (a, b) (3) seven roots in (a, b) (4) three roots in (a, b)

Q75.The number of real roots of the equation e4x + 2e3x −ex −6 = 0 is : (1) 0 (2) 1 (3) 4 (4) 2

Q76.Let slope of the tangent line to a curve at any point P(x, y) be given by xy2+yx x + 2y = 4 at x = −2, then the value of y, for which the point (3, y) lies on the curve, is : (1) −43 (2) 3518 (3) −1819 (4) −1811 −−

Q76.Let a vector →a be coplanar with vectors b = 2ˆi + ˆj + ˆk and →c= ˆi −ˆj + ˆk. If →a is perpendicular to → → → → → d = 3ˆi + 2ˆj + 6ˆk, and →a = √10. Then a possible value of [→a b →c] + [→a b d ] + [→a →c d ] is equal to: (1) −42 (2) −40 (3) −29 (4) −38 → → →

Q76.The area, enclosed by the curves 𝑦= sin𝑥+ cos𝑥 and 𝑦= | cos𝑥- sin𝑥| and the lines 𝑥= 0, 𝑥= 2, is : (1) 2√2 ( √2 + 1 ) (2) 2√2 ( √2 - 1 ) (3) 4 ( √2 - 1 ) (4) 2 ( √2 + 1 )

Q76.The value of ∫ −11 1 ) √2 (( x−1x+1 + ( x−1x+1 ) 2 −2) 2 √2 (1) loge 4 (2) 2 loge 16 + (3) loge 16 (4) 4 loge(3 2√2)

Q76.If In = ∫ π2 cotn xdx, then 4 (1) I2 + I4, (I3 + I5)2, I4 + I6 are in G. P. (2) I2 + I4, I3 + I5, I4 + I6 are in A. P. (3) 1 , 1 , 1 are in A. P. (4) 1 , 1 , 1 are in G. P. I2+I4 I3+I5 I4+I6 I2+I4 I3+I5 I4+I6 is equal to lim n1 + (n+1)2n + (n+2)2n + … + (2n−1)2n ]

Q76.The area (in sq. units) of the part of the circle 𝑥2 + 𝑦2 = 36, which is outside the parabola 𝑦2 = 9𝑥, is equal to (1) 12𝜋+ 3√3 (2) 24𝜋+ 3√3 (3) 24𝜋- 3√3 (4) 12𝜋- 3√3

Q76.Let C1 be the curve obtained by the solution of differential equation 2xy dxdy = y2 −x2, x > 0 . Let the curve C2 be the solution of x2−y22xy = dxdy . If both the curves pass through (1, 1), then the area (in sq. units) enclosed by the curves C1 and C2 is equal to : (1) π −1 (2) π2 −1 (3) π + 1 (4) π4 + 1 → → = 3 and

Q77.Let α be the angle between the lines whose direction cosines satisfy the equations l + m −n = 0 and l2 + m2 −n2 = 0. Then the value of sin4 α + cos4 α is : (1) 5 (2) 1 8 2 (3) 3 (4) 3 8 4