Practice Questions

Q86.Let f : [−1, 1] →R be defined as f(x) = ax2 + bx + c for all x ∈[−1, 1], where a, b, c ∈R such that f(−1) = 2, f ′(−1) = 1 and for x ∈(−1, 1) the maximum value of f ′′(x) is 21 . If f(x) ≤α, x ∈[−1, 1], then the least value of α is equal to x )ndx, where n ∈N . If (20)I10 = αI9 + βI8, for natural numbers α and β, then α −β

Q86.The number of elements in the set {𝐴= 𝑎 𝑏 𝑎, 𝑏, 𝑑∈{ - 1, 0, 1} and (𝐼- 𝐴) 0 𝑑: is 2 × 2 identity matrix, is .

Q86.Let the mean and variance of four numbers 3, 7, x and y (x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x −y is ______.

Q86.If the system of equations kx + y + 2z = 1 3x −y −2z = 2 −2x −2y −4z = 3 has infinitely many solutions, then k is equal to ______ .

Q86.If R is the least value of a such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and S is the greatest value of a such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then the value of |R −S| is is + x )dx

Q86.Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (−5, 0). If the locus of the point P is a circle of radius r, then 4r2 (in the nearest integer) is equal to __________.

Q86.The value of the integral ∫π0 |sin 2x|dx is ________.

Q86. x + a −c x + b x + a Let a, b, c, d be in arithmetic progression with common difference λ. If x −1 x + c x + b = 2 , then x −b + d x + d x + c value of λ2 is equal to________.

Q86.The total number of 3 × 3 matrices A having enteries from the set (0, 1, 2, 3) such that the sum of all the diagonal entries of AAT is 9, is equal to

Q86.Let f : [0, 3] →R be defined by f(x) = min{x −[x], 1 + [x] −x} where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x ∈[0, 3] where f is discontinuous, and Q denote the set containing all x ∈(0, 3) where f is not differentiable. Then the sum of number of elements in P and Q is equal to _____.

Q86.If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to ___ . dx is

Q86.The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is 5x8+7x6

Q86.Let a, b ∈R, b ≠0 . Defined a function, f(x) = tansin2x−sinπ 2x , for x > 0 {a bx3 If f is continuous at x = 0, then 10 −ab is equal to x = 0 is equal to

Q86.Let P(a sec θ, b tan θ) and Q(a sec ϕ, b tan ϕ) where θ + ϕ = π2 , be two points on the hyperbola x2a2 −y2b2 If the ordinate of the point of intersection of normals at P and Q is −k( a2+b22b ), then k is equal to

Q86.If the system of linear equations 2x + y −z = 3 x −y −z = α 3x + 3y + βz = 3 has infinitely many solutions, then |α + β −αβ| is equal to __________. + αx dydx + βy = 0, then |α −β| is equal to _______.

Q86.Let X1, X2, … , X18 be eighteen observations such that ∑18i=1(Xi −α) = 36 and ∑18i=1 (Xi −β)2 = 90 , where α and β are distinct real numbers. If the standard deviation of these observations is 1 , then the value of |α −β| is _______. Q87. ⎡ 1 0 0 ⎤ ⎡1 0 0⎤ If the matrix A = 0 2 0 satisfies the equation A20 + αA19 + βA = 0 4 0 for some real numbers ⎣ 3 0 −1 ⎦ ⎣0 0 1⎦ α and β, then β −α is equal to ______.

Q86.Consider the following frequency distribution : class 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 Frequency 𝛼 110 54 30 𝛽 If the sum of all frequencies is 584 and median is 45, then |𝛼- 𝛽| is equal to . JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper

Q86.Let 𝑓( 𝑥) = 𝑥6 + 2𝑥4 + 𝑥3 + 2𝑥+ 3, 𝑥∈R. Then the natural number 𝑛 for which lim 𝑥𝑛𝑓( 1 ) - 𝑓( 𝑥) = 44 is 𝑥→1 𝑥- 1 _____ . 2

Q87.The minimum value of 𝛼 for which the equation sin𝑥+ 1 - sin𝑥= 𝛼 has at least one solution in 0, 2 is______.

Q87.Let A be a 3 × 3 real matrix. If det (2 Adj (2 Adj (Adj (2A)))) = 241, then the value of det (A2) equals ______.

Q87.The value of ∫2−2 3x2 −3x −6

Q87.Let F : [3, 5] →R be a twice differentiable function on (3, 5) such that F(x) = e−x ∫x3 (3t2 + 2t + 4F ′(t))dt. If F ′(4) = αeβ−224 , then α + β is equal to _____. (eβ−4)2

Q87.An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper candidates is 15. If μ is the average marks of girls and σ2 is the variance of marks of 50 candidates, then μ + σ2 is equal to Q88. ∫2ex+3e−x4ex+7e−x dx = 141 (ux + v loge(4ex + 7e−x)) + C , where C is a constant of integration, then u + v is equal to

Q87.Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by |α + β + γ| is equal the tangent T , ellipse E , lines x = 1 and x = √5 is α√5 + β + γ cos−1( √51 ), then to______. →

Q87.The number of points, at which the function f(x) = |2x + 1| −3|x + 2| + x2 + x −2 , x ∈R is not differentiable, is