Practice Questions

Q86.The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5 . Then, the correct variance is equal to _____.

Q86.Let the hyperbola H : x2 −y2 = 1 and the ellipse E : 3x2 + 4y2 = 12 be such that the length of latus rectum a2 of H is equal to the length of latus rectum of E . If eH and eE are the eccentricities of H and E respectively, then the value of 12(e2H + e2E) is equal to _____.

Q86.Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62 , and their variance is 20 . A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is where i = √−1. Then, the number of elements in the set

Q86.Let 𝐴= 1, 2, 3, 4, 5, 6, 7 and 𝐵= 3, 6, 7, 9. Then the number of elements in the set 𝐶⊆𝐴: 𝐶∩𝐵≠𝜙 is ______

Q86.If f(θ) = sin θ + ∫ −π2 2 (sin θ + t cos θ) ⋅f(t)dt, then ∫ 0 2 f(θ)dθ is 9−x2

Q86.Let the abscissae of the two points 𝑃 and 𝑄 be the roots of 2𝑥2 - 𝑟𝑥+ 𝑝= 0 and the ordinates of 𝑃 and 𝑄 be the roots of 𝑥2 - 𝑠𝑥- 𝑞= 0. If the equation of the circle described on 𝑃𝑄 as diameter is 2𝑥2 + 𝑦2 - 11𝑥- 14𝑦- 22 = 0, then 2𝑟+ 𝑠- 2𝑞+ 𝑝 is equal to ______.

Q86.Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 −2x, and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ______.

Q86.Let A = {n ∈ N : H. C. F. (n, 45) = 1} and let B = {2k : k ∈{1, 2, … , 100}} . Then the sum of all the elements of A ∩B is _____.

Q86.The sum of the maximum and minimum values of the function f(x) = |5x −7| + [x2 + 2x] in the interval [ 54 , 2], where [t] is the greatest integer ≤t, is ______.

Q86.Let 𝑓𝑥= 2𝑥2 + 1 and 𝑔𝑥= 2𝑥- 3, 𝑥< 0 , where 𝑡 is the greatest integer ≤𝑡. Then, in the open interval 2𝑥+ 3, 𝑥≥0 -1, 1, the number of points where fog is discontinuous is equal to ______.

Q86.Let S = [−π, π2 ) −{−π2 , −π4 , −3π4 , π4 }. Then the number of elements in the set A = ∈S : tan + √5 = √5 {θ θ(1 tan(2θ)) −tan(2θ)} is _____ .

Q87.Let c, k ∈R. If f(x) = (c + 1)x2 + (1 −c2)x + 2k and f(x + y) = f(x) + f(y) −xy, for all x, y ∈R, then the value of |2(f(1) + f(2) + f(3) + … … + f(20))| is equal to ______. √2y π dy + = xetan−1(√2 cot 2x), 0 < x <

Q87.Let Max Min Max , = α1 + α2 loge( 158 ), then { 9−x25−x } 5−x } { 9−x25−x x}dx = β. If ∫2α−1β−83 0⩽x⩽2 = α and 0⩽x⩽2{ α1 + α2 is equal to ______

Q87.Let f(x) = max{|x + 1|, |x + 2|, … , |x + 5|} . Then ∫0−6 f(x)dx is equal to ______.

Q87.Let the mean and the variance of 20 observations x1, x2, … x20 be 15 and 9, respectively. For α ∈R, if the mean of (x1 + α)2, (x2 + α)2, … , (x20 + α)2 is 178, then the square of the maximum value of α is equal to JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper ______.

Q87.Let A = (1−i+ i 10 ) {n ∈{1, 2, … . , 100} : An = A} is

Q87.Let f : R →R be a function defined f(x) = e2x+e2e2x . Then f( 1001 ) + f( 1002 ) + f( 1003 ) + … + f( 10099 ) is equal to ______.

Q87.Let 𝑓𝑥= 𝑥- 1𝑥2 - 2𝑥- 3 + 𝑥- 3, 𝑥∈ℝ. If 𝑚 and 𝑀 are respectively the number of points of local minimum and local maximum of 𝑓 in the interval 0, 4, then 𝑚+ 𝑀 is equal to _____.

Q87.For k ∈R, let the solutions of the equation cos(sin−1(x cot(tan−1(cos(sin−1 x))))) = k, 0 < |x| < 1 be α √2 and β, where the inverse trigonometric functions take only principal values. If the solutions of the equation 1 and α , then b is equal to ______. x2 −bx −5 = 0 are 1 + β α2 β2 k2

Q87.Let 𝐴= 1 -1 and 𝐵= 𝛽1 , 𝛼, 𝛽∈𝑅. Let 𝛼1 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐴2 + 2 2 and 2 𝛼 1 0 2 2 𝛼2 be the value of 𝛼 which satisfies 𝐴+ 𝐵2 = 𝐵2. Then 𝛼1 - 𝛼2 is equal to

Q87.Let the function f(x) = 2x2 −loge x, x > 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y2 = 4ax at a point P on it passes through the point (8a, 8a −1) but does not pass through the point (−1a , 0). If the equation of the normal at P is αx + βy = 1 , then α + β is equal to n ∈N is equal to _______.

Q87.The number of matrices 𝐴= 𝑎 𝑏 where 𝑎, 𝑏, 𝑐, d ∈-1, 0, 1, 2, 3, … … , 10, such that 𝐴= 𝐴-1, is ______. 𝑐 𝑑,

Q87.Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve 4x3 −3xy2 + 6x2 −5xy −8y2 + 9x + 14 = 0 at the point (−2, 3) be A . Then 8A is equal to _______.

Q87.Let f and g be twice differentiable even functions on (−2, 2) such that f( 41 ) = 0, f( 21 ) = 0, f(1) = 1 and g( 34 ) = 0, g(1) = 2 Then, the minimum number of solutions of f(x)g′′(x) + f ′(x)g′′(x) = 0 in (−2, 2) is equal to _____.

Q87.Let 𝐴 be a 3 × 3 matrix having entries from the set -1, 0, 1. The number of all such matrices 𝐴 having sum of all the entries equal to 5, is _____ Q88. 1 𝑥25 Let 𝑓: 𝑅→𝑅 be a function defined by 𝑓𝑥= 21 - 2 + 𝑥25 50. If the function 𝑔𝑥= 𝑓𝑓𝑓𝑥+ 𝑓𝑓𝑥, then the 2 greatest integer less than or equal to 𝑔1 is ______.