Practice Questions

Q85.Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is

Q85.The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If 𝜎 is the standard deviation of the data after omitting the two wrong observations from the data, then 38𝜎2 is equal to _______. JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper

Q86.The number of distinct real roots of the equation x5(x3 −x2 −x + 1) + x(3x3 −4x2 −2x + 4) −1 = 0 is

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 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.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.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.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 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.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.For the curve C : (x2 + y2 −3) + (x2 −y2 −1) 5 = 0 , the value of 3y′ −y3y′′ , at the point (α, α), α > 0 , on C , is equal to ________.

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 the mirror image of a circle c1 : x2 + y2 −2x −6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx +10fy + 38 = 0. If r is the radius of circle c2 , then α + 6r2 is equal to ______

Q86.Let the equation of two diameters of a circle 𝑥2 + 𝑦2 - 2𝑥+ 2𝑓𝑦+ 1 = 0 be 2𝑝𝑥- 𝑦= 1 and 2𝑥+ 𝑝𝑦= 4𝑝. Then the slope 𝑚∈0, ∞ of the tangent to the hyperbola 3𝑥2 - 𝑦2 = 3 passing through the centre of the circle is equal to _____. Q87. 2 -1 -1 √3i - 1 Let 𝐴= 1 0 -1 and 𝐵= 𝐴- 𝐼. If 𝜔= , then the number of elements in the set 2 1 -1 0 𝑛∈1, 2, … , 100: 𝐴𝑛+ 𝜔𝐵𝑛= 𝐴+ 𝐵 is equal to _____ .

Q86.Let S be the set containing all 3 × 3 matrices with entries from {−1, 0, 1} . The total number of matrices A ∈S such that the sum of all the diagonal elements of ATA is 6 is ______.

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 S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S →S as f(n) = { 2n2n,−11 ifif nn == 1,6, 2,7, 3,8, 4,9, 510 + 1 , if n is odd Let g : S ≥S be a function such that fog(n) = , then {nn −1 , if n is even g(10)(g(1) + g(2) + g(3) + g(4) + g(5)) 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 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.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.A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is tan−1 34 . Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, 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 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 <