Practice Questions

Q85.Let the lines y + 2x = √11 + 7√7 and 2y + x = 2√11 + 6√7 be normal to a circle C , then the value of C : (x −h)2 + (y −k)2 = r2 . If the line √11y −3x = 5√773 + 11 is tangent to the circle (5h −8k)2 + 5r2 is equal to ______.

Q85.Let H : x2 −y2 = 1, a > 0, b > 0 , be a hyperbola such that the sum of lengths of the transverse and the a2 b2 H is √11 + , then value of a2 + b2 is equal to ______. 2 conjugate axes is 4(2√2 √14). If the eccentricity ) + 2 Q86. 50 tan(3 tan−1( 21 cos−1( √51 ))+4√2 tan( 21 tan−1(2√2)) is equal to ______.

Q85.The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _______.

Q85.The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x −y = 3 respectively. If its orthocentre is (2, a), −12 < a < 2 , then p is equal to

Q85.The number of functions f , from the set A = {x ∈N : x2 −10x + 9 ≤0} to the set B = {n2 : n ∈N} such that f(x) ≤(x −3)2 + 1 , for every x ∈A , is _______.

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.If 40C0 + 41C1 + 42C2 + ⋯+ 60C20 = mn × 60C20 where m & n are co-prime, then m + n is equal to and let L2 be the line passing through the origin and

Q85.If 1 + (2 + 49C1 + 49C2 + … . +49C49)(50C2 + 50C4 + … . . +50C50) is equal to 2n. m, where m is odd, then n + m is equal to _____ .

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

Q87.Let R1 and R2 be relations on the set {1, 2, … , 50} such that R1 ={ (p, pn) : p is a prime and n ≥0 is an integer} and R2 ={ (p, pn) : p is a prime and n = 0 or 1 }. Then, the number of elements in R1 −R2 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.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 A = (1−i+ i 10 ) {n ∈{1, 2, … . , 100} : An = A} is

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

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 _______.