The area bounded by the curve y = x2 -1, tangent to it at (2, 3) and y-axis is a) 2/3b) 4/3c) 8/3d) 1Solution:Equation of tangent at (2, 3) is(y+3)/2 = 2x-1y+3 = 4x-24x-y = 5Required area = Ar(PAB) - = 8-(16/3)= 8/3Answer (c)... View Article
Find common tangent of the two curves y2 = 4x and x2+y2-6x = 0. a) y = (x/3)+3b) y = (x/√3) -√3c) y = (x/3)-3d) y = (x/√3) +√3Solution:Equation of tangent to the parabola y2 = 4x is y =... View Article
A plane parallel to y-axis passing through line of intersection of planes x+y+z =1 and 2x+3y-z-4 = 0 which of the point lie on the plane a) (3,2,1)b) (−3,0,1)c) (−3,1,1)d) (3,1,−1)Solution:Required plane is π1+ λπ2 = 0(x+y+z-1)+λ(2x+3y-z-4) =... View Article
Consider the curves y = x2 +2 and y = 10-x2. Let θ be the angle between both the curves at point of intersection, then find ǀtan θǀ a) 8/15b) 5/17c) 3/17d) 8/17Solution:x2 +2 = 10-x2x = ±2 and y = 4Point of intersection of curves = (±2, 4)y = x2+2; y =... View Article
Find the equation of line through (−4,1,3) and parallel to the plane x + y + z = 3 while the line intersects another line whose equation is x + y – z = 0 = x + 2y -3z + 5 a) Answer (c)... View Article
3 circles of radii a, b, c (a < b < c) touch each other externally and have x – axis as a common tangent then a) a, b, c are in A.P. b) 1/√b = (1/√a) + (1/√c) c) √a, √b, √c are in A.P. d) (1/√c) + (1/√b) = (1/√a) Solution: PQ +QR = PR √(4ac) +... View Article
The value of 3(cos θ – sin θ)4 + 6(sin θ + cos θ)2+4 sin6θ is, where θ∈ (π/4, π/2) a) 13-4 cos4θb) 13-4 cos6θc) 13-4 cos6θ+2 sin4θ cos2θd) 13-4 cos4θ+2 sin4θ cos2θSolution:3(cos2 θ+ sin2 θ -sin 2θ)2 +6(sin2 θ+ cos2 θ+ sin 2θ)... View Article
Consider the system of equations x+y+z = 1, 2x+3y+2z =1, 2x+3y+(a2 -1)z = a+1 then a) system has a unique solution for ǀaǀ = √3b) system is inconsistence for ǀaǀ = √3c) system is inconsistence for a = 4d) system is... View Article
If px+qy+r = 0 represent a family of straight lines such that 3p+2q+4r = 0 then a) All lines are parallelb) All lines are inconsistentc) All lines are concurrent at (3/4, 1/2)d) All lines are concurrent at... View Article
If cos-1(2/3x) + cos-1(3/4x) = π/2, x>3/4 then x = a) √145/11b) √145/12c) √146/10d) √146/11Solution:cos-1 (2/3x) = π/2 - cos-1(3/4x)cos-1 (2/3x) = cos-1... View Article
If slant height of a right circular cone is 3 cm then the maximum volume of cone is a) 2√3 πcm3b) 4√3 πcm3c) (2+√3) πcm3d) (2-√3) πcm3Solution:l = 3r2 + h2 = 9r2 = 9 -h2V = (1/3)πr2h= (1/3)πh(9 -h2) V =... View Article
If eccentricity of the hyperbola [latex]\frac{x^{2}}{\cos ^{2}\theta }-\frac{y^{2}}{\sin ^{2}\theta } =1[/latex] is more than 2 when ∈(0, π/2) then values of length of latus rectum lies in the interval a) (3, ∞)b) (1, 3/ 2)c) (2, 3)d) (-3, -2)Solution:For hyperbola, e2 = 1+(b2/a2)= 1+tan2 θ= sec2θe>2 ⇒ sec θ > 2∈(π/3, π/2)Length of... View Article
From a well shuffled deck of cards, 2 cards are drawn with replacement. If x represent numbers of times ace coming, then value of P(x = 1) + P(x = 2) is a) 25/169b) 24/169c) 49/169d) 23/169Solution:P(x = 1) = 2C1×(4/52)×(48/52) = 24/169P(x = 2) = 2C2×(4/52)2= 1/169P(x = 1) + P(x... View Article
If y (x) is solution of x(dy/dx) +2y = x2, y(1) = 1, then value of y(1/2) a) -49/16b) 49/16c) 45/8d) -45/8Solution:x(dy/dx) + 2y = x2(dy/dx) + 2y/x = xThis is linear differential equation.I.F =e ∫(2/x)dx = e2 log x... View Article
If (A⊕B)∧(∼ A⊝ B) = [latex]Awedge B[/latex] what should be proper symbol in place of [latex]oplus[/latex] and ⊝ to hold the equation a) Solution:By inspection ⊕ represents ∧ and ⊝ represents ∨ABA∧B~A~A∨B(A∧B)∧ (~A∨B)TTTFTTTFFFFFFTFTTFFFFTTF(A∧B)∧(~A∨B)≡ A∧BAnswer: (a)... View Article
Let A = [latex]\begin{bmatrix} \cos \theta &-\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}[/latex] find the value of A-50 at θ = π/12. a) Answer: (b)... View Article
Find sum of all possible values of θ in the interval (-π/2, π) for which [latex]frac{3+2isin theta }{1-2isin theta}[/latex] is purely imaginary; a) π/3b) πc) 2π/3d) π/2Solution: sin2 θ = 3/4As θ∈ (-π/2,π ) = ±π/3, 2π/3Sum of all values of θ = 2π/3Answer: (c)... View Article
There is a parabola having axis as x-axis, vertex is at a distance of 2 unit from origin & focus is at (4, 0). Which of the following point does not lie on the parabola. a) (6, 8)b) (5, 2√6)c) (8, 4√3)d) (4,-4)Solution:The equation of parabola is y2 = 8(x-2)(6,8) does not lie on this curve.Answer: (a)... View Article
[latex]\lim_{y\to 0}\frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}[/latex] = a) 1/4√2b) 1/2√2c) 1/2√2(1+√2)d) does not existSolution:Rationalising numerator,Answer: (a)... View Article
{2403/15} = k/15 then find k. (where {.} denotes fractional part function) a) 2b) 8c) 1d) 4Solution:Solution:24 ≡ 1 (mod 15)2400 ≡ 1 (mod 15)2403 ≡ 8(mod 15){2403/15} = 8/15k = 15Answer: (b)... View Article
