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If α = cos^-1(3/5), β = tan^-1 (1/3), where 0<α, β < π/2, then α-β is equal to:

A) tan-1(9/5√10)B) sin-1(9/5√10)C) tan-1(9/15)D) cos-1(9/5√10)Solution:α = cos-1(3/5) = tan-1(4/3)β = tan-1... View Article

Find the magnitude of projection of vector [latex]2hat{i}+3hat{j}+hat{k}[/latex] on a vector which is perpendicular to the plane containing vectors [latex]hat{i}+hat{j}+hat{k}[/latex] and [latex]hat{i}+2hat{j}+3hat{k}[/latex]

A) √3/2B) √(3/2)C) 3√6D) √6Solution:The vector perpendicular to given vectors is = 2-6+1/√6= 3/√6= √(3/2)Answer: B... View Article

If the tangents on the ellipse 4x²+y² = 8 at the points (1,2) and (a,b) areperpendicular to each other, then a² is equal to

A) 2/17B) 64/17C) 128/17D) 4/17Solution:The equation of the ellipse 4x2+y2 = 8dy/dx = -4x/yThe tangent at (1,2) and (a,b) are... View Article

The sum of the solutions of the equation |√x-2|+√x(√x-4)+2 = 0, (x>0) is equal to

A) 4B) 10C) 9D) 12Solution:Given equation is |√x-2|+√x(√x-4)+2 = 0, (x>0)Put √x = t|t-2|+t(t-4)+2 = 0If t ≥2t-2+t2-4t+2 = 0t2-3t = 0t(t-3) =... View Article

If a plane passes through intersection of planes 2x-y-4 = 0 and y+2z-4 = 0 and also passes through the point (1, 1, 0). Then the equation of the plane is

A) x-y-z = 0B) 2x-z = 0C) x+2z-1 = 0D) x-z-1 = 0Solution:Required equation of the plane is (2x-y-4)+λ(y+2z-4) = 0Given it pass through... View Article

The perpendicular distance of point ( 2,-1,4) from the line [latex]frac{x+3}{10}=frac{y-2}{-7}=frac{z}{1}[/latex] lies between

A) (2, 3)B) (3, 4)C) (4, 5)D) (1, 2)Solution: Let A be (2,-1,4).Direction ratio of AB = (10λ-5, -7λ+3, λ-4)Drs of given line = (10, -7,... View Article

A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in

A) 1st, 2nd and 4th quadrantsB) 1st and 2nd quadrantsC) 4th quadrantsD) 1st quadrantsSolution:3x+5y = 15 is the equation of the straight line.If... View Article

Let vertices of the triangle ABC is A(0, 0), B(0, 1) and C(x, y) and perimeter is 4 then locus of ‘C’ is

A) 9x2 + 8y2+8y = 16B) 8x2 + 9y2+9y = 16C) 9x2 + 8y2-8y = 16D) 8x2 + 9y2-9x = 16Solution: Given AB+AC+BC = 41+ = 4√(x2+y2)2 =... View Article

Let f:[0,2] -> R be a twice differentiable functions such that f’’(x) >0 for all x ∈ (0,2). If f(x) = f(x)+f(2-x). then f is:

A) Increasing in (0, 1) and decreasing in (1, 2)B) Decreasing in (0, 1) and increasing in (1, 2)C) Increasing in (0, 2)D) Decreasing in (0,... View Article

Let S₁ is set of minima and S₂is set of maxima for the curve y = 9x⁴+12x³-36x²-25 then

A) S1 = {-2,-1} S2 = {0}B) S1 = {-2,1} S2 = {0}C) S1 = {-2,1} S2 = {-1}D) S1 = {-2,2} S2 = {0}Solution:y = 9x4+12x3-36x2-25dy/dx = 36x3 +... View Article

Let [latex]2y={{\left[ {{\cot }^{-1}}\left( \frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x} \right) \right]}^{2}}[/latex] then [latex]\frac{dy}{dx}=[/latex]

A) x-(π/6)B) x+(π/6)C) 2x-(π/6)D) 2x-(π/3)Solution: 2y = [(π/2) -(π/3)+x]22y = [(π/6)-x]2Differentiating we get,2 dy/dx =... View Article

Find the value of ‘c’ for which the following equations have non-trivial solutions: cx-y-z = 0, -cx+y-cz = 0 and x+y-cz = 0.

A) 1/2B) -1C) 2D) 0Solution:For non trivial solutions, c(-c+c) +1(c2+c)-1(-c-1) = 0c2+c+c+1 = 0(c+1)2 = 0c+1 = 0c = -1Answer: B... View Article

If f(x) = log((1-x)/(1+x)) then f(2x/1+x²) is equal to

A) f(x)B) 2f(x)C) -2f(x)D) [f(x)]2Solution:f(x) = = 2 f(x)Answer: B... View Article

Let y = y(x) be the solution of the differential equation, (x²+1)²dy/dx + 2x(x²+1)y = 1 such that y(0) = 0. If √a y(1) = π/32, then the value of a is:

A) 1/2B) 1/4C) 1/16D) 1Solution:(x2+1)2dy/dx + 2x(x2+1)y = 1 dy/dx +Py = QP = 2x/x2+1Q = 1/(x2+1)2 I.F = = 1+x2 The general solution of linear... View Article

If α and β are the roots of x²-2x+2 = 0 then the minimum value of n such that (α/β)ⁿ = 1

A) 4B) 3C) 2D) 5Solution:x2-2x+2 = 0x2-2x+1+1 = 0 (x-1)2 = -1(x-1) = ±ix = 1+i or 1-iLet α = 1+iβ = 1-i(α/β) =... View Article

If mean and variance of 7 variates are 8 and 16 respectively and five of them are 2, 4, 10, 12, 14 then the product of remaining two variates is

A) 49B) 48C) 45D) 40Solution:Let the remaining two variates be x and yGiven mean = 8(x+y+2+4+10+12+14)/7 = 8x+y+42 = 56x+y = 14 ..(1)Given... View Article

How many 9 digit numbers can be formed by using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 such that odd numbers occur at even places.

A) 160B) 175C) 180D) 220Solution: Even places = 4Odd numbers = 3Even numbers = 6Therefore number of numbers = Answer: C... View Article