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 (1/3) α-β =... View Article
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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) √3/2 B) √(3/2) C) 3√6 D) √6 Solution: The vector perpendicular to given vectors is \(\begin{array}{l}\vec{a}=\begin{vmatrix} \hat{i} & \hat{j}... 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) 2/17 B) 64/17 C) 128/17 D) 4/17 Solution: The equation of the ellipse 4x2+y2 = 8 dy/dx = -4x/y... View Article
The sum of the solutions of the equation |√x-2|+√x(√x-4)+2 = 0, (x>0) is equal to
A) 4 B) 10 C) 9 D) 12 Solution: Given equation is |√x-2|+√x(√x-4)+2 = 0, (x>0) Put √x = t... 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 = 0 B) 2x-z = 0 C) x+2z-1 = 0 D) x-z-1 = 0 Solution: Required equation of... 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... 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 quadrants B) 1st and 2nd quadrants C) 4th quadrants D) 1st quadrants Solution: 3x+5y =... 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 = 16 B) 8x2 + 9y2+9y = 16 C) 9x2 + 8y2-8y = 16 D) 8x2... 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)... View Article
Let S1 is set of minima and S2 is 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 =... 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]2 2y = [(π/6)-x]2 Differentiating we get, 2... 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) 1/2 B) -1 C) 2 D) 0 Solution: For non trivial solutions, \(\begin{array}{l}\left| \begin{matrix} c & -1 & -1... 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)]2 Solution: f(x) = \(\begin{array}{l}\log \left( \frac{1-x}{1+x} \right)\end{array} \) f(2x/1+x2) = \(\begin{array}{l}\log \left[... 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) 1/2 B) 1/4 C) 1/16 D) 1 Solution: (x2+1)2dy/dx + 2x(x2+1)y = 1 \(\begin{array}{l}\frac{dy}{dx}+\frac{2x}{{{x}^{2}}+1}.\,y=\frac{1}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\end{array} \) dy/dx +Py... 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) 4 B) 3 C) 2 D) 5 Solution: x2-2x+2 = 0 x2-2x+1+1 = 0 (x-1)2 = -1 (x-1) =... 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) 49 B) 48 C) 45 D) 40 Solution: Let the remaining two variates be x and y Given mean... View Article
Sum of natural numbers between 100 and 200 whose HCF with 91 should be more than 1.
A) 1121 B) 3210 C) 3121 D) 1520 Solution: Given numbers: 101, 102, 103, … 198, 199 91=13 × 7... View Article
The sum of the series 2. 20C0 + 5.20C1+8.20C2+62.20C20 is equal to
A) 16.222 B) 8.222 C) 8.221 D) 16.221 Solution: 2. 20C0 + 5.20C1+8.20C2+62.20C20 = \(\begin{array}{l}\sum\limits_{r=0}^{20}{\left( 3r+2 \right)20{{C}_{r}}}\end{array} \) = \(\begin{array}{l}3\sum\limits_{r=0}^{20}{r.\,20{{C}_{r}}}+2.\sum\limits_{r=0}^{20}{20{{C}_{r}}}.\end{array}... View Article
Let g(x) is equal to ln x and f(x) is equal to (1-xcos x) by (1+xcos x) then solve the integral.
A) ln 1 B) ln 2 C) ln e D) ln 4 Solution: g(f(x)) = \(\begin{array}{l}g\left[ \frac{1-x\cos x}{1+x\cos x} \right]\end{array}... 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) 160 B) 175 C) 180 D) 220 Solution: Even places = 4 Odd numbers = 3 Even numbers =... View Article
