If the lines\(\stackrel{→}{r}= ( \hat{i} - \hat{j} + \hat{k} ) + λ (\hat{3j} - \hat{k} )= ( \hat{i} - \hat{j} + \hat{k} ) + λ (\hat{3j} - \hat{k} )\)and\(\stackrel{→}{r} = ( \alpha \hat{i} - \hat{j} ) + μ( \hat{2j} - \hat{3k} )\)are co-planer , then the distance of the plane containing these two lines from the point \(( α , 0 , 0 )\) is :
Let f : R → R be a differentiable function such that\(f(\frac{π}{4})=\sqrt2,f(\frac{π}{2})=0 \) and \(f′(\frac{π}{2})=1\)and let\(g(x) = \int_{x}^{\frac{\pi}{4}} \left(f'(t)\sec(t) + \tan(t)\sec(t)f(t)\right) \, dt\)for\( x∈(\frac{π}{4},\frac{π}{2})\) Then \(\lim_{{x \to \frac{\pi}{2}^-}} g(x)\)is equal to
Let S be the set of all the natural numbers, for which the line \(\frac{x}{a}+\frac{y}{b}=2 \)is a tangent to the curve\((\frac{x}{a})^n+(\frac{y}{b})^n=2 \)at the point (a, b), ab ≠ 0. Then :
The number of 5-digit natural numbers, such that the product of their digits is 36, is _____ .
Letƒ : R → Rbe defined as f(x) = x -1 andg : R - { 1, -1 } → Rbe defined asg(x) = \(\frac{x²}{x² - 1}\)Then the function fog is :
\(\lim_{{x \to 0}} \limits\) \(\frac{cos(sin x) - cos x }{x^4}\) is equal to :
If m is the slope of a common tangent to the curves\(\frac{x²}{16} + \frac{y²} {9} = 1\)and x2 + y2 = 12, then 12m2 is equal to:
If y = y(x) is the solution of the differential equation
\(2x^2\frac{dy}{dx}-2xy+3y^2=0\) such that \(y(e)=\frac{e}{3},\)
then y(1) is equal to
Let ƒ :R→R be a function defined by \(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\)Then \(f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)\) is equal to ________.
LetA=\(\begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}\)If B = I – 5C1(adjA) + 5C2(adjA)2 – …. – 5C5(adjA)5, then the sum of all elements of the matrix B is