Let f : R → R be a continuous function satisfying f(x) + f(x + k) = n, for all x ∈ R where k > 0 and n is a positive integer. If \(l_1 = \int_{0}^{4nk} f(x) \, dx\) and \(l_2 = \int_{-k}^{3k} f(x) \, dx\), then
Let for n = 1, 2, …, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is \(\frac{1}{(n+1)^2}\) . Then the value of \(\frac{1}{26} + \sum_{n=1}^{50} \left(S_n+\frac{2}{n+1}-n-1 \right)\) is equal to ________.
Let \(f(x)=max\left\{|x+1|,|x+2|,……,|x+5|\right\} \)Then \(\int_{-6}^{0} f(x) \, dx\)is equal to_______
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 _____ .
Let A = {n∈N : H.C.F. (n, 45) = 1} andLet B = {2k :k∈ {1, 2, …,100}}. Then the sum of all the elements of \(A∩B\) is ___________
If A =\(\sum_{n=1}^{\infty}\)\(\frac{1}{( 3 + (-1)^n)^n}\) and B = \(\sum_{n=1}^{\infty}\) \(\frac{(-1)^n}{( 3 + (-1)^n)^n}\) , then A/B is equal to :
\(\lim_{{x \to 0}} \limits\) \(\frac{cos(sin x) - cos x }{x^4}\) is equal to :