Let,\(\vec{a}=a\hat{i}+2\hat{j}−\hat{k}\) and \(\vec{b}=−2\hat{i}+α\hat{j}+\hat{k}\), where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15(α^2+4)}\) , then the value of \(2|\vec{a}|^2+(\vec{a}⋅\vec{b})|\vec{b}|^2 \)is equal to :
Let f(x) = [2x2 + 1] and \(g(x)=\left\{\begin{matrix} 2x-3,\,x<0&\\2x+3, x≥0 &\end{matrix}\right.\)where [t] is the greatest integer ≤ t. Then, in the open interval (–1, 1), the number of points where fog is discontinuous is equal to _______.
Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then\((\frac{1}{p} + \frac{1}{q} )^{-2}\)is equal to _______.
\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)
Then
The area bounded by the curves y = |x2 – 1| and y = 1 is
If α, β are the roots of the equation\(x^2-(5+3^{\sqrt{log_35}}-5^{\sqrt{log_53}})+3(3^{(log_35)^{\frac{1}{3}}}-5^{(log_53)^{\frac{2}{3}}}-1) = 0\)then the equation, whose roots are α + 1/β and β + 1/α , is
Let \(\vec{a}, \vec{b}, \vec{c}\)be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and \((\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168\), then \(|\vec{a}| + |\vec{b}| + |\vec{c}|\)| is equal to :
\(\begin{array}{l} \frac{2^3-1^3}{1\times7}+\frac{4^3-3^3+2^2-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\cdots+\frac{30^3-29^3+28^3-27^3+\cdots+2^3-1^3}{15\times63}\end{array}\)
is equal to _______.
