Mathematics Tes...

Find the vector equation of the line that passes through the points \(A(1,0,2)\) and \(B(3,9,6)\).

\(\frac{\sin 7 x+6 \sin 5 x+17 \sin 3 x+12 \sin x}{\sin 6 x+5 \sin 4 x+12 \sin 2 x}\) equals:

Let \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & -i \\ 0 & 1\end{array}\right]\), where \(i=\sqrt{-1}\). If \(M=A^T B A\), then the inverse of the matrix \(A M^{2023} A^T\) is:

Let two vertices of a triangle \(\mathrm{ABC}\) be \((2,4,6)\) and \((0,-2,-5)\), and its centroid be \((2,1,-1)\). If the image of the third vertex in the plane \(x+2 y+4 z=11\) is \((\alpha, \beta, \gamma)\), then \(\alpha \beta+\beta \gamma+\gamma \alpha\) is equal to :

Let \(p(n)=x\left(x^{n-1}-n \cdot a^{n-1}+a^{n}(n-1)\right)\) is divisible by \((x-a)^{2}\) for:

If \(A=\left[\begin{array}{cc}4 & x+2 \\ 2 x-3 & x+1\end{array}\right]\) is symmetric, then what is \(x\) equal to:

If \(f(x)=\frac{\sin \left(e^{x-2}-1\right)}{\log (x-1)}, x \neq 2\) and \(f(x)=k\).Then, the value of k for which f will be continuous at x = 2 is:

If the planes \(2 x-y-3 z-7=0\) and \(4 x-2 y+5 k z+9=0\) are parallel, then \(5 k+7\) is: