Let \(S ={ (\begin{matrix} -1 & 0 \\ a & b \end{matrix}), a,b, ∈(1,2,3,.....100)}\) and let \(T_n = {A ∈ S : A^{n(n + 1)} = I}. \)Then the number of elements in \(\bigcap_{n=1}^{100}\) \(T_n \) is
The value of the integral \(\frac{48}{\pi^4} \int_{0}^{\pi}(\frac{3\pi x^2}{2} - x^3) \frac{ \sin(x)}{1 + \cos^2x} \, dx\) is equal to ________
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 ___________
2sin(\(\frac{\pi}{22}\))sin(\(\frac{3\pi}{22}\))sin(\(\frac{5\pi}{22}\))sin(\(\frac{7\pi}{22}\))sin(\(\frac{9\pi}{22}\)) is equal to
The number of matrices\(A=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\), where a,b,c,d ∈−1,0,1,2,3,…..,10such that A = A-1, is ______.
If the sum of solutions of the system of equations 2sin2θ – cos2θ = 0 and 2cos2θ + 3sinθ = 0 in the interval [0, 2π] is kπ, then k is equal to _______.
Let\(A = \{z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1\}\)and\(B = \{z \in \mathbb{C} : \text{arg}(\frac{z-1}{z+1}) = \frac{2\pi}{3}\}\)Then \(A∩B\) is :
The number of functions f, from the set\(A = {x∈N: x^2-10x+9≤0} \)to the set \(B = {n62:n∈N}\)such that\(f(x)≤(x-3)^2+1\), for every \(x∈A,\)is ______.
\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)
Then
Let the abscissae of the two points P and Q on a circle be the roots of x2 – 4x – 6 = 0 and the ordinates of P and Q be the roots of y2 + 2y – 7 = 0. If PQ is a diameter of the circle x2 + y2 + 2ax + 2by + c = 0, then the value of (a + b – c) is
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 _______.