If \(\alpha\) and \(\beta\) are the roots of the equation \(2^{6 x}-3\left(2^{3 x+2}\right)+32\) \(=0\) with \(\beta < 1\), then \(2 \alpha+3 \beta=\)

If \(A=\left[a_{i j}\right], 1 \leq i, j \leq n\) with \(\mathrm{n} \geq 2\) and \(a_{i j}=i+j\) is a matrix, then the rank of \(A\) is

If \(1, \alpha_1, \alpha_2, \ldots ., \alpha_{n-1}\) are the \(n\)th roots of unity and \(n\) is an even natural number, then \(\left(l+\alpha_1\right)\left(l+\alpha_2\right) \ldots .\left(l+\alpha_{n-1}\right)=\)

If \(l, m\) represent any two elements (identical or different) of the set \(\{1,2,3,4,5\), \(6,7\}\), then the probability that \(l x^2+m x+1>0 \forall x \in R\) is

The number of six digit natural numbers that can be formed with the digits \(2,3,4,0,5,6,7,8\) is

Two equal sides of an isosceles triangle are given by \(7 x-y+3=0\) and \(x+y-3=0\). If the slope \(m\) of the third side is an integer, then \(m=\)

If \(a x^2+2 h x y-2 a y^2+3 x+15 y-9=0\) represents a pair of lines intersecting at \((1,1)\), then ah \(=\)

If \(a\) and \(c\) are complex numbers and \(b\) is a real number in the Argand plane, then the perpendicular distance from \(c\) to the line \(a \bar{z}+\bar{a} z+b=0\) is

If uncertainity in the measurement of position and momentum of a microscopic object of mass $\text{'}\mathrm{m}\text{'}$ are equal, then the uncertainity in the measurement of velocity is given by the expression

Let \(\mathbf{A}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) and \(\mathbf{B}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\). If \(C\) is a vector such that \(\mathbf{A} \cdot \mathbf{C}=|\mathbf{C}|,|\mathbf{C}-\mathbf{A}|=2 \sqrt{2}\) and the angle between \(\mathbf{A} \times \mathbf{B}\) and \(\mathbf{C}\) is \(30^{\circ}\), then the value of \(|(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}|\) is

When an inductor of inductance \(\frac{6}{\pi} \mathrm{H}\), a capacitor of capacitance \(\frac{50}{\pi} \mu \mathrm{F}\) and resistor of resistance \(R\) are connected in series with an AC supply of rms voltage \(220 \mathrm{~V}\) and frequency \(50 \mathrm{~Hz}\), the rms current through the circuit is \(440 \mathrm{~mA}\). Match the inductive reactance, \(X_L\) the capacitive reactance, \(X_C\) the resistance \(R\) and the impedance \(Z\) of the circuit given in List-I with the corresponding values given in List-II.
\(\begin{array}{llll} \hline & \text { List- I } & & \text { List- II } \\ \hline \text {(A) } & X_L & \text { (i) } & 200 \Omega \\ \hline \text {(B) } & X_C & \text { (ii) } & 300 \Omega \\ \hline \text {(C) } & R & \text { (iii) } & 500 \Omega \\ \hline \text {(D) } & Z & \text { (iv) } & 600 \Omega \\ \hline \end{array}\)

How many numbers between 10 and 10000 can be formed by using the digits \(1,2,3,4,5\), if no digit is repeated in any number?

In the complex plane \(C\), the set \(\left\{z \in C: \arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}\right\}\) represents