A pair of perpendicular lines passes through the origin and also through the points of intersection of the curve \(x^2+y^2=4\) with \(x+y=a\), where \(a>0\). Then \(a\) is equal to

Let A and B be two events in a random experiment. If \(\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=0.1\), \(\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=0.2\) and \(\mathrm{P}(\mathrm{B})=0.5\) then \(\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}})=\)

If a function \(f:(-1,1) \rightarrow B(\subseteq R)\) is defined as \(f(x)=x+x^2+x^3+\ldots \infty\), then in order to have the inverse function of \(f, B\) is equal to

If the equation of the circle passing through the points \((-1,0),(-1,1),(1,1)\) is \(a x^2+a y^2+2 g x+2 f y-2=0\) then \(a=\)

\(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) are any four points. If E and F are mid points of AC and BD respectively, then \(\overline{\mathrm{AB}}+\overline{\mathrm{CB}}+\overline{\mathrm{CD}}+\overline{\mathrm{AD}}=\)

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If ( \(h, k\) ) is the internal centre of similitude of the circles \(x^2+y^2+2 x-6 y+1=0\) and \(x^2+y^2-4 x+2 y+4=0\), then \(4 h=\)

Observe the following species (i) \(\mathrm{NH}_3\) (ii) \(\mathrm{AlCl}_3\) (iii) \(\mathrm{SnCl}_4\) (iv) \(\mathrm{CO}_2\) (v) \(\mathrm{Ag}^{+}\) (vi) \(\mathrm{HSO}_4^{-}\)
How many of the above species act as Lewis acids?

If the vectors \(\mathrm{AB}=-3 \mathbf{i}+4 \mathbf{k}\) and \(\mathrm{AC}=5 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}\) are the sides of a \(\triangle A B C\), then the length of the median through \(A\) is

\(\lim _{x \rightarrow \infty}\left[\frac{x^2+x+3}{x^2-x+2}\right]^x\) is equal to

The following series \(L-C-R\) circuit, when driven by an emf source of angular frequency 70 kilo-radians per second, the circuit effectively behaves like

What is the maximum force $F$ that can be applied on block $m_1,$ so that both $m_1 \&{ m}_2$ will move together? There is no friction between $m_1$ and the horizontal table. The coefficient of friction between $\text{'}{m}_1\text{'}$ and $\text{'}{m}_2\text{'}$ is $\mu .$