A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

\(\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) d x\) is equal to

From a point $A\left(0,3\right)$ on the circle ${\left(x+2\right)}^2+{\left(y-3\right)}^2=4$, a chord $AB$ is drawn and it is extended to a point $Q$ such that $AQ=2 AB$. Then the locus of $Q$ is

If the order and degree of the differential equation corresponding to the family of curves \(y^2=4 a(x+a)\) ( \(a\) is parameter) are \(m\) and \(n\) respectively, then \(m+n^2=\)

\(\int_0^{\pi / 4} \frac{1}{5 \cos ^2 x+16 \sin ^2 x+8 \sin x \cos x} d x=\)

The equation of the circle of radius 5 and touching the co-ordinate axes in third quadrant is

\(A=\left[a_{i j}\right]\) is a \(3 \times 3\) matrix with positive integers as its elements. Elements of A are such that the sum of all the elements of each row is equal to 6 and \(a_{22}=2\).
If \(a_{i i}=\left\{\begin{array}{ll}a_{i j}+a_{j i}, & j=i+1 \text { when } i \lt 3 \\ a_{i j}+a_{j i}, & j=4-i \text { when } i=3\end{array}\right.\) for \(i=1,2,3\), then \(|A|=\)

An executive in a company makes on an average $5$ telephone calls per hour at a cost of Rs. $2$ per call. The probability that in any hour the cost of the calls exceeds a sum of Rs. $4$ is

In \(\triangle A B C\) if \(\angle C=\frac{\pi}{2}\) then
\(\tan ^{-1}\left(\frac{a}{b+c}\right)+\tan ^{-1}\left(\frac{b}{c+a}\right)+\tan ^{-1}\left(\frac{c}{a+b}\right)=\)

A plane passing through \((-1,2,3)\) and whose normal makes equal angles with the coordinate axes is

The equation of the plane passing through the points with position vectors \(A(2 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\), \(B(-3 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}-9 \hat{\mathbf{k}})\) and \(C(-5 \hat{\mathbf{i}}-6 \hat{\mathbf{k}})\) is

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 \(Z=\alpha+i \beta\) satisfies the equation \(|Z|-Z=1+2 i\) and \(|Z|=\sqrt{\alpha^2+\beta^2}\) then \(Z \bar{Z}=\)