Eliminating $a$ and $b$ from the relation $y=a \log x+b,$ we get ________

If a real valued function \(f:[-1,2] \rightarrow B\) defined by
\(f(x)=\left\{\begin{array}{l}
1-x, \quad \text { when }-1 \leq x \leq 1 \\
x-1, \quad \text { when } 1 < x \leq 2
\end{array}\right.\)
is a surjection, then \(\mathrm{B}=\)

A ray of light passing through the point \((2,3)\) reflects on \(Y\)-axis at a point \(P\). If the reflected ray passes through the point \((3,2)\) and \(\mathrm{P}=(a, b)\) then \(5 b=\)

If \(\sqrt{3} \cos \theta+\sin \theta>0\) then

If \((2-5 x)^{\frac{-1}{5}}=a_0+a_1 x+a_2 x^2+\ldots\), then \(\frac{a_1}{a_2}=\)

The square root of independent term in the expansion of \(\left(\frac{2 x^2}{5}+\sqrt{\frac{5}{x}}\right)^{10}\) is

The mean and variance of a binomial variable$X$ are $2$ and $1$ respectively. The probability that $X$ takes values greater than $1$, 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 $1,{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are the roots of $z^5-1=0$ and $\omega$ is a cube root of unity then $(\omega -1)\left(\omega -{\alpha }_1\right)\left(\omega -{\alpha }_2\right)\left(\omega -{\alpha }_3\right)\left(\omega -{\alpha }_4\right)+\omega$ is equal to

If ${\left(x^2+5x+5\right)}^{x+5}=1$ then the number of integers satisfying this equation is

In an atom the order of increasing energy of electrons with quantum numbers
(i) \(n=4, l=1\)
(ii) \(n=4, l=0\)
(iii) \(n=3, l=2\) and (iv) \(n=3, l=1\) is

The pole of the straight line \(9 x+y-28=0\) with respect to the circle \(2 x^2+2 y^2-3 x+5 y-7=0\) is

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft ) is