If the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin (a+1)x+\sin x}{x}& ,x<0\\ b& ,x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{\frac{3}{2}}}& ,x>0\end{array}\right.$ is continuous on $\mathrm{ℝ}$, then $a+b=$

If \(\quad d x+d y=(x+y)(d x-d y), \quad\) then \(\log (x+y)\) is equal to

\([\mathrm{x}]\) represents the greatest integer function.
If \(\lim _{x \rightarrow 0^{+}} \frac{\cos [x]-\cos (k x-[x])}{x^2}=5\) then \(k=\)

If \(f(x)=\left(p-x^n\right)^{1 / n}, p>0\) and \(n\) is a positive integer, then \(f[f(x)]\) is equal to

\(\mathrm{A}(4,3), \mathrm{B}(2,5)\) are two points. If P is a variable point on the same side as that of the origin with respect to the line AB and is at most at a distance of 5 units from the midpoint of \(A B\), then the locus of \(P\) is

\(\int \sqrt{x-1}(x \sqrt{x+1})^{-1} d x=\)

For \(a, b, c, d \in R\), if \(z_1=a+i b, z_2=c+i d\) are such that \(\left|z_1\right|=\left|z_2\right|=1\) and \(\operatorname{Re}\left(z_1 \bar{z}_2\right)=0\), then the pair of complex numbers \(w_1=a+i c\) and \(w_2=b+i d\) satisfy

Consider the following statements.
I. \(\mathrm{Cs}^{+}\)ion is more highly hydrated than other alkali metal ions.
II. Among the alkali metals, only lithium forms a stable nitride by direct combination with nitrogen.
III. Among the alkali metals \(\mathrm{Li}, \mathrm{Na}, \mathrm{K}, \mathrm{Rb}\), the metal, \(\mathrm{Rb}\) has the highest melting point.
IV. Among the alkali metals \(\mathrm{Li}, \mathrm{Na}, \mathrm{K}, \mathrm{Rb}\), only \(\mathrm{Li}\) forms peroxide when heated with oxygen.
Select the correct statement is.

Three charged particles of each mass \(0.1 \mathrm{~g}\) and charge ' \(\mathrm{q}\) ' are suspended from a common rigid point by insulated massless threads of each \(1 \mathrm{~m}\) long. If the three particles are in equilibrium and are located at the corners of an equilateral triangle of side \(3 \mathrm{~cm}\), the charge ' \(q\) ' on each particle is ____ \(\mathrm{nC}\). (The angle made by the line joining the centroid of the triangle and the point of suspension with the vertical is very small).
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) and \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\) )

The combined equation of two lines \(L\) and \(L_1\) is \(2 x^2+a x y+3 y^2=0\) and the combined equation of two lines \(L\) and \(L_2\) is \(2 x^2+b x y-3 y^2=0\). If \(L_1\) and \(L_2\) are perpendicular, then \(a^2+b^2=\)

A person who tosses an unbiased coin gains two points for turning up a head and loses one point for a tail. If three coins are tossed and the total score \(X\) is observed, then the range of \(x\) is

If $cosec\theta +\cot \theta =\frac{1}{3},$ then $\theta$ lies in the