The length of the chord joining points \((4 \cos \theta, 4 \sin \theta)\) and \(\left[4 \cos \left(\theta+60^{\circ}\right)\right.\), \(\left.4 \sin \left(\theta+60^{\circ}\right)\right]\) on the circle \(x^2+y^2=16\) is

Let \(g(x)=1+x-[x]\) and \(f(x)= \begin{cases}-1, & x < 0 \\ 0, & x=0 \\ 1, & x>0\end{cases}\) \([x]\) denotes the greatest integer less than or equal to \(x\). Then for all \(\mathrm{x}, \mathrm{f}(\mathrm{g}(\mathrm{x}))=\)

There are three families \(F_1, F_2, F_3. F_1\) has 2 boys and 1 girl; \(F_2\) has 1 boy and 2 girls; \(F_3\) has 1 boy and 1 girl. A family is randomly chosen and a child is chosen from that family randomly. If it is known that the child thus selected is a girl, then the probability that she is from \(\mathrm{F}_2\) is

If \(x \in R\), then the range of \(\frac{x}{x^2-5 x+9}\) is

If \(f(x)=\sin \left(\tan ^{-1} x\right)\), then \(\int_0^1 x f^{\prime \prime}(x) d x=\)

At any point for the curve \(3 y^2=(x+5)^3\), if ST represents the length of the subtangent and SN represents the length of the subnormal then \(9(\mathrm{ST})^2=\)

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?

\(100 \mathrm{~mL}\) of \(1.5 \%(\mathrm{w} / \mathrm{v})\) solution of urea is found to have an osmotic pressure of \(6.0 \mathrm{~atm}\) and \(100 \mathrm{~mL}\) of \(3.42 \%(w / v)\) solution of cane sugar is found to have an osmotic pressure of \(2.4 \mathrm{~atm}\). If the two solutions are mixed the osmotic pressure of the resulting solution in atm is
(Assume that there is no reaction between urea and cane sugar)

The general solution of \(2 \cos ^2 x-2 \tan x+1=0\) is

Which one of the following is a correct set?

The sum of the values of \(x\) satisfying the equation \(\sin ^{-1}\left(\frac{3 x}{5}\right)+\sin ^{-1}\left(\frac{4 x}{5}\right)=\sin ^{-1}(x)\), is

Which set of elements form electron precise hydrides?

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