If \(f(x)=3 x^{15}-5 x^{10}+7 x^5+50 \cos (x-1)\), then \(\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^3+3 h}=\)

If \(f(x)=\left|\begin{array}{ccc}2 \cos ^2 x & \sin 2 x & \sin x \\ \sin 2 x & 2 \sin ^2 x & -\cos x \\ \sin x & -\cos x & 0\end{array}\right|\) then
\[
\int_0^{\frac{\pi}{4}}\left(2|f(x)|+5 f^{\prime}(x)\right) d x=
\]

The solution of \(\cos y+(x \sin y-1) \frac{d y}{d x}=0\) is

If \(A(-1,3)\) and \(B(5,3)\) are points on a circle \(C\) and the chord \(A B\) subtends an angle \(\pi / 4\) at a point \(P\) on \(C\), then the equation of such a circle \(C\) is

If \(\cos x=\tan y, \cot y=\tan z\) and \(\cot z=\tan x\), then \(\sin x\) equals to

A rectangular hyperbola passing through \((3,2)\) has its asymptotes parallel to the coordinate axes. If \((1,1)\) is the point of intersection of the two perpendicular tangents of that hyperbola, then its equation 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?

Solution " \(X\) " contains \(\mathrm{Na}_2 \mathrm{CO}_3\) and \(\mathrm{NaHCO}_3\). \(20 \mathrm{~mL}\) of \(X\) when titrated using methyl orange indicator consumed \(60 \mathrm{~mL}\) of \(0.1 \mathrm{M}\) \(\mathrm{HCl}\) solution. In another experiment, \(20 \mathrm{~mL}\) of \(X\) solution when titrated using phenolphthalein consumed \(20 \mathrm{~mL}\) of \(0.1 \mathrm{M}\) \(\mathrm{HCl}\) solution. The concentrations (in \(\mathrm{mol} \mathrm{L}^{-1}\) ) of \(\mathrm{Na}_2 \mathrm{CO}_3\) and \(\mathrm{NaHCO}_3\) in \(X\) are respectively

For all real values of \(\mathrm{x}\), the minimum value of \(\frac{1-x+x^2}{1+x+x^2}\) is

If \(3 x+6 y+2=0, x+y+1=0,2 x-y+3=0\) are three given lines then the point \(\left(\frac{-4}{3}, \frac{1}{3}\right)\) is

In an ideal step up transformer, if the input voltage and input power are \(\mathrm{V}_1\) and \(\mathrm{P}_1\) respectively and the output voltage and output power are \(\mathrm{V}_2\) and \(\mathrm{P}_2\) respectively, then

If $y=\frac{x^2+14x+9}{x^2+2x+3}\forall x\in \mathrm{ℝ},$ then the interval of maximum length in which $y$ lies is

If \(\frac{x^2-2}{\left(x^2+1\right)\left(x^2+3\right)}=\frac{\mathrm{A} x+\mathrm{B}}{x^2+1}+\frac{\mathrm{C} x+\mathrm{D}}{x^2+3}\) then \(\mathrm{D}=\)