The principal solution of of \((5+3 \sin \theta)(2 \cos \theta+1)=0\) are
If \(y=\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)+\cot ^{-1}\left(\frac{3-2 x}{2+3 x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to
\(\int \mathrm{e}^x\left(\frac{x+5}{(x+6)^2}\right) \mathrm{d} x\) is
If Rolle's theorem holds for the function \(x^3+\mathrm{a} x^2+\mathrm{b} x, 1 \leq x \leq 2\) at the point \(\frac{4}{3}\), then the values of \(a\) and \(b\) are respectively
If \(\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) is
\[ \int_1^3 \frac{\log x^2}{\log \left(16 x^2-8 x^3+x^4\right)} d x= \]
The lines \(\frac{x-0}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda}\) and \(\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{\lambda}\) are coplanar and \(p\) is the plane containing these lines, then which of following point lie on the plane for all values of (\(\lambda\)).
\[ \int \frac{1}{\mathrm{e}^x+1} \mathrm{~d} x= \]
If \(\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{j}-\hat{k}\) then a vector \(\bar{c}\) such that \(\overline{\mathrm{a}} \times \overline{\mathrm{c}}=\overline{\mathrm{b}}\) and \(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=3\) is
The eccentricity of the curve represented by \(x=3(\cos t+\sin t), y=4(\cos t-\sin t)\) is
The rate at which the population of a city increases varies as the population. In a period of 20 years, the population increased from 4 lakhs to 6 lakhs. In another 20 years the population will be
The equation of the plane containing the line \(\frac{x+1}{2}=\frac{y+2}{1}=\frac{z-2}{3}\) and the point \((1,-1,3)\) is
The equation of the plane containing the line \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) and perpendicular to the plane containing the lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{1}\) and \(\frac{x}{3}=\frac{y}{2}=\frac{z}{1}\) is
Bag I contains 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is
If the tangent at the point \((2 \sec \theta, 3 \tan \theta)\) to the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+4=0\), then the value of \(\theta\) is
If \(\mathrm{f}(x)=3 x+10, \mathrm{~g}(x)=x^2-1\), then \((\mathrm{fog})^{-1}(x)=\)
If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that \(|\overrightarrow{\mathrm{a}}|=\sqrt{31}, 4|\overrightarrow{\mathrm{~b}}|=|\overrightarrow{\mathrm{c}}|=2\) and \(2(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=3(\overline{\mathrm{c}} \times \overline{\mathrm{a}})\) and if the angle between \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) is \(\frac{2 \pi}{3}\) then \(\left|\frac{\bar{a} \times \bar{c}}{\bar{a} \cdot \bar{b}}\right|^2=\)
Let \(\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}\) and vector \(\bar{c}\) be coplanar. If \(\bar{c}\) is perpendicular to \(\bar{a}\), then \(\bar{c}\) is
If \(x=-2+\sqrt{-3}\), then the value of \(2 x^4+5 x^3+7 x^2-x+38\) is equal to
The contrapositive of the statement \(\sim p \vee(q \wedge \sim r)\) is
If \(x=a \sin 2 t(1+\cos 2 t), y=b \cos 2 t(1-\cos 2 t)\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to
The graph with correct feasible region of L.P.P. for the constraints \(2 x+y \leqslant 10, y \leqslant x, y \leqslant 2, x, y \geqslant 0\) is …
If the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}\) intersect, then the value of k is