\[ \int_{\frac{\pi}{3}}^{\frac{2 \pi}{3}} \frac{x}{1+\sin x} d x= \]
If \(x=\operatorname{acos}^3 \theta y=\operatorname{asin}^3 \theta\)
Then \(\sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2}=\)
The population of a town increases at a rate proportional to the population at that time. If the population increases from forty thousand to eighty thousand in 20 years, then the population in another 40 years will be
The angle between the lines \(3 x=2 y=-\mathrm{z}\) and \(-x=6 y=-4 z\) is
If \(y+\frac{\mathrm{d}}{\mathrm{d} x}(x y)=x(\sin x+\log x)\) then
A coin is tossed until one head appears or a tail appears 4 times in succession. The probability distribution of the number of tosses is
\[ \int \frac{x^3}{(x+1)^2} d x= \]
A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is
If \(y=\mathrm{a}^x \cdot \mathrm{~b}^{2 x-1}\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) is equal to
In a culture bacteria count is \(1,00,000\) initially. The number increases by \(10 \%\) in first 2 hours. In how many hours will the count reach \(2,00,000\), if the rate of growth of bacteria is proportional to the number present?
If \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are mutually exclusive and exhaustive events of a sample space \(S\) such that \(P(B)=\frac{3}{2} P(A)\) and \(P(C)=\frac{1}{2} P(B)\), then \(P(A)=\)
The solution set for minimizing the function \(\mathrm{z}=x+y\) with constraints \(x+y \geqslant 2, x+2 y \leqslant 8, y \leqslant 3, x, y \geqslant 0\) contains
The abscissae of the points of the curve \(y=x^3\) are in the interval \([-2,2]\), where the slope of the tangents can be obtained by mean value theorem for the interval \([-2,2]\) are
\[ \int \cos \left(\frac{x}{16}\right) \cdot \cos \left(\frac{x}{8}\right) \cdot \cos \left(\frac{x}{4}\right) \cdot \sin \left(\frac{x}{16}\right) \mathrm{d} x= \]
The distance of the point \(\mathrm{P}(3,8,2)\) from the line \(\frac{x-1}{2}=\frac{y-3}{4}=\frac{z-2}{3}\) measured parallel to the plane \(3 x+2 y-2 z+15=0\) is
Let \(x\) be the length of each of the equal sides of an isosceles triangle and \(\theta\) be the angle between these sides. If \(x\) is increasing at the rate \(\frac{1}{12} \mathrm{~m} /\) hour and \(\theta\) is increasing at the rate \(\frac{\pi}{180} \mathrm{rad} /\) hour, then the rate at which area of the triangle is increasing when \(x=12 \mathrm{~m}\) and \(\theta=\frac{\pi}{4}\) is
\[ \mathop {\lim }\limits_{n \to \infty }\left[\frac{1}{1-n^4}+\frac{8}{1-n^4}+\ldots \ldots \ldots \ldots .+\frac{n^3}{1-n^4}\right]= \]
The angle between the lines \(\frac{x-1}{l}=\frac{y+1}{m}=\frac{z}{n}\) and \(\frac{x+1}{\mathrm{~m}}=\frac{y-3}{\mathrm{n}}=\frac{\mathrm{z}-1}{l}\), where \(l>\mathrm{m}>\mathrm{n}\) and \(1, \mathrm{~m}, \mathrm{n}\) are roots of the equation \(x^3+x^2-4 x-4=0\), is
If \(f(\theta)=\cos \theta_1 \cdot \cos \theta_2 \cdot \cos \theta_3\) .............. \(\cos \theta_n\), then \(\tan \theta_1+\tan \theta_2+\tan \theta_3+\). ............ \(+\tan \theta_{\mathrm{n}}=\)
In a triangle ABC , with usual notations, \(\tan \left(\frac{\mathrm{A}}{2}\right)=\frac{5}{6}, \tan \left(\frac{\mathrm{C}}{2}\right)=\frac{2}{5}\), then
If \(\{(\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\} \rightarrow \sim \mathrm{p} \vee \mathrm{q}\) has truth value false then truth values of the statements \(p, q, r\) are respectively
The angle between the lines whose direction cosines are \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\) and \(\frac{-\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\) is
With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If \(\mathrm{a}=1\), then \(\angle \mathrm{A}=\)
\[ \int_0^{\frac{\pi}{4}} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x= \]
The differential equation of all circles having their centres on the line \(y=5\) and touching ( X -axis) is \(\qquad\)