The negation of \((p \wedge \sim q) \rightarrow(p \vee \sim q)\) is
The approximate value of \(\cos \left(59^{\circ} 30^{\prime}\right)\) is (given \(1^{\circ}=0.0175^{\mathrm{c}}, \sin 60^{\circ}=0.8660\) )
The co-ordinates of the point where the line joining the points \((2,-3,1)\) and \((3,-4,-5)\) and intersects the plane \(2 x+y+z=7\) are
The distance of the point \((2,4,0)\) from the point of intersection of the lines \(\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}\) and \(\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}\) is
\[ \int \mathrm{e}^x \frac{(x-1)}{(x+1)^3} \mathrm{~d} x= \]
In a triangle \(A B C\), with usual notations, \(3 \mathrm{~b}=\mathrm{a}+\mathrm{c}\), then \(\cot \frac{\mathrm{A}}{2} \cdot \cot \frac{\mathrm{C}}{2}=\)
If \(\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}12 \\ 15 \\ 13\end{array}\right]\), then the value of \(x^2+y^2+z^2=\)
\[ \int_0^{\frac{\pi}{2}} \frac{d x}{1+(\cot x)^{101}}= \]
The solution of \(\left(1+y^2\right)+\left(x-\mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) is
The rate of reduction of a persons assets is proportional to the square root of the existing assets. The assets reduced from 25 lakhs to 6.25 lakhs in 2 years. This rate of reduction of his assets will make him bankrupt in
AOB is the positive quadrant of the ellipse \(\frac{x^2}{25}+\frac{y^2}{9}=1\) in which \(\mathrm{OA}=5, \mathrm{OB}=3\). The area between the arc AB and the chord AB of the ellipse in sq. units is
\[ \int_0^3 \frac{d x}{(x+2) \sqrt{x+1}}= \]
In a triangle ABC , the sides \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are such that they are the roots of the equation \(x^3-11 x^2+38 x-40=0\) Then
\[ \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}= \]
If \(2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x)\), then the value of \(x\) is
The possible values of \(\theta \in(0, \pi)\) such that \(\sin \theta+\sin (4 \theta)+\sin (7 \theta)=0\) are
In a triangle ABC with usual notations if \(\mathrm{a}=13\), \(b=14, c=15\) Then \(\sin A=\)
If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are unit vectors and \(\theta\) is the angle between them, then \(\tan \frac{\theta}{2}=\)
If \(2 \mathrm{f}(x)+3 \mathrm{f}\left(\frac{1}{x}\right)=x^2+1, x \neq 0\) and \(y=5 x^2 \mathrm{f}(x)\), then \(y\) is strictly increasing in
If the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x}{y}=\frac{\mathrm{a}}{y}\) where a is constant, represents a family of circles then the radius of the circle is \(\qquad\)
If the equation \(\mathrm{k} x y+10 x+8 y+16=0\) represents a pair of lines, then
A manufacturer produces \(x\) items per week at a total cost of ₹ \(\left(x^2+78 x+2500\right)\). The price per unit is given by \(8 x=600-\mathrm{p}\) where ' p ' is the price of each unit. Then the maximum profit obtained is
If the projection of \(\bar{a}\) on \(\bar{b}+\bar{c}\) is twice the projection of \(\bar{b}+\bar{c}\) on \(\bar{a}\) also if \(|\bar{b}|=2 \sqrt{2},|\bar{c}|=4\) and the angle between \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) is \(\frac{\pi}{4}\) then \(|\overline{\mathrm{a}}|=\)
\[ \int \log (2+x)^{2+x} d x= \]
If the equation of the median through vertex \(\mathrm{A}(3, \mathrm{k})\) of \(\triangle \mathrm{ABC}\) with vertices \(\mathrm{B}(2,1)\) and \(\mathrm{C}(-4,5)\) is \(x+4 y=\mathrm{p}\), then \(\mathrm{k}=\) where p and k are constants
The equation of the plane passing through the point of intersection of the planes \(2 x-y+z-3=0\) and \(4 x-3 y+5 z+9=0\) and parallel to the line \(\frac{x+1}{2}=\frac{y+3}{4}=\frac{z-3}{5}\) is \(\alpha x+\beta y+\gamma z+d=0\) Then \(\alpha+\beta+\gamma+d=\)