A rectangular block of surface area A emits energy E per second at \(27^{\circ} \mathrm{C}\). If length and breadth is reduced to half of initial value and temperature is raised to \(327^{\circ} \mathrm{C}\) then energy emitted per second becomes
Two very long straight conductors (wires) are set parallel to each other. Each carries a current ' I ' in the same direction and the separation between them is ' 2 r '. The intensity of the magnetic field at point ' P ' (as shown in the figure) ( \(\mu_0=\) permeability of free space) is
A pipe open at both ends of length 1.5 m is dipped in water at one end such that \(2^{\text {nd }}\) overtone of vibrating air column is resonating with a tuning fork of frequency 330 Hz . The length of the pipe immersed in water is (Speed of sound in air \(=330 \mathrm{~m} / \mathrm{s}\) ) (Neglect end correction)
A physical quantity A can be determined by measuring parameters \(\mathrm{B}, \mathrm{C}, \mathrm{D}\) and E using the relation \(A=\frac{B^a C^\beta}{D^\gamma E^\delta}\). If the maximum errors in the measurement are \(\mathrm{b} \%, \mathrm{c} \%, \mathrm{~d} \%\) and \(\mathrm{e} \%\) then maximum error in the value of A is
Two simple harmonic motions of angular frequency \(300 \mathrm{rad} / \mathrm{s}\) and \(3000 \mathrm{rad} / \mathrm{s}\) have same amplitude. The ratio of their maximum accelerations is
If the line \(\frac{x+1}{1}=\frac{y-k}{11}=\frac{z-4}{-5}\) lies in the plane \(2 x+\mathrm{p} y+7 z-41=0\) which is perpendicular to the plane \(x+4 y-2 z+13=0\) then \(\mathrm{k}=\)
A fair coin is tossed 99 times. If X is the number of times head occur then \(\mathrm{P}[\mathrm{X}=\mathrm{r}]\) is maximum when \(\mathrm{r}=\)
If \(\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})\) and \(\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})\) then the value of \((2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot[(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}})]=\)
A random variable \(X\) takes the values \(0,1,2,3\), \(\qquad\) with probability
\(\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x\), where k is a constant.
Then \(\mathrm{P}(\mathrm{X}=0)\) is
If \(u=\frac{\tan ^{-1} x}{\tan ^{-1} x+1}\) and \(v=\tan ^{-1}\left(\tan ^{-1} x\right)\) then \(\frac{d u}{d v}=\)
4 red balls and 5 green balls are selected from \(n\) balls. If the sum of both the selections is greater than \({ }^{n+1} C_4\) then the value of \(n\) is equal to
The value of \(\frac{(\cos \theta+i \sin \theta)^4}{(\sin \theta+i \cos \theta)^5}=\) where \(\mathrm{i}=\sqrt{-1}\)
Two numbers are selected at random, without replacement from the first 6 positive integers. Let \(X\) denote the larger of the two numbers. Then \(\mathrm{E}(\mathrm{X})=\)
If \(\mathrm{f}(x)=2 x^3+\mathrm{m} x^2-13 x+\mathrm{n}\) and 2,3 are the roots of the equation \(\mathrm{f}(x)=0\) then the value of \(4 m+5 n\) is
If \(x^y+y^x=\mathrm{a}^{\mathrm{b}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=1, y=2\) is
For \(\mathrm{k}=1,2,3\) the box \(\mathrm{B}_{\mathrm{k}}\) contains k red balls and \((k+1)\) white balls. Let \(P\left(B_1\right)=\frac{1}{2}, P\left(B_2\right)=\frac{1}{3}\) and \(\mathrm{P}\left(\mathrm{B}_3\right)=\frac{1}{6} . \mathrm{A}\) box is selected at random and a ball is drawn from it. If a red ball is drawn from it, then the probability that it comes from box \(\mathrm{B}_2\) is
\[ f(x)= \begin{cases}{\left[x^2\right]-\left[-x^2\right],} & x \neq 3 \\ k & , x=3\end{cases} \]
is continuous at \(x=3\), then \(\mathrm{k}=\) where \([\cdot]\) is greatest integer function
If the curve \(y=a x^2-6 x+b\) passes through \((0,4)\) and has its tangent parallel to the X-axis at \(x=\frac{3}{2}\), then the values of \(a\) and \(b\) respectively are
The equivalent statement of "If three vertices of a triangle are represented by cube roots of unity, then the triangle is an equilateral triangle" is
The negation of \((p \wedge \sim q) \rightarrow(p \vee \sim q)\) is
\(\mathop {\lim }\limits_{x \to 0} \frac{\left(7^x-1\right)^4}{\tan \left(\frac{x}{\mathrm{k}}\right) \cdot \log \left(1+\frac{x^2}{3}\right) \cdot \sin 4 x}=3(\log 7)^3\), then \(\mathrm{k}=\)
If \(\mathrm{a}\left(4+x^2\right)=x\) and \(y-x^3=\mathrm{a}^2\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=1\) is \(\qquad\)
\[ \int \mathrm{e}^x \frac{(x-1)}{(x+1)^3} \mathrm{~d} x= \]
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
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
