The maximum value and minimum value of the volume of the parallelopiped having coterminous edges \(\hat{\mathrm{i}}+x \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+x \hat{\mathrm{k}}\) and \(x \hat{\mathrm{i}}+\hat{\mathrm{k}}\) are respectively
Which of the following is not a homogeneous function?
A random variable X takes values \(0,1,2,3\), ........ with probabilities. \(\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{2}\right)^x, \mathrm{k}\) is a constant, then \(P(X=1)=\)
Let \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{c}}\) be unit vectors at an angle \(\frac{\pi}{3}\) with each other. If \((\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5\), then \([\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=\)
If \(\mathrm{f}(x)=\cos (\log x)\) then \(\mathrm{f}\left(x^2\right) \cdot \mathrm{f}\left(y^2\right)-\frac{1}{2}\left[\mathrm{f}\left(\frac{x^2}{y^2}\right)+\mathrm{f}\left(x^2 y^2\right)\right]\) has the value
A particular solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=(x+9 y)^2\), when \(x=0, y=\frac{1}{27}\) is
If a random variable X follows the Binomial distribution \(B(10, \quad p)\) such that \(5 \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)\), then the value of \(\frac{\mathrm{P}(\mathrm{X}=5)}{\mathrm{P}(\mathrm{X}=6)}\) is equal to
The shortest distance between the lines \(\bar{r}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k})\) and \(\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k})\) is
\[ \mathop {\lim }\limits_{x \to \infty } \frac{\mathrm{e}^{x^4}-1}{\mathrm{e}^{x^4}+1}= \]
If \(u=\frac{\tan ^{-1} x}{\tan ^{-1} x+1}\) and \(v=\tan ^{-1}\left(\tan ^{-1} x\right)\) then \(\frac{d u}{d v}=\)
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 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 \(\mathrm{k}=\)
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 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 \(\mathrm{r}=\)
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
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
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 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
\(\mathop {\lim }\limits_{x \to 0} \frac{\left(7^x-1\right)^4}{\tan \left(\frac{x}{\mathrm{k}}\right) \cdot \log \left(1+\frac{x^2}{3}\right) \cdot \sin 4 x}=3(\log 7)^3\), then \(\mathrm{k}=\)
\[ f(x)= \begin{cases}{\left[x^2\right]-\left[-x^2\right],} & x \neq 3 \\ k & , x=3\end{cases} \]
is continuous at \(x=3\), then \(\mathrm{k}=\) where \([\cdot]\) is greatest integer function
If \(\mathrm{a}\left(4+x^2\right)=x\) and \(y-x^3=\mathrm{a}^2\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=1\) is \(\qquad\)