Evaluate \[\int{\left[ \log (\log \,x)+\frac{1}{{{(\log \,x)}^{2}}} \right]}\,dx.\] |
OR |
Evaluate \[\int{\frac{\cos 2x-\cos 2\alpha }{\cos x-\cos \alpha }}\,dx.\] |
Find the angle between the line |
\[\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}\] and the plane \[10x+2y-11z=3.\] |
OR |
Find the equation of the plane which contains the line of intersection of |
planes \[\vec{r}\cdot (\hat{i}+2\hat{j}+3\hat{k})-4=0,\] |
\[\vec{r}\cdot (2\hat{i}+\hat{j}+\hat{k})-15=0\] and is perpendicular to the plane |
\[\vec{r}\cdot (5\hat{i}+3\hat{j}-6\hat{k})+8=0.\] |
It is known that 10% of certain articles manufactured are defective. What is probability that in a random sample such articles, 9 are defective? |
OR |
Consider the experiment of tossing a coin. If the coin shows tail, toss it again but if it shows head, then throw a die. Find the conditional probability off event that 'the die shows a number greater than 3' given that 'there is at least one head'. |
Villages | House calls | Letters | Announcements |
X | 400 | 300 | 100 |
Y | 300 | 250 | 75 |
Z | 500 | 400 | 150 |
If \[f:R-\{2\}\to R-\{3\}\] is defined by |
\[f(x)=\frac{3x+1}{x-2},\] where R is the set of real numbers, |
show that f is invertible and hence find the value of \[{{f}^{-1}}.\] |
OR |
Let \[f:N\to R\] be a function defined as \[f(x)=4{{x}^{2}}+12x+15.\] Show that \[f:N\to \] range f is invertible. Find the inverse of \[{{f}^{-1}}\]. |
A point on the hypotenuse of a right angled triangle is at distance of a units and b units from the sides. Show that the minimum length of hypotenuse is \[{{({{a}^{2/3}}+{{b}^{2/3}})}^{3/2}}.\] |
OR |
If the straight line \[x\cos \alpha +y\sin \alpha =P\] touches the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\] prove that \[{{P}^{2}}={{a}^{2}}{{\cos }^{2}}\alpha +{{b}^{2}}{{\sin }^{2}}\alpha .\] |
Find the image of the point \[2\hat{i}+3\hat{j}-4\hat{k}\] in the plane \[\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=3.\] |
OR |
Find the equation of the plane through the line of intersection of the planes \[x+y+z=1\] and \[2x+3y+4z=5,\] which is perpendicular to the plane \[x-y+z=0.\] Also, find the distance of the plane obtained above from the point (1, 1, 1). |
You need to login to perform this action.
You will be redirected in
3 sec