Evaluate \[\int{\frac{1}{3{{x}^{2}}+5x+7}\,dx.}\] |
OR |
Evaluate \[\int{\frac{x{{e}^{x}}}{{{(x+1)}^{2}}}\,dx.}\] |
A letter is known to have come from either TATANAGAR' or 'CALCUTTA?. On the envelope just two letters ?TA? are visible. What is the probability that the letter has come from |
(i) TATANAGAR? |
(ii) CALCUTTA? |
OR |
If A and B are two independent events such that \[P(\bar{A}\cap B)=\frac{2}{15}\] and \[P(A\cap \bar{B})=\frac{1}{6},\] then find P(A) and P(B). |
If \[f:R\to R\] is a function defined by \[f(x)=2{{x}^{3}}-5,\] then show that the function f is a objective function. |
OR |
Consider \[f:R\to R\] given by \[f(x)=4x+3.\]Show that f is invertible and find the inverse off. |
The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method, |
OR |
If prove that |
\[n\in N.\] |
A variable plane which remains at a constant distance 3p from the origin cut the coordinate axes at A, B and C. Show that the locus of the centroid of \[\Delta ABC\] is \[{{x}^{-2}}+{{y}^{-2}}+{{z}^{-2}}={{p}^{-2}}.\] |
OR |
If the lines \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\] and \[\frac{x-3}{2}=\frac{y-k}{2}=\frac{z}{1}\] intersect, find the value of k and hence, find the equation of the plane containing these lines. |
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