Prove that |
\[\int_{0}^{1}{{{\sin }^{-\,1}}\left( \frac{2x}{1+{{x}^{2}}} \right)}\,dx=\frac{\pi }{2}-\log 2.\] |
OR |
Evaluate \[\int_{0}^{2}{[{{x}^{2}}]}\,dx,\] where \[[.]\]the greatest integer function is. |
If \[\vec{a}=\hat{i}+\hat{j}+\hat{k}\] and \[\vec{b}=\hat{j}-\hat{k},\] then find a vector \[\vec{c}\] such that \[\vec{a}\times \vec{c}=\vec{b}\] and \[\vec{a}\cdot \vec{c}=3.\] |
OR |
If \[\vec{a}\] and \[\vec{b}\] are two non-zero vectors, then prove that \[{{(\vec{a}\times \vec{b})}^{2}}=\left| \begin{matrix} \vec{a}\cdot \vec{a} & \vec{a}\cdot \vec{b} \\ \vec{a}\cdot \vec{b} & \vec{b}\cdot \vec{b} \\ \end{matrix} \right|.\] |
Two dice are thrown simultaneously. Let X denotes the number of sixes. Find the probability distribution of X. |
OR |
A and B are two candidates seeking admission in a college. The probability that A is selected, is 0.7 and the probability that exactly one of them is selected, is 0.6. Find the probability that B is selected. |
(i) | (ii) | (iii) | |
X | 400 | 300 | 100 |
Y | 300 | 250 | 75 |
Z | 500 | 400 | 150 |
Let X be a non-empty set and P(X) be its power set. Let '*' be an operation defined on elements of P(X) by |
\[A*B=A\,\cap B,\] \[\forall A,\] \[B\in P(X).\] Then, |
(i) Prove that '*' is a binary operation in P(X). |
(ii) Prove that '*' is commutative. |
(iii) Prove that '*' is associative. |
(iv) If 'o' is another binary operation defriended on P(X) as \[AoB=A\cup B,\] then verify that 'o' distributes over '*'. |
OR |
Find \[{{f}^{-1}},\] If \[f:R\to (-\,1,\,1)\] is defined by |
\[f(x)=\frac{{{\sqrt{7}}^{x}}-{{\sqrt{7}}^{-x}}}{{{\sqrt{7}}^{x}}+{{\sqrt{7}}^{\,-\,x}}}.\] |
Prove that the surface area of a solid cuboid of square base and given volume is minimum when it is a cube. |
OR |
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is \[(\pi /3).\] |
Solve the following system of equations by matrix method when \[x\ne 0,\,\]\[y\ne 0\] and \[z\ne 0.\] |
\[\frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10,\] \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10\] |
and \[\frac{3}{x}-\frac{1}{y}+\frac{2}{z}=13\] |
OR |
The sum of three numbers is 6. Twice the third number when added to the first number gives 7. On adding the sum of the second and third numbers to thrice the first number, we get 12. Find the numbers, using matrix method. |
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