In Arjun's school, annual sports meet with the title 'Sports for Healthy Life' was being organised. On the last day of the meet, there was the event of hurdle race. In this hurdle, it was decided that a player has to cross 10 hurdles. |
From the past games experience it can be said that the probability of a player to clear each hurdle will be 5/6. Find the probability that a player will knock down fewer than 2 hurdles. |
OR |
A couple has 2 children. Find the probability that both are boys, if it is known that |
(i) One of them is a boy, |
(ii) The older child is a boy. |
If \[x=\sqrt{{{a}^{{{\sin }^{-1}}t}}}\] and \[y=\sqrt{{{a}^{{{\cos }^{-1}}t}}},\,\,a>0\] and \[-\,1<t<1,\] then prove that \[\frac{dy}{dx}=\frac{-\,y}{x}.\] |
OR |
In a given function \[f(x)={{x}^{3}}+b{{x}^{2}}+ax,\] \[x\in [1,\,\,3],\] Rolle's theorem holds with \[c=2+\frac{1}{\sqrt{3}}.\] Find the values of a and b. |
Evaluate \[\int{\frac{\sin x+\cos x}{9+16\sin 2x}dx.}\] |
OR |
Evaluate \[\int{\frac{{{x}^{2}}+1}{{{(x-1)}^{2}}(x+3)}dx.}\] |
Find the equation of tangent to the curve give b by \[x=a{{\sin }^{3}}t\] and \[y=b{{\cos }^{3}}t\] at a point \[t=\pi /2.\] |
OR |
A telephone company in a town has 500 subscribers on its list and collects fixed charge subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs. 1, one subscriber will |
Discontinue the service. Find what increase will bring maximum profit? |
Find the area of the region bounded by the curves |
\[{{y}^{2}}=4ax\] and \[{{x}^{2}}=4ay.\] |
OR |
Using integration, find the area of the triangular region whose sides have the equation \[y=2x+1,\]\[y=3x+1\] and x = 4. |
Using properties of determinants, prove that |
\[\left| \begin{matrix} {{a}^{2}}+1 & ab & ac \\ ab & {{b}^{2}}+1 & bc \\ ca & ca & {{c}^{2}}+1 \\ \end{matrix} \right|=1+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}.\]? |
OR |
Prove that |
\[\left| \begin{matrix} {{(b\,+c)}^{2}} & {{a}^{2}} & {{a}^{2}} \\ {{b}^{2}} & {{(c\,+a)}^{2}} & {{b}^{2}} \\ {{c}^{2}} & {{c}^{2}} & {{(a\,+b)}^{2}} \\ \end{matrix} \right|=2abc{{(a+b+c)}^{3}}.\] |
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