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Find the area of the parallelogram determined by the vectors \[\hat{i}+2\hat{j}+3\hat{k}\] and \[3\hat{i}-2\hat{j}+\hat{k}.\]
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If \[y={{\tan }^{-1}}\left( \frac{a+x}{1-ax} \right),\] find \[\frac{dy}{dx}.\]
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If \[A=\left[ \begin{matrix} 6 & -\,3 \\ -\,2 & 1 \\ \end{matrix} \right],\] show that \[{{A}^{-1}}\] does not exist.
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If \[\int_{0}^{1}{(3{{x}^{2}}+2x+K)}\,dx=0,\] find K.
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Evaluate \[\int_{\pi /4}^{\pi /2}{\cos 2x\log \sin x\,dx.}\]
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Show that the function \[f(x)=\left\{ \begin{matrix} 1+x & \text{if}\,\,x\le 2; \\ 5-x, & \text{if}\,\,x>2; \\ \end{matrix} \right.\] is not differentiable at x = 2.
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The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
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If \[\vec{a}\times \vec{b}=\vec{c}\times \vec{d}\] and \[\vec{a}\times \vec{c}=\vec{b}\times \vec{d},\] show that \[\vec{a}-\vec{d}\] is parallel to \[\vec{b}-\vec{c},\] where \[\vec{a}\ne \vec{d}\] and \[\vec{b}\ne \vec{c}.\]
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The odds against A solving a certain problem are. 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find the probability that the problem will be solved.
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Examine the continuity of \[f(x)=\left\{ \begin{matrix} \frac{\log x-\log 2}{x-2} & x>2 \\ \frac{1}{2}, & x=2\,\,\text{at}\,\,x=2. \\ 2\left( \frac{x-2}{{{x}^{2}}-4} \right), & x<2 \\ \end{matrix} \right.\]
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Evaluate the determinant \[\Delta =\left| \begin{matrix} {{\log }_{3}}512 & {{\log }_{4}}3 \\ {{\log }_{3}}8 & {{\log }_{4}}9 \\ \end{matrix} \right|.\]
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Differentiate \[{{\tan }^{-1}}\left( \frac{1+2x}{1-2x} \right)\] with respect to \[\sqrt{1+4{{x}^{2}}.}\]
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Form the differential equation of the family of hyperbolas having foci on y-axis and centre at origin.
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Evaluate \[\int{\frac{dx}{\sin (x-a)\cdot \cos (x-b)}}.\] |
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Evaluate \[\int{\frac{x{{e}^{2x}}}{{{(1+2x)}^{2}}}}\,dx.\] |
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Two bikers are running at the speed more than speed allowed on the road along lines \[\vec{r}=(3\hat{i}+5\hat{j}+7\hat{k})+\lambda (\hat{i}-2\hat{j}+\hat{k})\] and \[\vec{r}=(-\hat{i}-\hat{j}-\hat{k})+\mu (7\hat{i}-6\hat{j}+\hat{k}).\] Using shortest distance, check whether they meet to an accident or not.
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Let X denotes the number of hours, you study during a randomly selected school days. The probability that X can take the values x has the following form, where k is any unknown constant, \[P(x)=\left\{ \begin{matrix} 0.1 & \text{if}\,\,x=0 \\ k\,x, & \text{if}\,\,x=1\,\,or\,\,2 \\ k(5-x), & \text{if}\,\,x=3\,\,or\,\,4 \\ 0, & \text{otherwise} \\ \end{matrix} \right.\] (i) Find the value of k. (ii) What is the probability that you study (a) atleast 2 h? (b) exactly 2 h? Why early morning is considered as the best time to study, explain?
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A clever student used a biased coin so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find probability distribution and mean of numbers of tails.
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If y(x) is a solution of \[\left( \frac{2+\sin x}{1+y} \right)\,\frac{dy}{dx}=-\cos x\] and y(0) = 1, find the value of \[y\left( \frac{\pi }{2} \right).\]
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If \[{{a}_{1}},{{a}_{2}},{{a}_{3}}...,{{a}_{r}}\] are in GP, prove that the determinant is independent of r. |
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Evaluate |
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If \[{{\tan }^{-1}}x-{{\cot }^{-1}}x={{\tan }^{-1}}\frac{1}{\sqrt{3}},\] then find the value of x.
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Evaluate \[\int_{0}^{2\pi }{\frac{x{{\sin }^{2n}}x}{{{\sin }^{2n}}x+{{\cos }^{2n}}x}}\,dx\] |
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For x > 0, let \[f(x)=\int_{1}^{x}{\frac{{{\log }_{e}}t}{1+t}}\,dt.\] Find function \[f(x)+f\left( \frac{1}{x} \right)\] and show that \[f(e)+f\left( \frac{1}{e} \right)=\frac{1}{2}.\] |
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If \[\vec{a},\] \[\vec{b},\] and \[\vec{c}\] determine the vertices of a triangle, show that \[\frac{1}{2}[\vec{b}\times \vec{c}+\vec{c}\times \vec{a}+\vec{a}\times \vec{b}]\] gives the vector area of the triangle. Hence, deduce the condition that the three points \[\vec{a},\] \[\vec{b},\] and \[\vec{c}\] are collinear. Also, find the unit vector normal to the plane of the triangle.
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Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
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If \[{{R}_{1}}\] and \[{{R}_{2}}\] be two equivalence relations on a set A, prove that \[{{R}_{1}}\cap {{R}_{2}}\] is also an equivalence relation on A. |
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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 biliary operation in P(X), |
(ii) Is * commutative? |
(iii) Is * associative? |
(iv) Find the identity element in P(X) w.r.t. ?*?. |
(v) If o is another binary operation defined on P(X) as \[AoB=A\cup B,\] then verify that O distributes itself over ?*?. |
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Find the intervals in which the function given by \[f(x)=\frac{4\sin x-2x-x\cos x}{2+\cos x},\] \[0\le x\le 2\pi \] is (i) Strictly increasing and (ii) Strictly decreasing.
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A-fliet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods A and B are available at a cost of Rs. 4 and Rs. 3 per unit, respectively. 1 unit of food A contains 200 units of vitamins, 1 unit of minerals and 40 calories. Food B contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of food should be used to have the least cost. Why a proper diet is required for us?
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Find \[{{A}^{-1}},\] if \[A=\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{matrix} \right]\] and show that \[{{A}^{-1}}=\frac{{{A}^{2}}-3I}{2}.\] |
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If \[A=\left[ \begin{matrix} 1 & -\,1 & 1 \\ 2 & 1 & -\,3 \\ 1 & 1 & 1 \\ \end{matrix} \right],\] find \[{{A}^{-1}}\] and hence solve the system of linear equation \[x+2y+z=4,\] \[-\,x+y+z=0,\] \[x-3y+z=2.\] |
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Find the distance of the point \[(-\,2,\,\,3,\,\,-\,4)\] from the line |
\[\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}\] measured parallel to the plane \[4x+12y-3z+1=0.\] |
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Find the coordinates of foot of perpendicular drawn from the point |
(0, 2, 3) on the line \[\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}.\] Also, find the length of perpendicular. |
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Using integration, find the area of the region enclosed between the circles \[{{x}^{2}}+{{y}^{2}}=4\] and \[{{(x-2)}^{2}}+{{y}^{2}}=4.\]
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