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If the percentage error in the edge of a cube is 1, then find error in its volume.
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If A is a matrix of order \[2\times 3\] and B is a matrix of order \[3\times 5,\] then what is the order of matrix (AB)' or \[{{(AB)}^{T}}?\]
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Evaluate \[\int_{-2}^{1}{\frac{|x|}{x}\,}dx.\]
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Write the direction, cosines of vector \[2\hat{i}-\hat{j}+3\hat{k}.\]
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If \[A=\left[ \begin{matrix} 2 & 3 \\ -1 & 2 \\ \end{matrix} \right],\] find \[{{A}^{2}}-4A+I.\]
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Let A and B be two events of the same sample space S of an experiment, than prove that \[0\le P\left( A/B \right)\le 1,\] \[B\ne \phi .\]
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Differentiate \[{{\sin }^{2}}(3x+1)\] w.r.t.x.
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Find the point on the curve \[{{y}^{2}}=8x\] for which for the abscissa and ordinate change at the same rate.
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Find the slope of the tangent and normal to the curve \[y={{(\sin 2x+\cot x+2)}^{2}}\]At \[x=\frac{\pi }{2}.\]
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Evaluate \[\int{(x+1){{e}^{x}}\log \,\,(x{{e}^{x}})dx.}\]
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Use differentials, find the approximate value of \[{{(0.037)}^{1/2}}\]
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If the vector \[\vec{\alpha }=a\hat{i}+\hat{j}+\hat{k},\] \[\vec{\beta }=\hat{i}+b\hat{j}+\hat{k}\] and \[\vec{\gamma }=\hat{i}+\hat{j}+c\hat{k}\] are coplanar, then prove that. \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1,\] where \[\alpha \ne 1,\] \[b\ne 1\] \[c\ne 1.\]
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Evaluate \[\int{\frac{dx}{1-3\sin x}}\] |
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Evaluate \[\int{\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}}dx.\] |
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If \[y=a(1+\cos \theta )\] and \[x=a(\theta -\sin \theta ),\] find \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\,at\,\,\theta =\frac{\pi }{2}.\] |
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If \[\cos \frac{x}{2}\cdot \cos \frac{x}{4}\cdot \cos \frac{x}{8}...=\frac{\sin x}{x},\] prove that \[\frac{1}{{{2}^{2}}}{{\sec }^{2}}\frac{x}{2}+\frac{1}{{{2}^{4}}}{{\sec }^{2}}\frac{x}{4}+...=cose{{c}^{2}}x-\frac{1}{{{x}^{2}}}.\] |
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Find the equation of a curve passing through \[\left( 1,\,\,\frac{\pi }{4} \right),\] if the slop of the tangent to the curve at any point \[P\left( x,\text{ }y \right)\]is \[\frac{y}{x}-{{\cos }^{2}}\frac{y}{x}.\]
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Show that \[f(x)=2x+{{\cot }^{-1}}x+\log \,\,(\sqrt{1+{{x}^{2}}}-x)\] is Increasing in R.
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Mr. X has invested a part of his investment in 10% bond A and a part in 15% bond B. His interest income during first year is Rs. 4000. If he invests 20% more in 10% bond A and 10% more in 15% bond B, his income during second year increases by Rs. 500. Find the initial amount of investment in respective bonds, using matrix method. Is investment necessary a middle class?
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Solve the differential equation \[(x-y)(dx+dy)=dx-dy,\] \[y(0)=-1.\]
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If \[y={{\cot }^{-1}}(\sqrt{\cos x})-ta{{n}^{-1}}(\sqrt{\cos x}),\] prove that \[\sin \,\,y={{\tan }^{2}}\frac{x}{2}.\]
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A speaks the truth 8 times out of 10 times. He tossed a die. He reports that it was 5. What is the probability that it was actually 5? |
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A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. If there is atleast one girl on the committee, then calculate the probability that there are exactly 2 girls on the committee. |
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Using vectors, find the area of triangle with vertices A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).
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Find the angle between the lines \[\frac{2-x}{-\,5}=\frac{3+y}{3}=\frac{z}{2}\] and \[\frac{x+2}{-1}=\frac{3y-5}{2}=\frac{z-5}{4}.\]
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A coin is biased such that a head is three times as likely to occur as a tail. When it tossed twice, then find the probability distribution of number of heads. Also, find the mean and variance.
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Let \[A=R-\{3\},\] \[B=R-\{1\}.\] If \[f\,\,:\,\,A\to B\] be defined by \[f(x)=\frac{x-2}{x-3},\] \[\forall \,\,x\in A.\] Show that f is bijective and find the inverse of f.
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Find the angle of intersection of the curves \[{{y}^{2}}=x\] and \[{{x}^{2}}=y.\] |
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A metal box with a square base and vertical sides is to contain \[1024\,\,c{{m}^{3}}.\] If the material for the top and bottom costs Rs. 5 per \[c{{m}^{2}}.\] and the material for the sides Rs. 2.50 per \[c{{m}^{2}}\]. Then, find the least cost of the box. |
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Find the area bounded by the curve \[y=\cos x\]between x = 0 and \[x=2\,\pi .\] |
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Find the area bounded by the curve \[{{y}^{2}}=4{{a}^{2}}(x-1)\] and the lines \[x=1\] and \[y=4a.\] |
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If \[A=\left[ \begin{matrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 2 & 2 & 4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \\ \end{matrix} \right],\] find AB. Use this to solve the system of equations |
\[x-y=3,\] \[2x+3y+4z=17\] and \[y+2z=7.\] |
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By using elementary row operations, find the inverse of the matrix \[A=\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \\ \end{matrix} \right].\] |
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Two ships in the sea were reported missing their link with ground control suddenly. They were on the lines \[\overrightarrow{r}=(2+\lambda )\hat{i}-(3+\lambda )\hat{j}\,+\,(5+\lambda )\hat{k}\] and \[\overrightarrow{r}=(2\mu +1)\hat{i}\,+(4\mu -1)\hat{j}\,+\,(5-3\mu )\hat{k}.\] Using shortest distance formula, determine whether they met any mishappening. What value do you see in it.
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A dealer wishes to purchase a number of fans and radios. He has only Rs. 5760 to invest and has a space for at most 20 items. A fan costs him Rs. 360 and a radio Rs. 240. His expectation is that he can sell a fan at a profit of Rs 18. Assuming that, he can sell all the items that he buys, how should he invest his money for maximum profit? Translate the problem as LPP and solve it graphically.
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