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Evaluate \[\int_{-1}^{1}{x|x|dx.}\]
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If
and
find \[[A+2\,\,B]'.\]
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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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Write the cartesian equation of the line \[\vec{r}=(2\hat{i}+\hat{j})+\lambda (\hat{i}-\hat{j}+4\hat{k}).\]
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If A and B are square matrices such that \[B=-{{A}^{-1}}BA,\] Then find the value of \[{{(A+B)}^{2}}.\]
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Show that the lines \[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\] and \[\frac{x+1}{2}=\frac{y+2}{3}=\frac{z+3}{-\,2}\] are perpendicular.
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If \[y={{\sin }^{-1}}[\sqrt{x}\sqrt{1-{{x}^{2}}}-x\sqrt{1-x}),\] then find \[\frac{dy}{dx}.\]
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Find a point on the parabola \[y={{(x-3)}^{2}},\] where the tangent is parallel to the chord joining (3, 0) and (4, 1).
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The objective of A diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf' and eggs, and to vitamins A, B, C. The number of milligrams of each of these vitamins contained within A unit of each food is given below
vitamin | Litre of milk | Kg of beaf | Doze of eggs | Minimum daily requirements |
A | 1 | 1 | 10 | 1 mg |
B | 100 | 10 | 10 | 50 mg |
C | 10 | 100 | 10 | 10 mg |
Cost | Rs. 1.00 | Rs. 1.10 | Rs. 0.50 | |
What is the linear programming formulation for this problem?
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Find the slope of the tangent to the curve \[y={{x}^{3}}-3x+2\] at the point whose x coordinate is 3.
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Find the differential equation of the family of all straight lines passing through the origin.
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Three cards are drawn successively without replacement from a pack of 52 well-shuffled cards. What is the probability that first two cards are king and the third card drawn is an ace?
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Show that \[f(x)=\,\,|x-2|+|x-3|\] is not differentiable at x = 2.
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Find the angle between the lines whose direction cosines are given by the equations \[3l+m+5n=0\] and \[6mn-2nl+5lm=0.\]
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A trust fund has Rs. 30000 is to be invested in two different types of bonds. The first bond pays 9% interest per annum which will be given to orphanage and second bond pays 11% interest per annum which will be given to an NGO cancer aid society. Using matrix multiplication, determine how to divide Rs. 30000 among two types of bonds, if the trust fund obtains an annual total interest of Rs. 3060? What are the values reflected in the question?
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Prove that \[{{\tan }^{-1}}\left( \frac{\cos x}{1-\sin x} \right)=\left( \frac{\pi }{4}+\frac{x}{2} \right),\] \[x\in \left( \frac{-\,\pi }{2},\frac{\pi }{2} \right).\]
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Find the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
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Evaluate \[\int_{0}^{\pi /2}{\frac{{{\sin }^{4/5}}x}{{{\cos }^{4/5}}x+{{\sin }^{4/5}}x}\,dx.}\] |
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Evaluate integrals as a limit of sum \[\int_{2}^{4}{{{2}^{x}}\,dx.}\] |
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Evaluate \[\int{(x\,+1)}\sqrt{1-x-{{x}^{2}}}dx\]
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The probabilities of two students A and B coming to the school in time are \[\frac{3}{7}\] and \[\frac{5}{7},\] respectively. Assuming that the events, ?A coming in time' and 'B coming in time' are independent, find the probability, of only one of them coming to the school in time |
OR |
Find the probability that in 10 throws of a fair die, a score which is a multiple of 3 will be obtained in at least 8 of the throws. |
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A fruit grower can use two types of fertilisers in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and almost 310 kg of chlorine. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?
| Brand P | Brand Q |
Nitrogen | 3 | 3.5 |
Phosphoric acid | 1 | 2 |
Potash | 3 | 1.5 |
Chlorine | 1.5 | 2 |
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Find the equation of the plane passing through the intersection of the planes |
\[\vec{r}\cdot (\hat{i}+3\hat{j})-6=0\] and \[\vec{r}\cdot (3\hat{i}-\hat{j}-4\hat{k})=0,\] whose perpendicular distance from origin is unity. |
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If \[\vec{a}\cdot \vec{b}=\vec{a}\cdot \vec{c},\,\,\vec{a}\times \vec{b}=\vec{a}\times \vec{c}\] and \[\vec{a}\ne \vec{0},\] then prove that \[\vec{b}=\vec{c}.\] |
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An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
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Let A = {1, 2, 3,..., 9} and R be the relation in \[A\times A\] defined by (a, b)R(c, d), if \[a+d=b+c\]for (a, b), (c, d) in \[A\times A.\] Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]. |
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Prove that the relation R in set A = {1, 2, 3, 4, 5} given by \[R=\{(a,\,b):|a-b|\]is even}is even} is an equivalence relation. |
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Find the equation of tangents to the curve\[y=\cos (x+y),\] \[-\,2\pi \le x\le 2\pi ,\] that are parallel to the line \[x+2y=0.\] |
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Find the radius of the smallest circle with centre on Y-axis and passing through the point (7, 3). |
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Two groups of students representing 'SAVE MOTHER EARTH' and 'GO GREEN' are standing on two planes,, represented by the equations. \[\vec{r}\cdot (\hat{i}+\hat{j}+2\hat{k})=5\] and \[\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=8.\] What is the angle between the planes? Name few activities which should be taken up to save mother earth.
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Find the area of the region bounded by a circle \[4{{x}^{2}}+4{{y}^{2}}=9\] and the parabola \[{{y}^{2}}=4x.\]
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Solve the following initial value problem. |
\[2xy+{{y}^{2}}-2{{x}^{2}}\frac{dy}{dx}=0,\] \[y(1)=2\] |
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Solve the following differential equation. |
\[(1+y+{{x}^{2}}y)\,dx+(x+{{x}^{3}})dy=0\] |
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Let a, b and c denote the sides BC, CA and AB Respectively of \[\Delta ABC.\]If \[\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \\ \end{matrix} \right|=0,\] then find the value of \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C.\]
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