Recent questions tagged difficult

Solve system of linear equations, using matrix method:\[\] \[x-y+z=4\] \[2x+y-3z=0\] \[x+y+z=2\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method:\[\] \[2x+y+z=1\] \[x-2y-z = \frac{3}{2}\] \[3y-5z=9\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Prove that $x^2-y^2\;=\;c(x^2+y^2)^2$is the general solution of differential equation $(x^3-3x\;y^2)dx\;=\;(y^3-3x^2y)dy$ where $c$ is a parameter.

asked Nov 29, 2012 by sreemathi.v

1 answer

\[ \text{For the matrix A = } \begin{bmatrix} 1&1&1 \\ 1& 2&- 3 \\ 2 &-1& 3 \end{bmatrix} \] \[ \text{Show that } A^{3} - 6A^{2} + 5A + 11I = O. \text{ Hence, find } A^{-1}\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Show that the minimum of Z occurs at more than two points. Maximise $Z = x + y$, subject to$\; x $– $y\leq –1, -x + y \leq 0, x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Prove that \[\frac{9 \pi}{8} - \frac{9}{4} sin ^{-1}\frac{1}{3}= \frac{9}{4} sin^{-1}\frac{2\sqrt 2}{3}\]

asked Nov 29, 2012 by vaishali.a

1 answer

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2e^{-y} -1$, given that $y = 0$ when $x = 0$.

asked Nov 28, 2012 by sreemathi.v

1 answer

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time.If the population was 20,000 in 1999 and 25000 in the year of 2004,what will be the population of the village in 2009?

asked Nov 28, 2012 by sreemathi.v

1 answer

By using the properties of determinants show that \[\begin{vmatrix} x&x^2&yz \\ y&y^2&zx \\ z&z^2&xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx) \]

asked Nov 28, 2012 by balaji.thirumalai

1 answer

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\large\frac{2R}{\sqrt 3}$ . Also find the maximum volume.

asked Nov 28, 2012 by thanvigandhi_1

2 answers

A point on the hypotenuse of a triangle is at distance $a$ and $b$ from the sides of the triangle.Show that the maximum length of the hypotenuse is $ (a^{\large\frac{2}{3}} + $$b^{\large\frac{2}{3}})^{\Large\frac{3}{2}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Suppose we have four boxes A,B,C and D containing coloured marbles as given in the box below. One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A?, box B?, box C? \[\]$\begin{matrix} \text{Box} & &\text{Marble Colour} & \\ &\text{Red} &\text{White} & \text{Black}\\ A & 1 & 6 & 3\\ B & 6 & 2 &2 \\ C & 8 & 1 & 1\\ D & 0 & 6 & 4 \end{matrix}$

asked Nov 27, 2012 by balaji.thirumalai

1 answer

In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers TRUE; if it falls tails, he answers FALSE. Find the probability that he answers at least 12 questions correctly.

asked Nov 27, 2012 by balaji.thirumalai

1 answer

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is$\large\frac{8}{27}$ of the volume of the sphere.

asked Nov 27, 2012 by thanvigandhi_1

1 answer

if $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ prove that $A^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix} , n \in N$.

asked Nov 23, 2012 by pady_1

1 answer

If $A$ and $B$ are square matrices of the same order such that $AB = BA $, then prove by induction that $AB^n = B^nA $. Further, prove that $ (AB)^n = A^nB^n \: $ for all $ n ∈ N. $

asked Nov 22, 2012 by pady_1

1 answer

If $x$ and $y$ are connected parametrically by the equations given in $ x = \large\frac{\sin^3t}{\sqrt {\cos 2t}}, $$y = \large\frac{\cos^3t}{\sqrt {\cos 2t}} $ without eliminating the parameter, Find $\large\frac{ dy}{ dx}$.

asked Nov 20, 2012 by thanvigandhi_1

1 answer

Find $\large\frac{dy}{dx}$ of the functions given in $ x^{\large y }+ y^{\large x }= 1 $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x : $ $ (x \cos x )^{\large x} + (x\sin x)^{\Large\frac{1}{x}} $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

Differentiate the functions given in w.r.t. $x : $ $ x ^{\large\sin x} + (\sin x)^{\large\cos x} $

asked Nov 17, 2012 by thanvigandhi_1

1 answer

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

asked Nov 16, 2012 by fingeazy

1 answer

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table below. \[\] $\begin{matrix} \text{kg per bag:}& \text{Brand P} &\text{Brand Q} \\ \text {Nitrogen} &3 & 3.5\\ \text {Phosphoric Acid}& 1 & 2\\ \text {Potash}&3 &1.5 \\ \text {Chlorine}& 1.5 & 2 \end{matrix}$ \[\] 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?

asked Nov 16, 2012 by fingeazy

1 answer

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table below. If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden? \[\] $\begin{matrix} \text{kg per bag:}& \text{Brand P} &\text{Brand Q} \\ \text {Nitrogen} &3 & 3.5\\ \text {Phosphoric Acid}& 1 & 2\\ \text {Potash}&3 &1.5 \\ \text {Chlorine}& 1.5 & 2 \end{matrix}$

asked Nov 16, 2012 by fingeazy

1 answer

An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost? The distances (in km) between the depots and the petrol pumps is given in the following table: \[\] $\begin{matrix} & \underline{\text{Distance in Km}} & \\ \underline{\text{From} \setminus\text{To}}& \text{A}&& \text{B}\\ \text{D}& 7 &&3 \\ \text{E}& 6 &&4 \\ \text{F} & 3 && 2 \end{matrix}$

asked Nov 15, 2012 by fingeazy

1 answer

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively.

asked Nov 15, 2012 by fingeazy

1 answer

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

asked Nov 15, 2012 by fingeazy

1 answer

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

asked Nov 5, 2012 by thanvigandhi_1

1 answer

Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then $$1/a^2 + 1/b^2 + 1/c^2 = 1/p^2$$

asked Nov 2, 2012 by balaji.thirumalai

1 answer

Find the vector equation of the line passing through $(1, 2, 3)$ and parallel to the planes $\hat{r}. (\hat{i} - \hat{j} + 2\hat{k}) = 5$ and $\hat{r} (3\hat{i} + \hat{j} + \hat{k} ) = 6$

asked Nov 2, 2012 by balaji.thirumalai

1 answer

If the points $( 1, 1 , p)$ and $(– 3 , 0, 1)$ be equidistant from the plane $\hat{r} . (3\hat{i} +4\hat{j} -12\hat{k} +13) = 0$ then find the value of $p$.

asked Nov 2, 2012 by balaji.thirumalai

1 answer

If $ l_1, m_1, n_1$ and $l _2, m_2, n_2 $ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_1n_2-m_2n_1$, $n_1l_2-n_2l_1$, $l_1m_2-l_2m_1$

asked Nov 2, 2012 by balaji.thirumalai

1 answer

Ask Questions, Get Answers

Solve system of linear equations, using matrix method:\[\] \[x-y+z=4\] \[2x+y-3z=0\] \[x+y+z=2\]

Solve system of linear equations, using matrix method:\[\] \[2x+y+z=1\] \[x-2y-z = \frac{3}{2}\] \[3y-5z=9\]

Prove that \(x^2-y^2\;=\;c(x^2+y^2)^2\)is the general solution of differential equation \((x^3-3x\;y^2)dx\;=\;(y^3-3x^2y)dy\) where \(c\) is a parameter.

\[ \text{For the matrix A = } \begin{bmatrix} 1&1&1 \\ 1& 2&- 3 \\ 2 &-1& 3 \end{bmatrix} \] \[ \text{Show that } A^{3} - 6A^{2} + 5A + 11I = O. \text{ Hence, find } A^{-1}\]

Show that the minimum of Z occurs at more than two points. Maximise $Z = x + y$, subject to$\; x $– $y\leq –1, -x + y \leq 0, x, y \geq 0.$

Prove that \[\frac{9 \pi}{8} - \frac{9}{4} sin ^{-1}\frac{1}{3}= \frac{9}{4} sin^{-1}\frac{2\sqrt 2}{3}\]

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2e^{-y} -1$, given that $y = 0$ when $x = 0$.

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time.If the population was 20,000 in 1999 and 25000 in the year of 2004,what will be the population of the village in 2009?

By using the properties of determinants show that \[\begin{vmatrix} x&x^2&yz \\ y&y^2&zx \\ z&z^2&xy \end{vmatrix} = (x-y)(y-z)(z-x)(xy+yz+zx) \]

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \(\large\frac{2R}{\sqrt 3}\) . Also find the maximum volume.

A point on the hypotenuse of a triangle is at distance \(a\) and \(b\) from the sides of the triangle.Show that the maximum length of the hypotenuse is $ (a^{\large\frac{2}{3}} + $$b^{\large\frac{2}{3}})^{\Large\frac{3}{2}}$

In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers TRUE; if it falls tails, he answers FALSE. Find the probability that he answers at least 12 questions correctly.

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is\(\large\frac{8}{27}\) of the volume of the sphere.

if $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$ prove that $A^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix} , n \in N$.

If \(A\) and \(B\) are square matrices of the same order such that \(AB = BA \), then prove by induction that \(AB^n = B^nA \). Further, prove that \( (AB)^n = A^nB^n \: \) for all \( n ∈ N. \)

If $x$ and $y$ are connected parametrically by the equations given in $ x = \large\frac{\sin^3t}{\sqrt {\cos 2t}}, $$y = \large\frac{\cos^3t}{\sqrt {\cos 2t}} $ without eliminating the parameter, Find $\large\frac{ dy}{ dx}$.

Find $\large\frac{dy}{dx}$ of the functions given in $ x^{\large y }+ y^{\large x }= 1 $

Differentiate the functions given in w.r.t. $x : $ $ (x \cos x )^{\large x} + (x\sin x)^{\Large\frac{1}{x}} $

Differentiate the functions given in w.r.t. $x : $ $ x ^{\large\sin x} + (\sin x)^{\large\cos x} $

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively.

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then $$1/a^2 + 1/b^2 + 1/c^2 = 1/p^2$$

Find the vector equation of the line passing through $(1, 2, 3)$ and parallel to the planes $\hat{r}. (\hat{i} - \hat{j} + 2\hat{k}) = 5$ and $\hat{r} (3\hat{i} + \hat{j} + \hat{k} ) = 6$

If the points $( 1, 1 , p)$ and $(– 3 , 0, 1)$ be equidistant from the plane $\hat{r} . (3\hat{i} +4\hat{j} -12\hat{k} +13) = 0$ then find the value of $p$.

If $ l_1, m_1, n_1$ and $l _2, m_2, n_2 $ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_1n_2-m_2n_1$, $n_1l_2-n_2l_1$, $l_1m_2-l_2m_1$