Recent questions tagged q13

Principal and staff of a school decided to award prizes to their students for three values Honesty (x) cleanliness (y), Punctuality (z) in two groups. Group A ( upto class VIII) and group B ( IX - XII). It is decided that Rs. 12000 for three values (x, y, z) for 3, 4, 5 students of Group A and Rs. 11040 for 5, 4, 3 student of group B respectively. If all the three prizes together amount to Rs. 2880, then (i) Represent above situation by a matrix equation and form linear equation using matrices. (ii) Is it possible to solve the system of linear equations so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?

asked Jan 21, 2013 by thanvigandhi_1

0 answers

Form the differential equation having $y=(\sin^{-1}x)^2+A\cos^{-1}x+B,$where A and B are arbitrary constants,as its general solution.

asked Jan 18, 2013 by sreemathi.v

1 answer

(i) The cost of 5 kg of sugar, 5kg of wheat and 2kg of rice is Rs. 340. The cost of 4 kg of rice, 2kg of sugar and 5kg of wheat is Rs. 270. Rs. 264 is the cost of 3kg of wheat, 2kg of rice and 4 kg of sugar. Represent the above problem by a system of linear equations and find the cost of each item per kg by matix method. (ii) How increase in the prices of essential commodities affect economic weaker section of society?

asked Jan 17, 2013 by thanvigandhi_1

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Using vectors,find the area of the triangle ABC with vertices $A(1,2,3),B(2,-1,4)$ and $C(4,5,-1).$

asked Jan 17, 2013 by sreemathi.v

1 answer

(i) There are 2 men, 3 women and 3 children in family Y. The daily requirement of calories and protiens for men, women and children are 2000, 1800 and 1200 ; 50 g , 40 g and 35 g respectively. Represent the above information by matrices. Using matrix multiplication , calculate the total requirement of calories and protiens for each of the two families. (ii) Explain the importance of balance diet.

asked Jan 16, 2013 by thanvigandhi_1

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Find the area of the region bounded by $y=\sqrt x$ and $y=x.$

asked Jan 16, 2013 by sreemathi.v

1 answer

Naimish, a food supplier from Amritsar supplies three food products x, y and z to Shimla and Guwahati. His annual sale is given below: (a) If unit sale prices of x, y and z are Rs. 3.00, Rs. 2.50 and 1.00. Find the total revenue obtained from supply to Shimla and Guwahati. Which market generate more revenue? (b) If the unit costs of the above three food products are Rs. 2.00, Rs. 1.00 and Rs. 0.50 respectively, find the gross profit. Which supply has more profit? (C) In which direction from Delhi, Shimla and Guwahati lies. Which economic mode of transport do you suggest to your friend travelling to Shimla and Guwahati?

asked Jan 15, 2013 by thanvigandhi_1

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Evaluate the following $\int\frac{\sqrt{1+ x^2}}{x^4}dx$

asked Jan 14, 2013 by sreemathi.v

1 answer

If $ y=e^{\large a \: cos^{-1}\: x} ; -1 \: \leq \: x \leq \: 1$, show that $ (1+x^2) \large\frac{d^2y}{dx^2}-x\frac{dy}{dx}-$$a^2y=0.$

asked Jan 11, 2013 by thanvigandhi_1

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Prove that the curves $xy=4$ and $x^2+y^2=8$ touch each other.

asked Jan 11, 2013 by sreemathi.v

1 answer

Using properties of determinants, prove the following : $ \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{bmatrix} = (a-b)(b-c)(c-a). $

asked Jan 10, 2013 by thanvigandhi_1

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Using properties of determinants, prove the following : $ \begin{bmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{bmatrix} = 2(a+b+c)^3. $

asked Jan 10, 2013 by thanvigandhi_1

1 answer

Using properties of determinants show that: $ \begin{bmatrix} y+z & x & y \\ z+x & z & x \\ x+y & y & z \end{bmatrix} = (x+y+z)(x-z)^2.. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

If $ a, b, c$ are in A.P., show the following: $ \begin{bmatrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{bmatrix} = 0. $

asked Jan 10, 2013 by thanvigandhi_1

0 answers

Using properties of determinants prove that $ \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{bmatrix} = (a-b)(b-c)(c-a)(a+b+c). $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

Prove that $ \begin{bmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{bmatrix} = (5x+4)(4-x)^2. $

asked Jan 9, 2013 by thanvigandhi_1

0 answers

In a dice game, a player pays a stake of Re1 for each throw of a die.She receives Rs 5 if the die shows a 3,Rs 2 if the die shows a 1 or 6,and nothing otherwise.What is the player's expected profit per throw over a long series of throws?

asked Jan 8, 2013 by sreemathi.v

1 answer

Using properties of determinants, prove that : $ \begin{bmatrix} a^2+2a & 2a+1 & 1 \\ 2a+1 & a+2 & 1 \\ 3 & 3 & 1 \end{bmatrix} = (a-1)^3. $

asked Jan 8, 2013 by thanvigandhi_1

0 answers

Using the properties of determinants, prove that : $ \begin{bmatrix} x-3 & x-4 & x-\alpha \\ x-2 & x-3 & x-\beta \\ x-1 & x-2 & x-\gamma \end{bmatrix} = 0, \: where\: \alpha, \beta, \gamma \: are \: in \ A.P. $

asked Jan 7, 2013 by thanvigandhi_1

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A company manufactures two types of screws A and B.All the screws have to pass through a threading machine and a slotting machine.A box of type A screws requires 2 minutes on the threading machine and 3minutes on the slotting machine.A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine.In a week,each machine is available for 60 hours.On selling these screws,the company gets a profit of Rs100 per box on type A screws and Rs170 per box on type B screws.

asked Jan 7, 2013 by sreemathi.v

1 answer

Using properties of determinants, prove the following : $ \begin{bmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{bmatrix} = 2abc(a+b+c)^33 - 3abc. $

asked Jan 6, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $ \begin{bmatrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{bmatrix} = 2abc(a+b+c)^3. $

asked Jan 6, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following : $\begin{bmatrix} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2=b^2 \end{bmatrix} = (1 + a^2 + b^2)^3. $

asked Jan 4, 2013 by thanvigandhi_1

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Using properties of determinants, prove the following : $ \begin{bmatrix} 3a & -a+b & -a+c \\ a-b & 3b & c-b \\ a-c & b-c & 3c \end{bmatrix} = 3(a + b + c)(ab+bc+ca) $

asked Jan 4, 2013 by thanvigandhi_1

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If $ sin\: y = x\: sin (a+y)$ prove that $\large \frac{dy}{dx}=\frac{sin^2(a+y)}{sin\: a}$

asked Jan 4, 2013 by thanvigandhi_1

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If $ ( cos \: x )^y = (sin\: y)^y$ find $ \large\frac{dy}{dx} $

asked Jan 4, 2013 by thanvigandhi_1

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If a variable line in two adjacent positions has direction cosines l,m,n and $l+\delta l,m+\delta m,n+\delta n$,show that the small angle $\delta\theta$ between the two positions is given by \[\delta\theta^2=\delta l^2+\delta m^2+\delta n^2\]

asked Jan 3, 2013 by sreemathi.v

1 answer

If $ y = 3e^{2x} + 2e^{3x} $ prove that, $ \large\frac{d^2y}{dx^2} = $$5\large\frac{dy}{dx}$$+6y = 0.$

asked Jan 3, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, show that $\begin{bmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{bmatrix} = (a + b + c )^3 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, show that $ \begin{bmatrix} b+c & a & b \\ c+a & c & a \\ a+b & b & c \end{bmatrix} = (a + b + c )(a-c)^2 $

asked Jan 2, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove the following: $ \begin{bmatrix} a^2+1 \quad & ab \quad & ac \\ ab \quad & b^2+1 \quad & bc \\ ca \quad & cb \quad & c^2+1 \end{bmatrix} = 1+a^2+b^2+c^2. $

asked Dec 31, 2012 by thanvigandhi_1

0 answers

Find the value of $k$ in the following so that the function $f$ is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{\sqrt {1+kx}-\sqrt{1-kx}}{x}, & if\;-1\leq x< 0\\\large\frac{2x+1}{x-2}, & if\;0\leq x\leq 1\end{array}\right.\; at\; x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

If $ A = \begin{bmatrix} 1 & 2 & 2 \\[0.3em] 2 & 1 & 2 \\[0.3em] 2 & 2 & 1 \end{bmatrix}$ what is $ A^2-4A-5I=0$

asked Dec 29, 2012 by thanvigandhi_1

1 answer

If $ A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} $ show that $ A^2 - 5A - |4| = 0$. Hence find $ A^{-1} $

asked Dec 28, 2012 by thanvigandhi_1

0 answers

Using properties of determinants, prove that : $ \begin{bmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{bmatrix} = (1+a^2+b^2+c^2) $

asked Dec 26, 2012 by thanvigandhi_1

0 answers

Two farmers Mukul and Vipul cultivates only three crops potato, tomato and wheat. The annual sale of 2010, 2011 are given below : \[ Sale 2010 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 20000 & \quad 15000 & \quad 80000 & \quad Mukul \\ 30000 & \quad 10000 & \quad 90000 & \quad Vipul \end{bmatrix} \] \[ Sale 2011 = \begin{bmatrix} Potato & \quad Tomato & \quad Wheat \\ 30000 & \quad 15000 & \quad 60000 & \quad Mukul \\ 20000 & \quad 20000 & \quad 50000 & \quad Vipul \end{bmatrix} \]

asked Dec 25, 2012 by thanvigandhi_1

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The general solution of the differential equation $ \frac{dy}{dx} = e^{x+y} $is \[ \begin{array} (A)\: e^x+e^{-y}=C \quad & (B)\: e^x+e^y=C \\ (C)\: e^{-x}+e^y=C \quad & (D)\: e^{-x}+e^{-y}=C \end{array} \]

asked Dec 24, 2012 by thanvigandhi_1

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Find A,if $\begin{bmatrix}4\\1\\3\end{bmatrix}A=\begin{bmatrix}4 & 8 & 4\\1 & 2 &1\\3 & 6 & 3\end{bmatrix}$

asked Dec 22, 2012 by sreemathi.v

1 answer

Let f : N $ \rightarrow $ N be defined by : $ f(n) = \left\{ \begin{array}{l l} \large\frac{n+1}{2} & \quad \text{if $n$ is odd}\\ \large\frac{n}{2} & \quad \text{ if $n$ is even} \end{array} for \: all\: n\: \in \: N. \right.$ Find whether the function f is bijective.

asked Dec 21, 2012 by thanvigandhi_1

0 answers

Let $ A = \begin{bmatrix} 3 & 2 & 5 \\[0.3em] 4 & 1 & 3 \\[0.3em] 0 & 6 & 7 \end{bmatrix}$ Express $A$ as sum of two matrices such that one is symmetric and the other is skew symmetric.

asked Dec 20, 2012 by thanvigandhi_1

1 answer

If functions $f\;:\;A \rightarrow B\;and\;g\;:\;B \rightarrow A $ satisfy g o f =$I_A$ ,then is f one-one and g onto

asked Dec 18, 2012 by sreemathi.v

1 answer

Determine whether the given planes are parallel or perpendicular, and in case they are neither, then find the angles between them. $(a)\; 7x + 5y + 6z + 30 = 0 \: and \: 3x -y -10z + 4 = 0 $

asked Dec 14, 2012 by thanvigandhi_1

1 answer

Show that the lines $ \frac{ \large x - 5 }{ \large 7} = \frac{ \large y + 2}{ \large -5} = \frac{ \large z}{ \large 1}$ and $ \frac{ \large x }{ \large 1} = \frac{ \large y}{ \large 2} = \frac{ \large z}{ \large 3}$ are perpendicular to each other.

asked Dec 14, 2012 by thanvigandhi_1

1 answer

For the differential equations, find the particular solution satisfying the given condition: \[ \bigg[ x \: sin^2 \bigg( \frac{x}{y} - y \bigg) \bigg] dx + xdy = 0; y \frac{\large \pi}{\large 4} when \: x = 1\]

asked Dec 12, 2012 by thanvigandhi_1

1 answer

Find the value of \[ tan\frac{1}{2} \bigg[ sin^{-1} \frac{2x}{1+x^2} + cos^{-1} \frac{1-y^2}{1+y^2} \bigg], | x | < 1, y > 0 \: and \: xy < 1\]

asked Dec 10, 2012 by thanvigandhi_1

1 answer

Find the direction cosines of the vector joining the points $A(1, 2, –3)$ and $B(–1, –2, 1),$ directed from $A$ to $B$.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Find the integral $ \LARGE {\int } \normalsize \frac{\large x^3-x^2+x-1}{\large x-1}\;dx$

asked Dec 6, 2012 by sreemathi.v

1 answer

If $ \overrightarrow a, \overrightarrow b, \overrightarrow c, $are unit vectors such that $ \overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0$, find the value of $ \overrightarrow a ⋅ \overrightarrow b + \overrightarrow b ⋅ \overrightarrow c + \overrightarrow c ⋅ \overrightarrow a$.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Integrate the function $\frac{\large x^2}{(\large 2+3x^3)^3}$

asked Dec 6, 2012 by sreemathi.v

1 answer

Find the integrals of the functions\[\frac{\cos2x-\cos2\alpha}{\cos x-\cos\alpha}\]

asked Dec 5, 2012 by sreemathi.v

1 answer

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