Recent questions tagged bookproblem

Using properties of determinants, prove that: $\begin{vmatrix} 1&1+p&1+p+q\\ 2&3+2p&4+3p+2q\\ 3&6+3p&10+6p+3q \end{vmatrix}= 1.$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

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

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Using properties of determinants, prove that: $ \begin{vmatrix} x &x^2 &1 & px^3\\ y &y^2 &1 & py^3\\ z &z^2 &1 & pz^3 \end{vmatrix}= (1+pxyz) (x-z) (y-z)(z-x), $ where p is any scalar.

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Using properties of determinants, prove that: $\begin{vmatrix} \alpha & \alpha^2 & \beta+\gamma\\ \beta & \beta^2 & \alpha+\gamma\\ \gamma & \gamma^2 & \alpha+\beta \end{vmatrix} = (\beta-\gamma) (\gamma - \alpha) (\alpha-\beta) (\alpha+\beta+\gamma)$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Evaluate $\begin{vmatrix} 1 & x & y\\ 1 & x+y & y\\ 1 & x & x+y \end{vmatrix}$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Evaluate $ \begin{vmatrix} x & y & x+y\\ y & x+y & x\\ x+y & x & y \end{vmatrix}$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Let A = $\begin{bmatrix} 1 & -2 & 1\\ -2 & 3 & 1\\ 1 & 1 & 5 \end{bmatrix}$. Verify that \[\] $(i)\ \; \left [adj A \right ]^{-1} = adj(A^{-1}) \;\;\;\; $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

$If \; A^{-1}=\begin{bmatrix} 3 &-1 & 1\\ -15&6 &-5 \\ 5& -2&2 \end{bmatrix} \; and \; B = \begin{bmatrix} 1 & 2 & -2\\ -1 & 3 & 0\\ 0 & -2 & 1 \end{bmatrix}, \; find \; (AB)^{-1} $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Prove that $ \begin{vmatrix} a^2&bc&ac+c^2\\ a^2+ab& b^2& ac\\ ab&b^2+bc&c^2 \end{vmatrix}=4\,a^2\;b^2\;c^2. $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve the equation $ \begin{vmatrix} x+a & x & x \\ x & x+a & x\\ x & x & x+a \end{vmatrix}=0, \; a \neq 0. $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

If a, b and c are real numbers, and $\Delta = \begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & b+c\\ a+b & b+c & c+a \end{vmatrix}=0, $ show that either $a+b+c=0$ or $a=b=c.$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Evaluate $ \begin{vmatrix} cos\alpha \; cos\beta & cos\alpha \; sin\beta & -sin\alpha \\ -sin\beta & cos\beta & 0 \\ sin \alpha \; cos\beta & sin\alpha \; sin\beta & cos\alpha \end{vmatrix} $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Without expanding the determinant, prove that $ \begin{vmatrix} a &a^2 &bc \\ b &b^2 &ca \\ c &c^2 &ab \end{vmatrix}\; = \; \begin{vmatrix} 1 & a^2 & a^3 \\ 1 & b^2 & b^3 \\ 1 & c^2 & c^3 \end{vmatrix} $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

\[ \text{Prove that the determinant } \begin{vmatrix} x&sin\theta&cos\theta \\ -sin\theta&-x&1\\ cos\theta&1&x \end{vmatrix} \text{ is independent of } \theta \]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

asked Nov 29, 2012 by balaji.thirumalai

1 answer

If $A = \begin{bmatrix} 2 &-3 & 5\\ 3& 2 & -4\\ 1& 1& -2 \end{bmatrix}$, $\;$ find $A^{-1}.\;$ Using $ A^{-1}$ solve the system of equations: \[2x-3y+5z=11\] \[3x+2y-4z = -5\] \[x+y-2z = -3\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method:\[\] \[x-y+2z=7\] \[3x+4y-5z=-5\] \[2x-y+3z=12\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method:\[\] \[2x+3y+3z=5\] \[x-2y+z=-4\] \[3x-y-2z=3\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

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

verify that the given function (implicit or explicit) is a solution of the corresponding differential equation

asked Nov 29, 2012 by sreemathi.v

1 answer

Solve system of linear equations, using matrix method: $5x+2y=3$ $3x+2y=5$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method: $4x-3y=3$ $3x-5y=7$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method: $2x-y=-2$ $3x+4y = 3$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve system of linear equations, using matrix method: $5x+2y=4 $ $7x+3y = 5$

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad 5x -y +4z = 5 \] \[\quad2x + 3y + 5z= 2\] \[\quad 5x - 2y + 6z = -1\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad 3x -y -2z = 2 \] \[\quad 2y -z = -1\] \[\quad 3x -5y = 3\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad x + y + z = 1 \] \[\quad 2x + 3y +2z = 2\] \[\quad ax + ay +2az = 4\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad x + 3y = 5 \] \[\quad 2x + 6y = 8\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad 2x -y = 5 \] \[\quad x + y = 4\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Examine the consistency of the system of equations:\[\] \[\quad x + 2y = 2 \] \[\quad 2x + 3y = 3\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve the following Linear Programming Problems graphically: Maximise $Z = 3x + 4y$ . subject to the constraints : $x + y \leq 4, x \geq 0, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Form the differential equation representing the family of curves given by $(x-a)^2+2y^2\;=\;a^2$, where $a$ is an arbitrary constant

asked Nov 29, 2012 by sreemathi.v

1 answer

Solve the following Linear Programming Problems graphically: Minimise $Z =$ –$3x + 4 y$ subject to $ x + 2y \leq 8, 3x + 2y \leq 12, x \geq 0, y\geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Solve the following Linear Programming Problems graphically: Maximise $Z = 5x + 3y$ subject to $3x + 5y \leq 15, 5x + 2y \leq 10, x \geq 0, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Solve the following Linear Programming Problems graphically: Minimise $Z = 3x + 5y$. such that $ x + 3y \geq 3, x + y \geq 2, x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

If A is an invertible matrix of order 2, then det$(A^{-1})\;$ is equal to: $ (A)\; det\;(A) \quad (B)\; \frac{1}{det\;(A)} \quad (C)\; 1 \quad (D)\; 0 $

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Solve the following Linear Programming Problems graphically: Maximise $Z = 3x + 2y$ subject to $x + 2y \leq 10, 3x + y \leq 15, x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Solve the following Linear Programming Problems graphically: Minimise $Z = x + 2y$, subject to $2x + y \geq 3, x + 2y \geq 6, x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

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

Show that the minimum of $Z$ occurs at more than two points. Minimise and Maximise $Z = 5x + 10 y$ subject to $x + 2y \leq 120, x + y \geq 60, x – 2y \geq 0, x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

Let $A$ be a nonsingular square matrix of order $3 \times 3.$ Then $| adj \;A| $ is equal to:

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Show that the minimum of Z occurs at more than two points. Minimise and Maximise $Z = x + 2y$ subject to $ x + 2y \geq 100, 2x – y \leq 0, 2x + y \leq 200; x, y \geq 0.$

asked Nov 29, 2012 by fingeazy

1 answer

prove that \[tan^{-1} \left ( \frac {{\sqrt {1+x}}-{\sqrt {1-x}}}{{\sqrt {1+x}}+{\sqrt {1-x}}} \right ) = \frac {\pi}{4} -\frac{1}{2} cos^{-1}x,-\frac{1}{\sqrt2} \leq x \leq 1 \]

asked Nov 29, 2012 by vaishali.a

1 answer

Show that the minimum of Z occurs at more than two points. Maximise $z = – x + 2y$, subject to the constraints: $ x \geq 3, x + y \geq 5, x + 2y \geq 6, y \geq 0.$

asked Nov 29, 2012 by fingeazy

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

\[ \text{For the matrix A = } \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}, \text{ find the numbers a and b such that } A^{2} + aA + bI = O \]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

asked Nov 29, 2012 by sreemathi.v

1 answer

\[ \text{If A = } \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}, \text{show that } A^{2} -5A +7I = 0. \text{ Hence find } A^{-1}\]

asked Nov 29, 2012 by balaji.thirumalai

1 answer

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