X+ Y = 5 Y + Z = 3 X + Z = 4 - Mathematics

प्रश्न

x+ y = 5
y + z = 3
x + z = 4

उत्तर

These equations can be written as
x + y + 0z = 5
0x + y + z = 3
x + 0y + z = 4

$D = | \begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix} |$
$= 1 (1 - 0) - 1 (0 - 1) + 0 (0 - 1)$
$= 1 (1) - 1 (- 1) + 0$
$= 2$
$D_{1} = | \begin{matrix} 5 & 1 & 0 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \end{matrix} |$
$= 5 (1 - 0) - 1 (3 - 4) + 0 (0 - 4)$
$= 5 (1) - 1 (- 1)$
$= 6$
$D_{2} = | \begin{matrix} 1 & 5 & 0 \\ 0 & 3 & 1 \\ 1 & 4 & 1 \end{matrix} |$
$= 1 (3 - 4) - 5 (0 - 1) + 0 (0 - 4)$
$= 1 (- 1) - 5 (- 1)$
$= 4$
$D_{3} = | \begin{matrix} 1 & 1 & 5 \\ 0 & 1 & 3 \\ 1 & 0 & 4 \end{matrix} |$
$= 1 (4 - 0) - 1 (0 - 3) + 5 (0 - 1)$
$= 1 (4) - 1 (- 3) + 5 (- 1)$
$= 2$
Now,
$x = \frac{D_{1}}{D} = \frac{6}{2} = 3$
$y = \frac{D_{2}}{D} = \frac{4}{2} = 2$
$z = \frac{D_{3}}{D} = \frac{2}{2} = 1$
$∴ x = 3, y = 2 and z = 1$

shaalaa.com

Applications of Determinants and Matrices

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 14 Q 13 Q 15

अध्याय 6: Determinants - Exercise 6.4 [पृष्ठ ८४]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 6 Determinants
Exercise 6.4 | Q 14 | पृष्ठ ८४

वीडियो ट्यूटोरियलVIEW ALL [5]

view
वीडियो ट्यूटोरियल For All Subjects
Applications of Determinants and Matrices

video tutorial00:15:29
Applications of Determinants and Matrices

video tutorial00:27:00
Applications of Determinants and Matrices

video tutorial00:36:09
Applications of Determinants and Matrices

video tutorial00:11:34
Applications of Determinants and Matrices

video tutorial00:23:43

संबंधित प्रश्न

Find the value of a if $[\begin{matrix} a - b & 2 a + c \\ 2 a - b & 3 c + d \end{matrix}] = [\begin{matrix} - 1 & 5 \\ 0 & 13 \end{matrix}]$

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Solve the system of the following equations:

$\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$

$\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$

$\frac{6}{x} + \frac{9}{y} - \frac{20}{x} = 2$

Evaluate
$∆ = | \begin{matrix} 0 & \sin α & - \cos α \\ - \sin α & 0 & \sin β \\ \cos α & - \sin β & 0 \end{matrix} |$

Find the integral value of x, if $| \begin{matrix} x^{2} & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4 \end{matrix} | = 28 .$

For what value of x the matrix A is singular?

$A = [\begin{matrix} x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1 \end{matrix}]$

Without expanding, show that the value of the following determinant is zero:

$| \begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix} |$

Without expanding, show that the value of the following determinant is zero:

$| \begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix} |$

Evaluate the following:

$| \begin{matrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{matrix} |$

$I f ∆ = | \begin{matrix} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{matrix} |, ∆_{1} = | \begin{matrix} 1 & 1 & 1 \\ y z & z x & x y \\ x & y & z \end{matrix} |, then prove that ∆ + ∆_{1} = 0 .$

Prove the following identity:

$| \begin{matrix} 2 y & y - z - x & 2 y \\ 2 z & 2 z & z - x - y \\ x - y - z & 2 x & 2 x \end{matrix} | = {(x + y + z)}^{3}$

Solve the following determinant equation:

| \begin{matrix} x + a & b & c \\ a & x + b & c \\ a & b & x + c \end{matrix} | = 0

Solve the following determinant equation:

| \begin{matrix} 1 & x & x^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{matrix} | = 0, b \neq c

Solve the following determinant equation:

| \begin{matrix} 3 & - 2 & \sin (3 θ) \\ - 7 & 8 & \cos (2 θ) \\ - 11 & 14 & 2 \end{matrix} | = 0

Find the area of the triangle with vertice at the point:

(0, 0), (6, 0) and (4, 3)

If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.

Using determinants, find the value of k so that the points (k, 2 − 2 k), (−k + 1, 2k) and (−4 − k, 6 − 2k) may be collinear.

Prove that :

| \begin{matrix} 1 & b + c & b^{2} + c^{2} \\ 1 & c + a & c^{2} + a^{2} \\ 1 & a + b & a^{2} + b^{2} \end{matrix} | = (a - b) (b - c) (c - a)

Prove that :

| \begin{matrix} 1 & 1 + p & 1 + p + q \\ 2 & 3 + 2 p & 4 + 3 p + 2 q \\ 3 & 6 + 3 p & 10 + 6 p + 3 q \end{matrix} | = 1

x + y − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0

An automobile company uses three types of steel S₁, S₂ and S₃ for producing three types of cars C₁, C₂and C₃. Steel requirements (in tons) for each type of cars are given below :

	Cars C₁	C₂	C₃
Steel S₁	2	3	4
S₂	1	1	2
S₃	3	2	1

Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.

If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.

For what value of x, the following matrix is singular?

[\begin{matrix} 5 - x & x + 1 \\ 2 & 4 \end{matrix}]

If $| \begin{matrix} 2 x + 5 & 3 \\ 5 x + 2 & 9 \end{matrix} | = 0$

For what value of x is the matrix $[\begin{matrix} 6 - x & 4 \\ 3 - x & 1 \end{matrix}]$ singular?

The value of the determinant

| \begin{matrix} a^{2} & a & 1 \\ \cos n x & \cos (n + 1) x & \cos (n + 2) x \\ \sin n x & \sin (n + 1) x & \sin (n + 2) x \end{matrix} | is independent of

If x, y, z are different from zero and $| \begin{matrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{matrix} | = 0$ , then the value of x⁻¹ + y⁻¹ + z⁻¹ is

Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2

Show that the following systems of linear equations is consistent and also find their solutions:
x + y + z = 6
x + 2y + 3z = 14
x + 4y + 7z = 30

Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

Given $A = [\begin{matrix} 2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5 \end{matrix}], B = [\begin{matrix} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix}]$ , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17

The prices of three commodities P, Q and R are Rs x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. Cpurchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹2,200. School Q wants to spend ₹3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

If $[\begin{matrix} 1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ - 1 \\ z \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]$ , find x, y and z.

Show that $| \begin{matrix} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{matrix} | = (x + y + z) {(x - z)}^{2}$

The existence of unique solution of the system of linear equations x + y + z = a, 5x – y + bz = 10, 2x + 3y – z = 6 depends on

In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?

X+ Y = 5 Y + Z = 3 X + Z = 4 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Select a course

X+ Y = 5 Y + Z = 3 X + Z = 4 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Select a course

उत्तर