Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 5 Herrega kutaa 11: Veektaroota

Wanti fuula kana irra jiru hundi boqonnaa 5 kitaaba Herrega kutaa 11 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

3
Kaardii: 10
Gaaffii shaakalaa boqonnaa 5: 251–312
Fuula kitaabaa boqonnaa 5

Boqonnaan 5 maal haguuga

1.Revision on Vectors and Scalars
2.Representation of vectors
3.Vector Product
4.Application of Scalar and Cross Product
5.Application of Vectors
6.Applications

Yaadannoo gabaabaa boqonnaa 5

Kaardiiwwan fuula Herrega kutaa 11 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Baay'isa Iskaalaarii fi OrtogonalitiiFoormulaa

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_1 v_1 + u_2 v_2

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_1 v_1 + u_2 v_2

$\mathbf{u}, \mathbf{v}$: wektaroota kennaman $dimensionless$
$|\mathbf{u}|, |\mathbf{v}|$: dheerina wektarootaa $dimensionless$
$\theta$: kofa wektaroota lamaan gidduu $rad$
$u_1, u_2, v_1, v_2$: qaamolee iskaalaarii wektarootaa $dimensionless$

Yoom akka itti fayyadamtu

Baay'isa tuqaa shallaguuf, kofa wektaroota gidduu barbaaduuf, yookiin wektaroonni kof-qajeeloo ta'uu mirkaneessuuf itti fayyadami.

Of eeggadhu

Wektaroonni lama kof-qajeeloo kan ta'an baay'isni iskaalaarii isaanii zeeroo yoo ta'e qofaadha. Bu'aan iskaalaarii dha malee wektara miti.

Grade 11 Mathematics, fuula 251-312 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Sarara Tuqaa Geengoo Ka'umsa IrrattiFoormulaa

x x_1 + y y_1 = r^2

x x_1 + y y_1 = r^2

$x_1, y_1$: qindaa'inoota qabxii tuqaa $dimensionless$
$x, y$: qindaa'inoota qabxii kamiyyuu kan sarara tuqaa irraa $dimensionless$
$r$: raadiyeesii geengoo wiirtuun isaa ka'umsa ta'ee $dimensionless$

Yoom akka itti fayyadamtu

Yeroo wiirtuun geengoo ka'umsa (0, 0) ta'e herrega sarara tuqaa barbaaduuf itti fayyadami.

Of eeggadhu

Unki kun kan hojjetu yoo wiirtuun (0, 0) ta'e qofaadha. Wiirtuu biraatiif unka waliigalaa fayyadami.

Grade 11 Mathematics, fuula 251-312 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Sarara Tuqaa Geengoo WaliigalaaFoormulaa

(x - x_0)(x_1 - x_0) + (y - y_0)(y_1 - y_0) = r^2

(x - x_0)(x_1 - x_0) + (y - y_0)(y_1 - y_0) = r^2

$x_0, y_0$: qindaa'inoota wiirtuu geengoo $dimensionless$
$x_1, y_1$: qindaa'inoota qabxii tuqaa $dimensionless$
$x, y$: qindaa'inoota qabxii kamiyyuu kan sarara tuqaa irraa $dimensionless$
$r$: raadiyeesii geengoo $dimensionless$

Yoom akka itti fayyadamtu

Waliigaltee sarara tuqaa geengoo wiirtuun isaa (x_0, y_0) ta'ee fi qabxii tuqaa (x_1, y_1) irratti argamu barbaaduuf itti fayyadami.

Of eeggadhu

Qabxii tuqaa fi jijjiiramaa sararichaa wal hin makiin. Gara mirgaa irratti r^2 malee r hin ta'u.

Grade 11 Mathematics, fuula 251-312 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 5

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

When using Cramer's rule to solve a system of n linear equations in n variables, how many determinants must be evaluated?

A.n
B.n + 1
C.2n
D.n squared

Deebii fi sababa agarsiisi

Deebii: B n + 1

Sababa

The textbook states that solving a system of n linear equations in n variables using Cramer's rule requires evaluating n + 1 determinants.

Gaaffii 2

According to the textbook classification, which of the following is a vector quantity?

A.Mass
B.Time
C.Velocity
D.Temperature

Deebii fi sababa agarsiisi

Deebii: C Velocity

Sababa

Velocity requires both magnitude and direction, so it is classified as a vector quantity. Mass, time, and temperature are scalar quantities.

Gaaffii 3

Three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) in the xy-plane are collinear if and only if the determinant of the matrix with rows [x₁ y₁ 1], [x₂ y₂ 1], and [x₃ y₃ 1] is equal to what value?

A.0
B.1
C.-1
D.1/2

Deebii fi sababa agarsiisi

Deebii: A 0

Sababa

Three points in the xy-plane lie on the same straight line if and only if the determinant of the 3 by 3 matrix formed by their coordinates and ones is 0.

Gaaffii 4

If A is an invertible matrix of order n, what is the determinant of its inverse, det(A^-1)?

A.-det(A)
B.1 / det(A)
C.(det(A)) squared
D.0

Deebii fi sababa agarsiisi

Deebii: B 1 / det(A)

Sababa

By the properties of determinants in the textbook, if A is invertible, then det(A^-1) = 1 / det(A).

Gaaffii 5

What is the area of a triangle with vertices at (0, 5), (0, 0), and (5, 0) in the xy-plane?

A.5 square units
B.10 square units
C.12.5 square units
D.25 square units

Deebii fi sababa agarsiisi

Deebii: C 12.5 square units

Sababa

The determinant evaluating the vertices gives 25. The area is (1/2) * 25 = 12.5 square units.

Gaaffii 6

A student in Hawassa solves a linear system AX = B of n equations in n variables and finds |A| = 0. What can be concluded about the system?

A.It has a unique solution
B.It has either no solution or infinitely many solutions
C.It must have exactly two distinct solutions
D.Cramer's rule can still be applied directly

Deebii fi sababa agarsiisi

Deebii: B It has either no solution or infinitely many solutions

Sababa

When |A| = 0, Cramer's rule cannot be used. The textbook states there are two possibilities: the system has no solution or it has infinitely many solutions.

Gaaffii 7

Using determinants, what is the general equation of the line passing through the points P(2, 0) and R(0, 3)?

A.3x + 2y - 6 = 0
B.2x + 3y - 6 = 0
C.3x - 2y + 6 = 0
D.2x - 3y + 6 = 0

Deebii fi sababa agarsiisi

Deebii: A 3x + 2y - 6 = 0

Sababa

Setting the determinant of [x y 1; 2 0 1; 0 3 1] to 0 yields -2(y - 3) - 3x = 0, which simplifies to 3x + 2y - 6 = 0.

Gaaffii 8

What is the area of the triangle with vertices P(1, 1), Q(1, 3), and R(4, 5)?

A.1.5 square units
B.3 square units
C.4.5 square units
D.6 square units

Deebii fi sababa agarsiisi

Deebii: B 3 square units

Sababa

The determinant of the coordinate matrix is -6. Since area is positive, Area = -(1/2)(-6) = 3 square units.

Gaaffii 9

Let A and B be square matrices of order 3 with det(A) = 2 and det(B) = -3. What is the value of det(2AB)?

A.-12
B.-24
C.-48
D.48

Deebii fi sababa agarsiisi

Deebii: C -48

Sababa

For a 3 by 3 matrix, det(2M) = 2³ det(M) = 8 det(M). Thus, det(2AB) = 8 * det(A) * det(B) = 8 * 2 * (-3) = -48.

Gaaffii 10

An interpolating polynomial p(x) = a + bx + cx² fits the data points (0, 1), (1, 2), and (2, 22). What is the estimated value of p(3)?

A.22
B.43
C.54.5
D.61

Deebii fi sababa agarsiisi

Deebii: D 61

Sababa

Solving the system gives p(x) = 1 - 8.5x + 9.5x². Evaluating at x = 3 gives p(3) = 1 - 8.5(3) + 9.5(9) = 1 - 25.5 + 85.5 = 61.

Appiin boqonnaa kana irratti gaaffii 18 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.

Fuula kitaabaa boqonnaa 5

Boqonnaan 5 Maxxansa Ministeera Barnootaa keessatti fuula 251 hanga fuula 312 ti.

Maxxansa Ministeera Barnootaa, Ingiliffaa, Fuula 494

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Yaadannoo gabaabaa Herrega kutaa 11 hunda

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