MTH103
The equation of the circle centre (2, -3) and radius 4 is given as __________
*y2 + x2 – 4x + 6y - 3 = 0*
__________ is the equation of a line (-1, -4) whose gradient is 1?
*y = x – 3*
The equation of a straight line is of the form __________
*y = mx + c*
{100:SHORTANSWER:%100%1}1
{100:SHORTANSWER:%100%45 degree}45 degree
The distance between the pairs of points A(0, 1) and B(6, 9) is __________
*10*
Let the symbols i, j and k represent unit vectors in the direction OX, OZ and OZ respectively. If OP-=ai+bj+ck, then OP-= __________
*a2 + b2 + c2*
A point P(x, y) means that P is in the __________
*(x, y)-plane*
zn = [r(cosθ + isinθ )]n is called the __________
*De Moivre’s Theorem*
{100:SHORTANSWER:%100%First}First
{100:SHORTANSWER:%100%last}last
A __________ is also defined as a directed line segments
*Vector*
A __________ vector is a vector whose position in space is fixed in addition to its magnitude and direction
*Point located*
The direction cosines (l, m, n) are the cosines of the angles between the vectors and the axes __________ respectively
*OX, OY, OZ*
Two vectors a→ and b→ are said to be equal if they have the same __________
*magnitude and direction*
The subtraction of vectors can be considered as the addition of its __________
*Negatives*
Given that r1 = 3i + 5 j , r2 = −4i +19 j the modulus of 5r1 − r2.is __________
*19*
{100:SHORTANSWER:%100%-5}-5
{100:SHORTANSWER:%100%4}4
The angle between two lines in a plane is defined to be 0 if the lines are __________
*Parallel*
Force, velocity, acceleration are examples of __________
*Vectors*
If a= a1i+a2j+a3k and b= b1i+b2j+b3k, then ab = __________
*a1b1 + a2b2 + a3b3*
Mechanical force acting on a body is an example of __________ vector
*Line*
__________ is a vector which is not restricted in any way
*Free Vector*
__________ is the value of AB- + BC- + C D- + DE- + EF-
*AF-*
If PQRS is a square with T and U the mid points of SP and QR respectively, then __________= 2TU-
*PQ-+RS-*
The collection of objects called vectors which may be added together and multiplied by numbers is called __________
*Vector Space*
If Z1 = 3i + 5j and Z2 = 7i + 3j. __________ is Z1 + Z2
*10i + 8j*
If Z1 = 2i − 4j, Z2 = 2i + 6j, Z3 = 3i − j. __________ is Z1 + Z2 + Z3
*7i + j*
__________ is |P Q- | ifP Q- = 5i + 2j + 4k
5*3*
The focus in the equation y2 = 5x is __________
*54,0*
tan β is zero, then m1 = m2. This implies __________ lines
*Parallel*
When m1m2 = −1. This implies that the lines are __________
*Perpendicular*
If −3x+4y =8 and −2x − 8y − 14 = 0 are two intersecting lines, the angle formed is __________
*-14.63o*
__________ is the line equation through the point (−1, 2) which is parallel to y = 2x + 1
*3x + 5*
__________ is the parametric equation representing a circle with centre (2, -1) and radius 3
*y = −1 + 3 sin θ*
A __________ is defined as the locus of a point which moves so that it is always the same distance from the fixed point (called the focus) and a given line (called the directrix)
*Parabola*
{100:SHORTANSWER:%100%Horizontal}Horizontal
{100:SHORTANSWER:%100%vertical}vertical
In an axiomatic development of a branch of mathematics, one begins with __________
*Horizontal, vertical*
__________ in analytic geometry of vector are the cosines of angles between the vector and the three coordinate axes
*Direction Cosines*
The l,m,n of the vector r = 2i + 4j − 3k yields __________
*229,429,-329*
If a= a1i+a2j+a3k and b= b1i+b2j+b3k, then a x b = __________
a2b3- a3b2i- a1b3- a3b1j+ a1b2- a2b1k
The distance between x1,y1 and x2,y2 is given by __________
x2- x12+ y2- y12
Given a=10, b=13, θ= 59.5o, the scalar product of a and b is __________
*65.98*
When two vectors are at right angles to each other the dot product is __________
*0*
The cross product of a = (2, 3, 4) and b = (5, 6, 7) is __________
*(−3, 6, −3)*
The general equation of a circle is given by __________
x2+ y2+ 2gx+2fy+c=0
If i x j = k and j x k = i, then k x i = __________
*J*
The Locus of points at a distance of 3 from the point (0, 0) is given by the equation __________
*x2 + y2 = 9*
The coordinates of the midpoint of a line segment is __________ of the coordinates of the endpoints
*Average*
If A is (3, 6) and B is (4, 8). The coordinate of the midpoint of AB is __________
*(312 , 7)*
__________ of a line is defined as the ratio of the vertical distance the line rises or falls to the horizontal distance
*Gradient*
_________ is the center and radius of the circle x2 + y2 + 8x + 6y = 0
*Center = (−4 ,−3), r = 5*
A quantity that is defined by magnitude (size) but no direction is called _____
Scalar quantity
… is one that has both magnitude and direction in which it operates.
Vector quantity
Length, area, volume, mass, time etc, are examples of…
Scalar quantity
Force, velocity, acceleration, etc, are examples of …
Vector quantity
In geometrical representation of vectors in 1-3 dimensions, the length of the line denotes…
the magnitude of the quantity, according to a stated vector scale
An ellipse can be defined as the locus of all points that satisfies the equation
x2a2+ y2b2=1
Two vectors a and b are said to have the same magnitude and the same direction if and only if they…
equal
Which of the following is not a type of vector
Block Vector
For … AB to occur, point A has to be fixed.
Position Vector
… is one that can shift or slide along its line of action
Line Vector
Mechanical force acting on a body is an example of …
Line Vector
… is one which is not restricted in any way
Free Vector
… is defined by its magnitude and direction and can be drawn as any one of a set of equal length parallel lines
Free Vector
If a is a force of 30N, acting in the east direction. b is a force of 40N, acting in the north direction. Then the magnitude of the vector sum r of these forces will be
50N
Find AB- + BC- + C D- + DE- + EF-
AF-
ABCD is a square (quadrilateral) with T and U the mid points of SP and QR respectively. Which expressions is true?
PQ-+RS-=2TU-
In triangle ABC, the points L, M, N are the midpoints of the sides AB, BC and CA respectively. Which these expressions is true?
2AB-+3BC-+AC-=2LC-
… is a vector as a way of determining the magnitude and direction in its coordinate of x and y axis.
Component of a vector in terms of unit vector
… is a collection of objects called vectors which may be added together and multiplied by numbers.
Vector in space
If Z1 = 3i + 5j and Z2 = 7i + 3j. Find Z1 + Z2
10i + 8j
The gradient of the line AB in figure 8 is...
y-y1y2-y1=x-x1x2-x1
The equation of the circle centre (2, -3) and radius 4 is given as…
y2 + x2 – 4x + 6y - 3 = 0
What is the gradient of the straight line and the angle of inclination of the points, A(2, -3) and B(4, 5)?
gradient=1, θ = 45o
Let A, B, C be the points (2, 3), (3, -2) and (-1, 4). The length of AB yields…
26
In the x-y plane the equation of the hyperbola is given by...
x2a2- y2b2=1
If r is the hypothenus of a right-angled triangle with sides a and b and if i and j are unit vectors in the horizontal and vertical directions, then r is given by
ai+ bj
… is also defined as a directed line segment.
vector
Polar form of a complex number is …………………
r(cosθ+ isinθ)
a2 + b2 is equal to ……………
(a+ib)(a-ib)
Given that x + 2y = 3, 3x + 4y =1. What is x and y?
-5, 4
The coordinates of the midpoint of a line segment is
the average of the coordinates of the endpoints
If A is (3, 6) and B is (4, 8). Find the coordinate of the midpoint of AB.
[312 , 7]
Consider the diagram below:
The angle of slope in the above diagram is...
tan θ
The equation of a straight line in a gradient - intercept form is...
y = mx + c
Using the diagram above, the equation of a straight line of gradient and one point form when a straight line passes through a given point p (x1, y1) is ...
y − y1 = m(x − x1)
Vector P Q- is defined by …
its magnitude (r) and its angle (θ)
If r is the Hypothenus of a right-angled triangle with sides a and b, with r = ai + bj, then
i and j are unit vectors in the horizontal and vertical directions
____ in analytic geometry of vector are the cosines of angles between the vector and the three coordinate axes .
Direction cosines
Find l,m,n of the vector r = 2i + 4j − 3k
The scalar product of a and b is defined as the scalar (number)…
abcos θ
If a=10, b=13, θ= 59.5o, the scalar product of a and b is…
65.98
When two vectors are at right angles to each other the dot product is ?
Zero
If a = a1i + a2j + a3k and b = b1i + b2j + b3k, a.b gives...
a1b1 + a2b2 + a3b3
Vector product of a and b is represented by ...
a x b
_______ is the locus of all points equidistant from a central point.
A parabola
The cross product of a = (2, 3, 4) and b = (5, 6, 7) is ____
(−3, 6, −3)
Let us consider a simplest case of a circle with centre at the origin and radius r as shown in figure 1. If the circle has a centre (0, 0) and radius r,
using Pythagoras theorem, the equation of the circle from the figure above is given as
x2 + y2 = r2
The Locus of points at a distance of 3 from the point (0, 0) is given by the equation ___
x2 + y2 = 9
Find the distance between the points A(4, 3) and B(6, 5).