Course Code
MTH103
Question
_____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
Answer
A parabola
Question
The equation of a parabola with vertical axis is given by
Answer
\(x^{2}=4py\)
Question
What is the parametric equation of a circle with centre\( (2, -1)\) and radius 3
Answer
\(y = -1 + 3 sin\theta\)
Question
Find the center and radius of the circle \(x^{2} + y^{2} + 8x + 6y = 0\)
Answer
\((-4 ,-3), r = 5\)
Question
Find the equation of the line through the point \((-1, 2) \)which is parallel to y = 2x + 1
Answer
y = 3x + 5
Question
What is the angle between the two lines\( -3x+4y =8\) and \(-2x - 8y - 14 = 0\)?
Answer
\(-14.63 degree\)
Question
Given that x + 2y + 3z =1, 3x + 2y + z = 4, x + 3y + 2z = 0. What is x,y and z?
Answer
(7/4,-3/4,1/4)
Question
if \(a=a_{1}i+a_{2}j+a_{3}k\) and \(b=b_{1}i+b_{2}j+b_{3}k\), then ab
Answer
\(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\)
Question
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
Answer
ai + bj
Question
What is the distance between the points x and y, given that x = 3 + 2i and y = 4 + 3i?
Answer
square root of 2
Question
Let A, B, C be the points \((2, 3), (3, -2)\) and \((-1, 4)\). The length of AB yields_____
Answer
square root of 26
Question
What is the gradient of the straight line and the angle of inclination of the points, A(2, -3) and B(4, 5)?
Answer
gradient=1, angle = 45 degree
Question
The coordinate of the centre and the radius of the circle \(y^{2}+x^{2}-14x-8y+56=0\) will be_______
Answer
(7, 4) are the coordinates or centre of the circle, while 3 is the radius
Question
____ is defined as the ratio of the vertical distance the line rises or falls to the horizontal distance.
Answer
Gradient of a line
Question
If A is (3, 6) and B is (4, 8). Find the coordinate of the midpoint of AB.
Answer
\( [3\frac{1}{2},7]\)
Question
The coordinates of the midpoint of a line segment is
Answer
the average of the coordinates of the endpoints
Question
In Cartesian Coordinates, the coordinates are given by the position
Answer
of a point in a plane
Question
The Locus of points at a distance of 3 from the point (0, 0) is given by the equation…
Answer
\(x^{2}+y^{2}=9\)
Question
If a = 2i + 4j + 3k and b = i + 5j - 2k. Find the vector product of a and b
Answer
\(-23i + 7j + 6k \)
Question
if a=3i+4j+5k and b=2i+j+7k. Find a.b
Answer
45
Question
If i.i = 1, then i.j = …
Answer
\(-1\)
Question
if \(a=a_{1}i+a_{2}j+a_{3}k\) and \(b=b_{1}i+b_{2}j+b_{3}k\) find a.b
Answer
\(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\)
Question
When two vectors are at right angles to each other the dot product is ?
Answer
Zero
Question
If a=10, b=13 and the angle between a and b is 59.5 degree, what is the scalar product of a and b
Answer
65.98
Question
Find l,m,n of the vector \(r = 2i + 4j - 3k\)
Answer
\(2\sqrt{29}, 4\sqrt{29}, -3\sqrt{29}\)
Question
____ is the component contributions of the basis to the unit vector
Answer
Direction Cosines
Question
____ in analytic geometry of vector are the cosines of angles between the vector and the three coordinate axes
Answer
Direction Cosines
Question
___ is a vector as a way of determining the magnitude and direction in its coordinate of x and y axis
Answer
Component of a vector in terms of unit vector
Question
In triangle ABC, the points L, M, N are the midpoints of the sides AB, BC and CA respectively. Which these expressions is true?
Answer
\(\overline{2AB}+\overline{3BC}+\overline{AC}=2\overline{LC}\)
Question
ABCD is a square (quadrilateral) with T and U the mid points of SP and QR respectively. Which expressions is true?
Answer
PQ+RS =2TU
Question
_____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
Answer
A parabola
Question
The equation of a parabola with vertical axis is given by
Answer
\(x^{2}=4py\)
Question
What is the parametric equation of a circle with centre\( (2, -1)\) and radius 3
Answer
\(y = -1 + 3 sin\theta\)
Question
Find the center and radius of the circle \(x^{2} + y^{2} + 8x + 6y = 0\)
Answer
\((-4 ,-3), r = 5\)
Question
Find the equation of the line through the point \((-1, 2) \)which is parallel to y = 2x + 1
Answer
y = 3x + 5
Question
What is the angle between the two lines\( -3x+4y =8\) and \(-2x - 8y - 14 = 0\)?
Answer
\(-14.63 degree\)
Question
Given that x + 2y + 3z =1, 3x + 2y + z = 4, x + 3y + 2z = 0. What is x,y and z?
Answer
(7/4,-3/4,1/4)
Question
if \(a=a_{1}i+a_{2}j+a_{3}k\) and \(b=b_{1}i+b_{2}j+b_{3}k\), then ab
Answer
\(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\)
Question
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
Answer
ai + bj
Question
What is the distance between the points x and y, given that x = 3 + 2i and y = 4 + 3i?
Answer
square root of 2
Question
Let A, B, C be the points \((2, 3), (3, -2)\) and \((-1, 4)\). The length of AB yields_____
Answer
square root of 26
Question
What is the gradient of the straight line and the angle of inclination of the points, A(2, -3) and B(4, 5)?
Answer
gradient=1, angle = 45 degree
Question
The coordinate of the centre and the radius of the circle \(y^{2}+x^{2}-14x-8y+56=0\) will be_______
Answer
(7, 4) are the coordinates or centre of the circle, while 3 is the radius
Question
____ is defined as the ratio of the vertical distance the line rises or falls to the horizontal distance.
Answer
Gradient of a line
Question
If A is (3, 6) and B is (4, 8). Find the coordinate of the midpoint of AB.
Answer
\( [3\frac{1}{2},7]\)
Question
The coordinates of the midpoint of a line segment is
Answer
the average of the coordinates of the endpoints
Question
In Cartesian Coordinates, the coordinates are given by the position
Answer
of a point in a plane
Question
The Locus of points at a distance of 3 from the point (0, 0) is given by the equation…
Answer
\(x^{2}+y^{2}=9\)
Question
If a = 2i + 4j + 3k and b = i + 5j - 2k. Find the vector product of a and b
Answer
\(-23i + 7j + 6k \)
Question
if a=3i+4j+5k and b=2i+j+7k. Find a.b
Answer
45
Question
If i.i = 1, then i.j = …
Answer
\(-1\)
Question
if \(a=a_{1}i+a_{2}j+a_{3}k\) and \(b=b_{1}i+b_{2}j+b_{3}k\) find a.b
Answer
\(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\)
Question
When two vectors are at right angles to each other the dot product is ?
Answer
Zero
Question
If a=10, b=13 and the angle between a and b is 59.5 degree, what is the scalar product of a and b
Answer
65.98
Question
Find l,m,n of the vector \(r = 2i + 4j - 3k\)
Answer
\(2\sqrt{29}, 4\sqrt{29}, -3\sqrt{29}\)
Question
____ is the component contributions of the basis to the unit vector
Answer
Direction Cosines
Question
____ in analytic geometry of vector are the cosines of angles between the vector and the three coordinate axes
Answer
Direction Cosines
Question
___ is a vector as a way of determining the magnitude and direction in its coordinate of x and y axis
Answer
Component of a vector in terms of unit vector
Question
In triangle ABC, the points L, M, N are the midpoints of the sides AB, BC and CA respectively. Which these expressions is true?
Answer
\(\overline{2AB}+\overline{3BC}+\overline{AC}=2\overline{LC}\)
Question
ABCD is a square (quadrilateral) with T and U the mid points of SP and QR respectively. Which expressions is true?
Answer
PQ+RS =2TU
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