1)Given the vectors a=3i-j+2k,b=4i+2j-k and p=2a-b.Express p in terms of i,hand k.
2)Hence calculate /p/ correct to 3 significant figures.
3)The point A,B and C lies on a straight line.The position vectors of A and C are 2i+3j+9k and 5i-3j+4k respectively;B divides AC internally in the ratio 2:1.Find;
a)Position vector of B
4)Given that the position vectors of point P is p=5a-2b where a=(3/2) and b=(4/1)Find ;
a)Column vector p
b)P' the image of P under translation vector (-4/-6).
5)A translation maps point A(6,4) onto A'(10,8);
a)Determine the translation vector
b)A point B' is the image of B(4,10) under the same translation.Find the length A'B' leaving your answer in surd form.
6)P and Q are two points such that OP=i+2j+3k,OQ=4i+5j-3k.M is the point that divides PQ externally in the ratio 3:2.Find the coordinate of M.
7)The position vector of point x,y and z are x=ti+2ji,y=2i+3j;z=3i+4j and respectively.If the points x,y and z are collinear, find the value of t.
8)A sector of a circle of radius 42 cm subtends an angle of 120° at the centre of the circle. The sector is folded into an inverted right cone.Calculate;
a)The base radius of the cone.
b)To one decimal place the vertical height of the cone.
9)A metal sphere has a radius of 5cn and density of 2.4g/cm^3;
a)Calculate the mass of the ball in kg.
b)The ball is dropped into a cylindrical container which is fully submerged and the cylindrical has a radius of 8 cn.Calculate the chanege in water level.
c)The sphere is melted down to form a metal cylinder of base radius of 5 cm.Calculate the height of the cylinder formed.
10)Simplify completely;
3x^2-5xy-2y^2/y^2-9x^2
11)A model of a solid structure in the shape of a cone with hemispherical top is represented. Where the diameter of the hemispherical part is 70cm and is equal to the diameter of the top of the frustum.The frustum has a base diameter of 28cm and a slant height of 60cm.Calculate,taking π=22/7.
a)The area of the hemispherical surface.
b)The surface area of the curved surface area.
c)The total surface area of the model
1) We have p = 2a - b.
Substituting the values of a and b, we get:
p = 2(3i - j + 2k) - (4i + 2j - k)
= 6i - 2j + 4k - 4i - 2j + k
= (6i - 4i) + (-2j - 2j) + (4k + k)
= 2i - 4j + 5k
Therefore, p = 2i - 4j + 5k.
2) To find the magnitude of p, we calculate the square root of the sum of the squares of its components:
|p| = sqrt((2)^2 + (-4)^2 + (5)^2)
= sqrt(4 + 16 + 25)
= sqrt(45)
Rounding to 3 significant figures, |p| ≈ 6.71.
3) Let B = x. Then, we have the following equation using the ratio of 2:1:
AB/BC = 2/1
Using the position vectors:
(AB = B - A and BC = C - B)
(B - A) / (C - B) = 2/1
Substituting the given values:
(2i + 3j + 9k - x) / (5i - 3j + 4k - (x)) = 2/1
Cross multiplying:
(2i + 3j + 9k - x) = 2(5i - 3j + 4k - x)
Simplifying:
2i + 3j + 9k - x = 10i - 6j + 8k - 2x
Collecting like terms:
2x - 8i + 9j +k = 0
Therefore, the position vector of B is 2x - 8i + 9j + k.
4) Using the given values, we can calculate p as follows:
a = (3/2) = (3/2)i + 0j + 0k
b = (4/1) = (4/1)i + 0j + 0k
p = 5a - 2b
= 5((3/2)i + 0j + 0k) - 2((4/1)i + 0j + 0k)
= (15/2)i + 0j + 0k - (8/1)i + 0j + 0k
= (15/2 - 8/1)i + 0j + 0k
= (15/2 - 16/2)i + 0j + 0k
= (-1/2)i + 0j + 0k
Therefore, the column vector p is (-1/2)i.
To calculate P' (the image of P under translation vector (-4/-6)), we subtract the translation vector from p:
P' = p - (-4i - 6j)
= (-1/2)i - (-4i) + 0j - (-6j)
= (-1/2 + 4)i - 6j
= (7/2)i - 6j
Therefore, the image of P is (7/2)i - 6j.
5) a) To determine the translation vector, we subtract the coordinates of the original point from the coordinates of the image point:
Translation vector = (10-6)i + (8-4)j
= 4i + 4j
Therefore, the translation vector is 4i + 4j.
b) To find the length A'B', we use the distance formula:
AB' = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((10-4)^2 + (8-10)^2)
= sqrt(6^2 + (-2)^2)
= sqrt(36 + 4)
= sqrt(40)
= 2sqrt(10)
Therefore, the length A'B' is 2sqrt(10).
6) Let M = (x, y, z). According to the given ratios, we have:
PM:MQ = 3:2
Using the position vectors:
PM/MQ = (M - P)/(Q - M) = 3/2
Substituting the given values:
[(x, y, z) - (1, 2, 3)] / [(4, 5, -3) - (x, y, z)] = 3/2
Simplifying:
[(x-1, y-2, z-3)] / [(4-x, 5-y, -3-z)] = 3/2
Cross multiplying:
2(x-1, y-2, z-3) = 3(4-x, 5-y, -3-z)
Expanding:
2x - 2 + 2y - 4 + 2z - 6 = 12 - 3x, 15 - 3y, -9 - 3z
Collecting like terms:
5x + 5y + 5z = 27
Therefore, the coordinates of M are (x, y, z) = (27/5, 27/5, 27/5).
7) Since the points x, y, and z are collinear, the cross product of their position vectors should be zero:
[x, y, z] = 0
Using the given values:
[ti + 2ji, 2i + 3j, 3i + 4j]
= (4ti + 8tj)i - (2i + 3j + 2tj + 3ti)j + (3i + 4j)i - (3i + 4j)j
Collecting like terms:
(4ti + 11tj)i - (5 + 5t)j = 0
Matching the coefficient of i and j:
4ti + 11tj = 0
-5 - 5t = 0
From the first equation, we get:
4t = 0
t = 0
Substituting t = 0 in the second equation:
-5 - 5(0) = 0
-5 = 0
However, -5 does not equal zero, so there is no value of t that satisfies the condition. Therefore, the points x, y, and z are not collinear.
8) a) The base radius of the cone is equal to the radius of the semicircle, which is half the circle. Therefore, the base radius of the cone is 42/2 = 21 cm.
b) The slant height of the cone is the same as the radius of the semicircle. Therefore, the slant height is 42 cm.
c) The vertical height of the cone can be found using the Pythagorean theorem:
vertical height = sqrt((slant height)^2 - (base radius)^2)
= sqrt(42^2 - 21^2)
= sqrt(1764 - 441)
= sqrt(1323)
≈ 36.39 cm
Therefore, the vertical height of the cone is approximately 36.39 cm.
9) a) The formula to calculate the mass of a sphere is density multiplied by volume:
Mass = Density * Volume
The formula for the volume of a sphere is:
Volume = (4/3) * π * radius^3
Substituting the given values:
Mass = 2.4g/cm^3 * (4/3) * π * (5cm)^3
= 2.4 * (4/3) * 22/7 * 5^3
= 3.2 * 22 * 125
= 8800g
Converting grams to kilograms:
Mass = 8800/1000 = 8.8kg
Therefore, the mass of the ball is 8.8kg.
b) To calculate the change in water level, we need to find the volume of the ball and then calculate the difference in volumes between the ball and the cylinder.
The formula for the volume of a cylinder is:
Volume = π * radius^2 * height
Substituting the given values:
Volume of the cylinder = π * (8cm)^2 * h
The volume of the sphere is the same as the volume of the cylinder.
Therefore, we have:
(4/3) * π * (5cm)^3 = π * (8cm)^2 * h
Simplifying:
h = (4/3) * (5cm)^3 / (8cm)^2
= 250/64
≈ 3.91 cm
Therefore, the change in water level is approximately 3.91 cm.
c) The radius of the cylinder is the same as the radius of the sphere, which is 5 cm.
The formula for the volume of a cylinder is:
Volume = π * radius^2 * height
Substituting the given values:
Volume = π * (5cm)^2 * h
Since we have found the value of h in part b), we can substitute it into the formula to calculate the height of the cylinder:
Volume = π * (5cm)^2 * 3.91 cm
= 20π * 3.91 cm
≈ 246.26 cm^3
Therefore, the height of the cylinder formed is approximately 246.26 cm.
10) To simplify the expression (3x^2 - 5xy - 2y^2) / (y^2 - 9x^2), we can factor numerator and denominator:
Numerator: 3x^2 - 5xy - 2y^2
= (3x + y)(x - 2y)
Denominator: y^2 - 9x^2
= (y + 3x)(y - 3x)
Therefore, the expression simplifies to:
(3x + y)(x - 2y) / (y + 3x)(y - 3x)
11) a) The area of the hemispherical surface is given by the formula:
Area = 2πr^2
Substituting the given values:
Area = 2π(35cm)^2
= 2π * 1225
= 2450π
≈ 7682.79 cm^2
Therefore, the area of the hemispherical surface is approximately 7682.79 cm^2.
b) The slant height of the frustum can be found using the Pythagorean theorem:
Slant height = sqrt((total height)^2 + (radius1 - radius2)^2)
= sqrt((60cm)^2 + (35cm - 14cm)^2)
= sqrt(3600cm^2 + 21cm^2)
≈ sqrt(3636cm^2)
≈ 60.3 cm
The surface area of the curved surface area of the frustum is given by the formula:
Surface Area = π(R1 + R2) * slant height
Substituting the given values:
Surface Area = π(35cm + 28cm) * 60.3cm
= π(63cm) * 60.3cm
= 189π * 60.3cm
≈ 35865.97 cm^2
Therefore, the surface area of the curved surface area is approximately 35865.97 cm^2.
c) The total surface area of the model is the sum of the areas of the hemispherical surface and the curved surface area of the frustum:
Total Surface Area = Area of hemispherical surface + Surface Area of frustum
= 7682.79 cm^2 + 35865.97 cm^2
≈ 43548.76 cm^2
Therefore, the total surface area of the model is approximately 43548.76 cm^2.