1)Solve;2x+1/6=1/2x
2)Solve for y in the equation;6y-4/3-2y-1/2=6-5y/6.
3)Solve the equation: 4 sin^2titre+4cos titre=5 for 0° <\=titre<\=360°.Give your answer in degrees
4)Solve the equation; 2cos 2 titre=1 for 0°<\=titre<\=360°.
5)(a)Find the inverse of the matrix;
[7 4]
[3 2]
6)Using matrix method,solve the simultaneous equations;
7x+4y=14
3x+2y=8
7)Determine the value of x for which the matrix below is a singular(has no inverse);[x 4]
[1 x-3]
8)Two similar solids have masses of 80kg and 270kg respectively. Find the surface area of the larger solid if the smaller solid has a surface area of 48cm^2
1) To solve the equation 2x + 1/6 = 1/2x, we can start by getting rid of the fractions by multiplying both sides of the equation by the least common denominator (LCD), which is 6.
6(2x + 1/6) = 6(1/2x)
12x + 1 = 3
Next, isolate the variable x by subtracting 1 from both sides:
12x = 3 - 1
12x = 2
Finally, divide both sides by 12 to solve for x:
x = 2/12
Simplified, x = 1/6
2) To solve for y in the equation 6y - 4/3 - 2y - 1/2 = 6 - 5y/6, we can first combine like terms on both sides of the equation.
6y - 2y - 5y/6 = 6 + 4/3 + 1/2
(6 - 2 - 5/6)y = 6(6) + 4(6)/3 + 1(6)/2
(4 - 5/6)y = 36 + 8/3 + 3/2
Next, simplify the right side of the equation:
(4 - 5/6)y = (18/3) + (16/3) + (9/6)
(4 - 5/6)y = (34/3) + (9/6)
(4 - 5/6)y = (68/6) + (9/6)
(4 - 5/6)y = (68 + 9)/6
(4 - 5/6)y = 77/6
To isolate y, multiply both sides of the equation by (6/(4 - 5/6)):
y = (77/6)(6/(4 - 5/6))
Simplified, y = 77/(4/6)
y = (77 * 6)/4
y = 462/4
Finally, simplify the fraction:
y = 231/2
3) To solve the equation 4sin^2(t) + 4cos(t) = 5, we can use the Pythagorean identity sin^2(t) + cos^2(t) = 1 to replace sin^2(t) with 1 - cos^2(t):
4(1 - cos^2(t)) + 4cos(t) = 5
Expand and simplify:
4 - 4cos^2(t) + 4cos(t) = 5
Combine like terms:
-4cos^2(t) + 4cos(t) = 5 - 4
-4cos^2(t) + 4cos(t) = 1
Rearrange to get a quadratic equation in terms of cos(t):
4cos^2(t) - 4cos(t) + 1 = 0
This equation doesn't factor easily, so we can use the quadratic formula to solve for cos(t):
cos(t) = (-b ± √(b^2 - 4ac))/2a
In this case, a = 4, b = -4, and c = 1. Plugging these values into the formula:
cos(t) = (-(-4) ± √((-4)^2 - 4(4)(1)))/(2(4))
cos(t) = (4 ± √(16 - 16))/8
cos(t) = (4 ± √0)/8
cos(t) = (4 ± 0)/8
cos(t) = 4/8
cos(t) = 1/2
Since cos(t) = 1/2, we can now find the corresponding angle values using inverse cosine:
t = cos^(-1)(1/2)
The inverse cosine of 1/2 is 60 degrees, so t = 60 degrees.
Therefore, the solution to the equation is t = 60 degrees.
4) To solve the equation 2cos(2t) = 1, we can first divide both sides by 2 to isolate the cosine term:
cos(2t) = 1/2
Next, we can use the double-angle identity for cosine:
cos(2t) = 2cos^2(t) - 1
This gives us the equation:
2cos^2(t) - 1 = 1/2
Rearrange to get a quadratic equation in terms of cos(t):
2cos^2(t) = 1 + 1/2
2cos^2(t) = 3/2
Divide both sides by 2:
cos^2(t) = 3/4
Take the square root of both sides:
cos(t) = ±√(3/4)
cos(t) = ±√3/2
Since cosine is positive in the first and fourth quadrants, we only consider the positive value:
cos(t) = √3/2
t = cos^(-1)(√3/2)
Using inverse cosine, we find:
t = 30 degrees or t = 330 degrees
Therefore, the solutions to the equation are t = 30 degrees and t = 330 degrees.
5) To find the inverse of the matrix [7 4] [3 2], we can use the following formula for a 2x2 matrix:
1 | 2 -4 |
--- | ------ |
det | -3 7 |
First, calculate the determinant of the original matrix:
det = (7 * 2) - (4 * 3) = 14 - 12 = 2
Next, swap the positions of the numbers in the matrix:
[2 -4]
[-3 7]
Finally, divide each number by the determinant:
[2/2 -4/2]
[-3/2 7/2]
Simplify the fractions:
[1 -2]
[-3/2 7/2]
Therefore, the inverse of the matrix [7 4] [3 2] is [1 -2] [-3/2 7/2].
6) To solve the simultaneous equations 7x + 4y = 14 and 3x + 2y = 8 using the matrix method, we can set up the following augmented matrix:
[7 4 | 14]
[3 2 | 8]
We can use row operations to manipulate the matrix and solve for x and y.
First, multiply the second row by -7/3:
[7 4 | 14]
[(-7/3)(3) (-7/3)(2) | (-7/3)(8)]
[7 4 | 14]
[-7 -14/3 | -56/3]
Add the second row to the first row:
[7 + (-7) 4 + (-14/3) | 14 + (-56/3)]
[0 -2/3 | -14/3]
Next, divide the second row by -2/3:
[0 (-2/3)(-3/2) | (-14/3)(-3/2)]
[0 1 | 7/2]
Therefore, we have the simplified augmented matrix:
[1 0 | -7/2]
[0 1 | 7/2]
This tells us that x = -7/2 and y = 7/2.
Therefore, the solution to the simultaneous equations is x = -7/2 and y = 7/2.
7) To determine the value of x for which the matrix [x 4] [1 x-3] is singular (has no inverse), we need to calculate the determinant of the matrix.
The determinant of a 2x2 matrix [a b] [c d] is given by ad - bc.
In this case, the matrix [x 4] [1 x-3] has a determinant of (x)(x - 3) - (4)(1) = x^2 - 3x - 4.
For the matrix to be singular (has no inverse), the determinant must be equal to 0.
So, we need to solve the equation x^2 - 3x - 4 = 0.
To factor this quadratic equation, we find two numbers that multiply to -4 and add to -3. These numbers are -4 and 1.
Therefore, we can factor the equation as (x - 4)(x + 1) = 0.
Setting each factor equal to 0:
x - 4 = 0 or x + 1 = 0
Solving for x:
x = 4 or x = -1
So, the two values of x for which the matrix is singular are x = 4 and x = -1.
8) If two similar solids have masses of 80kg and 270kg respectively, we can use the fact that the ratio of the volumes of similar solids is equal to the ratio of their masses.
Let the surface area of the smaller solid be A1 and the surface area of the larger solid be A2.
We are given that A1 = 48 cm^2.
Since the solids are similar, the ratio of their surface areas is equal to the square of the ratio of their lengths.
Therefore, we can set up the following proportion:
(A2/A1) = (L2/L1)^2
Let L1 be the length of the smaller solid and L2 be the length of the larger solid.
The ratio of their masses is given as 270 kg/80 kg = 27/8.
Since the ratio of their lengths is equal to the cube root of the ratio of their masses, we have:
(L2/L1) = (27/8)^(1/3)
To find A2, we can substitute the values:
(A2/48) = [(27/8)^(1/3)]^2
(A2/48) = [(27/8)^(2/3)]
To find A2, we can cross multiply and solve for A2:
A2 = 48 * [(27/8)^(2/3)]
A2 = 48 * [(27^2)/(8^2/3)]
A2 = 48 * [(27^2)/(2^6)]
A2 = 48 * (729/64)
Simplifying the fraction:
A2 = 54 * 729
A2 = 39,366 cm^2
Therefore, the surface area of the larger solid is 39,366 cm^2.