Algebra
Solve the equation for x.
log5x−log54=2+log53
x=300
Given a rectangle with a length of 10 units and a width of 6 units, find the length of the diagonal.
You can use the Pythagorean Theorem to find the length of the diagonal. The length of the diagonal is √(10^2 + 6^2), which is approximately 11.66 units.
What is the sine of a 45-degree angle?
The sine of a 45-degree angle is √2 / 2 or approximately 0.7071
A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective?
1/5
Let f'(x) = 6x2 + 2x - 1. Given that f(2) = 5, find f(x).
f(x) = 2x3 + x2 - x - 13
An arithmetic sequence is given by 3, 5, 7…
Given that un=253, find the value of n.
We have u1=3 and u2=5. Hence the value of d is 2
un = u1 + (n - 1)*d
n=126
Calculate the area of a regular hexagon with a side length of 6 units.
To find the area of a regular hexagon, you can divide it into equilateral triangles. The area is (6 x 6) x (sqrt(3)/4) x 6, which simplifies to 18√3 square units.
Given a right triangle with an angle of 30 degrees, a hypotenuse of 10 units, and you want to find the length of the opposite side. What is the length of the opposite side?
The length of the opposite side can be found using the sine function: opposite side = hypotenuse * sin(angle). So, it is 10 * sin(30 degrees), which is 5 units.
If three coins are tossed simultaneously, then the probability of getting at least two heads, is
1/2
Let f(x) = 2x2 - 5x + 7. The line L intersects at P(3, 10) and is perpendicular to the tangent to the curve at P. Find the equation of L in the form y = mx + c.
y = -1/7 x + 73/7
Consider the infinite geometric sequence 9000, −7200, 5760, −4608, ...
Find the exact sum of the infinite sequence.
Using the sum of an infinite geometric sequence formula S∞= u1 / (1-r), we get
S∞=9000/(1−(−4/5))
=5000
The length of a rectangle is 3 more inches than its width. The area of the rectangle is 40. What is the perimeter of the rectangle?
b2 + 3b = 40
b = 5
26
Calculate the exact value of tan(60 degrees) without using a calculator.
The tangent of 60 degrees is √3.
100 cards are numbered from 1 to 100. Find the probability of getting a prime number.
1/4
What is the derivative of (x + 1) sin x?
sin x + (x + 1) cos x
9x + 2⋅3x+1 = 1
Solve for x.
x = log3(−3+√10)
The parallelogram ABCD has a perimeter of 44 and an area of 28√2. The line AB is 3x+2, the line BC is 5x+4
Find angle ABC in degrees.
x = 2
8*14*sin(ABC) * 0.5 = 64
Solve the equation cos2x−sin^2x=cos^2x+3cosx
for 0≤x≤2π
x=2π/3,4π/3
A 2 by 2 square is divided into four 1 by 1 squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares.Two dice each with faces marked are thrown simultaneously. Find the probability of the sum of the numbers coming up 7.
If a green square cannot share its top or right side with a red square, then a red square can not share its bottom or left side with a green square. Let us split this up into several cases.
Case 1: There are no green squares. This can be done in 1 way.
Case 2: There is one green square and three red squares. This can only be done when the green square's top and right edges are against the edge, so there is 1 way.
Case 3: There are two green squares and two red squares. This happens when the two green squares are in the two top squares or two right squares, so there are 2 ways.
Case 4: There are three green squares and one red square. Similar to case 2, this happens when the red square's left and bottom edges are against the edge, so there is 1 way.
Case 5: There are four green squares and zero red squares. 1 way.
=6
A curve is given by the equation y=cos(2πsinx).
Find the coordinates of all the points on the curve for which dy/dx=0, 0≤x≤π.
(Hint: there are 5 points)
(0,1),(π/6,−1),(π/2,1),(5π/6,−1),(π,1)
Consider f(x) = logk(8x - 2x2), for 0 < x < 4, where k > 0.
The equation f(x) = 3 has exactly one solution. Find the value of k.
If we rewrite the equation f(x) = 3 in the form ax2+bx+c = 0, we have
logk (8x - 2x2) =3
0 = 2x2-8x+k3
Hence the quadratic function g(x) = 2x2 - 8x + k2 has two equal roots.
Therefore we get
b2-4ac=0
k3 = 8
k = 2
A circle is inscribed in a regular hexagon with a radius of 12 units. Find the area of the hexagon.
side = 8√3
Area of a hexagon = 3√3/2*(side)2
288√3
1/4 = log4√(2)
log4(3√2sinx)=log4(3-cos2x)
double angle formula,
3√2sinx = 3-(1-2sin2x)
sinx = (√2)/2
x = π/4 or -π/4, but since 0<x<90
x = π/4 (45 degrees)
There are 31 different flavors of ice cream at the shop, how many different two-scoop combinations are possible? Assume vanilla:strawberry is the same as strawberry:vanilla.
465
A particle moves in a straight line such that at time t seconds (t≥0), its velocity is given by v=18t^3*e^(−3t^2). Find the exact distance traveled by the particle in the first two seconds. (In terms of e)
1-13e^-12