Algebra
Find a, b and c so that the graph of the quadratic function f(x) = ax2 + bx + c has a vertex at (-2 , 1) and passes through the point (0 , -3)
f(x) = a(x + 2)2 + 1 : equation of parabola in vertex form
f(0) = -3 = 4a + 1
a = -1 : solve for a
f(x) = -(x + 2)2 + 1 = -x2 - 4x - 3
a = -1 , b = -4 and c = -3
19-18=18
Draw one line to make the equation true.
19-10/10=18
Find all rational zeros of P(x) = x3 - 7x + 6
x = 1, x = 2 and x = -3
Point E is between points G and T.
GE = 2x, ET = 3x-1, and GT = 14.
What is the value of x?
3
Find this equation's derivative:
4x3 + 7x2 + 2x + 3
12x2+14x+2
If x2 - y2 = -12 and x + y = 6, find x and y.
x2 - y2 = (x - y)(x + y) = -12 : given
6(x - y) = -12 : substitute x + y by 6
(x - y) = -2 : solve for x - y
(x - y) = -2 and x + y = 6 : 2 by 2 system.
x = 2 , y = 4
3▢ x 5▢ = 6▢
what is ▢?
!
What is the last digit of 20192019?
Since we only care about the ones digit, it suffices to see what happens to 9 when we take subsequent powers. We observe that 91ends in a 9, 92ends in a 1, 93ends in a 9, 9 4ends in a 1, and so on. Since the power 2019 is odd, 20192019 ends in a 9.
At a certain time of day, a 25-foot telephone pole casts a 10-foot shadow. At that same time, how high would a tree have to be in order to cast a 25-food shadow?
62.5 feet. Proportional reasoning with similar triangles.
Find this equation's integral:
5x2-8x+5
5x3/3-4x2+5x+C
Andy rides his bike from his house to the store at 6 miles per hour. Upon arriving at the store, he realizes it’s closed and immediately heads back, riding his bike back home at a speed of 2 miles per hour. What is Andy’s average speed in miles per hour over the entire trip?
Let d be the distance from the house to the store. Let t1 be the amount of time it took to go from the house to the store and let t2 be the amount of time it took to go from the store to the house. Using d = vt, we get the equations:
d = 6t1; d = 2t2
which together, implies that t2 = 3t1.
Since average speed is total distance divided by total time, we get:
average speed = 2d t1 + t2 = 2(6t1) t1 + 3t1 = 12t1 4t1 = 3 mph
MON=3
TUE=5
WED=4
THU=?
6
Order from greatest to least
a) 25100
b) 2300
c) 3400
d) 4200
e) 2600
25100
2300 = (23)100 = 8100
3400 = (34)100 = 81100
4200 = (42)100 = 16100
2600 = (26)100 = 64100
From greatest to least: c, e, a, d, b
What is the area of a circle in which a 3 by 4 rectangle is inscribed?
25pi. The length of a diagonal of a 6 by 8 rectangle is 10 (pythagorean theorem). The diagonal of the rectangle is the diameter of the circle inscribed, and A = pi*r^2
Given f(x)=2x2-x+1, find:
the gradient of the tangent to the curve at x=1:
3
Let n ≥ 2 be a natural number (i.e. n = 2, 3, 4, ...). For how many n is logn(n + 1) a rational number?
Suppose, by contradiction, that logn (n + 1) = p q for some integers p, q. with q 6= 0. Since n + 1 > n, we have logn (n + 1) > 1 and thus, p/q > 1, and p, q > 0. Then, we have:
np/q= n + 1
n p = (n + 1)q
However, n and n + 1 have opposite parity (i.e. one is even and one is odd), and for any positive integers p and q, n p has the same parity as n and (n + 1)q has the same parity as (n + 1), so n p =/= (n + 1)qand we have a contradiction.
Thus, logn(n + 1) is not rational and the answer is 0.
115 15 16 19 x
What is x?
151
Solve for x the equation log9(x3) = log2(8)
log9(x3) = log2(8) : given
log2(23) = 3 : simplify right hand side of given equation.
log9(x3) = 3 : rewrite the above equation
log9(x3) = log9(93) : rewite 3 as a log base 9.
x3 = 93
x = 9
What is the number of sides of a polygon in which the sum of the degree measures of the interior angles is 4 times the sum of the degree measures of the exterior angles?
Sum of the degree measures of exterior angles is 360. S = (n-2)180 = 4 x 360, where n=number of sides. Solving for (n-2) = 8
n=10.
A rectangular plot of land is bounded on one side by a wall. Determine the largest area that can be enclosed using the wall and 800m of fencing
2y+x=800-> x=800-2y
xy=A -> (800-2y)y=A -> 800y-2y2=A
800-4y=0 -> y=200 and x=400
200*400 = 80000m2
Working alone, Bill takes three times the amount of time to paint a fence that Mike takes. If Bill and Mike work together, they can paint the fence in 2 hours. If Mike works at the same rate as when he worked with Bill, how long would it take Mike working alone to paint the same fence?
2 hours and 40 minutes. If x represents the number of hours it takes Mike working alone to paint the fence, then 3x is the number of hours it takes Bill working alone to paint the fence. So 2(1/x) + 2(1/3x) = 1 gives x=8/3 hours.
A B
x C
______
D E
+ F G
______
H I
Substitute numbers from 1 to 9 into each alphabet to complete the equation
1, 7, 4, 6, 8, 2, 5, 9, 3
How many zeroes in a row occur at the end of the number 100!
Each factor of 10 = 2 · 5 adds a zero, so we need to count the number of 2’s and 5’s in 100!. There are considerably more 2’s, so we need to determine how many 5’s appear as factors in the numbers from 1 to 100. There are 20 numbers that have at least one factor of 5 (5, 10, 15, ..., 100) and 4 numbers that have an additional factor of 5 (25, 50, 75, 100), so there are 24 5’s in 100! and thus, 24 zeroes.
Four circles of radius one are centered at the points (1, 1),(1, −1),(−1, −1), and (1, −1). A fifth circle is drawn centered at the origin such that it is tangent to the other four circles. What is the radius of the fifth circle?
1 + √ 2
A 25m long industrial ladder is leaning against a wall on a building site. The bottom of the ladder starts to slip along the floor at a rate of 0.2m/s. Determine how fast the top of the ladder is moving down the wall at the instant when it is 20m above the floor.
x2+y2=625
2x(dx/dt)+2y(dy/dt)=0
y=20m and dx/dt=0.2 -> given
x2+y2=252 -> x= 15
2(15)(0.2)+2(20)(dy/dt)=0
Thus, dy/dt=-0.15m/s