Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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-3
votes
1answer
37 views

Grade 11 question that requries using exponential decay fucntions. [on hold]

At $11 \text{pm}$ inspector Mortade was called to investigate a suspected murder on Brixton road. She measured the temperature of the victim and it was $32^\circ C$. She new normal body temperature is ...
0
votes
1answer
24 views

Geometric Progression of Air removed by an Air Pump

If one third of the air in the tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes Answer is 0.0877 I was thinking this was some sort of ...
3
votes
2answers
36 views

Sum of a Finite Sequence of Terms:$18, 25, 32, 39, … ,67$

Ok I know this question maybe too easy. What is the sum of a finite sequence of terms? $$18, 25, 32, 39, ... ,67$$ The answer is $340$. I use the formula: $${ S = \frac{n}{2} \times (a_1 + a_n) ...
0
votes
1answer
45 views

Doubt on a simple aptitude question

A mixture is composed of $8$ parts of brandy and $3$ parts of water. After adding $28$ litres of water, if the mixture contains brandy one half as much as water, then how many litres of brandy does it ...
0
votes
4answers
39 views

What is the equation of a 3D line which represents the intersection between two 3D planes?

The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. What is the equation ...
1
vote
1answer
31 views

Distance between two 3D lines

What is the distance between the 3D lines $x = \begin{pmatrix} 1 \\ 2 \\ -4 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix} t$ and $y = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix} + ...
-3
votes
1answer
30 views

Factoring trinomials. [on hold]

A student factored $m^2 + 12mn + 144n^2$ as shown. I know that since $m^2$ squares = $m^4$ and $144n^2$ squared = $12n$, the first and third terms of the trinomial are perfect squares. This means ...
2
votes
4answers
54 views

Roots of $f(x)=x-2+\frac{a-3}{x}$

I wanted to find the values of (a) for which the function $f(x)=x-2+\frac{a-3}{x}$ has more than one root. I know that the equation needs to be set equal to zero, from that step onward I have no idea ...
3
votes
1answer
37 views

I dont see how this algebraic manipulation is valid (trig functions)

So they have two equations: $v_{x}=V_0 \cos\theta-2\Omega V_o \sin\lambda \sin \theta *t$ $v_{y}=-V_0 \sin\theta -2 \Omega V_o \sin\lambda \cos \theta *t$ And they say "to lowest order in $\Omega$, ...
0
votes
2answers
89 views

How do you solve the equation $6g + 8 = 9g - 25$? [closed]

$6g+8=9g-25$ Can you simply solve for $g$? I'm having trouble with the steps.
1
vote
2answers
37 views

how to find log base 2 of decimal number without calculator

As with calculator things are simple but I don't know how to calculate log base 2 of decimal number without calculator. like $\log_2(0.25)$ etc.
3
votes
10answers
105 views

Point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and the $x$-axis

How can I algebraically (without looking at the graph) find the point of intersection of $f(x)=\sin(2x)+\cos(2x)$ and $x$-axis, in the interval $[0, \pi]$?
0
votes
4answers
34 views

Probability question with two conditions

New car registrationsplates contain two letters followed by two numerals followed by two more letters. Letters and numbers may be repeated. Which expression gives the number of car registration ...
0
votes
1answer
51 views

geometric mean of negative numbers is positive or negative?

What is the geometric mean of $-1$ and $-16$? Should it give $-4$ or $+4$?I think it should be $-4$ because a mean should always be greater than the least number and less than the greatest number. But ...
2
votes
2answers
85 views

Solve for $x : A^x- B ^x= C $ How to approach this kind of equation

Solve for $ x $in the below equation. I have an equation in the below format $ A^x- B^x= C $ How to approach this kind of equation and solve for $x$. I took log on both sides and it doesn't work out ...
0
votes
1answer
78 views

Factorizing a cubic polynomial

This is the result of determinant evaluation: $$p(x) = (x-3)((x-1)(x-2)-1)+1$$ How can I factor this polynomial?
-3
votes
2answers
67 views

Wolfram Alpha formula not working [closed]

I'm trying to solve the following formula for X in Wolfram but when I input it I get an error saying it does not understand "solve". $0 = ...
2
votes
4answers
86 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
0
votes
2answers
18 views

Domain of existence for this ODE.

I think this is some pre-calculus concept that I've forgotten. I am supposed to solve this initial value problem and determine how the interval in which the solution exists depends on $a$. $$yy' + x ...
1
vote
1answer
46 views

How do I find this variable?

$$\ln\left(\frac{mg-bv}{mg}\right)=-\frac{bt}{m}$$ Okay, so I have to find $v$, but I have no idea how to go about it because there is an $\ln$ in it. (Also, I'm not sure on how I am supposed to ...
1
vote
2answers
29 views

Work Problem: Painting a Home

A and B working together can finish painting a home in six days. A working alone can finish it in 5 days less than B. How long will it take each of them to finish the work alone? The answer is $B = ...
0
votes
1answer
31 views

Converging Geometric Series of a a Bouncing Ball

A rubber ball was dropped from a height of 36m. and each time its strikes the ground it rebounds to a height of 2/3 from which it last fell. Find the total distance traveled by the ball before it ...
2
votes
1answer
35 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...
0
votes
0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
1
vote
1answer
92 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
1
vote
2answers
25 views

Methods for verifying correct factorisation of polynomials

In an attempt to factor using a GCF, Mia wrote $8x^2 + 4x = 4x(2x – 0)$, which is not correct. a. Explain how Mia could check her work. b. What error did Mia make? She didn't factor using the GFC ...
3
votes
2answers
58 views

changing the power of 2 to the power of 3

this is a really simple question, I'm solving a time complexity program, to find the order of the program, however when it gets down to simplifying the mathematical part, I get stuck. I want to get ...
0
votes
0answers
16 views

End-point as point of tangency

Can an end-point be a point of tangency? For example in the function $f(x)=8x^{3/2}$ can the point $(0,0)$ be a tangent point?
3
votes
3answers
71 views

The number of positive integral solutions to the system of equations.

The number of positive integral solutions to the system of equations $$\begin{align} & a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=47\\ &a_{1}+a_{2}=37,\ \ \{a_{1},a_{2},a_{3},a_{4},a_{5}\} \in ...
3
votes
2answers
55 views

Number of real solutions

Prove that the equation $\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 4x\rfloor+\lfloor 8x\rfloor+\lfloor 16x\rfloor+\lfloor 32x\rfloor = 12345$ does not have any real solution. ($\lfloor x\rfloor$ ...
0
votes
1answer
19 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
6
votes
5answers
79 views

Solve for $x$ : $\log_3(3x + 2) = \log_9(4x + 5)$

Solve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ...
2
votes
3answers
73 views

Is there no such identity as $\csc^2+\sec^2=1$?

$$\csc^2+\sec^2=1?$$ I thought I could just use reciprocal from the other formula $\sin^2+\cos^2=1$, can you explain what's wrong?
1
vote
2answers
20 views

To find the remainders

A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. When it is divided by 5 and 4, then respective remainders are : $1,2$ $2,3$ $3,2$ $4,1 $ I thought of ...
0
votes
0answers
33 views

f(x,y) - Some clarifications needed.

so I have run into this problem: http://i.imgur.com/tZODxYV.png I was unable to solve it and when looking over the answer some of it seemed unclear to me, could someone please clear it up for me? ...
2
votes
2answers
33 views

How to find the sum of the three digits of a number $N$ that gives the same remainder when $2272$ and $875$ are divided by it

On dividing $2272$ as well as $875$ by three digit number $N$, we get same remainder. What is the sum of the digits of $N$? I cannot start hit and trial method here, so how should I do this? ...
0
votes
1answer
13 views

Need to check an answer

At a fast food restaurant, a milk shake costs $r$. A chicken sandwich costs 3 times as much as the shake. A large order of French fries costs $\$3$. If $r-2$, how much do 3 chicken sandwiches and 2 ...
1
vote
4answers
59 views

Find all $2 \times 2$ matrices $A$ and $B$ such that $AB = BA$

Find all possible $2 \times 2$ matrices A that for any $2 \times 2$ matrix B, AB = BA. Hint: AB = BA must hold for all B. Try matrices B that have lots of zero entries. I'm clueless as to how to ...
0
votes
3answers
32 views

What is the solution to this system?

Capital letters indicate constants and lowercase letters indicate variables. I am interested in solving for $\{a,b,c,d,e,f\}.$ How would I go about doing this by hand / what is the solution? $$ ...
0
votes
1answer
32 views

Find all real solutions to the following system of equations (involving fixed point iteration)

From the 1996 Canada National Olympiad. I have emphasised the real point of the question. Find all real solutions to the following system of equations. Carefully justify your answer. ...
1
vote
5answers
57 views

How to divide certain polynomials?

Can somebody help me with this question? $$\frac{15p^3+16p^2+46}{3p+5}$$ For some reason I can't wrap my head around the process used to divide polynomials, I can do long division but every time ...
1
vote
4answers
135 views

Is $|z-i| = |z+i|$?

I computed a Mobius transformation $-\frac{z-i}{z+i}$ that maps the upper half plane to a disk, with i mapping to the center of the disk, $w = 0$. How do I know that the disk is a unit disk and not ...
0
votes
1answer
21 views

Equality of polynomials and their equivalent fraction forms

A polynomial of the form $\frac{p(t)\cdot (x+1)}{(x+1)}$ is obviously equal to $p(t)$ because the binomials cancel out, where $p(t)$ is just any arbitrary polynomial. But the first form is undefined ...
4
votes
2answers
68 views

Intersection of semicircle and parabola (Omar Khayyam)

(Source: The History of Mathematics 7th Edition, David Burton) I can't even get through question a.. Could someone give a hint? The only thing I can think of is the Pythagorean Theorem, but it ...
0
votes
1answer
21 views

How to find the resultant vector in a word problem

If the wind blows at 20 mph due West and an airplane heads South at 400 mph, what is the resultant speed and direction of the plane?
1
vote
7answers
133 views

How do I solve the equation $e^{\ln(2x+1)} = 5x$?

The problem is $$e^{\ln(2x+1)} =5x$$ I've tried using natural logs to both sides like.. $2x+1= \ln 5x $ But I'm not sure if $\ln$ and $e^{\ln}$ cancel out.
1
vote
2answers
39 views

$x^2 + (k-3)x + k = 0$, ranges of k for roots to be of same sign

I need some help on the following. The quadratic that I am dealing with is $x^2 + (k-3)x + k = 0$, and I need to find ranges of values of $k$, for which the roots will have the same sign. For the ...
2
votes
3answers
47 views

Is this the correct period?

What is the period for the following: $$ y = 10 \sin\Bigl(\frac{2\pi}{365}(x-50)\Bigr) $$ Is the period $$ \frac{2\pi}{\frac{2\pi}{365}} $$ which would be $365$?
0
votes
1answer
51 views

How do I use the bowtie method to multiply $(2x-27)(-x+15)$?

The bowtie method seems like an easy concept to have down, but how is it used to multiply binomials such as $(2x-27)(-x+15)$? Calculating the answer is not the problem, because I can get ...
1
vote
2answers
64 views

How many such polynomial exist?

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. I got: $$P(x) = ax^2 + bx + c \implies P(0) = c = 2010$$ Let $P(r_1, r_2) ...