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I have a problem with the following question: A small lottery consists of 200 tickets. There are two prizes. The first prize ticket is drawn and then, without replacement, the second prize ticket is drawn. If you buy two tickets, what are your chances of winning a) first prize b) both prizes...

That is only the first part of the following question: "Prove that cis(x) + cis (y) = 2 cos [(x-y)/2]cis[(x+y)/2]. Hence show that [(z+1)/(z-1)]5 = 1 has solutions of the form z = i cot [n(pi)/5] for n = 1, 2, 3, and 4. Honestly, this is an extremely hard qustion for me and I cannot even...

I have a problem with the following question: Two surveyors estimate the height of a nearby hill. One stands 5 m away from the other on horizontal ground holding a 3 m stick vertically. The other surveryor finds a "line of sight" to the top of the hill, and observes this line passes the...

I have difficulty working out the following question: Use the principle of mathematical induction to prove that: cos^2(x) + cos^2(2x) + cos^2(3x) + cos^2(4x) +.....+cos^2(nx) = (1/2)[n +(cos[(n + 1)x]sin(nx)]/sinx] for all positive integers n. My working is as follows: Proposition Pn is...

Annabel ant at A wishes to visit her friend Bertie beetle at B on the opposite corner of a block of cheese which is 30cm by 20cm by 20cm. What is the shortest distance Annabel must travel if she a. Loves to eat cheese b. Hates to eat cheese (It is not necessary, but you can see the diagram at...

Please see the attachment and have a look at the diagram for this question. This is a question to prove double angle formulae using the diagram of a semi-circle of radius 1 unit. As you can see, it says as follows: In the diagram, the semi-circle has radius 1 unit, and angle PAB = theta...

I have difficulty with the following question: The sequence of numbers {un} is defined by un=n*n!. 1) Let Sn= u1+u2+u3+...+un. Investigate Sn for several different values of n. 2) Hence conjecture an expression for Sn. 1) is easy and can be worked out as follows: S1 = 1*1! = 1 S2 = 1*1...

Although this is not a proper proof, I can explain it like this: Odd number/Odd number = Odd Even number/Odd number = Even So the numerator 2^n - (-1)^n has to be an odd number. Since 2^k is always an even number and (-1)^k gives always either +1 or -1, the result must be an odd number...

I am struggling with the following question: Prove the sum of (nCk)*(mCr-k) = (m+n)Cr? In other words, we are required to prove: (mC0)*(nCr) + (mC1)*(nCr-1) + (mC2)*(nCr-2) +....+ (mCr-1)*(nC1) + (mCr)*(nC0) = (m+n)Cr. I simply expanded it as follows: n!/r!(n-1)! + m*n!/(r-1)!(n-r+1)! +...

I am struggling with the following question, which is one of the questions in a book for one of the papers for Cambridge International AS and A Level Mathematics - "Mechanics 1" (Even though this is one of Mathematics papers, it is also pretty much related to Physics): A load of 7000 Newtons...

Please find the attached file and look at the diagram of a triangle. As you can see, we need to find the size of the angle marked "x" (i.e. angle DEF). You can identify the sizes of surrounding angles using your basic knowledge of geometry as follows: Angle ACD = 20 (Angle sum of triangle =...

The questions are as follows: On a shelf there are 4 different mathematics books and 8 different English books. i) The books are to be arranged so that the mathematics books are together. In how many different ways can this be done? Answer: 4!*9! = 8 709 120 I did not have any problem with...

The question that I have problem is: Peta deals with a hand of 10 cards from a well-shuffled pack of ordinary playing cards. Show that the probability that she deals exactly 5 spades is less than 5%. There are 52 cards all together, out of which 13 are spades so the probability of getting a...

I have a problem with the following question: Q: What is the set of values of p for which p(x^2+2) < 2x^2+6x+1 for all real values of x? My working is as follows: First, just expand the left-hand side of the inequality, px^2 + 2 p < 2x^2 + 6x + 1 Then, rearrange it to: (2 - p)x^2 + 6x + 1 -...

I have come across the following basic question about "Function": Determine whether or not this is a function: "x" is mapped to "the length of the line from the origin to (0, x) where x is Real. And the answer provided is as follows: Not a function, undefined when x < 0 I do not understand...

The question that I am struggling with is as follows: On the day of her birth, 1st January 1998, Mary's grandparents invested $x in a savings account. They continued to deposit $x on the first day of each month thereafter. The account paid a fixed rate of 0.4% interest per month. The interest...

I would appreciate if someone can help me with the question below: When Boris goes to school in winter, the probability that he wears a hat is 5/8. If he wears a hat, the probability that he wears a scarf is2/3. If he does not wear a hat, the probability that he wears a scarf is 1/6. If Boris...