Mathematics: Post your doubts here!

SubCifer

SubCifer

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A level doubt

Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.
(i) Find the probability that at least 2 of the 5 integers are less than or equal to 4. [3]
Robert now generates n random integers between 1 and 9 inclusive. The random variable X is the
number of these nintegers which are less than or equal to a certain integer k between 1 and 9 inclusive.
It is given that the mean of X is 96 and the variance of X is 32.
(ii) Find the values of n and k.

i cant seem to find k
n=144
p=2/3
 
Thought blocker

Thought blocker

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RadzMau said:
http://papers.xtremepapers.com/CIE/Cambridge International A and AS Level/Mathematics (9709)/9709_s12_qp_11.pdf

June 2012 Paper 11 question 8 :)
Click to expand...
Sorry for the delay, but first I lost practice in P1 so was revising and then I got it, net got disconnect. But, Here you go :¬
8 :
i)
We are provided with this equation : f(x) = x² - 4x + k and now we have to make it in perfect square format.
So, now the equation is in the format of ax² + bx + c where a = 1, b = -4 and c = k. So now going to convert in perfect square format.
Step i)
Always half the coefficient of b and and change x² to x and square the whole bracket, that is in this case, (x - 2)².
step ii)
Opening (x - 2)² you get x² - 4x + 4 yes ? and we know that x² - 4x + k is given to us. So we want to avoid +4 in the equation x² - 4x + 4, so we need to subtract 4 from the equation so that it becomes x² - 4x + 0 yes ?
step iii)
So now, closing the bracket we know it is (x - 2)² that is (x + a)² we got a = -2 and b is -4 coz we need to remove that +4 from equation so final equation would be (x - 2)² -4 + k.

ii)
Now we know the equation (x - 2)² -4 + k yes ? There is a rule, if the equation is perfect square i.e in form of (x + a)² + b, b is the range. so range will be -4 + k.
So the final answer will be f(x) > -4 + k. Why there is this sign ">" the reason is range is all values of y that graph covers, since its a curve it should be > not ≥ .

iii)
Here we need to find value of x, and for that value of x is simply that you equate whatever there is in the bracket to be zero. so here x will be 2 as 2 - 2 is zero.

iv)
Here I will make you understand step by step.
step i)
Change x to y and y to x, that is we know that (x - 2)² - 4 + k = y so now change x to y and y to x so that it is now (y - 2)² - 4 + k = x.
Step ii)
Make y as subject of formula, like this :
(y - 2)² = x + 4 - k
y - 2 = √(x + 4 - k)
y = √( x + 4 - k) + 2
step iii)
Change y to f⁻¹(x) , that is the final answer :
f⁻¹(x) = 2 + √( x + 4 - k).
 
RadzMau

RadzMau

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Thought blocker said:
Sorry for the delay, but first I lost practice in P1 so was revising and then I got it, net got disconnect. But, Here you go :¬
8 :
i)
We are provided with this equation : f(x) = x² - 4x + k and now we have to make it in perfect square format.
So, now the equation is in the format of ax² + bx + c where a = 1, b = -4 and c = k. So now going to convert in perfect square format.
Step i)
Always half the coefficient of b and and change x² to x and square the whole bracket, that is in this case, (x - 2)².
step ii)
Opening (x - 2)² you get x² - 4x + 4 yes ? and we know that x² - 4x + k is given to us. So we want to avoid +4 in the equation x² - 4x + 4, so we need to subtract 4 from the equation so that it becomes x² - 4x + 0 yes ?
step iii)
So now, closing the bracket we know it is (x - 2)² that is (x + a)² we got a = -2 and b is -4 coz we need to remove that +4 from equation so final equation would be (x - 2)² -4 + k.

ii)
Now we know the equation (x - 2)² -4 + k yes ? There is a rule, if the equation is perfect square i.e in form of (x + a)² + b, b is the range. so range will be -4 + k.
So the final answer will be f(x) > -4 + k. Why there is this sign ">" the reason is range is all values of y that graph covers, since its a curve it should be > not ≥ .

iii)
Here we need to find value of x, and for that value of x is simply that you equate whatever there is in the bracket to be zero. so here x will be 2 as 2 - 2 is zero.

iv)
Here I will make you understand step by step.
step i)
Change x to y and y to x, that is we know that (x - 2)² - 4 + k = y so now change x to y and y to x so that it is now (y - 2)² - 4 + k = x.
Step ii)
Make y as subject of formula, like this :
(y - 2)² = x + 4 - k
y - 2 = √(x + 4 - k)
y = √( x + 4 - k) + 2
step iii)
Change y to f⁻¹(x) , that is the final answer :
f⁻¹(x) = 2 + √( x + 4 - k).
Click to expand...

Thanks a lot for your help! :) (y) very well explained, I understood!
God Bless!
 
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