Prelim 2016 Maths Help Thread (1 Viewer)

eyeseeyou · Sep 14, 2016

How do I do this question

x^2+qx+r=0 is three times the other root. Prove 3q^2=16r

I'm kinda confused when doing the working to the question

InteGrand · Sep 14, 2016

eyeseeyou said:
How do I do this question

x^2+qx+r=0 is three times the other root. Prove 3q^2=16r

I'm kinda confused when doing the working to the question

$\noindent Let the roots be $a$ and $3a$. By sum of roots, $4a = -q$. By product of roots, $3a^2 = r$. So from these equations, $q^2 = 16a^2 = \frac{16}{3}r\Rightarrow 3q^2 = 16 r$.$

eyeseeyou · Sep 14, 2016

InteGrand said:
$\noindent Let the roots be $a$ and $3a$. By sum of roots, $4a = -q$. By product of roots, $3a^2 = r$. So from the previous equation, $q^2 = 16a^2 = \frac{16}{3}r\Rightarrow 3q^2 = 16 r$.$

For what values of x is 3x^2+14x-5<0

Not sure what to do

eyeseeyou · Sep 14, 2016

Given that x=2 is a root of the equation x^2+x+m=0 find the other root and the value of m

FInd the value of k for the equation x^2-(k+2)x+4k=0 if
a. the sum of the roots is 5
b. The product of the roots is 12
c. One root is 2 more than the other root

InteGrand · Sep 14, 2016

eyeseeyou said:
For what values of x is 3x^2+14x-5<0

Not sure what to do

$\noindent This is a standard sort of question. The quadratic factors as $(3x-1)(x+5)$, so the roots are $-5$ and $\frac{1}{3}$. Therefore (sketch the graph, noting the parabola is upwards-facing), the quadratic is negative precisely when $-5< x < \frac{1}{3}$.$

InteGrand · Sep 14, 2016

eyeseeyou said:
Given that x=2 is a root of the equation x^2+x+m=0 find the other root and the value of m.

$\noindent Since $2$ is a root, we have $4+2+m = 0 \Rightarrow m = -6$. Also, using sum of roots, the other root $a$ satisfies $a+2 = -1$, so the other root is $a = -3$.$

eyeseeyou · Sep 14, 2016

For the parabola y=x^2-(p+q)x+pq (where p>q) find the values of x for which y is greater or equal to 0

InteGrand · Sep 14, 2016

eyeseeyou said:
FInd the value of k for the equation x^2-(k+2)x+4k=0 if
a. the sum of the roots is 5
b. The product of the roots is 12
c. One root is 2 more than the other root

a) The sum of roots is 5, so k+2 = 5, whence k = 3.

b) The product of roots is 12, so 4k = 12, which implies k = 3.

c) Let the roots be a and a+2, then use sum and product of roots to obtain two equations in two unknowns.

eyeseeyou · Sep 14, 2016

InteGrand said:
$\noindent Since $2$ is a root, we have $4+2+m = 0 \Rightarrow m = -6$. Also, using sum of roots, the other root $a$ satisfies $a+2 = -1$, so the other root is $a = -3$.$

Sum of the roots is alpha+beta, right?

InteGrand · Sep 14, 2016

eyeseeyou said:
Sum of the roots is alpha+beta, right?

Yeah.

InteGrand · Sep 14, 2016

eyeseeyou said:
For the parabola y=x^2-(p+q)x+pq (where p>q) find the values of x for which y is greater or equal to 0

$\noindent The roots are $p$ and $q$. So (noting the parabola is upwards-facing), the answer is $x \leq q$ or $x \geq p$.$

eyeseeyou · Sep 14, 2016

I don't get this Q

What is the relation btwn a and b if the graph of y=x^2+ax+b:

a) touches the x axis
b) cuts the x-axis at 2 points

eyeseeyou · Sep 14, 2016

InteGrand said:
$\noindent The roots are $p$ and $q$. So (noting the parabola is upwards-facing), the answer is $x \leq q$ or $x \geq p$.$

Huh I still don't get it

InteGrand · Sep 14, 2016

eyeseeyou said:
I don't get this Q

What is the relation btwn a and b if the graph of y=x^2+ax+b:

a) touches the x axis
b) cuts the x-axis at 2 points

$\noindent a) Discriminant is $0$, i.e. $a^2 -4b =0 \Longleftrightarrow a = \pm 2b$.$

$\noindent b) Discriminant is positive, i.e. $a^2 > 4b$.$

eyeseeyou · Sep 14, 2016

If it says show y>0 do you use delta=b^2-4ac?

InteGrand · Sep 14, 2016

eyeseeyou said:
Huh I still don't get it

$\noindent The roots are $p$ and $q$, and $q < p$, so when we sketch the parabola, because it's upwards-facing, it'll be above the $x$-axis to the left of the lower root (when $x < q$) and above the $x$-axis when we're to the right of the higher root ($x > p$). (Just sketch the parabola to see this.) We use $\leq$ and $\geq$ though due to what the question asked.$

InteGrand · Sep 14, 2016

eyeseeyou said:
If it says show y>0 do you use delta=b^2-4ac?

$\noindent Yeah you can do that (show the discriminant is negative), after noting the parabola is upwards-facing.$

eyeseeyou · Sep 14, 2016

Show that the roots of a equation (3m-5)x^2-3m^2x+5m^2=0 for m is in not equal to 3 are rational if m is rational

SHow that the roots of the equarion ax^2-(a+b)x+b=0 for a is not equal to 0 are rational for all rational values of a and b

For what values of m does the quadratic equation x^2+mx+(m+1)^2=0 have 2 roots

InteGrand · Sep 14, 2016

eyeseeyou said:
Show that the roots of a equation (3m-5)x^2-3m^2x+5m^2=0 for m is in not equal to 3 are rational if m is rational

SHow that the roots of the equarion ax^2-(a+b)x+b=0 for a is not equal to 0 are rational for all rational values of a and b

For what values of m does the quadratic equation x^2+mx+(m+1)^2=0 have 2 roots

For the first two, show that the discriminant is a perfect square.

For the last one, if we want distinct roots, we should set the discriminant to be positive and solve for m.

eyeseeyou · Sep 14, 2016

Without actually solving the simultaneous equations, state whether the following pairs of lines intersect are parrallel or coincide

a) x+y-7=0 and x+y-8=0

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