Admission Tests Practice Maths

Question 1

Given that one of the values of x satisfying the quadratic equation
x2−(2a+1)x+(a2+2)=0
is twice the other, find a.

a) give the optimal way to answer this question and b) complete it, explaining your steps

Answer 1

• equate coefficients, by stating that the equation equals (x-b)(x-2b) and then expand it, equating coefficients of x terms as it must be true for all x

If x^2−(2a+1)x+a^2+2=0 has one root double the other, then we can call the roots b and 2b, and write
x^2−(2a+1)x+a^2+2≡(x−b)(x−2b),
as the quadratic on the right hand side has roots b and 2b.

Multiplying out we have
x^2−(2a+1)x+a^2+2≡x^2−3bx+2b^2.

Since this holds for all x (it is an identity), we can equate coefficients, giving us 2a+1=3b and a^2+2=2b^2.

The first equation gives b=2a+13, so we can substitute this into the second equation to obtain
a^2+2=2(2a+13)2.

Expanding the brackets and rearranging yields a^2−8a+16=0, or (a−4)2=0, so a=4.

Checking this in the original equation: x^2−(2×4+1)x+42+2=(x−3)(x−6).

Question 2

write the ax^2 + bx + c in its most general factorised form

Answer 2

a(x-x1)(x-x2)
where x1 and x2 are the two solutions

• this form is particularly useful when equating coefficients - EQUATING COEFFICIENTS IS EXTREMELY USEFUL WHEN SOLVING QUADRATIC PROBLEMS

Question 3

does a^2 > b^2 imply that a>b?

Answer 3

no, as a or b could be negative
as squaring is not a monotonically increasing function, it doesn’t preserve the inequality

Question 4

if you are given this question:
Solve the inequality
square root (x+2) - square root (x-1) > square root (2x-3)
for real values of x and of the square roots. (taking only positive values of the roots)
and you get a quadratic (x^2 + x -6), and find the inequality of that quadratic is the question complete - why/why not?

Answer 4

no its not - as the square root can only accept positive expressions (in this case as stated in the question), so although the maximum value stated by the inequality is correct, the minimum may be too low
- this means that the actual set of solutions is smaller than the set you find using the quadratic
- so you need to find the minimum value of x which makes the expressions of x equal to 0 (and hence real) to find the true minimum value for the inequality
- you should get the answer instead that 3/2 <= x< 2
(underground maths reference R6387)

Question 5

If α and β are the roots of the equation
2x^2+3x+2=0,
find the equation whose roots are α+1 and β+1.

explain how to do this problem (complete it if you want)

Answer 5

this is simply a translation of the graph 1 unit towards the positive direction (which is a translation f(x-1))
- so just sub in x = x-1!
- the answer is x^2 - x + 1

Question 6

when you square an equation a) what do you need to be careful of b) why? c) how do you prevent issues arising from this?

Answer 6

• creating extra false solutions
• a function which has a variable squared has a 2:1 ratio of x to y values, whereas for a linear function the ratio is 1:1, so you end up with extra solutions for a given y value
• for example 2 = x+1 has one solution, whereas if you square that you get two solutions (positive and negative)
• sub your solutions back into the original equation to check you haven’t created false ones, if false ones are present then discard them

Question 7

what is the shape of the graph y = square root (2-x^2)

Answer 7

a semicircle with the origin as the centre (as it is a circle but only defined in the positive y)

Question 8

the point on the circle
(x−5)^2+(y−4)^2=4
which is closest to the circle
(x−1)^2+(y−1)^2=1
is

(3.4,2.8),
(3,4),
(5,2),
(3.8,2.4).
do this question in your head!!

Answer 8

• find vector connecting the two circles
• find the distance between the two circles
• find how far along the vector the point on the edge of the circle is (radius/magnitude of vector)
• add this scaled vector to the centre of the circle
  (the answer is 3.4, 2.8)

Question 8

can you write log2 (3^2) as 2log 2 (3)? why/why not?

Answer 8

yes, as if 2^x = 3 then 2^2x = 3^2