Mathematics level 1 – test (examples)

1. If x+y=5 and x-y=3, then x^{2}-y^{2}=

(A)  9

(B)  15

(C)  16

(D)  25

(E)  34

(B) x^{2}-y^{2}=(x-y)(x+y)=3\cdot 5=15.

2. If k^{2}-4=4-k^{2}, what are possible values of k?

(A) 0 only

(B) 2 only

(C) 4 only

(D) -2 and 2 only

(E) -2, 0 and 2


    \begin{align*} k^{2}-4&=4-k^{2}\\ 2k^{2}&=8\\ k^{2}&=4\\ k&\in\{-2,2\} \end{align*}

3. If y=x^{3}-1.5, for what value of x is y=2?

(A) 0.79

(B) 1.14

(C) 1.52

(D) 1.87

(E) 6.50


    \begin{align*} 2&=x^{3}-1.5\\ x^{3}&=3.5\\ x&=1.51829\ldots\\ x&=1.52 \end{align*}

4. For which of the following equations is it true that the sum of the roots equals the product of the roots?

(A) x^{2}-4=0

(B) x^{2}-2x+1=0

(C) x^{2}-4x+4=0

(D) x^{2}-5x+6=0

(E) x^{2}+4x+4=0

(C)  x_{1}+x_{2}=x_{1}x_{2}=4.

5. What is the least integer value of k such that x^{2}(3k+1)-6x+2=0 has no real roots?

(A)  5

(B)  2

(C)  1

(D)  -1

(E)  -2


    \begin{align*} \Delta<0\iff &36-8\cdot(3k+1)<0\\ &-24k<-28\\ &k>\frac{7}{6}\\ &k\in\{2\} \end{align*}

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