Szach-math

SAT

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 $2$

(D)

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

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) $6.50$

(C)

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

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) $-2$

(B)

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

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