Calculate the value of \(\dfrac{a^2-b^2}{a-b}-\dfrac{a^3-b^3}{a^2-b^2}\) for \(a=10.2\), \(b=-0.2\).
Solution
\(\dfrac{a^2-b^2}{a-b}=a+b\), and \(\dfrac{a^3-b^3}{a^2-b^2}=\dfrac{a^2+ab+b^2}{a+b}\). For \(a=10.2\), \(b=-0.2\), we get \(a+b=10\) and \(a^2+ab+b^2=102.04\). Thus \(10-102.04/10=-0.204\). Answer: \(-0.204\).
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2
Task
One correct answer
Find the value of \(\dfrac1b-\dfrac1a\) if \(\dfrac{\sqrt3a-\sqrt3b}{ab}=\sqrt{12}\).
Solution
\(\dfrac{\sqrt3a-\sqrt3b}{ab}=\sqrt3\left(\dfrac1b-\dfrac1a\right)\). Since \(\sqrt{12}=2\sqrt3\), we get \(\dfrac1b-\dfrac1a=2\). Answer: C.
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3
Task
Matching
Match expressions (1–4) with their values when \(x=0.5\) (A–E).
А
\(-2.5\) Б
\(-0.25\) В
\(0.25\) Г
\(1.5\) Д
\(2.5\)
Solution
1) \(x-3=-2.5\) — A. 2) The second expression equals \(0.25\) — C. 3) \(x+1=1.5\) — D. 4) \(\dfrac3{8(x-2)}=-0.25\) — B. Matching: 1–A, 2–C, 3–D, 4–B.
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4
Task
One correct answer
Simplify \(\dfrac{b^2\cdot b^{10}}{b^4}\), where \(b\ne0\).
Solution
Using exponent rules, \(\dfrac{b^2b^{10}}{b^4}=b^{2+10-4}=b^8\). Answer: B.
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5
Task
Numeric answer
Find the value of the expression for \(m=4.25\): \[\dfrac{m+4}{m^2-6m+9}\cdot\dfrac{2m-6}{m^2-16}-\dfrac2{m-4}.\]
Solution
Factor: \(m^2-6m+9=(m-3)^2\), \(m^2-16=(m-4)(m+4)\), and \(2m-6=2(m-3)\). The expression simplifies to \(-\dfrac2{m-3}\). For \(m=4.25\): \(-2/1.25=-1.6\). Answer: -1.6.
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6
Task
Matching
Match each expression (1–4) with an identically equal expression (A–E), where \(a\ne0\), \(a\ne1\), \(a\ne-1\).
А
\(a-1\) Б
\(-a-1\) В
\(-\dfrac1{a+1}\) Г
\(-\dfrac1{a-1}\) Д
\(a+1\)
Solution
1) \(a-1\) — A. 2) \(-a-1\) — B. 3) \(-1/(a+1)\) — C. 4) \(-1/(a-1)\) — D. Matching: 1–A, 2–B, 3–C, 4–D.
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7
Task
One correct answer
Simplify \(\dfrac{9-x^2}{x^2+6x+9}\).
Solution
\(9-x^2=(3-x)(x+3)\) and \(x^2+6x+9=(x+3)^2\). Hence the expression is \(\dfrac{3-x}{x+3}\). Answer: A.
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8
Task
One correct answer
Simplify \(\dfrac{3x^2y}{9xy^3}\).
Solution
Cancel common factors: \(\dfrac{3x^2y}{9xy^3}=\dfrac{x}{3y^2}\). Answer: E.
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9
Task
One correct answer
Simplify \(\dfrac{a^2-1}{1-\frac1a}\).
Solution
\(1-1/a=(a-1)/a\) and \(a^2-1=(a-1)(a+1)\). Therefore the expression simplifies to \(a(a+1)\). Answer: E.
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10
Task
One correct answer
Simplify \(\dfrac{3x+12}{x^2-16}\).
Solution
\(3x+12=3(x+4)\) and \(x^2-16=(x-4)(x+4)\). Cancelling \(x+4\) gives \(\dfrac3{x-4}\). Answer: C.
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11
Task
One correct answer
If \(a=1-\dfrac{b}{c}\), then \(b=\)
Solution
From \(a=1-\dfrac{b}{c}\), we get \(\dfrac{b}{c}=1-a\). Multiplying by \(c\) gives \(b=c(1-a)\). Answer: A.
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12
Task
Numeric answer
Calculate the value of \(\dfrac{10a+b}{b^2-4a^2}+\dfrac{4a+2b}{b^2+4ab+4a^2}\) for \(a=0.25\), \(b=4.5\).
Solution
Factor the denominators: \(b^2-4a^2=(b-2a)(b+2a)\) and \(b^2+4ab+4a^2=(b+2a)^2\). Also, \(4a+2b=2(b+2a)\). The expression simplifies to \(\dfrac{3}{b-2a}\). For \(a=0.25\), \(b=4.5\): \(\dfrac{3}{4.5-0.5}=\dfrac34=0.75\). Answer: 0.75.
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13
Task
Numeric answer
Calculate the value of \(\dfrac{a^2-6ab+9b^2}{a^2+4ab}:\dfrac{5a-15b}{a+4b}\) for \(a=0.1\), \(b=3.7\).
Solution
We have \(a^2-6ab+9b^2=(a-3b)^2\), \(a^2+4ab=a(a+4b)\), and \(5a-15b=5(a-3b)\). Thus the expression becomes \(\dfrac{a-3b}{5a}\). For \(a=0.1\), \(b=3.7\): \(\dfrac{0.1-11.1}{0.5}=-22\). Answer: -22.
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14
Task
One correct answer
Simplify \(\dfrac{a}{b(a-b)}-\dfrac{b}{a(a-b)}\).
Solution
Using the common denominator \(ab(a-b)\), we get \(\dfrac{a^2-b^2}{ab(a-b)}=\dfrac{(a-b)(a+b)}{ab(a-b)}=\dfrac{a+b}{ab}\). Answer: A.
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15
Task
One correct answer
Simplify \(\dfrac1{x-5}-\dfrac{2x-5}{x(x-5)}\).
Solution
With a common denominator: \(\dfrac{x-(2x-5)}{x(x-5)}=\dfrac{5-x}{x(x-5)}=-\dfrac1x\). Answer: A.
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16
Task
One correct answer
Simplify \(\dfrac{a^2+16}{a-4}-\dfrac{8a}{a-4}\).
Solution
The fractions have the same denominator: \(\dfrac{a^2+16-8a}{a-4}=\dfrac{(a-4)^2}{a-4}=a-4\). Answer: B.
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17
Task
One correct answer
Reduce the fraction \(\dfrac{10ab^3}{5a^2b}\).
Solution
Cancel coefficients and powers: \(10/5=2\), \(a/a^2=1/a\), and \(b^3/b=b^2\). Thus \(\dfrac{10ab^3}{5a^2b}=\dfrac{2b^2}{a}\). Answer: A.
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18
Task
One correct answer
Simplify \(\dfrac{(a-b)^2-b^2}{a}\).
Solution
Expand the square: \((a-b)^2-b^2=a^2-2ab=a(a-2b)\). Dividing by \(a\) gives \(a-2b\). Answer: B.
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19
Task
One correct answer
Simplify \(\dfrac{a^{24}}{(a^4)^2}\).
Solution
\((a^4)^2=a^8\), so \(\dfrac{a^{24}}{a^8}=a^{24-8}=a^{16}\). Answer: E.
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20
Task
One correct answer
Reduce the fraction \(\dfrac{a^2-b^2}{a^2-ab}\).
Solution
Factor: \(a^2-b^2=(a-b)(a+b)\) and \(a^2-ab=a(a-b)\). Canceling \(a-b\) gives \(\dfrac{a+b}{a}\). Answer: A.
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21
Task
One correct answer
Simplify \(\dfrac{(2x^2)^3}{4x^9}\).
Solution
\((2x^2)^3=8x^6\), so \(\dfrac{8x^6}{4x^9}=2x^{-3}=\dfrac{2}{x^3}\), with \(x\ne0\). Answer: A.
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22
Task
One correct answer
Add the fractions: \(\dfrac{a-2}{2a}+\dfrac{a-2}{a^2}\).
Solution
Using the common denominator \(2a^2\): \(\dfrac{a(a-2)+2(a-2)}{2a^2}=\dfrac{(a-2)(a+2)}{2a^2}=\dfrac{a^2-4}{2a^2}\), where \(a\ne0\). Answer: C.
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23
Task
One correct answer
Multiply the fractions: \(\dfrac{p+2}{2m-4}\cdot\dfrac{m-2}{p^2-4}\).
Solution
Factor the denominators: \(2m-4=2(m-2)\), \(p^2-4=(p-2)(p+2)\). Canceling common factors gives \(\dfrac1{2(p-2)}=\dfrac1{2p-4}\). Answer: B.
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24
Task
One correct answer
Write as a single fraction: \(\dfrac{2x+1}{x-3}-\dfrac{x-2}{3-x}\).
Solution
Since \(3-x=-(x-3)\), we have \(-\dfrac{x-2}{3-x}=+\dfrac{x-2}{x-3}\). Hence \(\dfrac{2x+1+x-2}{x-3}=\dfrac{3x-1}{x-3}\), with \(x\ne3\). Answer: D.
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25
Task
One correct answer
Perform the division: \(\dfrac{12ab}{5}:(4a^2)\).
Solution
Dividing by \(4a^2\) means multiplying by \(\dfrac1{4a^2}\): \(\dfrac{12ab}{5}\cdot\dfrac1{4a^2}=\dfrac{3b}{5a}\). Answer: B.
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26
Task
One correct answer
Simplify \(\dfrac{(3mn)^4}{m^2n}\).
Solution
\((3mn)^4=81m^4n^4\). Dividing by \(m^2n\) gives \(81m^{4-2}n^{4-1}=81m^2n^3\). Answer: D.
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27
Task
Matching
Match each expression (1–4) with its value when \(x=-2.5\) (A–E).
А
\(-5.5\) Б
\(-4.5\) В
\(-0.5\) Г
\(0.5\) Д
\(4.5\)
Solution
1) The first expression equals \(-1/2=-0.5\) — C. 2) \(x-3=-5.5\) — A. 3) \(x+3=0.5\) — D. 4) \(x-2=-4.5\) — B. Matching: 1–C, 2–A, 3–D, 4–B.
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28
Task
Numeric answer
Find the value of \(\dfrac{2ab+a^2+b^2}{a^2+b^2-2ab}\) for \(a=13.4\), \(b=12.4\).
Solution
The numerator is \(a^2+2ab+b^2=(a+b)^2\), and the denominator is \(a^2-2ab+b^2=(a-b)^2\). Thus \(\dfrac{25.8^2}{1^2}=665.64\). Answer: \(665.64\).
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29
Task
One correct answer
Simplify \(\dfrac{(a^2-1)(a+1)}{a^2+2a+1}\).
Solution
\(a^2-1=(a-1)(a+1)\) and \(a^2+2a+1=(a+1)^2\). For \(a\ne-1\), \(\dfrac{(a-1)(a+1)^2}{(a+1)^2}=a-1\). Answer: C.
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30
Task
One correct answer
Simplify the expression \(\dfrac{25-x^2}{x^2+5x}\).
Solution
Factor the numerator and denominator: \(25-x^2=(5-x)(5+x)\), \(x^2+5x=x(x+5)\). Thus, for \(x\ne0,-5\), \(\dfrac{(5-x)(5+x)}{x(x+5)}=\dfrac{5-x}{x}\). Answer: E.
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31
Task
One correct answer
Simplify the expression \(\dfrac{4x^2-25}{(2x-5)^2}\).
Solution
\(4x^2-25=(2x-5)(2x+5)\). For \(x\ne\dfrac52\), \(\dfrac{(2x-5)(2x+5)}{(2x-5)^2}=\dfrac{2x+5}{2x-5}\). Answer: A.
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32
Task
Matching
Match each expression (1–4) with the identically equal expression (A–E), where \(a\ne0\), \(a\ne1\), and \(a\ne-1\).
А
\(a-1\) Б
\(-a-1\) В
\(-\dfrac1{a+1}\) Г
\(-\dfrac1{a-1}\) Д
\(a+1\)
Solution
1) \(\dfrac{a}{a+1}\cdot\dfrac{a^2-1}{a}=a-1\) — A. 2) \(a^2+\dfrac{a^3-1}{1-a}=-a-1\) — B. 3) \(\dfrac{1-a}{a}:\dfrac{a^2-1}{a}=-\dfrac1{a+1}\) — C. 4) \(\dfrac{a-2}{a-1}-1=-\dfrac1{a-1}\) — D. Matching: 1–A, 2–B, 3–C, 4–D.
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33
Task
One correct answer
Simplify the expression \(\dfrac{9x^2-16}{(3x+4)^2}\).
Solution
Factor the numerator as a difference of squares: \(9x^2-16=(3x-4)(3x+4)\). Thus \(\dfrac{(3x-4)(3x+4)}{(3x+4)^2}=\dfrac{3x-4}{3x+4}\). Answer: D.
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34
Task
One correct answer
Simplify the expression \(\dfrac{a^2-4b^2}{3a+6b}\).
Solution
\(a^2-4b^2=(a-2b)(a+2b)\) and \(3a+6b=3(a+2b)\). Cancelling the common factor gives \(\dfrac{a-2b}{3}\). Answer: B.
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35
Task
One correct answer
Simplify the expression \(\dfrac{2-x}{x^2+x-6}\), where \(x\ne2\) and \(x\ne-3\).
Solution
Factor the denominator: \(x^2+x-6=(x-2)(x+3)\), and \(2-x=-(x-2)\). Therefore \(\dfrac{2-x}{x^2+x-6}=-\dfrac1{x+3}\). Answer: B.
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36
Task
One correct answer
Simplify the expression \(\dfrac{a^2+ab}{(a+b)^2}-\dfrac{2a+b}{a+b}\).
Solution
For \(a+b\ne0\), \(a^2+ab=a(a+b)\), so the first fraction is \(a/(a+b)\). Then \(\dfrac{a-(2a+b)}{a+b}=\dfrac{-(a+b)}{a+b}=-1\). Answer: E.
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37
Task
One correct answer
Simplify the expression \(\dfrac{2a^2-50b^2}{5b+a}\).
Solution
\(2a^2-50b^2=2(a-5b)(a+5b)\), and \(5b+a=a+5b\). For \(a+5b\ne0\), cancel the common factor to get \(2(a-5b)=2a-10b\). Answer: E.
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38
Task
One correct answer
Simplify the expression \(\dfrac{27a^3b-12ab}{9a^3b^2-6a^2b^2}\).
Solution
Factor: the numerator is \(3ab(3a-2)(3a+2)\), and the denominator is \(3a^2b^2(3a-2)\). Cancelling common factors gives \(\dfrac{3a+2}{ab}\) for admissible variable values. Answer: C.
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39
Task
One correct answer
Simplify the expression \(\dfrac{(2x-3)^2-9}{x}\).
Solution
\((2x-3)^2-9=4x^2-12x=4x(x-3)\). For \(x\ne0\), \(\dfrac{4x(x-3)}x=4x-12\). Answer: A.
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40
Task
One correct answer
Reduce the fraction \(\dfrac{4b^2+20b}{25+10b+b^2}\).
Solution
\(4b^2+20b=4b(b+5)\), and \(25+10b+b^2=(b+5)^2\). For \(b\ne-5\), \(\dfrac{4b(b+5)}{(b+5)^2}=\dfrac{4b}{b+5}\). Answer: E.
