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Simboli privi di significato e calcoli errati

Simboli privi di significato e calcoli errati. 120 domande

I seguenti  Otto Test, composti da 126 domande, presentano una serie di scritture (simboli, espressioni, formule, identità, equazioni, ecc.) vere o prive di significato matematico, mischiate con altre in cui sono stati commessi degli errori grossolani.
Il lettore individui quelle corrette e quelle errate e dica perché?

Questo test è utile per migliorare la comprensione dei simboli matematici, e per ripassare le proprietà degli oggetti matematici.

Test 1.- Stabilire quali sono le relazioni corrette e quali errate.

  1. E’ vero che:\[-\frac{1}{3}=\frac{-1}{3}=\frac{1}{-3}=\frac{-1}{-3}=\]
  2. E’ vero che: \[\frac{1}{3}x=\frac{x}{3}=\frac{1}{3x}\]
  3. E’ vero che:\[\frac{2a+b}{2c}=\frac{a+b}{c}=\frac{a}{c}+\frac{b}{2c}\]
  4. E’ vero che:\[\frac{-2a+b}{2c}=-\frac{2a-b}{c}=-\frac{a}{c}+\frac{b}{2c}\]
  5. E’ vero che: \[\left ( ab \right )^{2}=a^{2}b^{2}\]
  6. E’ vero che: \[log^{2}\, x=\left ( log\, x \right )^{2}=log(x^{2})\]
  7. E’ vero che: \[2log_{a}\, (-2)=log_{a}\left ( -2 \right )^{2}=log_{a}(4)\]
  8. E’ vero che:\[log_{3}\, \frac{1}{2}=-log_{3}\left ( 2 \right )=log_{3}(-2)\]
  9. E’ vero che:\[log_{-2}\, 4=2\Leftrightarrow (-2)^{2}=4\]
  10. E’ vero che: \[log_{2}\, \left (-\frac{1}{8} \right )=-3\, \, in\, quanto \, \, (2)^{-3}=-\frac{1}{8}\]
  11. E’ vero che: \[x^{2}-1>0\Rightarrow x^{2}>1\Rightarrow x>\pm 1\]
  12. E’ vero che: \[\sqrt{3600}=\sqrt{4}\cdot \sqrt{9}\cdot \sqrt{100}\]
  13. E’ vero che: \[\sqrt{3+7}=\sqrt{3}+\sqrt{7}\]
  14. E’ vero che: \[\sqrt{\left ( -1 \right )^{2}}=-1\]
  15. E’ vero che: \[\left ( 1-\sqrt{2} \right )\sqrt{a^{2}}=\sqrt{a^{2}\left ( 1-\sqrt{2} \right )^{2}}\]

Test 2.- Stabilire quali sono le relazioni corrette e quali errate.

  1. E’ vero che: \[\left ( 1-\frac{5}{4} \right )\sqrt{a}=\sqrt{a\left ( 1-\frac{5}{4} \right )^{2}}\]
  2. E’ vero che: \[\sqrt[6]{a^{3}+a^{9}}=\sqrt{1+a^{3}}\]
  3. E’ vero che:\[\left ( -\frac{1}{2} \right )^{\frac{1}{3}}=\sqrt[3]{-\frac{1}{2}}\]
  4. E’ vero che: \[\left ( \pi \sqrt{\left ( 3^{\pi } \right )^{e}} \right )^{0}=1\]
  5. E’ vero che \[\left ( \pi \sqrt{\left ( 3^{0 } \right )^{e}} \right )^{2}=\pi ^{2}\]
  6. E’ vero che \[\frac{2}{\sqrt[6]{8}}=\frac{2}{\sqrt{2}}\]
  7. E’ vero che \[\frac{3}{\sqrt{3}}=\sqrt{3}\]
  8. E’ vero che \[\frac{a}{\sqrt[n]{a}}=\sqrt[n]{a^{n-1}}\]
  9. E’ vero che \[\left ( -3 \right )^{\frac{1}{3}}=\sqrt[3]{-3}\]
  10. E’ vero che \[\left ( 3 \right )^{\frac{1}{2}}=\sqrt[2]{3}\]
  11. E’ vero che \[\left ( a^{x} \right )^{2}=a^{x^{2}}\]
  12. E’ vero che \[\left ( 3^{2} \right )^{x^{4}}=3^{8x}\]

Test 3.- Stabilire quali sono le relazioni corrette e quali errate.

  1. E’ vero che: \[log(-4)+log(-2)=log\, 8\]
  2. E’ vero che:\[3^{log_{3}4}=4\]
  3. E’ vero che:\[3!\cdot 4!=(3\cdot 4)!=12!\]
  4. E’ vero che:\[sen(30^{\circ})=\frac{1}{2}\Rightarrow sen2(30^{\circ})=2\cdot \frac{1}{2}=1\]
  5. E’ vero che:\[cos(180^{\circ})=cos\pi \Rightarrow 180^{\circ}=\pi\]
  6. E’ vero che:\[4x^{2}+y^{2}+2xy=(2x+y)^{^{2}}\]
  7. E’ vero che:\[x^{3}+1=(x+1)^{3}\]
  8. E’ vero che:\[x^{2}+1=(x+1)(x-1)\]
  9. E’ vero che:\[cos(30^{\circ}+45^{\circ})=cos\, 30^{\circ}+cos\, 45^{\circ}\]
  10. E’ vero che:\[1-senx=cosx\]
  11. E’ vero che:\[2senx=2senxcosx\]
  12. E’ vero che:\[\frac{1}{sen\, x}=\frac{cot\, x}{cos\, x}\]
  13. E’ vero che:\[120^{2}-60^{2}=180\cdot 60\]
  14. E’ vero che:\[10^{2}+5^{2}=(10-5)\cdot 5^{2}\]
  15. E’ vero che:\[x^{2}+y^{2}=(x-y)\cdot y^{2}\]
  16. E’ vero che:\[10^{3}+20^{3}=30\cdot \left ( 100-200+400 \right )\]
  17. E’ vero che\[(10+20-30)^{17}=\left ( -20+20 \right )^{156}\]
  18. E’ vero che\[\frac{1}{(10+20-30)^{17}}=\frac{1}{\left ( -20+20 \right )^{17}}\]
  19. E’ vero che\[\left ( 100-200+300+1 \right )^{2}=\left ( 199+2 \right )^{2}\]

Test 4.- Stabilire quali sono le relazioni corrette e quali errate.

  1. E’ vero che: \[x^{10}<10^{x}\]
  2. E’ vero che:\[\left ( a-b+c-1 \right )^{2}=\left ( a-b-1+c+2 \right )^{2}\]
  3. E’ vero che:\[-x^{2}+2x-1=\left ( x-1 \right )^{2}\]
  4. E’ vero che:\[x^{2}-1=0\Rightarrow x=\pm 1\]
  5. E’ vero che:\[\left | x \right |>1\Rightarrow x>1\]
  6. E’ vero che:\[\left | x \right |<1\Rightarrow -1<x<1\]
  7. E’ vero che: \[\left | -x \right |^{3}>0\Leftrightarrow x>0\]
  8. E’ vero che semplificando per 2 l’indice di radice 6 e il radicando 4 si ha:$\displaystyle \sqrt[6]{4}=\sqrt[3]{2}$
  9. E’ vero che semplificando per 2 l’indice del radicale e l’esponente del radicando ha: $\displaystyle \sqrt[6]{2^{2}}=\sqrt[3]{2}$
  10. E’ vero che:$\displaystyle cos^{2}30^{\circ}-cos60^{\circ}=sen^{2}30^{\circ}$
  11. E’ vero che:$\displaystyle cos^{2}x-cos2x=sen^{2}x$

Test 5.- Stabilire quali sono le relazioni corrette e quali errate.

  1. $\displaystyle 1:0=0$
  2. $\displaystyle 1:0=1$
  3. $\displaystyle 1:0=2$
  4. $\displaystyle 0,\overline{9}=1$
  5. $\displaystyle \sqrt{a^{2}\pi^{2} }=a\pi ,\, \, \, con\, \, a>0$
  6. $\displaystyle \sqrt{a^{2}+\pi^{2} }=a+\pi ,\, \, \, con\, \, a>0$
  7. $\displaystyle \sqrt{-7+\pi^{2} }=x+1i ,\, \, \, con\, \, x\, incognita$
  8. \[\sqrt[3]{-7+\pi }=x+1, \, \, con\, \, x\, \, incognita\]
  9. $\displaystyle i^{2}=1$
  10. $\displaystyle \frac{1}{i}=1$
  11. $\displaystyle i^{21}=1$
  12. $\displaystyle i^{324521}=-i$
  13. $\displaystyle log_{(-3)}\, \, x=1$
  14. $\displaystyle log_{3}\, \, (-x^{2})=1$
  15. $\displaystyle log_{3}\, \, (x)=1$
  16. $\displaystyle (-4)^{x}=3$
  17. $\displaystyle (4)^{x}=1$
  18. $\displaystyle (4)^{x^{2}}=4^{2x}$
  19. $\displaystyle (e)^{lnx}=x$
  20. $\displaystyle (a)^{log_{a}x}=x$

Test 6.- Stabilire quali sono le relazioni corrette e quali errate.

  1. $\displaystyle log_{a}a^{x}=x$
  2. $\displaystyle log_{1}(3)=\frac{1}{3}$
  3. $\displaystyle log_{2}(6)=log_{2}(-2)\cdot log_{2}(-3)$
  4. $\displaystyle log_{5}(-25)=-log_{5}(25)$
  5. $\displaystyle log_{5}\, 5=5$
  6. $\displaystyle log_{5}\, \frac{1}{5}=-1$
  7. $\displaystyle \left ( -3 \right )^{\frac{1}{3}}\left ( -3 \right )^{\frac{1}{2}}=\left ( -3 \right )^{\frac{1}{3}+\frac{1}{2}}$
  8. $\displaystyle \left ( -3 \right )^{3}\left ( -3 \right )^{5}=\left ( -3 \right )^{8}$
  9. $\displaystyle \left ( 3 \right )^{-33}\cdot \left ( 3 \right )^{55}=\left ( 3 \right )^{22}$
  10. $\displaystyle \left ( 2 \right )^{120}: \left ( 2 \right )^{-180}=\left ( 2 \right )^{-60}$
  11. $\displaystyle \left ( 5 \right )^{6}: \left ( 6 \right )^{5}=\left ( 11 \right )^{11}$
  12. $\displaystyle x+\frac{1}{x-x}=9$

Test 7.- Stabilire quali sono le relazioni corrette e quali errate.

  1. $\displaystyle \lim_{x\rightarrow +\infty }log_{x}x=1$
  2. $\displaystyle \lim_{x\rightarrow +\infty }log_{x}x=-1$
  3. $\displaystyle \lim_{x\rightarrow -\infty }log_{-x}(-x)=1$
  4. $\displaystyle \lim_{x\rightarrow -\infty }\sqrt{x-3}=-2$
  5. $\displaystyle \lim_{x\rightarrow 0 }\frac{\pi +3\sqrt{2}}{1-\sqrt{7}}=0$
  6. $\displaystyle \lim_{n\rightarrow -\infty }\left ( n^{4}+n^{4}+1 \right )=+\infty$, n numero naturale
  7. $\displaystyle D\left ( \frac{1}{n} \right ) , n\in N, \, D\, simbolo\, di \, derivata$
  8. $\displaystyle \left [D\left ( \frac{1}{x-1} \right ) \right ]_{x=1} =0,\, \, D\, simbolo\, di \, derivata$
  9. $\displaystyle \left [ D^{10}x^{10} \right ]_{x=10}=10^{10}$
  10. $\displaystyle D\left ( \frac{e^{2}\pi \sqrt{3}+7\sqrt{2}}{\sqrt{3}+\sqrt{5}} \right )=0$
  11. $\displaystyle D\left ( \frac{e^{2}\pi \sqrt{3}+7x\sqrt{3}}{\sqrt{3}} \right )=7$
  12. $\displaystyle \int Dx^{2}dx=x^{2}+c$

Test 8.- Stabilire quali sono le relazioni corrette e quali errate.

  1. $\displaystyle D\int xdx=x+c$
  2. $\displaystyle \int 2xdx=\int xdx+\int xdx$
  3. $\displaystyle \int 5xdx=5\int xdx$
  4. $\displaystyle \int 5^{4}xdx=4\int 5xdx$
  5. $\displaystyle \int x^{-1}dx=\frac{1}{x^{-2}}+c$
  6. $\displaystyle \int x^{-2}dx=\frac{1}{x^{-1}}+c$
  7. $\displaystyle \int x^{-2}dx=-x^{-1}+c$
  8. $\displaystyle \int \frac{e^{2x}}{2}=e^{4x}+c$
  9. $\displaystyle \int \sqrt{x}dx=\frac{1}{2\sqrt{x}}+c$
  10. $\displaystyle \int lnx\, dx=\frac{1}{x}+c$
  11. $\displaystyle \int \frac{1+\pi }{3-e^{2}}xdx=\frac{1}{2}\cdot \frac{1+\pi }{3-e^{2}}\cdot x^{2}+c$
  12. $\displaystyle \int_{0}^{2}\sqrt{x-1}dx$
  13. $\displaystyle \int_{1}^{4}\frac{1}{x}dx=\int_{1}^{2}\frac{1}{x}dx+\int_{2}^{3}\frac{1}{x}dx+\int_{3}^{4}\frac{1}{x}dx$
  14. $\displaystyle e^{\pi \int_{4}^{4}x^{-1}dx}=1$
  15. $\displaystyle \int_{-1}^{0}\frac{1}{x}dx$
  16. $\displaystyle \int_{21}^{21}\frac{1}{x}dx=21$
  17. $\displaystyle \int_{-11}^{-11}\frac{1}{x^{2}}dx=0$
  18. $\displaystyle e^{i\pi }=i^{2}$
  19. $\displaystyle e^{2i\pi }=1$
  20. $\displaystyle e^{i\pi }+1=0$