Difference between revisions of "Specimen Cyprus Seniors Provincial/2nd grade/Problems"

m (Fixed typo and latex errors)
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Let <math>\text{A}\text{B}\Gamma\Delta</math> be a parallelogram. Let <math>(\epsilon)</math> be a straight line passing through <math>\text{A}</math> without cutting <math>\text{A}\text{B}\Gamma\Delta</math>. If <math>\text{B} ', \Gamma ', \Delta ' </math> are the projections of <math>\text{B}, \Gamma, \Delta</math> on <math>(\epsilon)</math> respectively, show that
 
Let <math>\text{A}\text{B}\Gamma\Delta</math> be a parallelogram. Let <math>(\epsilon)</math> be a straight line passing through <math>\text{A}</math> without cutting <math>\text{A}\text{B}\Gamma\Delta</math>. If <math>\text{B} ', \Gamma ', \Delta ' </math> are the projections of <math>\text{B}, \Gamma, \Delta</math> on <math>(\epsilon)</math> respectively, show that
  
a) the distance of <math>\Gamma</math> from <math>(\epsilon)</math> is equal to the sum of the distances <math>\Beta, \Delta</math> from <math>(\epsilon)</math>.
+
a) the distance of <math>\Gamma</math> from <math>(\epsilon)</math> is equal to the sum of the distances <math>\text{B}</math> , <math>\Delta</math> from <math>(\epsilon)</math>.
  
 
b) <math>\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')</math>.
 
b) <math>\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')</math>.

Latest revision as of 19:19, 18 January 2021

Problem 1

Let $\text{A}\text{B}\Gamma\Delta$ be a parallelogram. Let $(\epsilon)$ be a straight line passing through $\text{A}$ without cutting $\text{A}\text{B}\Gamma\Delta$. If $\text{B} ', \Gamma ', \Delta '$ are the projections of $\text{B}, \Gamma, \Delta$ on $(\epsilon)$ respectively, show that

a) the distance of $\Gamma$ from $(\epsilon)$ is equal to the sum of the distances $\text{B}$ , $\Delta$ from $(\epsilon)$.

b) $\text{Area}(\text{B}\Gamma\Delta)=\text{Area}(\text{B} '\Gamma '\Delta ')$.

Solution

Problem 2

If $\alpha=\sin x_{1}$,$\beta=\cos x_{1}\sin x_{2}$, $\gamma=\cos x_{1}\cos x_{2} \sin x_{3}$ and $\delta=\cos x_{1}\cos x_{2}\cos x_{3}$ prove that $\alpha^2+\beta^2+\gamma^2+\delta^2=1$

Solution

Problem 3

Prove that if $\kappa, \lambda, \nu$ are positive integers, then the equation $x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0$ has irrational roots.

Solution

Problem 4

If $\rho_{1}, \rho_{2}$ are the roots of equation $x^2-x+1=0$ then:

a) Prove that $\rho_{1}^3=\rho_{2}^3 = -1$ and

b) Calculate the value of: $\rho_{1}^{2006} + \rho_{2}^{2006}$.

Solution

See also

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