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Test-1-Solve-equation-k-2-1-x-2-k-1-x-k-1-0-k-R-30-2-Prove-sin-x-1-cos-x-1-cos-x-sin-x-35-3-P-x-2x-3-2x-2-x-2409-Find-P-11-35-Evaluate-other-answers-and-give-mark




Question Number 13491 by 433 last updated on 20/May/17
Test  1. Solve equation  (k^2 −1)x^2 +(k−1)x+(k+1)=0  k∈R  (30)  2. Prove  ((sin(x))/(1+cos(x)))=((1−cos(x))/(sin(x)))  (35)  3.P(x)=−2x^3 −2x^2 −x+2409  Find P(−11)  (35)    Evaluate other answers and give marks  I want to see how math teachers evaluate in other countries  Sorry foy my english
$${Test} \\ $$$$\mathrm{1}.\:{Solve}\:{equation} \\ $$$$\left({k}^{\mathrm{2}} −\mathrm{1}\right){x}^{\mathrm{2}} +\left({k}−\mathrm{1}\right){x}+\left({k}+\mathrm{1}\right)=\mathrm{0}\:\:{k}\in\mathbb{R} \\ $$$$\left(\mathrm{30}\right) \\ $$$$\mathrm{2}.\:{Prove} \\ $$$$\frac{{sin}\left({x}\right)}{\mathrm{1}+{cos}\left({x}\right)}=\frac{\mathrm{1}−{cos}\left({x}\right)}{{sin}\left({x}\right)} \\ $$$$\left(\mathrm{35}\right) \\ $$$$\mathrm{3}.{P}\left({x}\right)=−\mathrm{2}{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} −{x}+\mathrm{2409} \\ $$$${Find}\:{P}\left(−\mathrm{11}\right) \\ $$$$\left(\mathrm{35}\right) \\ $$$$ \\ $$$${Evaluate}\:{other}\:{answers}\:{and}\:{give}\:{marks} \\ $$$${I}\:{want}\:{to}\:{see}\:{how}\:{math}\:{teachers}\:{evaluate}\:{in}\:{other}\:{countries} \\ $$$${Sorry}\:{foy}\:{my}\:{english} \\ $$
Answered by 433 last updated on 20/May/17
1. If k=1   2=0  If k=−1  −2x=0⇒x=0  If k≠±1  Δ=(k−1)^2 −4(k^2 −1)(k+1)  =(k−1)((k−1)−4(k+1)^2 )  =(k−1)(k−1−4k^2 −8k−4)  (k−1)(−4k^2 −7k−5)  Δ′=7^2 −20×4=49−80<0  −4k^2 −7k−5<0 ∀k  If k>1 ⇒ Δ<0  If k∈(−∞,−1)∪(−1,1) ⇒ Δ>0  x_(1,2) =((1−k±(√((k−1)(−4k^2 −7k−5))))/(2(k^2 −1)))    2. ((sin(x))/(1+cos(x)))=((1−cos(x))/(sin(x))) ⇔  sin^2 (x)=1−cos^2 (x)  sin^2 (x)+cos^2 (x)=1    3. Q(x)=x+11  P(x)=Q(x)R(x)+u  P(−11)=Q(−11)R(−11)+u  Q(−11)=−11+11=0  P(−11)=u  (−2x^3 −2x^2 −x+2409)=(x+11)(−2x^2 +20x−221)+4840  P(−11)=4840
$$\mathrm{1}.\:{If}\:{k}=\mathrm{1}\: \\ $$$$\mathrm{2}=\mathrm{0} \\ $$$${If}\:{k}=−\mathrm{1} \\ $$$$−\mathrm{2}{x}=\mathrm{0}\Rightarrow{x}=\mathrm{0} \\ $$$${If}\:{k}\neq\pm\mathrm{1} \\ $$$$\Delta=\left({k}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{4}\left({k}^{\mathrm{2}} −\mathrm{1}\right)\left({k}+\mathrm{1}\right) \\ $$$$=\left({k}−\mathrm{1}\right)\left(\left({k}−\mathrm{1}\right)−\mathrm{4}\left({k}+\mathrm{1}\right)^{\mathrm{2}} \right) \\ $$$$=\left({k}−\mathrm{1}\right)\left({k}−\mathrm{1}−\mathrm{4}{k}^{\mathrm{2}} −\mathrm{8}{k}−\mathrm{4}\right) \\ $$$$\left({k}−\mathrm{1}\right)\left(−\mathrm{4}{k}^{\mathrm{2}} −\mathrm{7}{k}−\mathrm{5}\right) \\ $$$$\Delta'=\mathrm{7}^{\mathrm{2}} −\mathrm{20}×\mathrm{4}=\mathrm{49}−\mathrm{80}<\mathrm{0} \\ $$$$−\mathrm{4}{k}^{\mathrm{2}} −\mathrm{7}{k}−\mathrm{5}<\mathrm{0}\:\forall{k} \\ $$$${If}\:{k}>\mathrm{1}\:\Rightarrow\:\Delta<\mathrm{0} \\ $$$${If}\:{k}\in\left(−\infty,−\mathrm{1}\right)\cup\left(−\mathrm{1},\mathrm{1}\right)\:\Rightarrow\:\Delta>\mathrm{0} \\ $$$${x}_{\mathrm{1},\mathrm{2}} =\frac{\mathrm{1}−{k}\pm\sqrt{\left({k}−\mathrm{1}\right)\left(−\mathrm{4}{k}^{\mathrm{2}} −\mathrm{7}{k}−\mathrm{5}\right)}}{\mathrm{2}\left({k}^{\mathrm{2}} −\mathrm{1}\right)} \\ $$$$ \\ $$$$\mathrm{2}.\:\frac{{sin}\left({x}\right)}{\mathrm{1}+{cos}\left({x}\right)}=\frac{\mathrm{1}−{cos}\left({x}\right)}{{sin}\left({x}\right)}\:\Leftrightarrow \\ $$$${sin}^{\mathrm{2}} \left({x}\right)=\mathrm{1}−{cos}^{\mathrm{2}} \left({x}\right) \\ $$$${sin}^{\mathrm{2}} \left({x}\right)+{cos}^{\mathrm{2}} \left({x}\right)=\mathrm{1} \\ $$$$ \\ $$$$\mathrm{3}.\:{Q}\left({x}\right)={x}+\mathrm{11} \\ $$$${P}\left({x}\right)={Q}\left({x}\right){R}\left({x}\right)+{u} \\ $$$${P}\left(−\mathrm{11}\right)={Q}\left(−\mathrm{11}\right){R}\left(−\mathrm{11}\right)+{u} \\ $$$${Q}\left(−\mathrm{11}\right)=−\mathrm{11}+\mathrm{11}=\mathrm{0} \\ $$$${P}\left(−\mathrm{11}\right)={u} \\ $$$$\left(−\mathrm{2}{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} −{x}+\mathrm{2409}\right)=\left({x}+\mathrm{11}\right)\left(−\mathrm{2}{x}^{\mathrm{2}} +\mathrm{20}{x}−\mathrm{221}\right)+\mathrm{4840} \\ $$$${P}\left(−\mathrm{11}\right)=\mathrm{4840} \\ $$

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