Using the discriminant, give the nature of the roots of . Also solve the equation.
b) Find the cubic equation whose roots are the cubes of the roots of .
c) Obtain the resolvent cubics, by Descartes’ method and by Ferrari’s method, of the equation . Are the cubics the same? Further, use either method to obtain the roots of this equation.
Consider first the equation (7x^{3}+x^{2}-35x-5=0). Dividing by 7 gives (x^{3}+\frac{1}{7}x^{2}-5x-\frac{5}{7}=0). For a cubic (x^{3}+px^{2}+qx+r=0), the discriminant is (\Delta=18pqr-4p^{3}r+p^{2}q^{2}-4q^{3}-27r^{2}). Here (p=\frac17,; q=-5,; r=-\frac57). Substituting, [ \Delta =18!\left(\frac17\right)(-5)!\left(-\frac57\right) -4!\left(\frac17\right)^{3}!\left(-\frac57\right) +\left(\frac17\right)^{2}(25) -4(-125) -27!\left(\frac{25}{49}\right). ] Simplifying term by term gives [ \Delta=\frac{450}{49}+\frac{20}{343}+\frac{25}{49}+500-\frac{675}{49}. ] Combining the fractional terms, [ \frac{450+25-675}{49}+\frac{20}{343} ______ ___ _______ _________ ______ _______ ____ ____ ____ ________.
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Give the following:
i) a matrix;
ii) the transpose of the matrix in (i) above;
iii) a system of linear equations represented by , where A is the matrix in (ii) above.
Using the method of substitution, obtain the solution set in , of the following:
i)
ii)
iii)
निम्नलिखित दीजिए :
i) एक आव्यूह;
ii) उपरोक्त (i) के आव्यूह का परिवर्त;
iii) , द्वारा निरूपित रैखिक समीकरणों का एक निकाय, जहां A उपरोक्त (ii) में दिया आव्यूह है।
ख) रैखिक निकाय
पर विचार कीजिए। इस निकाय में क्रेमर नियम क्यों लागू होता है, इसके दो कारण दीजिए। साथ ही, निकाय को हल करने के लिए, इस नियम का प्रयोग कीजिए।
एक आव्यूह;
उपरोक्त (i) के आव्यूह का परिवर्त;
, द्वारा निरूपित रैखिक समीकरणों का एक निकाय, जहां A उपरोक्त (ii) में दिया आव्यूह है।
ख) रैखिक निकाय
Find , and the number of elements in it, where
, and
.