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} =-\frac{200}{49}+\frac{20}{343} =-\frac{1380}{343}. ] Hence [ \Delta=500-\frac{1380}{343}=\frac{170120-1380}{343} =\frac{168740}{343}>0. ] Therefore the equation has three distinct real roots. Trying simple rational roots in the original _____ _________ _____ _____ _________ ___ _______.
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मान लीजिए कि . दर्शाइए कि
7. क) के वे मान ज्ञात कीजिए, जिनके लिए ai समीकरण
का एक हल है। साथ ही, इस समीकरण के सभी मूल भी ज्ञात कीजिए।
यहाँ पांचवीं छवि में दिए गए पाठ का लिखित रूप है:
ख) 3i - 3 के सभी आठवें मूलों को ज्ञात कीजिए। साथ ही, इनमें से किसी एक को एक आरगां आलेख में दर्शाइए।
वह त्रिघात समीकरण ज्ञात कीजिए जिसके मूल के मूलों के घन हैं।
In the context of your IGNOU studies, give the following:
i) an example of an implication;
ii) the converse of your statement in (i) above;
iii) the contrapositive of your statement in (i) above;
iv) a statement using ;
v) a statement using .
Find , and the number of elements in it, where
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.
Given any two sets C and D, under what conditions on them will and
have the same number of elements? Give reasons for your answer.