मान लीजिए कि . दर्शाइए कि
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Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) Eliminating z from ,
and
gives
.
ii) The roots of are given by
.
iii) .
iv) Given any n positive numbers in , the product of their harmonic mean and their arithmetic mean is 1.
v) If A and B are two sets such that is empty, then either
or
.
vi) For any .
vii) The geometrical representation of the set is a point.
viii) Any finite set is a subset of .
ix) Every biquadratic equation has at least one real root.
x) The converse of the statement, ‘Every student of MTE-04 has completed FST-01’, is ‘Every student of FST-01 has completed MTE-04’.
निहितार्थ का एक उदाहरण;
क) 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।
वास्तविक जीवन की स्थिति की एक ऐसी समस्या दीजिए, जिसका गणितीय सूत्रीकरण है
यदि A और B क्रमशः समपूर्णांकों और विषम पूर्णांकों के समुच्चय हैं, तो और
ज्ञात कीजिए।