निम्नलिखित आंकड़ों के लिए, गाउस पश्चांतर विधि का प्रयोग करके को अंतर्वेशी करने वाला बहुपद प्राप्त कीजिए :
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f (x) | 1.40 | 1.56 | 1.76 | 2.00 | 2.28 |
अतः, f(0.45) का मान ज्ञात कीजिए।
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Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
Let G be an infinite group such that for any non-trivial subgroup H of . Then prove that
i) or H is infinite;
ii) If , then o(g) is infinite.
List two distinct cosets of in D10, where r is a reflection in D10.
Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
यदि , तो C1 और C2।
Construct a field with 125 elements.
Is , for any two ideals I and J of a ring R? Give reasons for your answer.