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Help Please..

1) A polynomial \(p(x)={ x }^{ 4 }+a{ x }^{ 3 }+b{ x }^{ 2 }+cx+d\) has roots \(\sqrt { 2 } ,e\) and \(\pi\) and no other roots. Let \(I=\int _{ \sqrt { 2 } }^{ e }{ p(x)dx } \) and \(J=\int _{ e }^{ \pi }{ p(x)dx. } \) Then,

(A) \(I\) and \(J\) must have opposite signs.

(B) \(I\) and \(J\) can be both positive but not both negative.

(C) \(I\) and \(J\) can be both negative but not both positive.

(D) We do not have enough information to compare the signs of \(I\) and \(J\).

2) All the inner angles of a -7 gon are obtuse , their sizes in degree being distinct integers divisible by 9. What is the sum(in degree) of the largest two angles?

(A) 300




Note by Anandhu Raj
11 months ago

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1.C we can verify it by graph

2.B because the angles would be 99,108,117,126,135,153,162 as all angles are obtuse and divisible by 9 so sum of 2 largest angles =153+162=315

Btw U too preparing for kvpy?? Naman Kapoor · 11 months ago

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@Naman Kapoor Yup!! By the way could you please help with these also? The positive integer k for which \(\frac { { 101 }^{ \frac { k }{ 2 } } }{ k! } \) is a maximum is? Anandhu Raj · 11 months ago

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  1. C and 2. B
Parv Mor · 11 months ago

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@Parv Mor Could you please explain how it come through? Anandhu Raj · 11 months ago

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@Anandhu Raj For the first ques as it has three roots and no other root and also the function is bi quadratic so there must be a repeated root. Considering three different cases just plot the graph and check the sign of integral. Parv Mor · 11 months ago

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