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

(B)315

(C)330

(D)335

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## Comments

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Could you please explain how it come through?

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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.

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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??

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Yup!! By the way could you please help with these also? The positive integer

kfor which \(\frac { { 101 }^{ \frac { k }{ 2 } } }{ k! } \) is a maximum is?Log in to reply