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

Note by Anandhu Raj
2 years, 8 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??

- 2 years, 8 months ago

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?

- 2 years, 8 months ago

1. C and 2. B

- 2 years, 8 months ago

Could you please explain how it come through?

- 2 years, 8 months ago

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.

- 2 years, 8 months ago