1
IIT-JEE 1998
MCQ (Single Correct Answer)
+2
-0.5
Let $$n$$ be an odd integer. If $$\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $$ for every value of $$\theta ,$$ then
A
$${b_0} = 1,\,b = 3$$
B
$${b_0} = 0,\,{b_1} = n$$
C
$${b_0} = - 1,\,{b_1} = n$$
D
$${b_0} = 0,\,{b_1} = {n^2} - 3n + 3$$
2
IIT-JEE 1998
Subjective
+8
-0
Let $$p$$ be a prime and $$m$$ a positive integer. By mathematical induction on $$m$$, or otherwise, prove that whenever $$r$$ is an integer such that $$p$$ does not divide $$r$$, $$p$$ divides $${}^{np}{C_r},$$

[Hint: You may use the fact that $${\left( {1 + x} \right)^{\left( {m + 1} \right)p}} = {\left( {1 + x} \right)^p}{\left( {1 + x} \right)^{mp}}$$]

3
IIT-JEE 1998
MCQ (More than One Correct Answer)
+2
-0.5
The number of common tangents to the circles $${x^2}\, + \,{y^2} = 4$$ and $${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$ is
A
0
B
1
C
3
D
4
4
IIT-JEE 1998
MCQ (Single Correct Answer)
+2
-0.5
If $${a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} } $$ equals
A
$$\left( {n - 1} \right){a_n}$$
B
$$n{a_n}$$
C
$${1 \over 2}n{a_n}$$
D
None of the above
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