1
IIT-JEE 1999
MCQ (Single Correct Answer)
+2
-0.5
If the roots of the equation $${x^2} - 2ax + {a^2} + a - 3 = 0$$ are real and less than 3, then
A
$$a < 2$$
B
$$2 \le a \le 3$$
C
$$3 < a \le 4$$
D
$$a > 4$$
2
IIT-JEE 1999
MCQ (Single Correct Answer)
+2
-0.5
If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the coefficients of $$x$$ and $${x^2}$$ are $$3$$ and $$-6$$ respectively, then $$m$$ is
A
6
B
9
C
12
D
24
3
IIT-JEE 1999
Subjective
+10
-0
Let $$n$$ be any positive integer. Prove that $$$\sum\limits_{k = 0}^m {{{\left( {\matrix{ {2n - k} \cr k \cr } } \right)} \over {\left( {\matrix{ {2n - k} \cr n \cr } } \right)}}.{{\left( {2n - 4k + 1} \right)} \over {\left( {2n - 2k + 1} \right)}}{2^{n - 2k}} = {{\left( {\matrix{ n \cr m \cr } } \right)} \over {\left( {\matrix{ {2n - 2m} \cr {n - m} \cr } } \right)}}{2^{n - 2m}}} $$$

for each non-be gatuve integer $$m \le n.$$ $$\,\left( {Here\left( {\matrix{ p \cr q \cr } } \right) = {}^p{C_q}} \right).$$

4
IIT-JEE 1999
MCQ (Single Correct Answer)
+2
-0.5
Let $${a_1},{a_2},......{a_{10}}$$ be in $$A,\,P,$$ and $${h_1},{h_2},......{h_{10}}$$ be in H.P. If $${a_1} = {h_1} = 2$$ and $${a_{10}} = {h_{10}} = 3,$$ then $${a_4}{h_7}$$ is
A
2
B
3
C
5
D
6
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