1
IIT-JEE 1999
MCQ (Single Correct Answer)
+2
-0.5
$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$$ is equal to
A
$$1 - i\sqrt 3 $$
B
$$ - 1 + i\sqrt 3 $$
C
$$i\sqrt 3 $$
D
$$ - i\sqrt 3 $$
2
IIT-JEE 1999
MCQ (More than One Correct Answer)
+3
-0.75
For a positive integer $$\,n$$, let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$$ Then
A
$${f_2}\left( {{\pi \over {16}}} \right) = 1$$
B
$${f_3}\left( {{\pi \over {32}}} \right) = 1$$
C
$${f_4}\left( {{\pi \over {64}}} \right) = 1$$
D
$${f_5}\left( {{\pi \over {128}}} \right) = 1$$
3
IIT-JEE 1999
Subjective
+10
-0
For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$ z = w\,or\,z\overline {\,w} = 1$$.
4
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
$$a > 4$$
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