1
IIT-JEE 2010 Paper 1 Offline
Numerical
+4
-0
The maximum value of the expression $${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$ is
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2
IIT-JEE 2010 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $${{z_1}}$$ and $${{z_2}}$$ be two distinct complex number and let z =( 1 - t)$${{z_1}}$$ + t$${{z_2}}$$ for some real number t with 0 < t < 1. IfArg (w) denote the principal argument of a non-zero complex number w, then
A
$$\left| {z - {z_1}} \right| + \left| {z - {z_2}} \right| = \left| {{z_1} - {z_2}} \right|$$
B
Arg $$(z - {z_1})$$ = Arg$$(z - {z_2})$$
C
$$\left| {\matrix{ {z - {z_1}} & {\overline z - {{\overline z }_1}} \cr {{z_2} - {z_1}} & {{{\overline z }_2} - {{\overline z }_1}} \cr } } \right|$$ = 0
D
Arg $$(z - {z_1})$$ = Arg$$({z_2} - {z_1})$$
3
IIT-JEE 2010 Paper 1 Offline
MCQ (Single Correct Answer)
+3
-0.75
Let $$p$$ and $$q$$ be real numbers such that $$p \ne 0,\,{p^3} \ne q$$ and $${p^3} \ne - q.$$ If $${p^3} \ne - q.$$ and $$\,\beta $$ are nonzero complex numbers satisfying $$\alpha \, + \beta = - p\,$$ and $${\alpha ^3} + {\beta ^3} = q,$$ then a quadratic equation having $${\alpha \over \beta }$$ and $${\beta \over \alpha }$$ as its roots is
A
$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} + 2q} \right)x + \left( {{p^3} + q} \right) = 0$$
B
$$\left( {{p^3} + q} \right){x^2} - \left( {{p^3} - 2q} \right)x + \left( {{p^3} + q} \right) = 0$$
C
$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} - 2q} \right)x + \left( {{p^3} - q} \right) = 0$$
D
$$\left( {{p^3} - q} \right){x^2} - \left( {5{p^3} + 2q} \right)x + \left( {{p^3} - q} \right) = 0$$
4
IIT-JEE 2010 Paper 1 Offline
Numerical
+4
-0
Let $${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $$\,{{k - 1} \over {k\,!}}$$ and the common ratio is $${1 \over k}$$. Then the value of $${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{k = 1}^{100} {\left| {({k^2} - 3k + 1)\,\,{S_k}} \right|\,\,} $$ is
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