1
IIT-JEE 2012 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
Let $${a_1},{a_2},{a_3},.....$$ be in harmonic progression with $${a_1} = 5$$ and $${a_{20}} = 25.$$ The least positive integer $$n$$ for which $${a_n} < 0$$ is
A
22
B
23
C
24
D
25
2
IIT-JEE 2012 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
Let $${{a_n}}$$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let $${{b_n}}$$ = the number of such n-digit integers ending with digit 1 and $${{c_n}}$$ =the number of such n-digit integers ending with digit 0.

Which of the following is correct?

A
$${a_{17}} = {a_{16}} + {a_{15}}$$
B
$${c_{17}} \ne {c_{16}} + {c_{15}}$$
C
$${b_{17}} \ne {b_{16}} + {c_{16}}$$
D
$${a_{17}} = {c_{17}} + {b_{16}}$$
3
IIT-JEE 2012 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

If P is a 3 $$\times$$ 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 $$\times$$ 3 identity matrix, then there exists a column matrix $$X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$ such that

A
$$PX = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]$$
B
PX = X
C
PX = 2X
D
PX = $$-$$X
4
IIT-JEE 2012 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1

Let $$\alpha$$(a) and $$\beta$$(a) be the roots of the equation $$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1) = 0$$ where $$a > - 1$$. Then $$\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a)$$ and $$\mathop {\lim }\limits_{a \to {0^ + }} \beta (a)$$ are

A
$$ - {5 \over 2}$$
B
$$ - {1 \over 2}$$
C
$$ - {7 \over 2}$$
D
$$ - {9 \over 2}$$
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