1
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
2
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
$$ - {9 \over 2}$$
3
IIT-JEE 2012 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1

For every integer n, let an and bn be real numbers. Let function f : R $$\to$$ R be given by

$$f(x) = \left\{ {\matrix{ {{a_n} + \sin \pi x,} & {for\,x \in [2n,2n + 1]} \cr {{b_n} + \cos \pi x,} & {for\,x \in (2n - 1,2n)} \cr } } \right.$$, for all integers n. If f is continuous, then which of the following hold(s) for all n ?

A
an $$-$$ 1 $$-$$ bn $$-$$ 1 = 0
B
an $$-$$ bn = 1
C
an $$-$$ bn $$+$$ 1 = 1
D
an $$-$$ 1 $$-$$ bn = $$-$$1
4
IIT-JEE 2012 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1

If the ad joint of a 3 $$\times$$ 3 matrix P is $$\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]$$, then the possible value(s) of the determinant of P is(are)

A
$$-$$2
B
$$-$$1
C
1
D
2
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