1
IIT-JEE 2007
+3
-0.75
The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$- {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is
A
zero
B
one
C
two
D
three
2
IIT-JEE 2007
+3
-0.75
Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$$ Which one of the following is correct ?
A
$$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a = \overrightarrow 0$$
B
$$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a \ne \overrightarrow 0$$
C
$$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow a \times \overrightarrow c \ne \overrightarrow 0$$
D
$$\overrightarrow a \times \overrightarrow b ,b \times \overrightarrow c ,\overrightarrow c \times \overrightarrow a$$ are muturally perpendicular
3
IIT-JEE 2007
+3
-0.75
Consider the planes $$3x-6y-2z=15$$ and $$2x+y-2z=5.$$

STATEMENT-1: The parametric equations of the line of intersection of the given planes are $$x=3+14t,y=1+2t,z=15t.$$ because

STATEMENT-2: The vector $${14\widehat i + 2\widehat j + 15\widehat k}$$ is parallel to the line of intersection of given planes.

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
C
Statement-1 is True, Statement-2 is False
D
Statement-1 is False, Statement-2 is True.
4
IIT-JEE 2007
+3
-0.75
Let the vectors $$\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$$ and $$\overrightarrow {UP} ,$$ represent the sides of a regular hexagon.

STATEMENT-1: $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 .$$ because
STATEMENT-2: $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0$$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$$

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False
D
Statement-1 is False, Statement-2 is True.
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