1
AIEEE 2005
MCQ (Single Correct Answer)
+4
-1
Let $$a, b$$ and $$c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ lie in a plane, then $$c$$ is :
A
the Geometric Mean of $$a$$ and $$b$$
B
the Arithmetic Mean of $$a$$ and $$b$$
C
equal to zero
D
the Harmonic Mean of $$a$$ and $$b$$
2
AIEEE 2005
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar vectors and $$\lambda $$ is a real number then

$$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right)\,\,\,\,\,\,\,\,{\lambda ^2}\overrightarrow b \,\,\,\,\,\,\,\,\lambda \overrightarrow c } \right] = \left[ {\overrightarrow a \,\,\,\,\,\,\,\,\overrightarrow b + \overrightarrow c \,\,\,\,\,\,\,\,\overrightarrow b } \right]$$ for :
A
exactly one value of $$\lambda $$
B
no value of $$\lambda $$
C
exactly three values of $$\lambda $$
D
exactly two values of $$\lambda $$
3
AIEEE 2005
MCQ (Single Correct Answer)
+4
-1
For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}$$ is equal to :
A
$$3{\overrightarrow a ^2}$$
B
$${\overrightarrow a ^2}$$
C
$$2{\overrightarrow a ^2}$$
D
$$4{\overrightarrow a ^2}$$
4
AIEEE 2005
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Let $$\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1 - x} \right)\widehat k$$ and $$\overrightarrow c \,\, = \,\,y\widehat i + x\widehat j + \left( {1 + x - y} \right)\widehat k.$$ Then $$\left[ {\overrightarrow a ,\overrightarrow b ,\overrightarrow c } \right]$$ depends on :
A
only $$y$$
B
only $$x$$
C
both $$x$$ and $$y$$
D
neither $$x$$ nor $$y$$
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