Test for Copl...

Test for Coplanarity

Trending Questions

If $A = [\begin{matrix} 1 & - 2 \\ 4 & 5 \end{matrix}]$ and $f (t) = t^{2} - 3 t + 7$ , then $f (A) + [\begin{matrix} 3 & 6 \\ - 12 & - 9 \end{matrix}] =$

$[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$
$[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$
$[\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}]$

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If $a f (x) + b f (\frac{1}{x}) = x - 1, x \neq 0$ , and $a \neq b$ , then $f (2) =$

$[\frac{(2 a + b)}{2 (a^{2} + b^{2})}]$
$[\frac{(2 a + b)}{2 (a^{2} - b^{2})}]$
$[\frac{(a + 2 b)}{(a^{2} + b^{2})}]$
$[\frac{a}{(a^{2} - b^{2})}]$

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Q. Four points given by position vectors

2 \to a + 3 \to b - \to c, \to a - 2 \to b + 3 \to c, 3 \to a + 4 \to b - 2 \to c a n d \to a - 6 \to b + 6 \to c

are coplanar, where

\to a, \to b a n d \to c

are non-coplanar vectors.

True
False

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Q. Let

α \in R

and the three vectors

\to a = α^i +^j + 3^k, \to b = 2^i +^j - α^k

and

\to c = α^i - 2^j + 3^k

. Then the set

S = {α : \to a, \to b and \to c are coplanar}

is empty
is singleton
contains exactly two positive numbers
contains exactly two numbers only one of which is positive

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Q. Let a, b, c be distinct non - negative numbers. If the vectors

a^i + a^j + c^k,^i +^k

and

c^i + c^j + b^k

, are coplanar then c is

AM of a and b
GM of a and b
HM of a and b
None of the above

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Q. Let

\to c

be a vector perpendicular to the vectors

\to a =^i +^j -^k

and

\to b =^i + 2^j +^k .

\to c \cdot (^i +^j + 3^k),

then the value of

\to c \cdot (\to a \times \to b)

is equal to

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For any sets $A$ and $B$ , $P (A)$ is union of $P (B) = P (A u n i o n B)$ ? (True/False) Justify your answer.

True
False

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Q. If the vectors

\to a + λ \to b + 3 \to c, - 2 \to a + 3 \to b - 4 \to c

and

\to a - 3 \to b + 5 \to c

are coplanar, then the value of

λ

2
-1
1
-2

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Q. The points

2^i -^j -^k,^i +^j +^k, 2^i + 2^j +^k, 2^j + 5^k

are

collinear
coplanar but not collinear
noncoplanar
none

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Q. Find the value of

λ

so that the points

P, Q, R

and

S

on the sides

O A, O B, O C

and

A B

respectively of a regular tetrahedron

O A B C

are coplanar. It is given that

\frac{O P}{O A} = \frac{1}{3}, \frac{O Q}{O B} = \frac{1}{2}

and

\frac{O S}{A B} = λ

$λ = \frac{1}{2}$
$λ = - 1$
$λ = 0$
For no value of $λ$

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Q. If non-zero vectors

\to a

and

\to b

are equally inclined to coplanar vector

\to c

, then

\to c

can be

$\frac{| \to a |}{| \to a | + 2 | \to b |} \to a + \frac{| \to b |}{| \to a | + | \to b |} \to b$
$\frac{| \to b |}{| \to a | + | \to b |} \to a + \frac{| \to a |}{| \to a | + | \to b |} \to b$
$\frac{| \to a |}{| \to a | + 2 | \to b |} \to a + \frac{| \to b |}{| \to a | + 2 | \to b |} \to b$
$\frac{| \to b |}{2 | \to a | + | \to b |} \to a + \frac{| \to a |}{2 | \to a | + | \to b |} \to b$

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Q. The vectors

\to a = x^i + (x + 1)^j + (x + 2)^k, \to b = (x + 3)^i + (x + 4)^j + (x + 5)^k

and

\to c = (x + 6)^i + (x + 7)^j + (x + 8)^k

are coplanar for

all values of x
x < 0
x > 0
None of these

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Q. Let

\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k

and

\to c = - 2^i + 3^j + 6^k

. Let

{\to a}_{1}

be the projection of

\to a

\to b

and

{\to a}_{2}

be the projection of

{\to a}_{1}

\to c

. Then, which of the following is true?

$\to a$ and ${\to a}_{2}$ are collinear
${\to a}_{1}$ and $\to c$ are collinear
$\to a, {\to a}_{1}$ and $\to b$ are coplanar
$\to a, {\to a}_{1}$ and $\to c$ are coplanar

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Q. If

\to a, \to b

and

\to c

are three non – coplanar vectors, then

(\to a + \to b + \to c) . ((\to a + \to b) \times (\to a + \to c))

equals

$\to 0$
$[\to a \to b \to c]$
$2 [\to a \to b \to c]$
$- [\to a \to b \to c]$

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$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to

Zero
$[\to a \to b \to c]$
$- [\to a \to b \to c]$
None of these

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Zero
$[\to a \to b \to c]$
$- [\to a \to b \to c]$
None of these

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Q. The sum of the distinct real values of

μ,

for which the vectors,

μ^i +^j +^k,^i + μ^j +^k,^i +^j + μ^k

are co-planar, is:

$0$
$- 1$
$1$
$2$

View Solution