Fundamental T...

Fundamental Theorem in 2D

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\frac{2}{3}

and 1 are the solutions of equations

m x^{2} + n x + 1 = 0

. Find the value of m and n

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Q. The lines whose vector equations are

\to r = \to a + t \to b

and

\to r = \to c + s \to d

are coplanar if..

$(\to b - \to c) . (\to a \times \to d) = 0$
$(\to a - \to b) . (\to c \times \to d) = 0$
$(\to a - \to c) . (\to b \times \to d) = 0$
$(\to b - \to d) . (\to a \times \to c) = 0$

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Q. The vector is one of the vectors that are linearly dependent with the vector

2^i + 3^j

$^i +^j$
$4^i -^j$
$4^i + 6^j$

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Q. If the position vector of the centroid of tetrahedron whose position vector of vertices are

^i +^j + 3^k, 2^i -^j, - 3^i + 2^j + 8^k

and

3^i + 2^j +^k

x^i + y^j + z^k .

Then the value of

4 (x - y + z) =

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Q. Find the scalar and vector components of the vector with initial point

(2, 1)

and terminal point

(- 5, 7)

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Q. Find the value of

k_{1}

and

k_{2}

for which

\to V

can be represented as

\to V = k_{1} \to A + k_{2} \to B

. Where,

\to V = 2 i - j

\to A = 3 i - 2 j

and

\to B = 3 i + 3 j .

$k_{1} = 1$ and $k_{2} = 1$
$k_{1} = \frac{3}{5} and k_{2} = \frac{1}{2}$
$k_{1} = \frac{- 2}{3} and k_{2} = \frac{2}{5}$
$k_{1} = \frac{3}{5} and k_{2} = \frac{1}{15}$

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If $(2 k - 1, k)$ is solution of the equation $10 x - 9 y = 12$ , then find the value of $k$ is

$12$
$6$
$- 2$
$2$

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$y = \log x$ is a solution of

$x y_{2} - y_{1} = 0$
$x y_{2} + y = 0$
$x y_{1} + y_{2} = 0$
$x y_{2} + y_{1} = 0$

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Q. Any vector

\to r

in the plane of two non-zero, non collinear vectors

\to a

and

\to b

can be expressed uniquely as a linear combination

x \to a + y \to b

\to a a n d \to b,

where

x \to a a n d y \to b

are the components of vector

\to r

True
False

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Q. Which of the following statements is / are true?
1.Set of two vectors

\to a, \to b

is linearly dependent if and only if either any of

\to a and \to b

is zero or they are parallel
2.

\to a, \to b and \to c

are linearly dependent

\Leftrightarrow \to a, \to b and \to c

are coplanar.

Both $1$ and $2$ are false
$1$ is true and $2$ is false
Both $1$ and $2$ are true
$2$ is true and $1$ is false

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Q. Which of the following statements is / are true?
1.Set of two vectors

\to a, \to b

is linearly dependent if and only if either any of

\to a and \to b

is zero or they are parallel
2.

\to a, \to b and \to c

are linearly dependent

\Leftrightarrow \to a, \to b and \to c

are coplanar.

Both $1$ and $2$ are false
$1$ is true and $2$ is false
Both $1$ and $2$ are true
$2$ is true and $1$ is false

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Q. Any vector

\to r

in the plane of two non-zero, non collinear vectors

\to a

and

\to b

can be expressed uniquely as a linear combination

x \to a + y \to b

\to a a n d \to b,

where

x \to a a n d y \to b

are the components of vector

\to r

True
False

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Q. Any vector

\to r

in the plane of two non-zero, non collinear vectors

\to a

and

\to b

can be expressed uniquely as a linear combination

x \to a + y \to b

\to a a n d \to b,

where

x \to a a n d y \to b

are the components of vector

\to r

True
False

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Q. Which of the following statements is / are true?
1.Set of two vectors

\to a, \to b

is linearly dependent if and only if either any of

\to a and \to b

is zero or they are parallel
2.

\to a, \to b and \to c

are linearly dependent

\Leftrightarrow \to a, \to b and \to c

are coplanar.

Both $1$ and $2$ are false
$1$ is true and $2$ is false
Both $1$ and $2$ are true
$2$ is true and $1$ is false

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Evaluate the expression when $k = - 3$ and $m = - 6$
$| m - k |$

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Q. Any vector

\to r

in the plane of two non-zero, non collinear vectors

\to a

and

\to b

can be expressed uniquely as a linear combination

x \to a + y \to b

\to a a n d \to b,

where

x \to a a n d y \to b

are the components of vector

\to r

True
False

View Solution