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Vectors Test - 12

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Vectors Test - 12
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  • Question 1
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A): The position vector of a point say and its magnitude is

    Reason (R): If then coefficient of are called the direction ratios of vector

    Solution

    Assertion (A) and Reason (R) both are individually correct.

  • Question 2
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A) : The area of parallelogram with diagonals

    Reason (R): If represent the adjacent sides of a triangle, then the area of triangle can be obtained by evaluating

    Solution

    If represent the adjacent sides of a triangle, then the area of triangle can be obtained by evaluating

  • Question 3
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A): The direction of cosines of vector

    Reason (R): A vector having zero magnitude and arbitrary direction is called ‘zero vector’ or ‘null vector’.

    Solution

    Assertion (A) is correct.

    Direction cosines of

  • Question 4
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A): The position of a particle in a rectangular coordinate system is (3, 2, 5). Then its position vector be

    Reason (R): The displacement vector of the particle that moves from point P(2, 3, 5) to point Q(3, 4, 5) is

    Solution

    Assertion (A) is wrong.

    The position of a particle in a rectangular coordinate system is (3, 2, 5). Then its position vector be

    Reason (R) is correct.

    The displacement vector of the particle that moves from point P(2, 3, 5) to point Q(3, 4, 5)

  • Question 5
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A): The vectors which can undergo parallel displacement without changing its magnitude and direction are called free vectors.

    Reason (R):

    Solution

    Assertion (A) and Reason (R) both are individually correct.

    Reason (R) is the distributive property of dot product.

  • Question 6
    1 / -0.25

    Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as

    Assertion (A): For any two vectors we always have

    Reason (R): The given inequality holds trivially when either in such a case

    Then consider

    So, let us check it for

    For cos θ ≤ 1, we have :

    or

    or

  • Question 7
    1 / -0.25

     

    If   are any two vectors, then

     

    Solution

     

     

    |a+b| ≤|a| + |b|
    Let us take an example : a = 1, b= 2
    |1 + 2| ≤|1| + |2|
    |3| ≤|3|
    Hence, proved

     

     

  • Question 8
    1 / -0.25

     

    The angle between the vectors is:   is :

     

    Solution

     

     

    a = 6i - 3j + 2k    b = 2i + j - 2k
    a.b = 12 - 3 - 4 = 5
    |a| = [(6)2 + (-3)2 + (2)2 ]1/2  
    |a| = [36 + 9 + 4]½
    |a| = (49)½
    |a| = 7
    |b| = [(2)2 + (1)2 + (-2)2 ]½
    |b| = [4 + 1 + 4]½
    |b| = 3
    Cos θ= (a.b)/|a||b|
    = 5/(7)(3)
    = 5/21
    θ= cos-1 (5/21)

     

     

  • Question 9
    1 / -0.25

     

    If    are two vectors, such that   , then  = ……​

     

    Solution

     

     

     |a - b|2 = |a|2 + |b|2 - 2|a||b|
    |a - b|2  = (3)2 + (2)2 - 2(5)
    |a - b|2  = 9 + 4 - 10
    |a - b|2  = 3  
    |a - b|  = (3)½.

     

     

  • Question 10
    1 / -0.25

     

    The projection of the vector    on the vector is:​

     

    Solution

     

     

    Projection = (A.B)/|B|
    = [(i + 2j + k) . (2i + 3j + 2k)]/[(2)2 + (3)2 + (2)2 ]½
    = (2 + 6 + 2)/[4 + 9 + 4]½
    = 10/(17)1/2

     

     

  • Question 11
    1 / -0.25

     

    The angle between two non-zero vectors   is given by

     

    Solution

     

     

    A sequence is a function whose domain is the set of natural numbers or a subset of the natural numbers. We usually use the symbol an to represent a sequence, where n is a natural number and an is the value of the function on n. A sequence may be finite or infinite.

     

     

  • Question 12
    1 / -0.25

     

    If | a | = 2, | b | = 5 and | a ×b | = 8, thencan be equal to

     

    Solution

     

     

     

     

  • Question 13
    1 / -0.25

     

    If  are unit vectors along X-axis, Y-axis and Z-axis respectively, then

     

    Solution

     

     

    As we know that,

     

     

  • Question 14
    1 / -0.25

     

    The area of triangle whose adjacent sides are is :

     

    Solution

     

     

    Area of triangle = ½(a * b)
    a = (1, 0, -2)  b = (2, 3, 1)
    = i(0 + 6) + j(-4 - 1) + k(3 - 0)
    = 6i - 5j + 3k
    |a * b| = (36 + 25 + 9)½
    |a * b| = (70)½
    Area of triangle = ½(a * b)
    = [(70)½ ]/2

     

     

  • Question 15
    1 / -0.25

     

    The value of   is:

     

    Solution

     

     

     i.(j * k) + j.(k * i) + k(i * j)
    = i.(i) + j.(j) + k.(k)
    = 1 + 1 + 1  
    = 3

     

     

  • Question 16
    1 / -0.25

     

    The area of parallelogram whose sides areis:​

     

    Solution

     

     

    A = {(1, -3, 4)}   B = {(3, 1, -2)}
    A * B = {(i^(6-4) - j^(-2 -12) + k^ (1+9)} 
    = 2i^ + 14j^ + 10k^
    Area = [(2)2 + (14)2 + (10)2 ]½
    = (300)½
    = 10(3)½ sq unit

     

     

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