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Mathematics Test 126
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Mathematics Test 126

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  • Question 1/10
    4 / -1

    If z is a complex number such that |z| + z = 3 + 4i then z is equal to

    Solutions

     

  • Question 2/10
    4 / -1

    If z and w are two complex number satisfying |z – 1| = 2 and |w – 5| = 3, then the maximum value of |z – 4w| is

    Solutions

    |4w – 20| = 12 let z1 = 4w
    ⇒ |z1 – 20| = 12

    so max value of |z – z1| is 33.

     

  • Question 3/10
    4 / -1

    Equation of plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 which is parallel to x-axis is

    Solutions

    Let equation of plane is p1 + λp2 = 0
    or (x + y + z – 1) + λ(2x + 3y – z + 4) = 0 
    which is parallel to x-axis or perpendicular yz plane 
    x(1 + 2λ) + y(1 + 3λ) + z(1 – λ) + – 1 + 4λ = 0 
    1.x + 0.y + 0.z = 0 
    ∴ (1 + 2λ).1 + (1 + 3λ).0 + (1 – λ).0 = 0 

    or y – 3z + 6 = 0

     

  • Question 4/10
    4 / -1

    A plane passes through the point (1, –2, 3) and is parallel to the plane 2x – 2y + z = 0. The distance of the point (–1, 2, 0) from the plane is

    Solutions

    Equation of plane parallel to 2x – 2y + z = 0 is
    2x – 2y + z + λ = 0
    which is passing through (1, –2, 3)
    ∴ λ = –9
    ∴ 2x – 2y + z – 9 = 0
    Now distance from (–1, 2, 0)

     

  • Question 5/10
    4 / -1

    A point z moves such that |z – 3 – i| + |z – 1 – 3i| = 3, then locus of z is -

    Solutions

    Clearly PA + PB = 3
    where A ≡ (3,1) & B ≡ (1,3)
    ∴ P moves on an ellipse whose focii are A & B.

     

  • Question 6/10
    4 / -1

    If z1, z2 are two non-zero complex numbers such that |19z1 – 31z2|2 = |19z1|2 + |31z2|2 then

    Solutions

    |α – β|2 = |α|2 + |β|2

     

  • Question 7/10
    4 / -1

    Two lines L1 : x = 5,  and L2 : x = α,  are coplanar. Then α can take value(s)

    Solutions

     

  • Question 8/10
    4 / -1

    A line ℓ passing through the origin is perpendicular to the lines

    ℓ1 : (3 + t)î + (–1 + 2t)ĵ + (4 + 2t), –∞ < t < ∞
    ℓ2 : (3 + 2s)î + (3 + 2s)ĵ + (2 + s), –∞ < s < ∞

    Then, the coordinate(s) of the point(s) on ℓ2 at a distance of  from the point of intersection of ℓ and ℓ1 is(are) -

    Solutions

     

  • Question 9/10
    4 / -1

    Consider planes P1 & P2 given by P1 : x + y + z = 3 and P2 : x – 2y + z = 3.

    Line of intersection of P1 & P2 is given by -

    Solutions

     

  • Question 10/10
    4 / -1

    Consider planes P1 & P2 given by P1 : x + y + z = 3 and P2 : x – 2y + z = 3.

    Equation of a plane which is perpendicular to P1 and parallel to the line of intersection of P1 & P2 is given by -

    Solutions

    Required plane must be parallel to P2

    ∵ P2 is ⊥ P1

     

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