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Vector Algebra Test - 5

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Vector Algebra Test - 5
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  • Question 1
    2 / -0.83

    The degree of the differential equation satisfying  

    Solution

    Putting x  = sin A and y = sin B in the given relation, we get
    cos A + cos B = a(sin A - sin B)

    Clearly, it is a differential equation of degree one

  • Question 2
    2 / -0.83

    The differential equation whose solution is Ax2 + By2 = 1, where A and B are arbitrary constants, is of

    Solution



    Again differentiating w.r.t. x, we get

  • Question 3
    2 / -0.83

    Differential equation of the family of circles touching the line y = 2 at (0, 2) is  

    Solution

    Options A, B, C –all have a family of circles with centre (0,2).
    But according to given conditions (0,2) is a point on the circle.
    So, none of the options are correct.
     

  • Question 4
    2 / -0.83

    The differential equation of all parabolas whose axis are parallel to the y-axis is   

    Solution

    The equation of a member of the family of parabolas having axis parallel toy-axis is
    y —Ax2 + Bx+ C    ......(1)
    where A, B, and C are arbitrary constants
    Differentiating equation (1) w.r.t. x, we get    ......(2)
    which on again differentiating w.r.t, x gives  ....(3)
    Differentiating (3) w.r.t. x, we get   

  • Question 5
    2 / -0.83

    The differential equation of all circles which pass through the origin and whose centers lie on the y-axis is  

    Solution

    If (0, k) be the centre ony-axis then its radius will be k as it passes through origin. Hence its equation is


  • Question 6
    2 / -0.83

    The differential equation whose general solution is given by, y =  , where c1 , c2 , c3 , c4 , c5 are arbitrary constants, is  

    Solution


    where l, m, n are arbitrary constant


  • Question 7
    2 / -0.83

    The equation of the curves through the point (1, 0) and whose slope is  

    Solution


  • Question 8
    2 / -0.83

    The solution of the equation log(dy/dx) = ax + by is  

    Solution

  • Question 9
    2 / -0.83

    Solution of differential equation dy –sin x sin ydx = 0 is

    Solution



  • Question 10
    2 / -0.83

    The solution of the equation  

    Solution

    Putting u = x - y, we get du/dx  = 1 - dy/dx. The given equation can be written as 1 - du/dx = cos u

  • Question 11
    2 / -0.83

    Solution


  • Question 12
    2 / -0.83

    The general solution of the differential equation  

    Solution


  • Question 13
    2 / -0.83

    The solutions of (x + y + 1) dy = dx is

    Solution

    Putting x + y + 1 = u, we have du = dx + dy and the given equation reduces to u(du - dx) = dx

  • Question 14
    2 / -0.83

    The slope of the tangent at (x, y) to a curve passing through  then the  equation of the curve is  

    Solution



  • Question 15
    2 / -0.83

    The solution of (x2 + xy)dy = (x2 + y2 )dx is  

    Solution




  • Question 16
    2 / -0.83

    The solution of (y + x + 5)dy = (y –x + 1) dx is  

    Solution

    The intersection of y –x + 1 = 0 and y + x + 5 = 0 is (-2, -3).
    Put x = X - 2, y = Y - 3  
    The given equation reduces to  

  • Question 17
    2 / -0.83

    The slope of the tangent at (x, y) to a curve passing through a point (2, 1) is  then the equation  of the curve is  

    Solution




  • Question 18
    2 / -0.83

    The solution of  satisfying y(1) = 1 is given by  

    Solution

    Rewritting the given equation as  




    Hence y2 = x(1 + x) - 1 which represents a system  of hyperbola.

  • Question 19
    2 / -0.83

    Solution of the equation  

    Solution

    The given differential equation can be written as    which is linear differential  equation of first order. 




  • Question 20
    2 / -0.83

    A function y = f(x) satisfies  then f(x) is  

    Solution


  • Question 21
    2 / -0.83

    The general solution of the equation  

    Solution


  • Question 22
    2 / -0.83

    The curve satisfying the equation  and passing through the point (4, - 2) is  

    Solution




  • Question 23
    2 / -0.83

    The solution of the differential equation  

    Solution


  • Question 24
    2 / -0.83

    Which of the following is not the differential equation of family of curves whose tangent from an angle of π/4 with the hyperbola xy = c2 ? 

    Solution



  • Question 25
    2 / -0.83

    Tangent to a curve intercepts the y-axis at a point P. A line perpendicular to this tangent through P passes through another point (1, 0). The differential equation of the curve is  

    Solution


    The equation of the tangent at the point

    The coordinates of the point


    which is the required differential equation to the curve at y =f(x).

  • Question 26
    2 / -0.83

    The curve for which the normal at any point (x, y) and the line joining the origin to that point from an isosceles triangle with the x-axis as base is

    Solution


    t is given that the triangle OPC is an isosceles triangle. 

    On integration, we get x2 -y2 = C, which is a rectangular hyperbola.

  • Question 27
    2 / -0.83

    A normal at P(x, y) on a curve meets the x-axis at Q and N is the foot of the ordinate at P. If NQ = then the equation of curve given that it passes through the point (3,1) is  

    Solution

    Let the equation of the  curve be y = f (x).



  • Question 28
    2 / -0.83

    The equation of the curve passing through (2, 7/2) and having gradient  

    Solution


  • Question 29
    2 / -0.83

    A normal at any point (x, y) to the curve y = f(x) cuts a triangle of unit area with the axis, the differential equation of the curve is

    Solution


    Equation of normal at point p is
    Y –y = (dx/dy)(X –x) 

  • Question 30
    2 / -0.83

    The differential equation of all parabola each of which has a latus rectum 4a and whose axis parallel to the x-axis is

    Solution

    Equation to the family of parabolas is (y - k)2  = 4a(x - h).


    Hence the order is 2 and the degree is 1.

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