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JEE Main Test 42
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JEE Main Test 42

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  • Question 1/10
    4 / -1

    In the rectangular Cartesian plane, the equation of a curve is given by the equation ( x2 + y2 ) 2 = 4x2y. The exhaustive set of y – coordinates of the points on this curve is given by

    Solutions

    The equation is x + 2 ( y2 - 2y ) x2 + y4 = 0. 

    Put x2 = t then t2 + 2 ( y2 - 2y ) t + y2 = 0 .............. ( i ) 

    Equation ( i ) will be non - negative roots if 

    4 ( y2 - 2y ) 2 - 4y2 ≥ 0 

    ⇒ y ≤ 1

    Also, y4≥ 0 and, 0 ≤ - ( y2 - 2y ) . 

    ⇒ 0 ≤ y ≤ 2. So y ∈ [ 0, 1 ]

    The correct answer is: [ 0, 1 ]

  • Question 2/10
    4 / -1

    If b > a, then the equation (x– a) (x– b) – 1 = 0 has

    Solutions

    Let, f (x) = (x - a) (x - b) - 1

    ⇒f (a) = -1 and f (b) = -1.

    Also, The coefficient x2 = 1 > 0

    Hence a and b both lie between the roots of the equation f (x) = 0.

    ∴ The equation ( x - a ) ( x - b) - 1 = 0 has one root

    In (- ∞, a) and other in ( b, ∞ ) [ ∵  b > a ]

    The correct answer is: one root in (- ∞, a) and other in (b, ∞).

  • Question 3/10
    4 / -1

    If a variable plane forms a tetrahedron of constant volume 64k3 with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

    Solutions

  • Question 4/10
    4 / -1

    Solutions

     

  • Question 5/10
    4 / -1

    The distance of the point (1, 0, -3) from the plane x - y - z = 9 measured parallel to the line 

    Solutions

     

  • Question 6/10
    4 / -1

    PQ is any focal chord of the parabola y2 = 32x. The length of PQ can never be less than

    Solutions

     

  • Question 7/10
    4 / -1

    If O is the origin and OP and OQ are the tangents from the origin to the circle x2 + y2 – 6x + 4y – 8 = 0 the circumcenter of the triangle OPQ is

    Solutions

    We note that PQ is the chord of contact of the tangents from the origin to the circle


    x2 + y2 - 6x + 4y + 8 = 0 ......(i)


    Equation of PQ is 3x - 2y - 8 = 0 ......(ii)


    Equation of a circle passing through the intersection of (i) and (ii) is


    x2 + y2 - 6x + 4y + 8 + λ(3x - 2y - 8) = 0 ......(iii)


    If this represents the circumcircle of the triangle OAB, it passes through O(0,0)


    so from (iii), λ = 1,


    then equation (iii) becomes x2 + y2 - 3x + 2y = 0


    so that the required coordinates of the centre are  

    The correct answer is: ( 3/2 , – 1 )

     

  • Question 8/10
    4 / -1

    If the tangent at the point P (x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, then the mid- point of QR is

    Solutions

     

  • Question 9/10
    4 / -1

    The pair of straight lines represented by the equation 2x2 + 2y2 + 8xy + 23√x+23√y + 1 = 0 intersect at the point

    Solutions

     

  • Question 10/10
    4 / -1

    if a2 + b2 – c2 – 2ab = 0 then the point of concurrency of family of straight lines ax + by + c = 0 lies on the line

    Solutions

    (a - b)2 - c2 = 0

    (a + b + c)(a - b - c ) = 0.

    Line ax + by + c = 0 passes through 

    either of two points (1, -1) and (-1, 1 ).

    The correct answer is: y = –x

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