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Mathematics Test - 15
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Mathematics Test - 15
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  • Question 1/15
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  • Question 2/15
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    Find the equation of the line passing through the intersection of the lines x + 2y - 3 = 0 and 4x – y + 7 = 0 and which is parallel to y – x + 10 = 0.

    Solutions

    Let the equation be x + 2y - 3 + k (4x - y + 7) = 0 ............ (i)

    i.e., (1 + 4k) x + (2 - k) y + 7k - 3 = 0

    i.e., y = (-1 - 4k)/(2-k) + 3 - 7k

    Since it is parallel to y - x + 10 = 0 (i.e., y = x - 10), slopes are equal.

    So, (- 1 - 4k)/(2 - k) = 1

    - 1 - 4k = 2 - k

    k = - 1

    Substituting this in eq (i), we get:

    The required line is x + 2y - 3 + (-1)(4x - y + 7) = 0

    i.e. - 3x + 3y - 10 = 0

  • Question 3/15
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  • Question 4/15
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    A line which is parallel to the x-axis and crosses the curve y =√x at an angle of 45o is

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  • Question 5/15
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    The locus of the point of intersection of the lines x sin θ + (1 – cos θ) y = a sin θ and x sin  – (1 + cos θ ) y + a sin θ = 0 is

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  • Question 6/15
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    The equation of a circle touching the coordinate axes and the line x cos α + y sin α = 2 is x2 + y2 – 2gx + 2gy + g2 = 0, where g is equal to

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  • Question 7/15
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    C1 and C2 are circles of unit radii with centres at (0, 0) and (1, 0), respectively. C3 is a circle of unit radius, which passes through the centres of the circles C1 and C2 and has its centre above the x-axis. The equation of the common tangent to C1 and C3, which does not pass through C2, is

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  • Question 8/15
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    A chord of the circle x2 + y2 – 4x – 6y = 0 passing through the origin subtends an angle tan–1(7/4) at the point where the circle meets the positive y-axis. The equation of the chord is

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  • Question 9/15
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    If a circle passes through (a, b) and cuts the circle x2 + y2 = 4 orthogonally, then the locus of its centre is

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  • Question 10/15
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    The circles x2 + y2 – 10x + 16 = 0 and x2 + y2 = r2 intersect each other at two distinct points if

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    Centres of the given circles are (5, 0) and (0, 0) and their radii are 3 and r, respectively. The two circles will intersect at two distinct points if the distance between their centres is greater than the difference of their radii and less than the sum of radii.

    ⇒ |3 – r| < 5 < 3 + r⇒ 2 < r < 8

  • Question 11/15
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  • Question 12/15
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    Let f(x) = sin x + cos x. Then,

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  • Question 13/15
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    The sides of the rectangle of the greatest area, that can be inscribed in the ellipse x2 + 2y2 = 8, are given by

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  • Question 14/15
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    (1) f(x) has one point of inflexion.

    (2) g(x) has one point of inflexion.

    (3) f(x) has one point of local minima.

    (4) g(x) has one point of local maxima.

    Which one of the following is correct regarding the above statements?

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  • Question 15/15
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