Self Studies

Tangents and it...

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  • Question 1
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    If the circle x2+y2+2gx+2fy+c=0x^2+y^2+2gx+2fy+c=0 is touched by y=xy=x at P such that OP = 626\sqrt{2}
    then the value of c is

  • Question 2
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    The angle made by the tangent of the curve x=a(t+sintcost);y=a(1+sint)2\displaystyle x = a(t + \sin t \cos t); y = a (1 + \sin t)^2 with the x-axis at any point on it is

  • Question 3
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    Equation of the line through the point (12,2)\left(\dfrac{1}{2}, 2 \right) and tangent to the parabola y=x22+2\displaystyle y = \frac {-x^2}{2}+2 and secant to the curve y=4x2\displaystyle y = \sqrt {4 - x^2} is :

  • Question 4
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    What is the minimum intercept made by the axes on the tangent to the ellipse x2a2+y2b2=1 \displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 ?

  • Question 5
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    Let CC be the curve y=x3\displaystyle y = x^3 (where xx takes all real values.) The tangent at AA meets the curve again at BB. If the gradient of the curve at BB is KK times the gradient at AA then KK is equal to

  • Question 6
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    The graphs y=2x34x+2y=2x^3-4x+2 and y=x3+2x1y=x^3+2x-1 intersect at exactly 3 distinct points. The slope of the line passing through two of these points

  • Question 7
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    Consider the curve represented parametrically by the equation
    x=t34t23t\displaystyle x = t^3 - 4t^2 - 3t and y=2t2+3t5\displaystyle y = 2t^2 + 3t - 5 where t ϵR\displaystyle t \: \epsilon \: R.
    If HH denotes the number of point on the curve where the tangent is horizontal and VV the number of point where the tangent is vertical then

  • Question 8
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    The number of points on the curve x3/2+y3/2=a3/2x^{3/2}+y^{3/2}=a^{3/2}, where the tangents are equally inclined to the axes, is

  • Question 9
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    Let  f(x)=lnmx(m>0)\displaystyle f(x) = ln \: mx (m > 0) and g(x)=pxg(x) = pxThen the equation  f(x)=g(x)\displaystyle |f(x)| = g(x) has only one solution for

  • Question 10
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    A point  P(a,an)\displaystyle P(a, a^n) on the graph of y=xn(nϵN)\displaystyle y = x^n (n \: \epsilon \: N) in the first quadrant a normal is drawn. The normal intersects the y-axis at the point (0,b)(0, b). If a0Limb=12\displaystyle _{a \rightarrow 0}^{Lim \: b} \textrm{=} \frac {1}{2}, then n equals

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