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question_answer1) If the line joining the points (0, 3) and (5, -2) is a tangent to the curve \[y=\frac{c}{x+1}\], then find the value of c.
question_answer2) Find the number of points to the curve \[{{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}\]where the tangents are equally inclined to the axes.
question_answer3) Find the length of the subtangent to the curve\[{{x}^{2}}+xy+{{y}^{2}}=7\text{ }at\left( 1,-3 \right)\].
question_answer4) A particle moves along the curve \[y={{x}^{3/2}}\]in the first quadrant in such a way that its distance from the origin increase at the rate of 11 units per second. Then find the value of\[\frac{dx}{dt}at\,x=3\].
question_answer5) Let \[f\left( x \right)={{x}^{3}}+ax+b\] with \[a\ne b\] and suppose the tangent lines to the graph of f at \[x=a\And x=b\] have the same gradient. Then find the value of \[f\left( 1 \right)\].
question_answer6) If the curves \[\frac{{{x}^{2}}}{{{a}^{2}}}=\frac{{{y}^{2}}}{4}=1\] and \[{{y}^{3}}=16x\] intersect at right angle, then find value of \[{{a}^{2}}\].
question_answer7) Find the length of the sub-tangent to the curve \[\sqrt{x}+\sqrt{y}=3\] at the point (4, 1).
question_answer8) The tangent to the curve \[2{{y}^{3}}=a{{x}^{2}}+{{x}^{3}}\]at (a, a) cuts of intercepts p and q on co-ordinate axes. If\[{{p}^{2}}+{{q}^{2}}=61\], then find value of a.
question_answer9) Find the area of the triangle formed by the tangent to the curve \[\sin y={{x}^{3}}-{{x}^{5}}\] at the point (1, 0) and the coordinate axes.
question_answer10) If tangent of acute angle between the curves \[y=\left| {{x}^{2}}-1 \right|\]and \[y=\sqrt{7-{{x}^{2}}}\]at their points of intersection is\[k\sqrt{3}\], then find k?
question_answer11) If Lagrange's mean value theorem is applicable for the function \[f(x)=\left\{ \begin{matrix} mx+c, & x<0 \\ {{e}^{x}}, & x\ge 0 \\ \end{matrix} \right.\] in \[[-\,2,\,\,2]\] then find the value of \[m+3c\].
question_answer12) If \[f\left( x \right)=x\left( x-2 \right)\left( x-4 \right),1\le \,\,x\,\,\le 4,\] then find a number satisfying the conditions of the mean value theorem.
question_answer13) Water is dripping out from a conical funnel of semi-vertical angle \[\frac{\pi }{4}\] at the uniform rate of \[2c{{m}^{3}}/\sec \] in its surface area through a tiny hole at the vertex in the bottom. When the slant height of the water is 4 cm, if the rate of decrease of the slant height of the water, is \[\frac{\sqrt{k}}{4\pi }\] cm/sec then find k.
question_answer14) If \[4{{x}^{2}}+p{{y}^{2}}=45\] and \[{{x}^{2}}\text{- }4{{y}^{2}}=\text{ }5\] cut orthogonally, then find the value of p.
question_answer15) Side of an equilateral triangle expands at the rate of 2 cm/sec. If the rate of increase of its area when each side is 10 cm is \[k\sqrt{3}c{{m}^{2}}/sec\] then find k.
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