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The \(\lim _{x \rightarrow \infty}\left(\frac{3 x-1}{3 x+1}\right)^{4 x}\) equals

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(\int_{\log _{e} 2}^{x}\left(e^{x}-1\right)^{-1} d x=\log _{e} \frac{3}{2}\) then the value of \(x\) is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The normal to a curve at \(P(x, y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\) then the curve is

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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Let \(g(x)=\int_{x}^{2 x} \frac{f(t)}{t} d t\) where \(x>0\) and \(f\) be continuous function and \(f(2 x)=f(x)\), then

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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The value of the integral \(\int_{-1 / 2}^{1 / 2}\left\{\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right\}^{1 / 2} d x\) is equal to

asked Dec 10, 2021 in Sets, relations and functions by kritika (90.1k points)

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1 answer

$$ \int_{1}^{3} \frac{|x-1|}{|x-2|+|x-3|} d x= $$

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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3 answers

If \(\mathrm{I}=\lim _{x \rightarrow 0} \sin \left(\frac{e^{x}-x-1-\frac{x^{2}}{2}}{x^{2}}\right)\), then limit

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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If \(\int \frac{\sin 2 x}{(a+b \cos x)^{2}} d x=\alpha\left[\log _{e}|a+b \cos x|+\frac{a}{a+b \cos x}\right]+c\), then \(\alpha=\)

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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2 answers

Let \(f: R \rightarrow R\) be such that \(f(0)=0\) and \(\left|f^{\prime}(x)\right| \leq 5\) for all \(x\). Then \(f(1)\) is in

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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1 answer

The equation \(6^{x}+8^{x}=10^{x}\) has

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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1 answer

Let \(f: D \rightarrow R\) where \(D=[0,1] \cup[2,4]\) be defined by \(f(x)=\left\{\begin{array}{l}x, \quad \text { if } x \in[0,1] \\ 4-x, \text { if } x \in[2,4]\end{array}\right.\). Then,

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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3 answers

For \(y=\sin ^{-1}\left\{\frac{5 x+12 \sqrt{1-x^{2}}}{13}\right\} ;|x| \leq 1\), if \(a\left(1-x^{2}\right) y_{2}+b x y_{1}=0\) then \((a, b)=\)

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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3 answers

Consider the functions \(f_{1}(x)=x, f_{2}(x)=2+\log _{e} x, x>0\). The graphs of the functions intersect.

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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\(f(x)\) is real valued function such that \(2 f(x)+3 f(-x)=15-4 x\) for all \(x \in R\). Then \(f(2)=\)

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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If from a point \(P(a, b, c)\), perpendiculars \(P A\) and \(P B\) are drawn to \(Y Z\) and \(Z X\) planes respectively, then the equation of the plane \(\mathrm{OAB}\) is

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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A line with positive direction cosines passes through the point \(P(2,-1,2)\) and makes equal angle with co-ordinate axes. The line meets the plane \(2 x+y+z=9\) at point \(Q\). The length of the line segment \(P Q\) equals.

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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1 answer

The locus of the center of a variable circle which always touches two given circles externally is

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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3 answers

The co-ordinate of a point on the auxiliary circle of the ellipse \(x^{2}+2 y^{2}=4\) corresponding to the point on the ellipse whose eccentric angle is \(60^{\circ}\) will be

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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The locus of the vertices of the family of parabolas \(6 y=2 a^{3} x^{2}+3 a^{2} x-12 a\) is

asked Dec 9, 2021 in Sets, relations and functions by kritika (90.1k points)

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