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Let \(P\) be a variable point on a circle \(C\) and \(Q\) be a fixed point outside \(C\). If \(R\) is the midpoint of the line segment \(P Q\), then locus of \(R\) is
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Dec 10, 2021
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A plane meets the co-ordinate axes at the points \(A, B, C\) respectively in such a way that the centroid of \(\triangle A B C\) is \(\left(1, r, r^{2}\right)\) for some real \(r\). If the plane passes through the point \((5,5,-12)\) then \(r=\)
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Let \(f(x)\) be a continuous periodic function with period \(T\). Let \(I=\int_{a}^{a+T} f(x) d x\). Then
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The differential of \(f(x)=\log _{e}\left(1+e^{10 x}\right)-\tan ^{-1}\left(e^{5 x}\right)\) at \(x=0\) and for \(d x=0.2\) is
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Dec 10, 2021
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If \(b=\int_{0}^{1} \frac{e^{t}}{t+1} d t\), then \(\int_{a-1}^{a} \frac{e^{-1}}{t-a-1}\) is
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The plane \(\ell x+m y=0\) is rotated about its line of intersection with the plane \(z=0\) through an angle \(\alpha\). The equation changes to
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Dec 10, 2021
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Let \(I=\int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} d x\). Then
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Dec 10, 2021
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The points of intersection of two ellipses \(x^{2}+2 y^{2}-6 x-12 y+20=0\) and \(2 x^{2}+y^{2}-10 x-6 y+15=0\) lie on a circle. The centre of the circle is
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Dec 10, 2021
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The determinant \(\left|\begin{array}{ccc}a^{2}+10 & a b & a c \\ a b & b^{2}+10 & b c \\ a c & b c & c^{2}+10\end{array}\right|\) is
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Dec 10, 2021
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Let \(R\) be the real line. Let the relations \(S\) and \(T\) on \(R\) be defined by \(S=\{(x, y): y=x+1,0<x<2\}, T=\{(x, y):(x-y)\) is an integer \(\}\). Then
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The co-efficient of \(a^{3} b^{4} c^{5}\) in the expansion of \((b c+c a+a b)^{6}\) is
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Three unequal positive numbers \(a, b, c\) are such that \(a, b, c\) are in G.P. while \(\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)\) are in A.P. Then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are the lengths of the sides of
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If \(a(\vec{\alpha} \times \vec{\beta})+b(\vec{\beta} \times \vec{\gamma})+c(\vec{\gamma} \times \vec{\alpha})=\overrightarrow{0}\), where \(a, b, c\) are non-zero scalars, then the vectors \(\vec{\alpha}, \vec{\beta}, \vec{\gamma}\) are
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Dec 10, 2021
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If the tangent at the point \(P\) with co-ordinates \((h, k)\) on the curve \(y^{2}=2 x^{3}\) is perpendicular to the straight line \(4 x=3 y\), then
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Dec 10, 2021
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Given that \(f: S \rightarrow R\) is said to have a fixed point at \(c\) of \(S\) if \(f(c)=c\). Let \(f:[1, \infty) \rightarrow R\) be defined by \(f(x)=1+\sqrt{x}\). Then
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Dec 10, 2021
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The area bounded by the parabolas \(y=4 x^{2}, y=\frac{x^{2}}{9}\) and the straight line \(y=2\) is
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Dec 10, 2021
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The \(\lim _{x \rightarrow \infty}\left(\frac{3 x-1}{3 x+1}\right)^{4 x}\) equals
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Dec 10, 2021
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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
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The differential equation of all the ellipses centred at the origin and have axes as the co-ordinate axes is
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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
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