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The area of the region \(\left\{(x, y): x^{2}+y^{2} \leq 1 \leq x+y\right\}\) is
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Dec 10, 2021
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Let \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+12\). Then
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Dec 10, 2021
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Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be twice continuously differentiable (or \(f\) " exists and is continuous) such that \(f(0)=f(1)=f^{\prime}(0)=0 .\) Then
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Dec 10, 2021
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If \(\lim _{x \rightarrow 0}\left(\frac{1+c x}{1-c x}\right)^{1 / x}=4\), then \(\lim _{x \rightarrow 0}\left(\frac{1+2 c x}{1-2 c x}\right)^{1 / x}\) is
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Dec 10, 2021
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Let \(y=f(x)=2 x^{2}-3 x+2\). The differential of \(y\) when \(x\) changes from 2 to \(1.99\) is
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Dec 10, 2021
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The domain of \(f(x)=\sqrt{\left(\frac{1}{\sqrt{x}}-\sqrt{x+1}\right)}\) is
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Dec 10, 2021
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Let \(f(x)=\sin x+\cos a x\) be periodic function. Then
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Dec 10, 2021
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The sine of the angle between the straight line \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) and the plane \(2 x-2 y+z=5\) is
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Dec 10, 2021
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The equation of the plane through the point \((2,-1,-3)\) and parallel to the lines \(\frac{x-1}{2}=\frac{y+2}{3}=\frac{z}{-4}\) and \(\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\) is
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Dec 10, 2021
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The equation of the latus rectum of a parabola is \(x+y=8\) and the equation of the tangent at the vertex is \(x+y=12\). Then the length of the latus rectum is
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Dec 10, 2021
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If \(B\) and \(B^{\prime}\) are the ends of minor axis and \(S\) and \(S^{\prime}\) are the foci of the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\), then the area of the rhombus SBS'B' will be
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Dec 10, 2021
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A double ordinate \(P Q\) of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is such that \(\Delta O P Q\) is equilateral, \(O\) being the centre of the hyperbola. Then the eccentricity e satisfies the relation
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Dec 10, 2021
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The length of the chord of the parabola \(y^{2}=4 a x(a>0)\) which passes through the vertex and makes an acute angle \(\alpha\) with the axis of the parabola is
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Dec 10, 2021
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The equation of circle of radius \(\sqrt{17}\) unit, with centre on the positive side of \(x\)-axis and through the point \((0,1)\) is
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Dec 10, 2021
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Area in the first quadrant between the ellipses \(x^{2}+2 y^{2}=a^{2}\) and \(2 x^{2}+y^{2}=a^{2}\) is
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Dec 10, 2021
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A straight line through the origin \(O\) meets the parallel lines \(4 x+2 y=9\) and \(2 x+y+6=0\) at \(P\) and \(Q\) respectively. The point \(\mathrm{O}\) divides the segment \(\mathrm{PQ}\) in the ratio
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Dec 10, 2021
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Let each of the equations \(x^{2}+2 x y+a y^{2}=0 \& a x^{2}+2 x y+y^{2}=0\) represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by
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Dec 10, 2021
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The locus of the centre of the circles which touch both the circles \(x^{2}+y^{2}=a^{2}\) and \(x^{2}+y^{2}=4 a x\) externally is
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Dec 10, 2021
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The equation \(r \cos \left(\theta-\frac{\pi}{3}\right)=2\) represents
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Dec 10, 2021
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The differential equation of the family of curves \(y=e^{x}(A \cos x+B \sin x)\) where \(A, B\) are arbitrary constants is
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Dec 10, 2021
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