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The general value of the real angle \(\theta\), which satisfies the equation, \((\cos \theta+i \sin \theta)(\cos 2 \theta+i \sin 2 \theta) \ldots \ldots .\) \((\cos n \theta+i \sin n \theta)=1\) is given by, (assuming \(k\) is an integer)
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Let \(z\) be a complex number such that the principal value of argument, \(\arg z>0\). Then \(\arg z-\arg (-z)\) is
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If \(\log _{2} 6+\frac{1}{2 x}=\log _{2}\left(2^{\frac{1}{x}}+8\right)\), then the values of \(x\) are
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The three sides of a right-angled triangle are in G.P (geometric progression). If the two acute angles be \(\alpha\) and \(\beta\), then \(\tan \alpha\) and \(\tan \beta\) are
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If the radius of a spherical balloon increases by \(0.1 \%\), then its volume increases approximately by
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Let \(P(4,3)\) be a point on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 .\) If the normal at \(P\) intersects the \(X\)-axis at \((16,0)\), then the eccentricity of the hyperbola is
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General solution of \((x+y)^{2} \frac{d y}{d x}=a^{2}, a \neq 0\) is (c is an arbitrary constant)
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The general solution of the differential equation \(\left(1+e^{\frac{x}{y}}\right) d x+\left(1-\frac{x}{y}\right) e^{\frac{x}{y}} d y=0\) is (c is an arbitrary constant)
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\(\lim _{n \rightarrow \infty} \frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+\ldots+\sqrt{\frac{n}{n+3(n-1)}}\right\}\)
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The value of the integeral \(\int_{-1}^{1}\left\{\frac{x^{2015}}{e^{|x|}\left(x^{2}+\cos x\right)}+\frac{1}{e^{|x|}}\right\} d x\) is equal to
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If \(\int 2^{2^{x}} \cdot 2^{x} d x=A \cdot 2^{2^{x}}+c\), then \(A=\)
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The value of \(\lim _{x \rightarrow 0} \frac{1}{x}\left[\int_{y}^{a} e^{\sin ^{2} t} d t-\int_{x+y}^{a} e^{\sin ^{2} t} d t\right]\) is equal to
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The value of the integration \(\int_{-\pi / 4}^{\pi / 4}\left(\lambda|\sin x|+\frac{\mu \sin x}{1+\cos x}+\gamma\right) d x\)
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\(y=\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x\) is an equation of a family of
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If \(\int \cos x \log \left(\tan \frac{x}{2}\right) d x=\sin x \log \left(\tan \frac{x}{2}\right)+f(x)\) then \(f(x)\) is equal to, (assuming \(c\) is a arbitrary real constant)
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\(\lim _{x \rightarrow 0+}\left(x^{n} \ln x\right), n>0\)
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Tangent is drawn at any point \(P(x, y)\) on a curve, which passes through \((1,1)\). The tangent cuts \(X\)-axis and \(Y\)-axis at \(A\) and \(B\) respectively. If \(A P: B P=3: 1\), then
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Let \(f(x)=\frac{1}{3} x \sin x-(1-\cos x)\). The smallest positive interger \(k\) such that \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{k}} \neq 0\) is
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Let \(y=\frac{x^{2}}{(x+1)^{2}(x+2)}\). Then \(\frac{d^{2} y}{d x^{2}}\) is
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Consider a tangent to the ellipse \(\frac{x^{2}}{2}+\frac{y^{2}}{1}=1\) at any point. The locus of the midpoint of the portion intercepted between the axes is
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