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Mathematics
List of top Mathematics Questions
Domain of the real-valued function \(f(x) = \log(x^2 - 1) + x \, \coth^{-1}x\) is
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Mathematics
Triangles
If \(|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5, |\vec{a} + \vec{b} + \vec{c}| = \sqrt{69}\) and angle between \((\vec{a}, \vec{b}) = \dfrac{\pi}{3}\), then angle between \((\vec{c}, \vec{a}) =\)
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Mathematics
Geometry and Vectors
One of the latus recta of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ subtends an angle $2 \tan^{-1} \left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2 = 36$ and $e$ is the eccentricity of the hyperbola, then find $\sqrt{a^2 + e^2}$.
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Mathematics
Geometry
If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola onto the normal $L$ is:
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Mathematics
Geometry
If the acute angle between the circles $S \equiv x^2 + y^2 + 2kx + 4y - 3 = 0$ and $S^1 \equiv x^2 + y^2 - 4x + 2ky + 9 = 0$ is $\cos^{-1}\left(\frac{3}{8}\right)$ and the centre of $S^1 = 0$ lies in the first quadrant, then the radical axis of $S = 0$ and $S^1 = 0$ is:
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Mathematics
Geometry
If \(\cos \alpha = \sec h \beta\), then \(\beta =\)
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Mathematics
Triangles
If \((1+x)^n = \sum_{r=0}^n \binom{n}{r} x^r\), then the value of \[ C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \cdots + (C_0 + C_1 + \cdots + C_n) \] is:
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Mathematics
Combinatorics
The area of the region (in sq. units) enclosed between the curves \( y = |x| \), \( y = [x] \) and the ordinates \( x = -1, x = 0, x = 1 \) is
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Mathematics
Integration
The general solution of the differential equation \(\frac{dy}{dx} + xy = 4x - 2y + 8\) is
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Mathematics
Differential Equations
The general solution of the differential equation \((x+2y)^3\frac{dy}{dx} = y = 0, y>0\) is
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Mathematics
Differential Equations
Evaluate the integral \[ \int_{\frac{1}{2}}^{\frac{\sqrt{3}}{2}} \frac{1}{\left(x + \sqrt{1 - x^2}\right) \cdot \left(1 - x^2\right)} \, dx = \]
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Mathematics
Integration
The general solution of the differential equation \(\frac{dy}{dx} = \frac{x + y + 1}{x - 3y + 5}\) is
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Mathematics
Differential Equations
\( \int \frac{x}{(1-x^2)\sqrt{2 - x^2
} dx = \)}
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Mathematics
Exponential and Logarithmic Functions
If \( \int \frac{dx}{(x-1)^2(x-3)^2
= \sqrt{f(x)} + c \), then \( f(-1) - f(0) = \)}
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Mathematics
Integration
Evaluate the integral \[ \int_{0}^{\pi} x \cdot \sin x \cdot \int_{x}^{5} \frac{\cos x}{x} \cdot dx = \]
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Mathematics
Limits and Exponential Functions
Evaluate the integral $\displaystyle \int_0^{\frac{\pi
{2}} \log(\tan x + \cot x)\, dx$}
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Mathematics
Differentiation
\( \int \frac{1 + x + \sqrt{x + x^2}}{\sqrt{x + \sqrt{1 + x}}} dx = \)
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Mathematics
Integration
If \( \int x^2 \cos^3 x\, dx = \frac{1}{6}f(x) + g(x) \sin 2x + h(x) \cos 2x + c \), then \( f(1) + g(2) + h\left(\frac{1}{2}\right) = \)
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Mathematics
Integration
If the curves
\(y^2 = 16x \text{ and } 9x^2 + \alpha y^2 = 25\)
intersect at right angles, then
\(\alpha =\)
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Mathematics
Geometry
If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is
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Mathematics
Geometry
If the function
\(y = \sin(x)(1 - \cos x)\)
is defined in the interval
\([-\pi, \pi]\),
then y is strictly increasing in the interval
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Mathematics
Geometry
If
\(y = (ax + b)\cos x\),
then
\(y_2 + y_1 \sin 2x + y(1 + \sin^2 x) = \)
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Mathematics
Differentiability
If the normal drawn at the point P on the curve
\(y = x \log x\)
is parallel to the line
\(2x - 2y = 3\),
then P =
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Mathematics
Geometry
If
\(x^2 + y^2 = \dfrac{1}{t} \text{ and } x^4 + y^4 = t^2 + \dfrac{1}{t^2},\)
then
\(\dfrac{dy}{dx} =\)
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Mathematics
Continuity
If a function defined by
\[ f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x<0
a, & x = 0
\dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x} - 4}}, & x>0 \end{cases} \]
is continuous at
\(x = 0\),
then
\(a =\)
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Mathematics
Continuity
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