The number of real roots of the equation, $$e^{4x} + e^{3x} - 4e^{2x} + e^x + 1 = 0$$ is:
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The number of real roots of the equation, $$e^{4x} + e^{3x} - 4e^{2x} + e^x + 1 = 0$$ is:
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Let $$z$$ be a complex number such that $$\left|\frac{z-i}{z+2i}\right| = 1$$ and $$|z| = \frac{5}{2}$$. Then, the value of $$|z + 3i|$$ is:
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If the number of five digit numbers with distinct digits and 2 at the $$10^{th}$$ place is $$336k$$, then $$k$$ is equal to:
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The product $$2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots$$ to $$\infty$$ is equal to:
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The value of $$\cos^3\left(\frac{\pi}{8}\right) \cdot \cos\left(\frac{3\pi}{8}\right) + \sin^3\left(\frac{\pi}{8}\right) \cdot \sin\left(\frac{3\pi}{8}\right)$$ is:
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A circle touches the y-axis at the point $$(0, 4)$$ and passes through the point $$(2, 0)$$. Which of the following lines is not a tangent to this circle?
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If $$e_1$$ and $$e_2$$ are the eccentricities of the ellipse $$\frac{x^2}{18} + \frac{y^2}{4} = 1$$ and the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$ respectively and $$(e_1, e_2)$$ is a point on the ellipse $$15x^2 + 3y^2 = k$$, then the value of $$k$$ is equal to:
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Negation of the statement: $$\sqrt{5}$$ is an integer or 5 is irrational is:
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Let the observation $$x_i(1 \le i \le 10)$$ satisfy the equations $$\sum_{i=1}^{10}(x_i - 5) = 10$$, $$\sum_{i=1}^{10}(x_i - 5)^2 = 40$$. If $$\mu$$ and $$\lambda$$ are the mean and the variance of the observations, $$x_1 - 3, x_2 - 3, \ldots, x_{10} - 3$$, then the ordered pair $$(\mu, \lambda)$$ is equal to:
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If $$A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3 \end{bmatrix}$$, $$B = adj \; A$$ and $$C = 3A$$, then $$\frac{|adj \; B|}{|C|}$$ is equal to:
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If for some $$\alpha$$ and $$\beta$$ in $$R$$, the intersection of the following three planes
$$x + 4y - 2z = 1$$
$$x + 7y - 5z = \beta$$
$$x + 5y + \alpha z = 5$$
is a line in $$R^3$$, then $$\alpha + \beta$$ is equal to:
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If $$f(x) = \begin{cases} \frac{\sin(a+2)x + \sin x}{x} & ; x < 0 \\ b & ; x = 0 \\ \frac{(x+3x^2)^{1/3} - x^{1/3}}{x^{1/3}} & ; x > 0 \end{cases}$$ is continuous at $$x = 0$$, then $$a + 2b$$ is equal to:
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Let $$f$$ be any function continuous on $$[a, b]$$ and twice differentiable on $$(a, b)$$. If all $$x \in (a, b)$$, $$f'(x) > 0$$ and $$f''(x) < 0$$, then for any $$c \in (a, b)$$, $$\frac{f(c) - f(a)}{f(b) - f(c)}$$ is:
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A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50 \; cm^3/min$$. When the thickness of ice is 5 cm, then the rate (in cm/min) at which the thickness of ice decreases, is:
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The integral $$\int \frac{dx}{(x+4)^{\frac{8}{7}}(x-3)^{\frac{6}{7}}}$$ is equal to: (where $$C$$ is a constant of integration)
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If for all real triplets $$(a, b, c)$$, $$f(x) = a + bx + cx^2$$; then $$\int_0^1 f(x) \; dx$$ is equal to:
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The value of $$\int_0^{2\pi} \frac{x \sin^8 x}{\sin^8 x + \cos^8 x} dx$$ is equal to:
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If $$f'(x) = \tan^{-1}(\sec x + \tan x)$$, $$-\frac{\pi}{2} < x < \frac{\pi}{2}$$ and $$f(0) = 0$$, then $$f(1)$$ is equal to:
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Let $$D$$ be the centroid of the triangle with vertices $$(3, -1)$$, $$(1, 3)$$ and $$(2, 4)$$. Let P be the point of intersection of the lines $$x + 3y - 1 = 0$$ and $$3x - y + 1 = 0$$. Then, the line passing through the points $$D$$ and $$P$$ also passes through the point:
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In a box, there are 20 cards, out of which 10 are labelled as $$A$$ and the remaining 10 are labelled as $$B$$. Cards are drawn at random, one after the other and with replacement, till a second $$A$$ card is obtained. The probability that the second $$A$$ card appears before the third $$B$$ card is:
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The number of distinct solutions of the equation, $$\log_{\frac{1}{2}}|\sin x| = 2 - \log_{\frac{1}{2}}|\cos x|$$ in the interval $$[0, 2\pi]$$, is ___________.
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The coefficient of $$x^4$$ in the expansion of $$(1 + x + x^2)^{10}$$ is ___________.
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If for $$x \ge 0$$, $$y = y(x)$$ is the solution of the differential equation,
$$(x + 1)dy = ((x + 1)^2 + y - 3)dx$$, $$y(2) = 0$$ then $$y(3)$$ is equal to ___________.
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If the vectors, $$\vec{p} = (a+1)\hat{i} + a\hat{j} + a\hat{k}$$, $$\vec{q} = a\hat{i} + (a+1)\hat{j} + a\hat{k}$$ and $$\vec{r} = a\hat{i} + a\hat{j} + (a+1)\hat{k}$$ $$(a \in R)$$ are coplanar and $$3(\vec{p} \cdot \vec{q})^2 - \lambda|\vec{r} \times \vec{q}|^2 = 0$$, then the value of $$\lambda$$ is ___________.
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The projection of the line segment joining the point $$(1, -1, 3)$$ and $$(2, -4, 11)$$ on the line joining the points $$(-1, 2, 3)$$ and $$(3, -2, 10)$$ is ___________.
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