The value of $$4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{4 + \ldots \infty}}}}$$ is:
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The value of $$4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{4 + \ldots \infty}}}}$$ is:
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The area of the triangle with vertices $$P(z)$$, $$Q(iz)$$ and $$R(z + iz)$$ is:
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Team 'A' consists of 7 boys and $$n$$ girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $$n$$ is equal to:
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If the fourth term in the expansion of $$\left(x + x^{\log_2 x}\right)^7$$ is 4480, then the value of $$x$$ where $$x \in N$$ is equal to:
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In a triangle $$PQR$$, the co-ordinates of the points $$P$$ and $$Q$$ are $$(-2, 4)$$ and $$(4, -2)$$ respectively. If the equation of the perpendicular bisector of $$PR$$ is $$2x - y + 2 = 0$$, then the centre of the circumcircle of the $$\triangle PQR$$ is:
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The line $$2x - y + 1 = 0$$ is a tangent to the circle at the point $$(2, 5)$$ and the centre of the circle lies on $$x - 2y = 4$$. Then, the radius of the circle is:
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Choose the incorrect statement about the two circles whose equations are given below:
$$x^2 + y^2 - 10x - 10y + 41 = 0$$ and $$x^2 + y^2 - 16x - 10y + 80 = 0$$
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The value of $$\lim_{x \to 0^+} \frac{\cos^{-1}(x - [x]^2) \cdot \sin^{-1}(x - [x]^2)}{x - x^3}$$, where $$[x]$$ denotes the greatest integer $$\leq x$$ is:
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If the Boolean expression $$(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$$ is a tautology, then the Boolean expression $$p * (\sim q)$$ is equivalent to:
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In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?
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If $$A = \begin{bmatrix} 0 & \sin\alpha \\ \sin\alpha & 0 \end{bmatrix}$$ and $$\det\left(A^2 - \frac{1}{2}I\right) = 0$$, then a possible value of $$\alpha$$ is:
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The system of equations $$kx + y + z = 1$$, $$x + ky + z = k$$ and $$x + y + zk = k^2$$ has no solution if $$k$$ is equal to:
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If $$\cot^{-1}(\alpha) = \cot^{-1} 2 + \cot^{-1} 8 + \cot^{-1} 18 + \cot^{-1} 32 + \ldots$$ upto 100 terms, then $$\alpha$$ is:
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The sum of possible values of $$x$$ for $$\tan^{-1}(x+1) + \cot^{-1}\left(\frac{1}{x-1}\right) = \tan^{-1}\left(\frac{8}{31}\right)$$ is:
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The inverse of $$y = 5^{\log x}$$ is:
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Which of the following statement is correct for the function $$g(\alpha)$$ for $$\alpha \in R$$ such that $$g(\alpha) = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin^\alpha x}{\cos^\alpha x + \sin^\alpha x} dx$$:
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Which of the following is true for $$y(x)$$ that satisfies the differential equation $$\frac{dy}{dx} = xy - 1 + x - y$$; $$y(0) = 0$$:
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Let $$\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$$ and $$\vec{b} = 7\hat{i} + \hat{j} - 6\hat{k}$$. If $$\vec{r} \times \vec{a} = \vec{r} \times \vec{b}$$, $$\vec{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = -3$$, then $$\vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})$$ is equal to:
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The equation of the plane which contains the $$y$$-axis and passes through the point $$(1, 2, 3)$$ is:
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Two dices are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is:
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If $$(2021)^{3762}$$ is divided by 17, then the remainder is ________.
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The minimum distance between any two points $$P_1$$ and $$P_2$$ while considering point $$P_1$$ on one circle and point $$P_2$$ on the other circle for the given circles' equations:
$$x^2 + y^2 - 10x - 10y + 41 = 0$$
$$x^2 + y^2 - 24x - 10y + 160 = 0$$ is ________.
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If $$A = \begin{bmatrix} 2 & 3 \\ 0 & -1 \end{bmatrix}$$, then the value of $$\det(A^4) + \det\left(A^{10} - (\text{Adj}(2A))^{10}\right)$$ is equal to ________.
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If the function $$f(x) = \frac{\cos(\sin x) - \cos x}{x^4}$$ is continuous at each point in its domain and $$f(0) = \frac{1}{k}$$, then $$k$$ is ________.
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If $$f(x) = \sin\left(\cos^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$$ and its first derivative with respect to $$x$$ is $$-\frac{b}{a}\log_e 2$$ when $$x = 1$$, where $$a$$ and $$b$$ are integers, then the minimum value of $$|a^2 - b^2|$$ is ________.
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The maximum value of $$z$$ in the following equation $$z = 6xy + y^2$$, where $$3x + 4y \leq 100$$ and $$4x + 3y \leq 75$$ for $$x \geq 0$$ and $$y \geq 0$$ is ________.
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If $$[\cdot]$$ represents the greatest integer function, then the value of $$\left|\int_0^{\sqrt{\frac{\pi}{2}}} \left[\left[x^2\right] - \cos x\right] dx\right|$$ is ________.
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If $$\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}$$, $$\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}$$ and $$\vec{c} = \hat{i} - 2\hat{j} - \hat{k}$$ such that $$\vec{a} \cdot \vec{b} = 1$$ and $$\vec{b} \cdot \vec{c} = -3$$, then $$\frac{1}{3}\left((\vec{a} \times \vec{b}) \cdot \vec{c}\right)$$ is equal to ________.
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If the equation of the plane passing through the line of intersection of the planes $$2x - 7y + 4z - 3 = 0$$, $$3x - 5y + 4z + 11 = 0$$ and the point $$(-2, 1, 3)$$ is $$ax + by + cz - 7 = 0$$, then the value of $$2a + b + c - 7$$ is ________.
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Let there be three independent events $$E_1, E_2$$ and $$E_3$$. The probability that only $$E_1$$ occurs is $$\alpha$$, only $$E_2$$ occurs is $$\beta$$ and only $$E_3$$ occurs is $$\gamma$$. Let '$$p$$' denote the probability of none of events occurs that satisfies the equations $$(\alpha - 2\beta)p = \alpha\beta$$ and $$(\beta - 3\gamma)p = 2\beta\gamma$$. All the given probabilities are assumed to lie in the interval $$(0, 1)$$. Then, $$\frac{\text{Probability of occurrence of } E_1}{\text{Probability of occurrence of } E_3}$$ is equal to ________.
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