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The number of integral values of $$k$$, for which one root of the equation $$2x^2 - 8x + k = 0$$ lies in the interval $$(1, 2)$$ and its other root lies in the interval $$(2, 3)$$, is:
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Let $$a, b$$ be two real numbers such that $$ab < 0$$. If the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a + ib$$ lies on the circle $$|z - 1| = |2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is:
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The sum $$\sum_{n=1}^{\infty} \frac{2n^2+3n+4}{(2n)!}$$ is equal to:
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Let $$P(x_0, y_0)$$ be the point on the hyperbola $$3x^2 - 4y^2 = 36$$, which is nearest to the line $$3x + 2y = 1$$. Then $$\sqrt{2}(y_0 - x_0)$$ is equal to:
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Which of the following statements is a tautology?
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Let $$9 = x_1 < x_2 < \ldots < x_7$$ be in an A.P. with common difference $$d$$. If the standard deviation of $$x_1, x_2, \ldots, x_7$$ is $$4$$ and the mean is $$\bar{x}$$, then $$\bar{x} + x_6$$ is equal to:
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Let $$P(S)$$ denote the power set of $$S = \{1, 2, 3, \ldots, 10\}$$. Define the relations $$R_1$$ and $$R_2$$ on $$P(S)$$ as $$AR_1B$$ if $$(A \cap B^c) \cup (B \cap A^c) = \phi$$ and $$AR_2 B$$ if $$A \cup B^c = B \cup A^c, \forall A, B \in P(S)$$. Then:
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If $$A = \frac{1}{2}\begin{bmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{bmatrix}$$ then,
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For the system of linear equations $$ax + y + z = 1$$, $$x + ay + z = 1$$, $$x + y + az = \beta$$, which one of the following statements is NOT correct?
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Let $$S = \left\{x \in R : 0 < x < 1 \text{ and } 2\tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$$. If $$n(S)$$ denotes the number of elements in $$S$$ then:
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Let $$f : R - \{0, 1\} \to R$$ be a function such that $$f(x) + f\left(\frac{1}{1-x}\right) = 1 + x$$. Then $$f(2)$$ is equal to:
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If $$y(x) = x^x, x > 0$$, then $$y''(2) - 2y'(2)$$ is equal to:
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The sum of the absolute maximum and minimum values of the function $$f(x) = |x^2 - 5x + 6| - 3x + 2$$ in the interval $$[-1, 3]$$ is equal to:
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The area of the region given by $$\{(x, y) : xy \leq 8, 1 \leq y \leq x^2\}$$ is:
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Let $$\alpha x = \exp(x^{\beta}y^{\gamma})$$ be the solution of the differential equation $$2x^2 y dy - (1 - xy^2)dx = 0$$, $$x \gt 0$$, $$y(2) = \sqrt{\log_e 2}$$. Then $$\alpha + \beta - \gamma$$ equals:
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Let $$\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$$ and $$\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$$ be two vectors. Then which one of the following statements is TRUE?
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Two dice are thrown independently. Let $$A$$ be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, $$B$$ be the event that the number appeared on the 1st die is even and that on the second die is odd, and $$C$$ be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then
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Number of integral solutions to the equation $$x + y + z = 21$$, where $$x \geq 1, y \geq 3, z \geq 4$$, is equal to ______.
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The total number of six digit numbers, formed using the digits $$4, 5, 9$$ only and divisible by $$6$$, is ______.
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The sum of the common terms of the following three arithmetic progressions.
$$3, 7, 11, 15, \ldots, 399$$
$$2, 5, 8, 11, \ldots, 359$$ and
$$2, 7, 12, 17, \ldots, 197$$, is equal to ______.
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If the term without $$x$$ in the expansion of $$\left(x^{\frac{2}{3}} + \frac{\alpha}{x^3}\right)^{22}$$ is $$7315$$, then $$|\alpha|$$ is equal to ______.
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Let the sixth term in the binomial expansion of $$\left(\sqrt{2^{\log_2(10-3^x)}} + \sqrt[5]{2^{(x-2)\log_2 3}}\right)^m$$ powers of $$2^{(x-2)\log_2 3}$$, be $$21$$. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is ______.
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If the $$x$$-intercept of a focal chord of the parabola $$y^2 = 8x + 4y + 4$$ is $$3$$, then the length of this chord is equal to ______.
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The line $$x = 8$$ is the directrix of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with the corresponding focus $$(2, 0)$$. If the tangent to $$E$$ at the point $$P$$ in the first quadrant passes through the point $$(0, 4\sqrt{3})$$ and intersects the $$x$$-axis at $$Q$$, then $$(3PQ)^2$$ is equal to ______.
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The value of the integral $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} dx$$ is:
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If $$\int_0^{\pi} \frac{5^{\cos x}(1+\cos x \cos 3x + \cos^2 x + \cos^3 x \cos 3x) dx}{1+5^{\cos x}} = \frac{k\pi}{16}$$, then $$k$$ is equal to ______.
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Let $$\vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k}$$, $$\vec{b} = \hat{i} + \hat{k}$$ and $$\vec{c} = \hat{i} + 2\hat{j} - 3\hat{k}$$ be three given vectors. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b} = 0$$, then $$|\vec{r}|$$ is equal to:
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Let the plane $$P$$ pass through the intersection of the planes $$2x + 3y - z = 2$$ and $$x + 2y + 3z = 6$$, and be perpendicular to the plane $$2x + y - z + 1 = 0$$. If $$d$$ is the distance of $$P$$ from the point $$(-7, 1, 1)$$, then $$d^2$$ is equal to:
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Let $$\alpha x + \beta y + \gamma z = 1$$ be the equation of a plane passing through the point $$(3, -2, 5)$$ and perpendicular to the line joining the points $$(1, 2, 3)$$ and $$(-2, 3, 5)$$. Then the value of $$\alpha \beta y$$ is equal to ______.
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The point of intersection $$C$$ of the plane $$8x + y + 2z = 0$$ and the line joining the points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ divides the line segment $$AB$$ internally in the ratio $$k : 1$$. If $$a, b, c$$ ($$|a|, |b|, |c|$$ are coprime) are the direction ratios of the perpendicular from the point $$C$$ on the line $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$, then $$|a + b + c|$$ is equal to ______.
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