The equation $$e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0$$, $$x \in R$$ has:
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The equation $$e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0$$, $$x \in R$$ has:
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The complex number $$z = \dfrac{i-1}{\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}}$$ is equal to:
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Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$a_7 = 3$$, the product $$(a_1 a_4)$$ is minimum and the sum of its first $$n$$ terms is zero then $$n! - 4a_{n(n+2)}$$ is equal to
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The coefficient of $$x^{-6}$$, in the expansion of $$\left(\dfrac{4x}{5} + \dfrac{5}{2x^2}\right)^9$$, is
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If $$^{2n+1}P_{n-1} : ^{2n-1}P_n = 11 : 21$$, then $$n^2 + n + 15$$ is equal to:
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The set of all values of $$a^2$$ for which the line $$x + y = 0$$ bisects two distinct chords drawn from a point $$P\left(\dfrac{1+a}{2}, \dfrac{1-a}{2}\right)$$ on the circle $$2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$$, is equal to:
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Let H be the hyperbola, whose foci are $$(1 \pm \sqrt{2}, 0)$$ and eccentricity is $$\sqrt{2}$$. Then the length of its latus rectum is:
$$\lim_{x \to \infty} \dfrac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3$$
The number of values of $$r \in \{p, q, \sim p, \sim q\}$$ for which $$((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$$ is a tautology, is:
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Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $$\alpha(> 0)$$, and the mean and standard deviation of marks of class B of $$n$$ students be respectively 55 and $$30 - \alpha$$. If the mean and variance of the marks of the combined class of $$100 + n$$ students are respectively 50 and 350, then the sum of variances of classes A and B is
Among the relations
$$S = \{(a,b): a, b \in R - \{0\}, 2 + \dfrac{a}{b} > 0\}$$ and $$T = \{(a,b): a, b \in R, a^2 - b^2 \in Z\}$$,
If a point $$P(\alpha, \beta, \gamma)$$ satisfying $$(\alpha \; \beta \; \gamma) \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = (0 \; 0 \; 0)$$ lies on the plane $$2x + 4y + 3z = 5$$, then $$6\alpha + 9\beta + 7\gamma$$ is equal to
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Let $$(a, b) \subset (0, 2\pi)$$ be the largest interval for which $$\sin^{-1}(\sin\theta) - \cos^{-1}(\sin\theta) > 0$$, $$\theta \in (0, 2\pi)$$, holds. If $$\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$$ and $$\alpha - \beta = b - a$$, then $$\alpha$$ is equal to:
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Let $$f: R - \{2, 6\} \to R$$ be real valued function defined as $$f(x) = \dfrac{x^2+2x+1}{x^2-8x+12}$$. Then range of $$f$$ is
The absolute minimum value, of the function $$f(x) = |x^2 - x + 1| + [x^2 - x + 1]$$, where $$[t]$$ denotes the greatest integer function, in the interval $$[-1, 2]$$, is
Let $$y = y(x)$$ be the solution of the differential equation $$(3y^2 - 5x^2)y\,dx + 2x(x^2 - y^2)\,dy = 0$$ such that $$y(1) = 1$$. Then $$|(y(2))^3 - 12y(2)|$$ is equal to:
Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = \hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{c} = 5\hat{i} - 3\hat{j} + 3\hat{k}$$, be three vectors. If $$\vec{r}$$ is a vector such that, $$\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{r} \cdot \vec{a} = 0$$, then $$25|\vec{r}|^2$$ is equal to
Let the plane $$P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0$$ be parallel to the line $$L: \dfrac{x+2}{2} = \dfrac{y-3}{3} = \dfrac{z+4}{5}$$. If the intercept of $$P$$ on the y-axis is 1, then the distance between $$P$$ and $$L$$ is
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Let $$P$$ be the plane, passing through the point $$(1, -1, -5)$$ and perpendicular to the line joining the points $$(4, 1, -3)$$ and $$(2, 4, 3)$$. Then the distance of $$P$$ from the point $$(3, -2, 2)$$ is
The foot of perpendicular from the origin $$O$$ to a plane $$P$$ which meets the co-ordinate axes at the point $$A, B, C$$ is $$(2, a, 4)$$, $$a \in N$$. If the volume of the tetrahedron $$OABC$$ is 144 unit$$^3$$, then which of the following points is NOT on $$P$$?
The sum $$1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$$ is ______.
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If the constant term in the binomial expansion of $$\left(\dfrac{x^{5/2}}{2} - \dfrac{4}{x^l}\right)^9$$ is -84 and the coefficient of $$x^{-3l}$$ is $$2^\alpha \beta$$ where $$\beta < 0$$ is an odd number, then $$|\alpha l - \beta|$$ is equal to ______.
Let $$S$$ be the set of all $$a \in N$$ such that the area of the triangle formed by the tangent at the point $$P(b, c)$$, $$b, c \in N$$, on the parabola $$y^2 = 2ax$$ and the lines $$x = b$$, $$y = 0$$ is 16 unit$$^2$$, then $$\sum_{a \in S} a$$ is equal to
Let $$A = [a_{ij}]$$, $$a_{ij} \in Z \cap [0, 4]$$, $$1 \le i, j \le 2$$. The number of matrices $$A$$ such that the sum of all entries is a prime number $$p \in (2, 13)$$ is ______.
Let $$A$$ be a $$n \times n$$ matrix such that $$|A| = 2$$. If the determinant of the matrix $$\text{Adj}\left(2 \cdot \text{Adj}(2A^{-1})\right)$$ is $$2^{84}$$, then $$n$$ is equal to ______.
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Let $$\alpha > 0$$. If $$\int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$$ then $$\alpha$$ is equal to :
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If $$\phi(x) = \dfrac{1}{\sqrt{x}} \int_{\pi/4}^{x} \left(4\sqrt{2}\sin t - 3\phi'(t)\right) dt$$, $$x > 0$$ then $$\phi'\left(\dfrac{\pi}{4}\right)$$ is equal to ______.
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Let the area of the region $$\{(x,y): |2x-1| \le y \le |x^2-x|, 0 \le x \le 1\}$$ be $$A$$. Then $$(6A + 11)^2$$ is equal to ______.
Let $$\vec{a}, \vec{b}, \vec{c}$$ be three vectors such that $$|\vec{a}| = \sqrt{31}$$, $$4|\vec{b}| = |\vec{c}| = 2$$ and $$2(\vec{a} \times \vec{b}) = 3(\vec{c} \times \vec{a})$$. If the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\dfrac{2\pi}{3}$$, then $$\left(\dfrac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$$ is equal to ______.
Let $$A$$ be the event that the absolute difference between two randomly chosen real numbers in the sample space $$[0, 60]$$ is less than or equal to $$a$$. If $$P(A) = \dfrac{11}{36}$$, then $$a$$ is equal to ______.
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