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Let $$s_1, s_2, s_3, \ldots, s_{10}$$ respectively be the sum of 12 terms of 10 A.P.s whose first terms are $$1, 2, 3, \ldots, 10$$ and the common differences are $$1, 3, 5, \ldots, 19$$ respectively. Then $$\sum_{i=1}^{10} s_i$$ is equal to
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Fractional part of the number $$\frac{4^{2022}}{15}$$ is equal to
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Let $$PQ$$ be a focal chord of the parabola $$y^2 = 36x$$ of length 100, making an acute angle with the positive $$x-$$axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that $$PM : MQ = 3 : 1$$. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $$PQ$$?
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Let the tangent and normal at the point $$(3\sqrt{3}, 1)$$ on the ellipse $$\frac{x^2}{36} + \frac{y^2}{4} = 1$$ meet the $$y-$$axis at the points $$A$$ and $$B$$ respectively. Let the circle $$C$$ be drawn taking $$AB$$ as a diameter and the line $$x = 2\sqrt{5}$$ intersect $$C$$ at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point $$(\alpha, \beta)$$, then $$\alpha^2 - \beta^2$$ is equal to
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The negation of the statement $$((A \wedge (B \vee C)) \Rightarrow (A \vee B)) \Rightarrow A$$ is
The number of symmetric matrices of order 3, with all the entries from the set $$\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$ is
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Let $$B = \begin{bmatrix} 1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4 \end{bmatrix}$$, $$\alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A| = 2$$. Then $$\begin{bmatrix} \alpha & -2\alpha & \alpha \end{bmatrix} B \begin{bmatrix} \alpha \\ -2\alpha \end{bmatrix}$$ is equal to
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For the system of linear equations
$$2x + 4y + 2az = b$$
$$x + 2y + 3z = 4$$
$$2x + 5y + 2z = 8$$
which of the following is NOT correct?
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For $$x \in \mathbb{R}$$, two real valued functions $$f(x)$$ and $$g(x)$$ are such that, $$g(x) = \sqrt{x} + 1$$ and $$fog(x) = x + 3 - \sqrt{x}$$. Then $$f(0)$$ is equal to
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For the differentiable function $$f : \mathbb{R} - \{0\} - \mathbb{R}$$, let $$3f(x) + 2f\left(\frac{1}{x}\right) = \frac{1}{x} - 10$$, then $$\left|f(3) + f'\left(\frac{1}{4}\right)\right|$$ is equal to
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$$\max_{0 \leq x \leq \pi} \left\{x - 2\sin x \cos x + \frac{1}{3}\sin 3x\right\} =$$
The set of all $$a \in \mathbb{R}$$ for which the equation $$x|x-1| + |x+2| + a = 0$$ has exactly one real root, is
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$$\int_0^{\infty} \frac{6}{e^{3x} + 6e^{2x} + 11e^x + 6} dx =$$
Among
$$(S1) : \lim_{n \to \infty} \frac{1}{n^2}(2 + 4 + 6 + \ldots + 2n) = 1$$
$$(S2) : \lim_{n \to \infty} \frac{1}{n^{16}}(1^{15} + 2^{15} + 3^{15} + \ldots + n^{15}) = \frac{1}{16}$$
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The area of the region enclosed by the curve $$f(x) = \max\{\sin x, \cos x\}$$, $$-\pi \leq x \leq \pi$$ and the $$x-$$axis is
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Let $$y = y_1(x)$$ and $$y = y_2(x)$$ be the solution curves the differential equation $$\frac{dy}{dx} = y + 7$$ with initial conditions $$y_1(0) = 0$$ and $$y_2(0) = 1$$ respectively. Then the curves $$y = y_1(x)$$ and $$y = y_2(x)$$ intersect at
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Let $$\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$$, $$\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$$ and $$\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$$. If a vector $$\vec{d}$$ satisfies $$\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{d} \cdot \vec{a} = 24$$, then $$|\vec{d}|^2$$ is equal to
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Let the equation of plane passing through the line of intersection of the planes $$x + 2y + az = 2$$ and $$x - y + z = 3$$ be $$5x - 11y + bz = 6a - 1$$. For $$c \in \mathbb{Z}$$, if the distance of this plane from the point $$(a, -c, c)$$ is $$\frac{2}{\sqrt{a}}$$, then $$\frac{a+b}{c}$$ is equal to
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The distance of the point $$(-1, 2, 3)$$ from the plane $$\vec{r} \cdot (\hat{i} - 2\hat{j} + 3\hat{k}) = 10$$ parallel to the line of the shortest distance between the lines $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{k})$$ and $$\vec{r} = (2\hat{i} - \hat{j}) + \mu(\hat{i} - \hat{j} + \hat{k})$$ is
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A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $$X$$ denotes the number of tosses of the coin, then the mean of $$X$$ is
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Let $$w = z\bar{z} + k_1z + k_2iz + \lambda(1+i)$$, $$k_1, k_2 \in \mathbb{R}$$. Let $$Re(w) = 0$$ be the circle $$C$$ of radius 1 in the first quadrant touching the line $$y = 1$$ and the $$y$$-axis. If the curve $$Im(w) = 0$$ intersects $$C$$ at $$A$$ and $$B$$, then $$30(AB)^2$$ is equal to _____.
The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.
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The sum to 20 terms of the series $$2 \cdot 2^2 - 3^2 + 2 \cdot 4^2 - 5^2 + 2 \cdot 6^2 - \ldots$$ is equal to _____.
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Let $$\alpha$$ be the constant term in the binomial expansion of $$\left(\sqrt{x} - \frac{6}{x^{3/2}}\right)^n$$, $$n \leq 15$$. If the sum of the coefficients of the remaining terms in the expansion is $$649$$ and the coefficient of $$x^{-n}$$ is $$\lambda\alpha$$, then $$\lambda$$ is equal to _____.
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Let $$m_1$$ and $$m_2$$ be the slopes of the tangents drawn from the point $$P(4, 1)$$ to the hyperbola $$H : \frac{y^2}{25} - \frac{x^2}{16} = 1$$. If $$Q$$ is the point from which the tangents drawn to $$H$$ have slopes $$|m_1|$$ and $$|m_2|$$ and they make positive intercepts $$\alpha$$ and $$\beta$$ on the $$x-$$axis, then $$\frac{(PQ)^2}{\alpha\beta}$$ is equal to _____.
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Let the mean of the data
| $$x$$ | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| Frequency ($$f$$) | 4 | 24 | 28 | $$\alpha$$ | 8 |
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If $$S = \left\{x \in \mathbb{R} : \sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right) - \sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right) = \frac{\pi}{4}\right\}$$ then $$\sum_{x \in S}\left(\sin\left((x^2+x+5)\frac{\pi}{2}\right) - \cos\left((x^2+x+5)\pi\right)\right)$$ is equal to _____.
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Let for $$x \in \mathbb{R}$$, $$S_0(x) = x$$, $$S_k(x) = C_k x + k\int_0^x S_{k-1}(t)dt$$ where $$C_0 = 1$$, $$C_k = 1 - \int_0^1 S_{k-1}(x)dx$$, $$k = 1, 2, 3, \ldots$$ Then $$S_2(3) + 6C_3$$ is equal to _____.
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Let $$\vec{a} = 3\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{c} = 2\hat{i} - 3\hat{j} + 3\hat{k}$$. If $$\vec{b}$$ is a vector such that $$\vec{a} = \vec{b} \times \vec{c}$$ and $$|\vec{b}|^2 = 50$$, then $$\left|72 - |\vec{b} + \vec{c}|^2\right|$$ is equal to _____.
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Let the image of the point $$\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$$ in the plane $$x - 2y + z - 2 = 0$$ be $$P$$. If the distance of the point $$Q(6, -2, \alpha)$$, $$\alpha > 0$$, from $$P$$ is $$13$$, then $$\alpha$$ is equal to _____.
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