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Let the solution curve of the differential equation $$xdy-ydx=\sqrt{x^{2}+y^{2}}dx,x>0$$, $$y(1) = 0$$ , be $$y$$ = $$y(x)$$ . Then $$y(3)$$ is equal to
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If the line $$\alpha x + 2y = 1$$, where $$\alpha \in R $$, does not meet the hyperbola $$x^{2}-9y^{2}=9$$, then a possible value of $$\alpha$$ is:
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The coefficient of $$x^{48}$$ in $$ (1+x) + 2(1+x)^{2}+3(1+x)^{3}+....+100(1+x)^{100} $$ is equal to
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Let $$ f: [1 , \infty ) \rightarrow R$$ be a differentiable function. If $$6 \int_{1}^{x} f(t)dt=3x f(x)+ x^{3}-4$$ for all $$x\geq 1$$ then the value of $$f(2)-f(3)$$ is
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If $$A=\begin{bmatrix}2 & 3 \\3 & 5 \end{bmatrix}$$, then the determinant of the matrix $$ (A^{2025}-3A^{2024}+ A^{2023})$$ is
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If a random variable x has the probability distribution
then $$ P(3< x\leq 6)$$ is equal to
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Let the relation R on the set $$ M=\left\{ 1,2,3,...,16 \right\}$$ be given by $$ R=\left\{ (x, y): 4y= 5x-3,x,y \text{ }\in \text{ }M\right\}$$.
Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to
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Let the set of all values of r, for which the circles $$ (x+1)^{2}+(y+4)^{2}=r^{2}$$ and $$ x^{2}+y^{2}-4x-2y-4=0$$ intersect at two distinct points be the interval $$( \alpha,\beta )$$. Then $$ \alpha\beta $$ is equal to
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The value of $$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1}{[x]+4}\right)dx $$ where [.]denotes the greatest integer function, is
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The number of solutions of $$ \tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6} $$, where $$ -\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}, $$ is equal to
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Let the line $$x = - 1$$ divide the area of the region $$ \left\{(x,y): 1+x^{2}\leq y \leq 3 -x\right\} $$ in the ratio m : n, gcd (m, n) = 1. Then m + n is equal to
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Two distinct numbers a and b are selected at random from 1, 2, 3, ... , 50. The probability, that their product ab is divisible by 3, is
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If the image of the point $$P(1 , 2, a)$$ in the line $$ \frac{x-6}{3} = \frac{y - 7}{2} = \frac{7 -z}{2}$$ is $$Q(5, b, c)$$, then $$ a^{2}+b^{2}+c^{2}$$ is equal to
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The number of distinct real solutions of the equation $$x\lvert x+4 \rvert + 3\lvert x+2 \rvert + 10 = 0$$ is
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If the domain of the function $$ \large f(x)=\sin ^{-1} \left( \frac{5-x}{3+2x} \right)+\frac{1}{\log_{e}{(10-x)}} $$ is $$ \large (-\infty,\propto] \cup [\beta,\gamma) - \left\{ \delta\right\} $$, then $$ \large 6(\alpha+ \beta+ \gamma+\delta) $$ is equal to
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If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is
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If the chord joining the points $$ P_{1}(x_{1}, y_{1}) $$ and $$P_{2}(x_{2},y_{2})$$ on the parabola $$y^{2}=12x$$ subtends a right angle at the vertex of the parabola, then $$ x_{1}x_{2}-y_{1}y_{2} $$ is equal to
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Let $$ P(\alpha,\beta, \gamma)$$ be the point on the line $$\frac{x-1}{2}=\frac{y+1}{-3}=z$$ at a distance $$4\sqrt{14}$$ from the point (1, -1, 0) and nearer to the origin. Then the shortest di stance, between the Lines $$\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}$$ and $$\frac{x+5}{2}= \frac{y-10}{1}=\frac{z-3}{1}$$, is equal to
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Let $$\overrightarrow{AB} = 2\widehat{i}+4\widehat{j}-5\widehat{k}$$ and $$ \overrightarrow{AD} = \widehat{i}+2\widehat{j}+\lambda\widehat{k}, \lambda\text{ }\epsilon \text{ } R$$. Let the projection of the vector $$ \overrightarrow{v}=\widehat{i}+\widehat{j}+\widehat{k}$$ on the diagonal $$\overrightarrow{AC}$$ of the parallelogram ABCD be of length one unit. If $$\alpha> \beta$$, be the roots of the equation $$\lambda^{2}x^{2}-6\lambda x+5=0$$, then $$2\alpha-\beta$$ is equal to
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Let $$ f(x)=x^{2025}-x^{2000}, x \text{ }\epsilon \text{ }[0,1] $$ and the minimmu value of the function $$ f(x)$$ in the interval [0, 1] be $$(80)^{80}(n)^{-81}$$. Then n is equal to
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If $$\int (\sin x) ^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}}dx= -\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}}-\frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}}-\frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}}+ \frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}}+C,\text{ where }p_{i} \text{ and } q_{i} $$ are positive integers with $$gcd(p_{i}, q_{i}) = 1$$ for i = 1, 2, 3, 4 and C is the constant of integration, then $$\frac{15p_{1}p_{2}p_{3}p_{4}}{q_{1}q_{2}q_{3}q_{4}} $$ is equal to ______
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If $$\dfrac{\cos^{2}48^{o}-\sin^{2}12^{o}}{\sin^{2}24^{o}-\sin^{2}6^{o}}=\dfrac{\alpha+\beta\sqrt{5}}{2}$$, where $$\alpha, \beta \text{ }\epsilon \text{ }N$$, then $$\alpha + \beta $$ is equal to ________
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Let $$ABC$$ be a triangle. Consider four points $$p_{1},p_{2},p_{3},p_{4}$$ on the side AB, five points $$p_{5},p_{6},p_{7},p_{8},p_{9}$$ on the side $$BC$$, and four points $$p_{10},p_{11},p_{12},p_{13}$$ on the side $$AC$$. None of these points is a vertex of the trinagle $$ABC$$. Then the total number of pentagons, that can be formed by taking all the vertices from the points $$p_{1},p_{2},... ,p_{13}$$, is_______
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Let $$\alpha = \frac{-1+i\sqrt{3}}{2}$$ and $$ \beta=\frac{-1-i\sqrt{3}}{2},i=\sqrt{-1}.$$
If $$(7-7\alpha+9\beta)^{20}+(9+7\alpha+7\beta)^{20}+(-7+9\alpha+7\beta)^{20}+(14+7\alpha+7\beta)^{20}=m^{10},$$ then $$m$$ is
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Let A be a $$3 \times 3$$ matrix such that A+ A^{T} = 0. If $$A\begin{bmatrix} 1 \\-1 \\ 0 \end{bmatrix}=\begin{bmatrix} 3 \\3 \\ 2 \end{bmatrix},A^{2}\begin{bmatrix} 1 \\-1 \\ 0 \end{bmatrix}=\begin{bmatrix} -3 \\19 \\ -24 \end{bmatrix}$$ and $$det(adj(2 adj(A+I))) = (2)^{\alpha }\cdot (3)^{\beta}\cdot (11)^{\gamma},\alpha,\beta,\gamma$$ are non-negative integers, then $$\alpha+\beta+\gamma$$ is equal to _____
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