Let $$\alpha, \beta$$ be the roots of the equation $$x^2 + 2\sqrt{2}x - 1 = 0$$. The quadratic equation, whose roots are $$\alpha^4 + \beta^4$$ and $$\frac{1}{10}(\alpha^6 + \beta^6)$$, is :
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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 + 2\sqrt{2}x - 1 = 0$$. The quadratic equation, whose roots are $$\alpha^4 + \beta^4$$ and $$\frac{1}{10}(\alpha^6 + \beta^6)$$, is :
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If the sum of the series $$\frac{1}{1 \cdot (1+d)} + \frac{1}{(1+d)(1+2d)} + \ldots + \frac{1}{(1+9d)(1+10d)}$$ is equal to $$5$$, then $$50d$$ is equal to :
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The coefficient of $$x^{70}$$ in $$x^2(1+x)^{98} + x^3(1+x)^{97} + x^4(1+x)^{96} + \ldots + x^{54}(1+x)^{46}$$ is $$^{99}C_p - ^{46}C_q$$. Then a possible value of $$p + q$$ is :
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Let $$|\cos\theta \cos(60° - \theta) \cos(60° + \theta)| \leq \frac{1}{8}$$, $$\theta \in [0, 2\pi]$$. Then, the sum of all $$\theta \in [0, 2\pi]$$, where $$\cos 3\theta$$ attains its maximum value, is :
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A ray of light coming from the point $$P(1, 2)$$ gets reflected from the point $$Q$$ on the $$x$$-axis and then passes through the point $$R(4, 3)$$. If the point $$S(h, k)$$ is such that PQRS is a parallelogram, then $$hk^2$$ is equal to :
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Let a circle passing through $$(2, 0)$$ have its centre at the point $$(h, k)$$. Let $$(x_c, y_c)$$ be the point of intersection of the lines $$3x + 5y = 1$$ and $$(2 + c)x + 5c^2y = 1$$. If $$h = \lim_{c \to 1} x_c$$ and $$k = \lim_{c \to 1} y_c$$, then the equation of the circle is :
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Let $$f(x) = x^2 + 9$$, $$g(x) = \frac{x}{x-9}$$ and $$a = f \circ g(10)$$, $$b = g \circ f(3)$$. If $$e$$ and $$l$$ denote the eccentricity and the length of the latus rectum of the ellipse $$\frac{x^2}{a} + \frac{y^2}{b} = 1$$, then $$8e^2 + l^2$$ is equal to :
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The frequency distribution of the age of students in a class of 40 students is given below.
If the mean deviation about the median is 1.25, then $$4x + 5y$$ is equal to :
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Let $$\lambda, \mu \in \mathbb{R}$$. If the system of equations
$$3x + 5y + \lambda z = 3$$
$$7x + 11y - 9z = 2$$
$$97x + 155y - 189z = \mu$$
has infinitely many solutions, then $$\mu + 2\lambda$$ is equal to :
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If the domain of the function $$f(x) = \sin^{-1}\left(\frac{x-1}{2x+3}\right)$$ is $$\mathbb{R} - (\alpha, \beta)$$, then $$12\alpha\beta$$ is equal to :
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Let $$f(x) = ax^3 + bx^2 + cx + 41$$ be such that $$f(1) = 40$$, $$f'(1) = 2$$ and $$f''(1) = 4$$. Then $$a^2 + b^2 + c^2$$ is equal to:
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A variable line $$L$$ passes through the point $$(3, 5)$$ and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :
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Let $$\int \frac{2 - \tan x}{3 + \tan x} dx = \frac{1}{2}(\alpha x + \log_e|\beta \sin x + \gamma \cos x|) + C$$, where $$C$$ is the constant of integration. Then $$\alpha + \frac{\gamma}{\beta}$$ is equal to :
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The parabola $$y^2 = 4x$$ divides the area of the circle $$x^2 + y^2 = 5$$ in two parts. The area of the smaller part is equal to:
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The solution curve, of the differential equation $$2y\frac{dy}{dx} + 3 = 5\frac{dy}{dx}$$, passing through the point $$(0, 1)$$ is a conic, whose vertex lies on the line:
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The solution of the differential equation $$(x^2 + y^2)dx - 5xy \, dy = 0$$, $$y(1) = 0$$, is :
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Let three vectors $$\vec{a} = \alpha\hat{i} + 4\hat{j} + 2\hat{k}$$, $$\vec{b} = 5\hat{i} + 3\hat{j} + 4\hat{k}$$, $$\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}$$ form a triangle such that $$\vec{c} = \vec{a} - \vec{b}$$ and the area of the triangle is $$5\sqrt{6}$$. If $$\alpha$$ is a positive real number, then $$|\vec{c}|^2$$ is equal to:
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Let $$\vec{OA} = 2\vec{a}$$, $$\vec{OB} = 6\vec{a} + 5\vec{b}$$ and $$\vec{OC} = 3\vec{b}$$, where $$O$$ is the origin. If the area of the parallelogram with adjacent sides $$\vec{OA}$$ and $$\vec{OC}$$ is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :
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Let the line L intersect the lines $$x - 2 = -y = z - 1$$, $$2(x + 1) = 2(y - 1) = z + 1$$ and be parallel to the line $$\frac{x-2}{3} = \frac{y-1}{1} = \frac{z-2}{2}$$. Then which of the following points lies on L?
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The shortest distance between the lines $$\frac{x-3}{4} = \frac{y+7}{-11} = \frac{z-1}{5}$$ and $$\frac{x-5}{3} = \frac{y-9}{-6} = \frac{z+2}{1}$$ is:
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The sum of the square of the modulus of the elements in the set $$\{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z - 1| \leq 1, |z - 5| \leq |z - 5i|\}$$ is ________
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The remainder when $$428^{2024}$$ is divided by 21 is __________
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Let the centre of a circle, passing through the points $$(0, 0)$$, $$(1, 0)$$ and touching the circle $$x^2 + y^2 = 9$$, be $$(h, k)$$. Then for all possible values of the coordinates of the centre $$(h, k)$$, $$4(h^2 + k^2)$$ is equal to _________
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Let $$\lim_{n \to \infty} \left(\frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \ldots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$$ be $$\frac{\pi}{k}$$, using only the principal values of the inverse trigonometric functions. Then $$k^2$$ is equal to ________
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Let $$A = \{2, 3, 6, 7\}$$ and $$B = \{4, 5, 6, 8\}$$. Let $$R$$ be a relation defined on $$A \times B$$ by $$(a_1, b_1) R (a_2, b_2)$$ if and only if $$a_1 + a_2 = b_1 + b_2$$. Then the number of elements in $$R$$ is _________
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Let $$A$$ be a non-singular matrix of order 3. If $$\det(3 \text{ adj}(2 \text{ adj}((\det A)A))) = 3^{-13} \cdot 2^{-10}$$ and $$\det(3 \text{ adj}(2A)) = 2^m \cdot 3^n$$, then $$|3m + 2n|$$ is equal to ________
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If a function $$f$$ satisfies $$f(m + n) = f(m) + f(n)$$ for all $$m, n \in \mathbb{N}$$ and $$f(1) = 1$$, then the largest natural number $$\lambda$$ such that $$\sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2$$ is equal to _________
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Let $$f : (0, \pi) \rightarrow \mathbb{R}$$ be a function given by
$$f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ (1 + |\cot x|)^{\frac{b}{|\tan x|}}, & \frac{\pi}{2} < x < \pi \end{cases}$$
where $$a, b \in \mathbb{Z}$$. If $$f$$ is continuous at $$x = \frac{\pi}{2}$$, then $$a^2 + b^2$$ is equal to ________
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Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1 + x(\lambda^2 - x^2))$$ satisfies $$\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0$$, be $$(\alpha, \beta)$$. Then $$\alpha^2 + \beta^2$$ is equal to _________
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Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $$1, 2, 3, 4$$. If the probability that $$ax^2 + bx + c = 0$$ has all real roots is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to ________
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