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Let $$\lambda \neq 0$$ be in $$R$$. If $$\alpha$$ and $$\beta$$ are the roots of the equation, $$x^2 - x + 2\lambda = 0$$ and $$\alpha$$ and $$\gamma$$ are the roots of the equation, $$3x^2 - 10x + 27\lambda = 0$$, then $$\frac{\beta\gamma}{\lambda}$$ is equal to:
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If $$a$$ and $$b$$ are real numbers such that $$(2 + \alpha)^4 = a + b\alpha$$, where $$\alpha = \frac{-1 + i\sqrt{3}}{2}$$, then $$a + b$$ is equal to:
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Let $$a_1, a_2, \ldots, a_n$$ be a given A.P. whose common difference is an integer and $$S_n = a_1 + a_2 + \ldots + a_n$$. If $$a_1 = 1, a_n = 300$$ and $$15 \leq n \leq 50$$, then the ordered pair $$(S_{n-4}, a_{n-4})$$ is equal to:
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If for some positive integer $$n$$, the coefficients of three consecutive terms in the binomial expansion of $$(1 + x)^{n+5}$$ are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is:
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If the perpendicular bisector of the line segment joining the points $$P(1, 4)$$ and $$Q(k, 3)$$ has $$y$$-intercept equal to $$-4$$, then a value of $$k$$ is:
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The circle passing through the intersection of the circles, $$x^2 + y^2 - 6x = 0$$ and $$x^2 + y^2 - 4y = 0$$ having its centre on the line, $$2x - 3y + 12 = 0$$, also passes through the point:
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Let $$x = 4$$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $$\frac{1}{2}$$. If $$P(1, \beta), \beta \gt 0$$ is a point on this ellipse, then the equation of the normal to it at P is:
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Contrapositive of the statement:
'If a function $$f$$ is differentiable at $$a$$, then it is also continuous at $$a$$', is
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The angle of elevation of a cloud $$C$$ from a point $$P$$, 200 m above a still lake is 30°. If the angle of depression of the image of $$C$$ in the lake from the point $$P$$ is 60°, then $$PC$$ (in m) is equal to
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Let $$\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$$, where each $$X_i$$ contains 10 elements and each $$Y_i$$ contains 5 elements. If each element of the set $$T$$ is an element of exactly 20 of sets $$X_i$$'s and exactly 6 of sets $$Y_i$$'s then $$n$$ is equal to:
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If the system of equations
$$x + y + z = 2$$
$$2x + 4y - z = 6$$
$$3x + 2y + \lambda z = \mu$$
has infinitely many solutions, then:
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Suppose the vectors $$x_1, x_2$$ and $$x_3$$ are the solutions of the system of linear equations, $$Ax = b$$ when the vector $$b$$ on the right side is equal to $$b_1, b_2$$ and $$b_3$$ respectively. If $$x_1 = \begin{bmatrix}1\\1\\1\end{bmatrix}$$, $$x_2 = \begin{bmatrix}0\\2\\1\end{bmatrix}$$, $$x_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}$$; $$b_1 = \begin{bmatrix}1\\0\\0\end{bmatrix}$$, $$b_2 = \begin{bmatrix}0\\2\\0\end{bmatrix}$$, $$b_3 = \begin{bmatrix}0\\0\\2\end{bmatrix}$$, then the determinant of $$A$$ is equal to
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The minimum value of $$2^{\sin x} + 2^{\cos x}$$ is:
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The function $$f(x) = \begin{cases} \frac{\pi}{4} + \tan^{-1}x, & |x| \leq 1 \\ \frac{1}{2}(|x| - 1), & |x| > 1 \end{cases}$$ is:
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Let $$f : (0, \infty) \to (0, \infty)$$ be a differentiable function such that $$f(1) = e$$ and $$\lim_{t \to x}\frac{t^2f^2(x) - x^2f^2(t)}{t - x} = 0$$. If $$f(x) = 1$$, then $$x$$ is equal to:
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The area (in sq. units) of the largest rectangle $$ABCD$$ whose vertices $$A$$ and $$B$$ lie on the $$x$$-axis and vertices $$C$$ and $$D$$ lie on the parabola, $$y = x^2 - 1$$ below the $$x$$-axis, is:
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The integral $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan^3 x \cdot \sin^2 3x(2\sec^2 x \cdot \sin^2 3x + 3\tan x \cdot \sin 6x)dx$$ is equal to:
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The solution of the differential equation $$\frac{dy}{dx} - \frac{y+3x}{\log_e(y+3x)} + 3 = 0$$ is (where C is a constant of integration)
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The distance of the point $$(1, -2, 3)$$ from the plane $$x - y + z = 5$$ measured parallel to the line $$\frac{x}{2} = \frac{y}{3} = \frac{z}{-6}$$ is:
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In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is:
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A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is __________
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Let PQ be a diameter of the circle $$x^2 + y^2 = 9$$. If $$\alpha$$ and $$\beta$$ are the lengths of the perpendiculars from P and Q on the straight line, $$x + y = 2$$ respectively, then the maximum value of $$\alpha\beta$$ is __________
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If the variance of the following frequency distribution:
is 50, then x is equal to __________
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Let $$\{x\}$$ and $$[x]$$ denote the fractional part of $$x$$ and the greatest integer $$\leq x$$ respectively of a real number $$x$$. If $$\int_0^n \{x\}dx$$, $$\int_0^n [x]dx$$ and $$10(n^2 - n)$$, $$(n \in N, n > 1)$$ are three consecutive terms of a G.P. then $$n$$ is equal to __________
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If $$\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$$, then the value of $$\left|\hat{i} \times (\vec{a} \times \hat{i})\right|^2 + \left|\hat{j} \times (\vec{a} \times \hat{j})\right|^2 + \left|\hat{k} \times (\vec{a} \times \hat{k})\right|^2$$, is equal to:
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