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If $$a \in R$$ and the equation $$-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$$ (where $$[x]$$ denotes the greatest integer $$\leq x$$) has no integral solution, then all possible values of $$a$$ lie in the interval:
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Let $$\alpha$$ and $$\beta$$ be the roots of equation $$px^2 + qx + r = 0$$, $$p \neq 0$$. If $$p$$, $$q$$, $$r$$ are in A.P. and $$\frac{1}{\alpha} + \frac{1}{\beta} = 4$$, then the value of $$|\alpha - \beta|$$ is:
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If $$z$$ is a complex number such that $$|z| \geq 2$$, then the minimum value of $$\left|z + \frac{1}{2}\right|$$:
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If $$(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \ldots + 10(11)^9 = k(10)^9$$, then $$k$$ is equal to:
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Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is:
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If the coefficients of $$x^3$$ and $$x^4$$ in the expansion of $$(1 + ax + bx^2)(1 - 2x)^{18}$$ in powers of $$x$$ are both zero, then $$(a, b)$$ is equal to:
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Let $$f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$$ where $$x \in R$$ and $$k \geq 1$$. Then $$f_4(x) - f_6(x)$$ equals:
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Let $$PS$$ be the median of the triangle with vertices $$P(2, 2)$$, $$Q(6, -1)$$ and $$R(7, 3)$$. The equation of the line passing through $$(1, -1)$$ and parallel to $$PS$$ is:
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Let $$a$$, $$b$$, $$c$$ and $$d$$ be non-zero numbers. If the point of intersection of the lines $$4ax + 2ay + c = 0$$ and $$5bx + 2by + d = 0$$ lies in the fourth quadrant and is equidistant from the two axes then:
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Let $$C$$ be the circle with center at $$(1, 1)$$ and radius = 1. If $$T$$ is the circle centered at $$(0, y)$$, passing through the origin and touching the circle $$C$$ externally, then the radius of $$T$$ is equal to:
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The locus of the foot of perpendicular drawn from the centre of the ellipse $$x^2 + 3y^2 = 6$$ on any tangent to it is:
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$$\lim_{x \to 0} \frac{\sin(\pi \cos^2 x)}{x^2}$$ is equal to:
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The statement $$\sim(p \leftrightarrow \sim q)$$ is:
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The variance of the first 50 even natural numbers is:
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A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45°. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to 30°. Then the speed (in m/s) of the bird is:
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If $$X = \{4^n - 3n - 1 : n \in N\}$$ and $$Y = \{9(n-1) : n \in N\}$$, where $$N$$ is the set of natural numbers, then $$X \cup Y$$ is equal to:
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If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA' = A'A$$ and $$B = A^{-1}A'$$, then $$BB'$$ equals, where $$X'$$ denotes the transpose of the matrix $$X$$.
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If $$\alpha$$, $$\beta \neq 0$$, $$f(n) = \alpha^n + \beta^n$$ and $$\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha - \beta)^2$$, then K is equal to:
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If $$g$$ is the inverse of a function $$f$$ and $$f'(x) = \frac{1}{1+x^5}$$, then $$g'(x)$$ is equal to:
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If $$f$$ & $$g$$ are differentiable functions in $$[0, 1]$$ satisfying $$f(0) = 2 = g(1)$$, $$g(0) = 0$$ & $$f(1) = 6$$, then for some $$c \in ]0, 1[$$:
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If $$x = -1$$ and $$x = 2$$ are extreme points of $$f(x) = \alpha \log|x| + \beta x^2 + x$$, then:
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The slope of the line touching both the parabolas $$y^2 = 4x$$ and $$x^2 = -32y$$ is:
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The integral $$\int\left(1 + x - \frac{1}{x}\right)e^{x+\frac{1}{x}} dx$$ is equal to:
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The integral $$\int_0^{\pi} \sqrt{1 + 4\sin^2\frac{x}{2} - 4\sin\frac{x}{2}} \, dx$$ equals:
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The area (in sq. unit) of the region described by $$A = \{(x, y) : x^2 + y^2 \leq 1$$ and $$y^2 \leq 1 - x\}$$ is:
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Let the population of rabbits surviving at a time $$t$$ be governed by the differential equation $$\frac{dp(t)}{dt} = \frac{1}{2}\{p(t) - 400\}$$. If $$p(0) = 100$$, then $$p(t)$$ equals:
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If $$\begin{bmatrix} \vec{a} \times \vec{b} & \vec{b} \times \vec{c} & \vec{c} \times \vec{a} \end{bmatrix} = \lambda \begin{bmatrix} \vec{a} & \vec{b} & \vec{c} \end{bmatrix}^2$$, then $$\lambda$$ is equal to:
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The image of the line $$\frac{x-1}{3} = \frac{y-3}{1} = \frac{z-4}{-5}$$ in the plane $$2x - y + z + 3 = 0$$ is the line:
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The angle between the lines whose direction cosines satisfy the equations $$l + m + n = 0$$ and $$l^2 = m^2 + n^2$$ is:
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Let $$A$$ and $$B$$ be two events such that $$P\left(\overline{A \cup B}\right) = \frac{1}{6}$$, $$P(A \cap B) = \frac{1}{4}$$ and $$P(\bar{A}) = \frac{1}{4}$$, where $$\bar{A}$$ stands for the complement of the event $$A$$. Then the events $$A$$ and $$B$$ are:
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