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If $$p$$ and $$q$$ are non-zero real numbers and $$\alpha^3 + \beta^3 = -p$$, $$\alpha\beta = q$$, then a quadratic equation whose roots are $$\frac{\alpha^2}{\beta}$$, $$\frac{\beta^2}{\alpha}$$ is :
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Let z satisfy $$|z| = 1$$ and $$z = 1 - \bar{z}$$.
Statement 1 : z is a real number.
Statement 2 : Principal argument of z is $$\frac{\pi}{3}$$.
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5-digit numbers are to be formed using 2, 3, 5, 7, 9 without repeating the digits. If $$p$$ be the number of such numbers that exceed 20000 and $$q$$ be the number of those that lie between 30000 and 90000, then $$p : q$$ is:
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Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is :
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The value of $$1^2 + 3^2 + 5^2 + \ldots + 25^2$$ is :
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If for positive integers $$r > 1$$, $$n > 2$$, the coefficients of the $$(3r)^{th}$$ and $$(r+2)^{th}$$ powers of $$x$$ in the expansion of $$(1 + x)^{2n}$$ are equal, then $$n$$ is equal to:
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Let $$A = \{\theta : \sin(\theta) = \tan(\theta)\}$$ and $$B = \{\theta : \cos(\theta) = 1\}$$ be two sets. Then :
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If the image of point P(2, 3) in a line L is Q(4, 5), then the image of point R(0, 0) in the same line is:
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Let $$x \in (0, 1)$$. The set of all $$x$$ such that $$\sin^{-1}x > \cos^{-1}x$$, is the interval:
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Statement 1: The only circle having radius $$\sqrt{10}$$ and a diameter along line $$2x + y = 5$$ is $$x^2 + y^2 - 6x + 2y = 0$$.
Statement 2: $$2x + y = 5$$ is a normal to the circle $$x^2 + y^2 - 6x + 2y = 0$$.
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If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle 60° on the circumference of the first circle, then the radius of the arc is:
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A point on the ellipse, $$4x^2 + 9y^2 = 36$$, where the normal is parallel to the line, $$4x - 2y - 5 = 0$$, is :
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Consider the system of equations : $$x + ay = 0$$, $$y + az = 0$$ and $$z + ax = 0$$. Then the set of all real values of 'a' for which the system has a unique solution is:
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Let $$p$$ and $$q$$ be any two logical statements and $$r : p \rightarrow (\sim p \vee q)$$. If $$r$$ has a truth value $$F$$, then the truth values of $$p$$ and $$q$$ are respectively:
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In a set of $$2n$$ observations, half of them are equal to 'a' and the remaining half are equal to '-a'. If the standard deviation of all the observations is 2; then the value of |a| is :
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A common tangent to the conics $$x^2 = 6y$$ and $$2x^2 - 4y^2 = 9$$ is:
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Let $$S = \left\{\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} : a_{ij} \in \{0, 1, 2\}, a_{11} = a_{22}\right\}$$. Then the number of non-singular matrices in the set S is :
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Consider the function : $$f(x) = [x] + |1 - x|$$, $$-1 \leq x \leq 3$$ where [x] is the greatest integer function.
Statement 1: $$f$$ is not continuous at $$x = 0, 1, 2$$ and 3.
Statement 2: f(x) =$$\begin{cases}-x, & -1 \le x < 0 \\1 - x, & 0 \le x < 1 \\1 + x, & 1 \le x < 2 \\2 + x, & 2 \le x \le 3\end{cases}$$
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A spherical balloon is being inflated at the rate of 35cc/min. The rate of increase in the surface area (in cm$$^2$$/min.) of the balloon when its diameter is 14 cm, is :
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Let $$f(1) = -2$$ and $$f'(x) \geq 4.2$$ for $$1 \leq x \leq 6$$. The possible value of $$f(6)$$ lies in the interval :
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If an equation of a tangent to the curve, $$y = \cos(x + y)$$, $$- 1 \leq x \leq 1 + \pi$$, is $$x + 2y = k$$ then $$k$$ is equal to :
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If the integral $$\int \frac{\cos 8x + 1}{\cot 2x - \tan 2x}dx = A\cos 8x + k$$ where k is an arbitrary constant, then A is equal to:
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For $$0 \leq x \leq \frac{\pi}{2}$$, the value of $$\int_0^{\sin^2 x} \sin^{-1}(\sqrt{t})dt + \int_0^{\cos^2 x} \cos^{-1}(\sqrt{t})dt$$ equals :
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Let $$f : [-2, 3] \rightarrow [0, \infty)$$ be a continuous function such that $$f(1-x) = f(x)$$ for all $$x \in [-2, 3]$$. If $$R_1$$ is the numerical value of the area of the region bounded by $$y = f(x)$$, $$x = -2$$, $$x = 3$$ and the axis of x and $$R_2 = \int_{-2}^{3} xf(x)dx$$, then :
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The equation of the curve passing through the origin and satisfying the differential equation $$(1 + x^2)\frac{dy}{dx} + 2xy = 4x^2$$ is
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Let $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$\vec{a} \cdot \vec{c} = |\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$\vec{a} \times \vec{b}$$ and $$\vec{c}$$ is 30°, then $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ equals:
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Let A(-3, 2) and B(-2, 1) be the vertices of a triangle ABC. If the centroid of this triangle lies on the line $$3x + 4y + 2 = 0$$, then the vertex C lies on the line :
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Let ABC be a triangle with vertices at points A(2, 3, 5), B(-1, 3, 2) and C($$\lambda$$, 5, $$\mu$$) in three dimensional space. If the median through A is equally inclined with the axes, then $$(\lambda, \mu)$$ is equal to:
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The equation of a plane through the line of intersection of the planes $$x + 2y = 3$$, $$y - 2z + 1 = 0$$, and perpendicular to the first plane is :
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If the events A and B are mutually exclusive events such that $$P(A) = \frac{3x+1}{3}$$ and $$P(B) = \frac{1-x}{4}$$, then the set of possible values of x lies in the interval :
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