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Let $$\lambda \neq 0$$ be a real number. Let $$\alpha, \beta$$ be the roots of the equation $$14x^2 - 31x + 3\lambda = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$35x^2 - 53x + 4\lambda = 0$$. Then $$\frac{3\alpha}{\beta}$$ and $$\frac{4\alpha}{\gamma}$$ are the roots of the equation:
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For two non-zero complex numbers $$z_1$$ and $$z_2$$, if $$\text{Re}(z_1 z_2) = 0$$ and $$\text{Re}(z_1 + z_2) = 0$$, then which of the following are possible?
(A) Im $$(z_1) > 0$$ and Im $$(z_2) > 0$$
(B) Im $$(z_1) < 0$$ and Im $$(z_2) > 0$$
(C) Im $$(z_1) > 0$$ and Im $$(z_2) < 0$$
(D) Im $$(z_1) < 0$$ and Im $$(z_2) < 0$$
Choose the correct answer from the options given below:
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Let $$f(\theta) = 3\left(\sin^4\left(\frac{3\pi}{2} - \theta\right) + \sin^4(3\pi + \theta)\right) - 2\left(1 - \sin^2 2\theta\right)$$ and $$S = \left\{\theta \in [0, \pi] : f'(\theta) = -\frac{\sqrt{3}}{2}\right\}$$. If $$4\beta = \sum_{\theta \in S} \theta$$ then $$f(\beta)$$ is equal to
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A light ray emits from the origin making angle $$30°$$ with the positive $$x$$-axis. After getting reflected by the line $$x + y = 1$$, if this ray intersects x-axis at Q, then the abscissa of Q is
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Let $$B$$ and $$C$$ be the two points on the line $$y + x = 0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y - 2x = 2$$ such that $$\triangle ABC$$ is an equilateral triangle. Then, the area of the $$\triangle ABC$$ is
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Let the tangents at the points $$A(4, -11)$$ and $$B(8, -5)$$ on the circle $$x^2 + y^2 - 3x + 10y - 15 = 0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to
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Let $$x = 2$$ be a root of the equation $$x^2 + px + q = 0$$ and $$f(x) = \begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x = 2p \end{cases}$$. Then $$\lim_{x \to 2p^+} [f(x)]$$
where $$[.]$$ denotes greatest integer function, is
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If $$p, q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p$$, $$q$$ and $$r$$ makes the logical expression $$\{(p \vee q) \wedge ((\neg p) \vee r)\} \to ((\neg q) \vee r)$$ false?
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Let $$\alpha$$ and $$\beta$$ be real numbers. Consider a $$3 \times 3$$ matrix $$A$$ such that $$A^2 = 3A + \alpha I$$. If $$A^4 = 21A + \beta I$$, then
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Consider the following system of equations
$$\alpha x + 2y + z = 1$$
$$2\alpha x + 3y + z = 1$$
$$3x + \alpha y + 2z = \beta$$
For some $$\alpha, \beta \in \mathbb{R}$$. Then which of the following is NOT correct.
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The domain of $$f(x) = \dfrac{\log_{(x+1)}(x-2)}{e^{2\log_e x} - (2x+3)}$$, $$x \in R$$ is
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Let $$f : R \to R$$ be a function such that $$f(x) = \frac{x^2+2x+1}{x^2+1}$$. Then
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Let $$f(x) = x + \frac{a}{\pi^2-4}\sin x + \frac{b}{\pi^2-4}\cos x$$, $$x \in \mathbb{R}$$ be a function which satisfies $$f(x) = x + \int_0^{\pi/2} \sin(x+y)f(y)dy$$. Then $$(a+b)$$ is equal to
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Let $$[x]$$ denote the greatest integer $$\leq x$$. Consider the function $$f(x) = \max\{x^2, 1 + [x]\}$$. Then the value of the integral $$\int_0^2 f(x) dx$$ is:
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Let $$A = \left\{(x,y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4-(x-1)^2}\right\}$$ and
$$B = \left\{(x,y) \in \mathbb{R} \times \mathbb{R} : 0 \leq y \leq \min\left\{2x, \sqrt{4-(x-1)^2}\right\}\right\}$$. Then the ratio of the area of $$A$$ to the area of $$B$$ is
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Let $$\Delta$$ be the area of the region $$\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}$$. Then $$\frac{1}{2}\left(\Delta - 21\sin^{-1}\frac{2}{\sqrt{7}}\right)$$ is equal to
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Let $$y = f(x)$$ be the solution of the differential equation $$y(x+1)dx - x^2 dy = 0$$, $$y(1) = e$$. Then $$\lim_{x \to 0^+} f(x)$$ is equal to
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If the vectors $$\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$$, $$\vec{b} = -2\hat{i} + 4\hat{j} - 2\hat{k}$$ and $$\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$$ are coplanar and the projection of $$\vec{a}$$ on the vector $$\vec{b}$$ is $$\sqrt{54}$$ units, then the sum of all possible values of $$\lambda + \mu$$ is equal to
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Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $$\mu$$ and $$\sigma^2$$ represent mean and variance of X, respectively, then $$10(\mu^2 + \sigma^2)$$ is equal to
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If all the six digit numbers $$x_1x_2x_3x_4x_5x_6$$ with $$0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$72^{th}$$ number is ______.
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Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is
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Let $$a_1, a_2, a_3, \ldots$$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is $$9$$ and the sum of fifth and seventh terms is $$24$$, then $$a_1a_9 + a_2a_4a_9 + a_5 + a_7$$ is equal to
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Let the coefficients of three consecutive terms in the binomial expansion of $$(1 + 2x)^n$$ be in the ratio $$2 : 5 : 8$$. Then the coefficient of the term, which is in the middle of these three terms, is
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If the co-efficient of $$x^9$$ in $$\left(\alpha x^3 + \frac{1}{\beta x}\right)^{11}$$ and the co-efficient of $$x^{-9}$$ in $$\left(\alpha x - \frac{1}{\beta x^3}\right)^{11}$$ are equal, then $$(\alpha\beta)^2$$ is equal to
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y) = f(x) + f(y) - 1, \forall x, y \in \mathbb{R}$$. If $$f'(0) = 2$$, then $$|f(-2)|$$ is equal to
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Suppose f is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x, y \in \mathbb{N}$$ and $$f(1) = \frac{1}{5}$$. If $$\sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$$ then m is equal to ______.
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero non-coplanar vectors. Let the position vectors of four points $$A$$, $$B$$, $$C$$ and $$D$$ be $$\vec{a} - \vec{b} + \vec{c}$$, $$\lambda\vec{a} - 3\vec{b} + 4\vec{c}$$, $$-\vec{a} + 2\vec{b} - 3\vec{c}$$ and $$2\vec{a} - 4\vec{b} + 6\vec{c}$$ respectively. If $$\vec{AB}$$, $$\vec{AC}$$ and $$\vec{AD}$$ are coplanar, then $$\lambda$$ is:
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Let the equation of the plane P containing the line $$x + 10 = \frac{8-y}{2} = z$$ be $$ax + by + 3z = 2(a+b)$$ and the distance of the plane P from the point $$(1, 27, 7)$$ be $$c$$. Then $$a^2 + b^2 + c^2$$ is equal to
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Let the co-ordinates of one vertex of $$\triangle ABC$$ be $$A(0, 2, \alpha)$$ and the other two vertices lie on the line $$\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. For $$\alpha \in \mathbb{Z}$$, if the area of $$\triangle ABC$$ is $$21$$ sq. units and the line segment $$BC$$ has length $$2\sqrt{21}$$ units, then $$\alpha^2$$ is equal to ______.
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