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Let $$O$$ be the origin and $$A$$ be the point $$z_1 = 1 + 2i$$. If $$B$$ is the point $$z_2$$, $$\text{Re}(z_2) < 0$$, such that $$OAB$$ is a right angled isosceles triangle with $$OB$$ as hypotenuse, then which of the following is NOT true?
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Consider two G.P.s $$2, 2^2, 2^3, \ldots$$ and $$4, 4^2, 4^3, \ldots$$ of $$60$$ and $$n$$ terms respectively. If the geometric mean of all the $$60 + n$$ terms is $$(2)^{\frac{225}{8}}$$, then $$\displaystyle\sum_{k=1}^{n} k(n-k)$$ is equal to:
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Let $$S = \{\theta \in [0, 2\pi] : 8^{2\sin^2\theta} + 8^{2\cos^2\theta} = 16\}$$. Then $$n(S) + \displaystyle\sum_{\theta \in S} \left(\sec\left(\dfrac{\pi}{4} + 2\theta\right) \cosec\left(\dfrac{\pi}{4} + 2\theta\right)\right)$$ is equal to:
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A point $$P$$ moves so that the sum of squares of its distances from the points $$(1, 2)$$ and $$(-2, 1)$$ is $$14$$. Let $$f(x, y) = 0$$ be the locus of $$P$$, which intersects the $$x$$-axis at the points $$A, B$$ and the $$y$$-axis at the point $$C, D$$. Then the area of the quadrilateral $$ACBD$$ is equal to
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Let the tangent drawn to the parabola $$y^2 = 24x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2x + 2y = 5$$. Then the normal to the hyperbola $$\dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1$$ at the point $$(\alpha + 4, \beta + 4)$$ does NOT pass through the point:
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The statement $$(\sim(p \Leftrightarrow \sim q)) \wedge q$$ is:
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Let $$A$$ be a $$2 \times 2$$ matrix with $$\det(A) = -1$$ and $$\det((A + I)(\text{Adj}(A) + I)) = 4$$. Then the sum of the diagonal elements of $$A$$ can be:
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If the system of linear equations
$$8x + y + 4z = -2$$
$$x + y + z = 0$$
$$\lambda x - 3y = \mu$$
has infinitely many solutions, then the distance of the point $$(\lambda, \mu, -\dfrac{1}{2})$$ from the plane $$8x + y + 4z + 2 = 0$$ is:
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$$\tan\left(2\tan^{-1}\dfrac{1}{5} + \sec^{-1}\dfrac{\sqrt{5}}{2} + 2\tan^{-1}\dfrac{1}{8}\right)$$ is equal to:
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to:
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If the function $$f(x) = \begin{cases} \dfrac{\log_e(1-x+x^2) + \log_e(1+x+x^2)}{\sec x - \cos x}, & x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) - \{0\} \\ k, & x = 0 \end{cases}$$ is continuous at $$x = 0$$, then $$k$$ is equal to:
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If $$f(x) = \begin{cases} x + a, & x \le 0 \\ |x - 4|, & x > 0 \end{cases}$$ and $$g(x) = \begin{cases} x + 1, & x < 0 \\ (x-4)^2 + b, & x \ge 0 \end{cases}$$ are continuous on $$\mathbb{R}$$, then $$(gof)(2) + (fog)(-2)$$ is equal to:
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Let $$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \le 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$$. Then the set of all values of $$b$$, for which $$f(x)$$ has maximum value at $$x = 1$$, is:
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If $$a = \displaystyle\lim_{n \to \infty} \sum_{k=1}^{n} \dfrac{2n}{n^2 + k^2}$$ and $$f(x) = \sqrt{\dfrac{1-\cos x}{1+\cos x}}$$, $$x \in (0, 1)$$, then:
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The odd natural number $$a$$, such that the area of the region bounded by $$y = 1$$, $$y = 3$$, $$x = 0$$, $$x = y^a$$ is $$\dfrac{364}{3}$$, is equal to:
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If $$\dfrac{dy}{dx} + 2y \tan x = \sin x$$, $$0 < x < \dfrac{\pi}{2}$$ and $$y\left(\dfrac{\pi}{3}\right) = 0$$, then the maximum value of $$y(x)$$ is
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Let $$\vec{a} = \alpha\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}$$, $$\alpha > 0$$. If the projection of $$\vec{a} \times \vec{b}$$ on the vector $$-\hat{i} + 2\hat{j} - 2\hat{k}$$ is $$30$$, then $$\alpha$$ is equal to
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The length of the perpendicular from the point $$(1, -2, 5)$$ on the line passing through $$(1, 2, 4)$$ and parallel to the line $$x + y - z = 0 = x - 2y + 3z - 5$$ is:
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The mean and variance of a binomial distribution are $$\alpha$$ and $$\dfrac{\alpha}{3}$$ respectively. If $$P(X = 1) = \dfrac{4}{243}$$, then $$P(X = 4 \text{ or } 5)$$ is equal to:
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Let $$E_1, E_2, E_3$$ be three mutually exclusive events such that $$P(E_1) = \dfrac{2+3p}{6}$$, $$P(E_2) = \dfrac{2-p}{8}$$ and $$P(E_3) = \dfrac{1-p}{2}$$. If the maximum and minimum values of $$p$$ are $$p_1$$ and $$p_2$$ then $$(p_1 + p_2)$$ is equal to:
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If for some $$p, q, r \in \mathbb{R}$$, all have positive sign, one of the roots of the equation $$(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0$$ is also a root of the equation $$x^2 + 2x - 8 = 0$$, then $$\dfrac{q^2 + r^2}{p^2}$$ is equal to ______.
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The number of 5-digit natural numbers, such that the product of their digits is 36, is ______.
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The series of positive multiples of 3 is divided into sets: $$\{3\}, \{6, 9, 12\}, \{15, 18, 21, 24, 27\}, \ldots$$ Then the sum of the elements in the $$11^{th}$$ set is equal to ______.
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If the coefficients of $$x$$ and $$x^2$$ in the expansion of $$(1 + x)^p(1 - x)^q$$, $$p, q \le 15$$, are $$-3$$ and $$-5$$ respectively, then the coefficient of $$x^3$$ is equal to ______.
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The equations of the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ are $$2x + y = 0$$, $$x + py = 15a$$ and $$x - y = 3$$ respectively. If its orthocentre is $$(2, a)$$, $$-\dfrac{1}{2} < a < 2$$, then $$p$$ is equal to ______.
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The number of distinct real roots of the equation $$x^5(x^3 - x^2 - x + 1) + x(3x^3 - 4x^2 - 2x + 4) - 1 = 0$$ is ______.
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Let the function $$f(x) = 2x^2 - \log_e x$$, $$x > 0$$, be decreasing in $$(0, a)$$ and increasing in $$(a, 4)$$. A tangent to the parabola $$y^2 = 4ax$$ at a point $$P$$ on it passes through the point $$(8a, 8a - 1)$$ but does not pass through the point $$\left(-\dfrac{1}{a}, 0\right)$$. If the equation of the normal at $$P$$ is $$\dfrac{x}{\alpha} + \dfrac{y}{\beta} = 1$$, then $$\alpha + \beta$$ is equal to ______.
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If $$n(2n + 1) \displaystyle\int_0^1 (1 - x^n)^{2n} dx = 1177 \int_0^1 (1 - x^n)^{2n+1} dx$$, $$n \in \mathbb{N}$$, then $$n$$ is equal to ______.
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Let a curve $$y = y(x)$$ pass through the point $$(3, 3)$$ and the area of the region under this curve, above the $$x$$-axis and between the abscissae $$3$$ and $$x (> 3)$$ be $$\left(\dfrac{y}{x}\right)^3$$. If this curve also passes through the point $$(\alpha, 6\sqrt{10})$$ in the first quadrant, then $$\alpha$$ is equal to ______.
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Let $$Q$$ and $$R$$ be two points on the line $$\dfrac{x+1}{2} = \dfrac{y+2}{3} = \dfrac{z-1}{2}$$ at a distance $$\sqrt{26}$$ from the point $$P(4, 2, 7)$$. Then the square of the area of the triangle $$PQR$$ is ______.
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