If $z \neq 0$ be a complex number such that $z - \frac{1}{z} = 2$, then the maximum value of $|z|$ is (1) $\sqrt{2}$ (2) 1 (3) $\sqrt{2} - 1$ (4) $\sqrt{2} + 1$
Let $S = \{z = x + iy : |z - 1 + i| \geq |z|, |z| < 2, |z + i| = |z - 1|\}$. Then the set of all values of $x$, for which $w = 2x + iy \in S$ for some $y \in \mathbb{R}$, is (1) $\left(-\sqrt{2}, \frac{1}{2\sqrt{2}}\right)$ (2) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right)$ (3) $\left(-\sqrt{2}, \frac{1}{2}\right)$ (4) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}\right)$
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence such that $a_0 = a_1 = 0$ and $a_{n+2} = 3a_{n+1} - 2a_n + 1, \forall n \geq 0$. Then $a_{25}a_{23} - 2a_{25}a_{22} - 2a_{23}a_{24} + 4a_{22}a_{24}$ is equal to (1) 483 (2) 528 (3) 575 (4) 624
The number of elements in the set $S = \left\{x \in \mathbb{R} : 2\cos\left(\frac{x^2 + x}{6}\right) = 4^x + 4^{-x}\right\}$ is (1) 1 (2) 3 (3) 0 (4) infinite
Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2 + 11a + 3(m_1^2 + m_2^2) = 220$. If one vertex of the square is $(10\cos\alpha - \sin\alpha, 10\sin\alpha + \cos\alpha)$, where $\alpha \in \left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10$, then $72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13$ is equal to (1) 119 (2) 128 (3) 145 (4) 155
Let $A(\alpha, -2)$, $B(\alpha, 6)$ and $C\left(\frac{\alpha}{4}, -2\right)$ be vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$ (1) area is 24 (2) perimeter is 25 (3) circumradius is 5 (4) inradius is 2
If the system of equations $x + y + z = 6$ $2x + 5y + \alpha z = \beta$ $x + 2y + 3z = 14$ has infinitely many solutions, then $\alpha + \beta$ is equal to (1) 8 (2) 36 (3) 44 (4) 48
Let the function $f(x) = \begin{cases} \frac{\log_e(1 + 5x) - \log_e(1 + \alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$ be continuous at $x = 0$. Then $\alpha$ is equal to (1) 10 (2) $-10$ (3) 5 (4) $-5$
If $[t]$ denotes the greatest integer $\leq t$, then the value of $\int_0^1 \left[2x - \left|3x^2 - 5x + 2\right| + 1\right] dx$ is (1) $\frac{\sqrt{37} + \sqrt{13} - 4}{6}$ (2) $\frac{\sqrt{37} - \sqrt{13} - 4}{6}$ (3) $\frac{-\sqrt{37} - \sqrt{13} + 4}{6}$ (4) $\frac{-\sqrt{37} + \sqrt{13} + 4}{6}$
Q75
First order differential equations (integrating factor)View
If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x + y - 2}{x - y}$ passes through the point $(2, 1)$ and $(k + 1, 2)$, $k > 0$, then (1) $2\tan^{-1}\left(\frac{1}{k}\right) = \log_e(k^2 + 1)$ (2) $\tan^{-1}\left(\frac{1}{k}\right) = \log_e(k^2 + 1)$ (3) $2\tan^{-1}\left(\frac{1}{k+1}\right) = \log_e(k^2 + 2k + 2)$ (4) $2\tan^{-1}\left(\frac{1}{k}\right) = \log_e\frac{k^2 + 1}{k^2}$
Q76
First order differential equations (integrating factor)View
Let $y = y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} + \frac{2x^2 + 11x + 13}{x^3 + 6x^2 + 11x + 6} y = \frac{x + 3}{x + 1}$, $x > -1$, which passes through the point $(0, 1)$. Then $y(1)$ is equal to (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{5}{2}$ (4) $\frac{7}{2}$
If $(2, 3, 9)$, $(5, 2, 1)$, $(1, \lambda, 8)$ and $(\lambda, 2, 3)$ are coplanar, then the product of all possible values of $\lambda$ is (1) $\frac{21}{2}$ (2) $\frac{59}{8}$ (3) $\frac{57}{8}$ (4) $\frac{95}{8}$
Let $\vec{a}, \vec{b}, \vec{c}$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $(\vec{a} \times \vec{b}) \cdot (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) \cdot (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) \cdot (\vec{a} \times \vec{b}) = 168$, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to (1) 10 (2) 14 (3) 16 (4) 18
Let $Q$ be the foot of perpendicular drawn from the point $P(1, 2, 3)$ to the plane $x + 2y + z = 14$. If $R$ is a point on the plane such that $\angle PRQ = 60^\circ$, then the area of $\triangle PQR$ is equal to