List of practice Questions

Let the product of $ \omega_1 = (8 + i) \sin \theta + (7 + 4i) \cos \theta $ and $ \omega_2 = (1 + 8i) \sin \theta + (4 + 7i) \cos \theta $ be $ \alpha + i\beta $, where $ i = \sqrt{-1} $. Let $ p $ and $ q $ be the maximum and the minimum values of $ \alpha + \beta $ respectively.

Among the statements: (S1): The set $ \{ z \in \mathbb{C} - \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \} $ contains exactly two elements.
(S2): The set $ \{ z \in \mathbb{C} - \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \} $ contains infinitely many elements. Then, which of the following is correct?

Let $ z \in \mathbb{C} $ be such that $ \frac{z+3i}{z-2+i} = 2+3i $. Then the sum of all possible values of $ z $ is

If $ z_1, z_2, z_3 \in \mathbb{C} $ are the vertices of an equilateral triangle, whose centroid is $ z_0 $, then $ \sum_{k=1}^{3} (z_k - z_0)^2 $ is equal to

If the locus of $ z \in \mathbb{C} $, such that $ \text{Re} \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{ \bar{z} - 1}{2 \bar{z} - i} \right) = 2, $ is a circle of radius $ r $ and center $ (a, b) $, then $ \frac{15ab}{r^2} \text{ is equal to:} $

If $ z $ is a complex number and $ k \in \mathbb{R} $, such that $ |z| = 1 $, \[ \frac{2 + k^2 z}{k + \overline{z}} = kz, \] then the maximum distance from $ k + i k^2 $ to the circle $ |z - (1 + 2i)| = 1 $ is:

Estimate the ratio of shortest wavelength of radio waves to the longest wavelength of gamma waves.

Which one of the following guides the entry of the pollen tube into the embryo sac?

The sum of all local minimum values of the function $ f(x) $ as defined below is:

\[ f(x) = \left\{ \begin{array}{ll} 1 - 2x & \text{if } x < -1 \\ \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2 \\ \frac{11}{18} (x-4)(x-5) & \text{if } x > 2 \end{array} \right. \]

Let $ f: \mathbb{R} \to \mathbb{R} $ be a function defined by $ f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 $. If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of $ 28 \sum_{i=1}^5 f(i) $ is:

The integral \[ 80 \int_0^{\frac{\pi}{4}} \frac{(\sin \theta + \cos \theta)}{(9 + 16 \sin 2\theta)} \, d\theta \] is equal to:

The least value of $ n $ for which the number of integral terms in the Binomial expansion of $ \left( \sqrt{7} + \sqrt{11} \right)^n $ is 183, is:

The area of the region enclosed by the curves $ y = e^x $, $ y = |e^x - 1| $, and the y-axis is:

In an arithmetic progression, if $ S_{40} = 1030 $ and $ S_{12} = 57 $, then $ S_{30} - S_{10} $ is equal to:

Let $ A = \{1, 2, 3, 4\} $ and $ B = \{1, 4, 9, 16\} $. Then the number of many-one functions $ f: A \to B $ such that $ 1 \in f(A) $ is equal to:

In $ I(m, n) = \int_0^1 x^{m-1} (1-x)^{n-1} \, dx $, where $ m, n > 0 $, then $ I(9, 14) + I(10, 13) $ is:

Let $ f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32} $. Then the value of \[ 8 \left( f\left( \frac{1}{15} \right) + f\left( \frac{2}{15} \right) + \dots + f\left( \frac{59}{15} \right) \right) \] is equal to:

Let ([x]) denote the greatest integer less than or equal to (x).Then the domain of $(f(x) =sec^{-1}2[x] + 1))$ is:

Let $ f: [0, 3] \to A $ be defined by $ f(x) = 2x^3 - 15x^2 + 36x + 7 $ and $ g: [0, \infty) \to B $ be defined by $ g(x) = \frac{x}{x^{2025} + 1}. $ If both functions are onto and $ S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} $, then $ n(S) $ is equal to:

Let $ f(x) = \log x $ and \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] Then the domain of $ f \circ g $ is:

The sum of all local minimum values of the function $ f(x) $ as defined below is:
\[ f(x) = \begin{cases} 1 - 2x & \text{if } x < -1, \\[10pt] \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2, \\[10pt] \frac{11}{18}(x-4)(x-5) & \text{if } x > 2. \end{cases} \]