List of practice Questions

Which one of the following series is convergent?

The mean and the standard deviation of weights of ponies in a large animal shelter are 20 kg and 3 kg, respectively. A pony is selected at random. Using Chebyshev's inequality, find the lower bound of the probability that its weight lies between 14 kg and 26 kg.

Let $E$ and $F$ be two events. Then which one of the following statements is NOT always TRUE?

Let $X$ be a random variable having Poisson(2) distribution. Then $E\left(\dfrac{1}{1+X}\right)$ equals

Carefully read the paragraph below:
A map is a useful metaphor for our brain when talking about __ because at its most basic level our brain__ to be our atlas of sorts, a system of routes __ to navigate us toward just one destination: staying alive!
From the options below, choose the set that MOST appropriately fills up the blanks.

Carefully read the paragraph below:
__, medicine has been operated by trial and error, in other words,___. We know by now that there can be entirely___ connections between symptoms and treatment, and some medications succeed in medical trials for mere random reasons.
From the options below, choose the one that MOST appropriately fills up the blanks.

I. Yes: Studies have shown that breathing in vehicular pollution reaches dangerous levels when the exposure is more than five hours a day.
II. No: Breathing in vehicular pollution for more than five hours a day does not lead to lung cancer as revealed in a study analyzing vehicular pollution.
Which statement is strong?

Create meaningful words from the following and then find the odd one out.

If $H<G=J$ and $J>K$, then which of the following is necessarily true?
(I) $H>K$
(II) $G>K$

If 8 g of a non-electrolyte solute is dissolved in 114 g of n-octane to reduce its vapour pressure to 80%, the molar mass (in g mol^-1) of the solute is
[Given that the molar mass of n-octane is 114 g mol^-1.]

Let $ T : \mathbb{R}^7 \to \mathbb{R}^7 $ be a linear transformation with $ \text{Nullity}(T) = 2. $ Then, the minimum possible value for $ \text{Rank}(T^2) $ is ............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function with $ f'(x) = f(x) $ for all $ x. $ Suppose that $ f(\alpha x) $ and $ f(\beta x) $ are two non-zero solutions of the differential equation \[ 4 \frac{d^2 y}{dx^2} - p \frac{dy}{dx} + 3y = 0 \] satisfying $ f(\alpha x)f(\beta x) = f(2x) $ and $ f(\alpha x)f(-\beta x) = f(x). $ Then, the value of $ p $ is ...........

The minimum value of the function $ f(x, y) = x^2 + xy + y^2 - 3x - 6y + 11 $ is ............

Let $ f(x) = \sqrt{x} + \alpha x, \; x > 0 $ and \[ g(x) = a_0 + a_1(x - 1) + a_2(x - 1)^2 \] be the sum of the first three terms of the Taylor series of $ f(x) $ around $ x = 1 $. If $ g(3) = 3 $, then $ \alpha $ is .............

Consider the expansion of the function $ f(x) = \dfrac{3}{(1 - x)(1 + 2x)} $ in powers of $ x $, valid in $ |x| < \dfrac{1}{2}. $ Then the coefficient of $ x^4 $ is ................

Let $ f(x, y) = 0 $ be a solution of the homogeneous differential equation $ (2x + 5y)dx - (x + 3y)dy = 0. $
If $ f(x + \alpha, y - 3) = 0 $ is a solution of $ (2x + 5y - 1)dx + (2 - x - 3y)dy = 0, $ then the value of $ \alpha $ is .............

Let $ S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\} $ and $ f : S \to \mathbb{R} $ be defined by $ f(x) = \frac{1}{x}. $ Then \[ \max \left\{ \delta : |x - \tfrac{1}{3}| < \delta \Rightarrow |f(x) - f(\tfrac{1}{3})| < 1 \right\} \] is ............. (rounded off to two decimal places).

Let $ f : \mathbb{R} \to \mathbb{R} $ be such that $ f, f', f'' $ are continuous with $ f > 0, f' > 0, f'' > 0. $ Then $ \displaystyle \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} $ is ............

Let $ f $ be a real-valued function of a real variable, such that $ |f^{(n)}(0)| \leq K $ for all $ n \in \mathbb{N} $, where $ K > 0. $ Which of the following is/are true?

Consider the following system of linear equations:
\[ \begin{cases} x + y + 5z = 3, \\ x + 2y + mz = 5, \\ x + 2y + 4z = k. \end{cases} \]

The system is consistent if

Let $ L[y] = x^2\dfrac{d^2y}{dx^2} + px\dfrac{dy}{dx} + qy, $ where $ p, q $ are real constants. Let $ y_1(x) $ and $ y_2(x) $ be two solutions of $ L[y] = 0, \, x > 0, $ that satisfy $ y_1(x_0) = 1, y_1'(x_0) = 0, y_2(x_0) = 0, y_2'(x_0) = 1 $ for some $ x_0 > 0. $ Then,

If $ s_n = \dfrac{(-1)^n}{2^n + 3} $ and $ t_n = \dfrac{(-1)^n}{4n - 1}, \, n = 0, 1, 2, \dots, $ then

Let $ S^1 = \{z \in \mathbb{C} : |z| = 1\} $ be the circle group under multiplication and $ i = \sqrt{-1}. $ Then the set $ \{\theta \in \mathbb{R} : (e^{i2\pi\theta}) \text{ is infinite}\} $ is

Let $ f : [0, 1] \to \mathbb{R} $ be a continuous function such that $ f\left(\dfrac{1}{2}\right) = -\dfrac{1}{2} $ and \[ |f(x) - f(y) - (x - y)| \le \sin(|x - y|^2) \] for all $ x, y \in [0, 1]. $ Then $ \int_0^1 f(x) \, dx $ is

Let $ D = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \le 1\} $ and $ f : D \to \mathbb{R} $ be a non-constant continuous function. Which of the following is TRUE?