List of practice Questions

Consider the following hierarchical cache system with the following access times:

\[ \begin{array}{|c|c|c|} \hline \textbf{Cache Level} & \textbf{Hit Rate} & \textbf{Access Time} \\ \hline L1 & 90\% & 1 \text{ ns} \\ L2 & 80\% & 10 \text{ ns} \\ L3 & 100\% & 100 \text{ ns} \\ \hline \end{array} \]

Find $ T_{avg} $ for hierarchical or simultaneous access.

How many triangles are there in the figure given below?

The emf of the cell $\mathrm{Ni}_{(s)} \left| \mathrm{Ni}^{2+}(0.16 \, \mathrm{M}) \right| \mathrm{Ag}^{+}(0.002 \, \mathrm{M}) \left| \mathrm{Ag}_{(s)} \right.$ is ($E^0_{\text{cell}} = 1.05 \, \mathrm{V}$):

Prove that $\int_0^{\pi/2} \log(\cos x) \, dx = -\frac{\pi}{2} \log 2$.

If $A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$ solve the following system of equations:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3.

Match the following:

List - I	List - II
A) Morphine	IV) Pain killer
B) Cannabinoids	V) Dopamine
C) Cocaine	I) Cardiovascular system
D) Benzodiazepines	II) Tranquilizers

A metal crystallizes in simple cubic lattice. The volume of one unit cell is $ 6.4 \times 10^{-7} \, \text{pm}^3 $. What is the radius of the metal atom in pm?

The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from $ \pm 15 \, {V} $ supplies. If a $ +1 \, {V} $ signal is applied at the input, the output voltage (rounded off to one decimal place) is:

Which fiber is known for its excellent moisture-wicking properties, making it ideal for active wear?

On prolonged heating with HI, glucose gives a compound 'C', which can be obtained by Wurtz reaction using sodium metal and compound 'D'. Identify 'D'

Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.

If the incentre of the triangle formed by lines $$ x - 2 = 0, \quad x + y - 1 = 0, \quad x - y + 3 = 0 $$ is $ (\alpha, \beta) $, then find $ \beta $.

If $ l, m $ represent any two elements (identical or different) of the set $ \{1, 2, 3, 4, 5, 6, 7\} $, then the probability that $ lx^2 + mx + 1>0 \,\, \forall x \in \mathbb{R} $ is

Which city will host the 2025 International Expo, focusing on sustainable innovation and technology?

The circle $ x^2 + y^2 + ax + by + \gamma = 0 $ is the image of the circle $ x^2 + y^2 - 6x - 10y + 30 = 0 $ across the line $ 3x + y = 2 $. The value of $ [\alpha + \beta + \gamma] $ is (where $ [.] $ represents the floor function)

Let $ g: \mathbb{R} \to \mathbb{R} $ and $ h: \mathbb{R} \to \mathbb{R} $ be two functions such that $ h(x) = s \cdot g \cdot n(g(x)) $, where $ \mathbb{R} $ denotes the set of all real numbers and $ s $ and $ n $ stand for the signum and the sine functions respectively. Then, select which of the following is not true?

If $ 8^{x-1} = \left( \frac{1}{4} \right)^x $, then the value of \[ \frac{1}{\log_{x+14} x - \log_{x+15}} \div \frac{1}{\log_{1-x} 4 - \log_{1-x} 5} \] is

Let $ A = (1, 2, 3, \dots, 20) $. Let $ R \subseteq A \times A $ such that $ R = \{(x, y) : y = 2x - 7 \} $. Then the number of elements in $ R $ is equal to:

Let $ A $ and $ B $ be two square matrices of the same order satisfying $ A^2 + 5A + 5I = 0 $ and $ B^2 + 3B + I = 0 $ respectively, where $ I $ is the identity matrix. Then the inverse of the matrix $ C = BA + 2B - 2A + 4I $ is:

Consider the sample space $ \Omega = \{(x, y): x, y \in \{1, 2, 3, 4\} \} $ where each outcome is equally likely. Let $ A = \{ x \geq 2 \} $ and $ B = \{ y > x \} $ be two events. Then which of the following is NOT true?

Which one of the following is NOT a correct statement?

The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:

The number of accidents per week in a town follows a Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is $ k e^{-6} $, then the value of $ k $ is:

The slope of the normal line to the curve $ x = t^2 + 3t - 8 $ and $ y = t^2 - 2t - 5 $ at the point $ (2, -1) $ is

If $ B = \sin^2y + \cos^4y $, then for all real $ y $,