All practice questions

The critical velocity $v_c$ of a fluid flowing through a cylindrical pipe of diameter $D$ depends on the dynamic viscosity $\eta$ of the fluid, its density $\rho$, and the pipe diameter $D$ according to the empirical relationship $v_c = R_e \eta^x \rho^y D^z$, where $R_e$ is the dimensionless Reynolds number. Using dimensional analysis, calculate the numerical value of the sum of the exponents $x + y + z$.

Given the simultaneous equations $2x - y = 4$ and $x^2 + y^2 = 13$, where both $x$ and $y$ are positive real numbers, calculate the value of $x + y$.

A curve has a gradient function defined by $\frac{dy}{dx} = 12x^3 - 6\sin(3x) + 4$. If the curve passes through the point $(0, 15)$, what is the value of the constant of integration $C$?

Given the matrices $A = \begin{pmatrix} x & 2 & 1 \\ 3 & 0 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -1 \\ 5 & 3 \end{pmatrix}$, find the value of $x$ if $\det(A) + \det(B) = 5$.

A business owner borrowed $\text{₦}150,000$ to expand his store at a compound interest rate of $8\%$ per annum, compounded annually. What is the total compound interest he will pay at the end of 2 years?

During Rutherford's alpha particle scattering experiment, the vast majority of alpha particles passed straight through the gold foil with little to no deflection. What conclusion about atomic structure was drawn directly from this specific observation?

A signal generator positioned inside a transparent, sealed glass chamber produces both sound waves and radio waves simultaneously. As an air pump gradually evacuates the chamber to create a vacuum, which of the following best describes what happens to the propagation of these two waves?

A research committee of $5$ members is to be formed from $6$ male scientists and $5$ female scientists. The committee must contain strictly more male scientists than female scientists. If one particular male scientist and one particular female scientist refuse to serve on the same committee together, in how many different ways can the committee be selected?

A binary operation $\star$ is defined on the set of real numbers $\mathbb{R}$ by $a \star b = a + b + 7$. What is the identity element of the operation?

Match each historical atomic model on the left with its defining postulate, experimental outcome, or theoretical limitation on the right.

A committee of $6$ delegates is to be seated around a circular table for a conference. If two specific delegates must always sit next to each other, in how many distinct ways can the delegates be arranged?

Match each physical quantity to its correct classification as either a fundamental or a derived physical quantity.

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 20\}$. Consider two subsets of $\mathcal{U}$ given by $A = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$ and $B = \{x \in \mathcal{U} : x \text{ is a factor of } 20\}$. What is the number of elements in the complement of $(A \cup B)$, denoted as $n((A \cup B)')$?

A small charged sphere of mass $2.0 \times 10^{-4}\text{ kg}$ carrying a positive charge of $+4.0 \times 10^{-8}\text{ C}$ is suspended by a light insulating string between two vertical parallel plates. When a uniform horizontal electric field of magnitude $E$ is applied between the plates, the string deflects and comes to equilibrium at an angle of $45^\circ$ to the vertical. Taking the acceleration due to gravity $g = 10\text{ m}\cdot\text{s}^{-2}$, calculate the magnitude of the electric field intensity $E$ in $\text{N}\cdot\text{C}^{-1}$.

In $\Delta PQR$, side $p = 6\text{ cm}$, side $q = 6\sqrt{2}\text{ cm}$, and $\angle P = 30^\circ$. If $\angle Q$ is an acute angle, what is the measure of $\angle Q$?

What is the value of the definite integral $\int_{1}^{3} (4x - 1) \, dx$?

A trader bought a set of solar panels for $\text{₦}160,000$. He marked up the cost price by $25\%$ to fix the marked price. During a trade fair, he offered a $10\%$ discount on the marked price. What is his net profit in Naira?

A binary operation $\oplus$ defined on the set of real numbers is given by $a \oplus b = a + b + kab$, where $k$ is a non-zero real constant. If the inverse of $3$ under this operation is $-\frac{1}{2}$, what is the inverse of $4$ under the same operation?

A pie chart illustrates the annual allocation of funds for an agricultural research station across four sectors: Crop Research, Livestock, Irrigation, and Equipment Maintenance. The sector angles for Crop Research, Livestock, and Irrigation are $135^\circ$, $90^\circ$, and $60^\circ$ respectively. If the total annual budget is $\text{N}7,200,000$, what is the amount allocated to Equipment Maintenance in Naira?

In a hydrogen atom modeled according to Bohr's theory, an electron undergoes a transition from an excited state with an energy of $-1.51\text{ eV}$ to a lower energy state of $-3.40\text{ eV}$. Calculate the energy of the emitted photon in electron-volts ($\text{eV}$).