1.
Solve for the roots of the quadratic equation: $x^2 - 7x + 12 = 0$.
2.
Find the roots of the equation $2x^2 + 5x - 3 = 0$ using the quadratic formula.
3.
If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x + 8 = 0$, find
the value of $\alpha^2 + \beta^2$.
4.
Solve the linear equation for $x$: $\frac{3x + 4}{2x - 1} = 5$.
5.
Write the binomial expansion for $(1 + x)^n$ up to the first three terms.
6.
Using binomial approximation, evaluate the value of $(1.02)^4$ up to three
decimal places.
7.
Evaluate $\sqrt{0.99}$ using binomial theorem approximation, assuming $x \ll 1$.
8.
Expand the expression $(1 - x)^{-2}$ up to three terms, given that $|x| <
1$.
9.
Write the formula for the sum of an infinite geometric progression (G.P.).
10.
Find the sum of the infinite series: $1 + \frac{1}{2} + \frac{1}{4} +
\frac{1}{8} + \dots \infty$.
11.
Convert $150^\circ$ into radians.
12.
Convert $\frac{5\pi}{6}$ radians into degrees.
13.
In a right-angled triangle, what are the standard physics values for
$\sin(37^\circ)$ and $\cos(37^\circ)$?
14.
Find the exact value of $\sin(120^\circ)$.
15.
Evaluate $\cos(210^\circ)$.
16.
Find the value of $\tan(315^\circ)$.
17.
If $\sin \theta = \frac{4}{5}$ and $\theta$ lies in the first quadrant, find
$\cos \theta$ and $\tan \theta$.
18.
Evaluate $\sin(15^\circ)$ using the compound angle formula for $\sin(A - B)$.
19.
State the formula for $\sin(2\theta)$ in terms of $\sin \theta$ and $\cos
\theta$.
20.
Write the three standard formulas for $\cos(2\theta)$.
21.
What is the approximate value of $\sin \theta$ and $\tan \theta$ if $\theta$
is very small (in radians)?
22.
What is the approximate value of $\cos \theta$ for very small angles?
23.
Simplify the expression: $\sin(180^\circ + \theta)$.
24.
Simplify the expression: $\cos(90^\circ + \theta)$.
25.
Solve for $\theta$ if $2\cos\theta - \sqrt{3} = 0$, given $0 \le \theta \le
90^\circ$.
26.
Evaluate the base-2 logarithm: $\log_2 64$.
27.
Find the value of $x$ if $\log_5 x = 3$.
28.
Simplify the expression: $\log_{10} 20 + \log_{10} 5$.
29.
Write the expression $3\ln(x) - 2\ln(y)$ as a single natural logarithm.
30.
Solve the exponential equation for $x$: $e^{3x} = 15$.
31.
What is the value of $\ln(e^4)$?
32.
If $\log_x 81 = 4$, determine the base $x$.
33.
What is the numerical value of $\ln(1)$ and $\log_{10}(1)$?
34.
Simplify the general logarithmic expression: $\log_a (a^x)$.
35.
State the relationship formula used to convert a natural logarithm ($\ln x$)
to a common logarithm ($\log_{10} x$).
36.
Find the slope ($m$) and y-intercept ($c$) of the straight line represented
by $4x + 3y = 24$.
37.
Write the equation of a straight line passing through the origin with a slope
of $-3$.
38.
Calculate the slope of the line passing through the points $(2, 3)$ and $(5,
9)$.
39.
Identify the geometric shape of the graph for the equation $y = 5x^2$. Where
does its vertex lie?
40.
Identify the geometric shape of the graph for the equation $x^2 + y^2 = 36$.
What is its radius?
41.
Write the standard equation of an ellipse centered at the origin with
semi-major axis $a$ and semi-minor axis $b$.
42.
If a physical quantity $y$ is inversely proportional to $x$ ($y \propto
\frac{1}{x}$), what is the shape of the graph plotted between $y$ and $x$?
43.
Find the straight-line distance between the points $(4, 3)$ and the origin
$(0, 0)$.
44.
Describe the shape of the exponential decay curve given by the equation $y =
e^{-x}$. Does it ever touch the x-axis?
45.
What is the equation of a parabola that opens leftwards and has its vertex at
the origin?
46.
Find the derivative $\frac{dy}{dx}$ if $y = x^6$.
47.
Differentiate $y = 4x^3 - 5x^2 + 7x - 2$ with respect to $x$.
48.
Find the derivative $\frac{d}{dx}(\sqrt{x})$.
49.
Differentiate $y = \frac{1}{x^3}$ with respect to $x$.
50.
Find the derivative of the trigonometric function $y = \sin x - \cos x$.
51.
Using the Product Rule, differentiate $y = x^2 \sin x$.
52.
Using the Quotient Rule, differentiate $y = \frac{\ln x}{x^2}$.
53.
Using the Chain Rule, find $\frac{dy}{dx}$ if $y = (3x + 4)^5$.
54.
Differentiate the composite function $y = \cos(x^3)$.
55.
Find the derivative of the exponential function $y = e^{4x}$.
56.
Calculate the double derivative $\frac{d^2y}{dx^2}$ for the function $y = x^4
- 2x^3$.
57.
The position of a moving particle is given by $x(t) = 3t^2 - 6t + 2$. Find
its velocity equation $v(t) = \frac{dx}{dt}$.
58.
For the particle in the previous question, calculate its acceleration $a(t) =
\frac{dv}{dt}$. Is the acceleration constant?
59.
Find the value of $x$ for which the function $y = x^2 - 6x + 9$ is at a
minimum. What is the minimum value?
60.
For what value(s) of $x$ does the function $y = x^3 - 12x$ have a local
maximum or minimum?
61.
Evaluate the indefinite integral: $\int x^4 \, dx$.
62.
Find $\int (3x^2 - 4x + 2) \, dx$.
63.
Evaluate the standard integral: $\int \frac{1}{x} \, dx$.
64.
Find the integral $\int \cos x \, dx$.
65.
Evaluate the integral with substitution: $\int \sin(3x) \, dx$.
66.
Find the integral of the exponential function: $\int e^{5x} \, dx$.
67.
Evaluate the definite integral: $\int_{1}^{2} x^3 \, dx$.
68.
Evaluate the definite trigonometric integral: $\int_{0}^{\pi} \sin x \, dx$.
69.
Calculate the area under the curve $y = 3x^2$ from $x = 0$ to $x = 2$.
70.
Evaluate $\int \left(x + \frac{1}{x}\right)^2 \, dx$ by first expanding the
algebraic expression.
71.
The velocity of a particle is given by $v(t) = 4t^3$. Find its displacement
equation $x(t)$ given that $x(0) = 0$.
72.
Evaluate the infinite definite integral: $\int_{0}^{\infty} e^{-x} \, dx$.
73.
Integrate the polynomial: $\int (6x^5 - 2x) \, dx$.
74.
Evaluate the integral involving roots: $\int \sqrt{x} \, dx$.
75.
Find the integral $\int \frac{1}{\sqrt{x}} \, dx$.
