List of top Mathematics Questions

Write all the factors of the following numbers:

1. 24
2. 15
3. 21
4. 27
5. 12
6. 20
7. 18
8. 23
9. 36

The value of \[ \int_1^4 \log(\lfloor x \rfloor) \, dx \] where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), is:

Some pictures (a) to (c) are given below. Tell which of them show:
\((a)\)

\((b)\)

\((c)\)

(i) 3 × \(\frac{1}{5}\) =\(\frac{3}{5}\)
(ii) 2 × \(\frac{1}{3}\) =\(\frac{2}{3}\)
(iii) 3 × \(\frac{3}{4}\) = \(\frac{21}{4}\)

Solve and check result: \(8x + 4 = 3(x - 1) + 7\)

If y=2x^3x,then \(\frac{dy}{dx}\) at x=1 is

Solve the equation: \[ x + \log_{15}(5 + 3x) = x \log_{15} 5 + \log_{15} 24 \]

If the angle \( \theta \) between the line \[ \frac{2t + 1}{1} = \frac{y - 1}{2} = \frac{z}{2} \] and the plane \( 2x - y\sqrt{7} + z + 4 = 0 \) is such that \( \sin \theta = \frac{8}{\sqrt{3}} \), then the value of the expression is:

If \( y = y(x) \) satisfies \[ \left( \frac{2 + \sin x}{1 + y} \right) \frac{dy}{dx} = -\cos x, \] such that \( y(0) = 2 \), then the value of \( y\left( \frac{\pi}{2} \right) \) is:

Evaluate the integral: \[ \int_{-1}^{1} \log\left(\frac{2 - x}{2 + x}\right) dx \]

(i) For any integer a, what is (–1) × a equal to?
(ii) Determine the integer whose product with (–1) is
(a) –22
(b) 37
(c) 0

If \[ f(x) = 2 \left( \cos x + i \sin x \right) \left( \cos 3x + i \sin 3x \right) \cdots \left( \cos (2n-1)x + i \sin (2n-1)x \right) \] where \( n \in \mathbb{N} \), then what is the value of \( f''(x) \) ?

A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are \(2\frac 12\) m apart, what is the length of the wood required for the rungs?

Find the area bounded between the parabola \( y^2 = 4x \) and the line \( y = 2x - 3 \).

The value of \[ \int_1^4 \log(\lfloor x \rfloor) \, dx \] where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), is equal to:

Let \[ f(x) = (\cos x + \sin x) \cdot \cos(3x + i \sin x) \cdot \left[ (2n - 1)x + i \sin((2n - 1)x) \right], \] where \( n \in \mathbb{N} \), and \( i = \sqrt{-1} \). Then: \[ f''(x) = \, ? \]

A population \( P(t) \) of 1000 bacteria introduced to a nutrient medium grows according to the relation \[ P(t) = \frac{1000t + 1000t}{100 + t^2} \] The maximum size of this bacterial population is:

If \[ [2\vec{p} - 3\vec{q} \ \vec{q} \ \vec{s}] + [3\vec{p} + 2\vec{q} \ \vec{r} \ \vec{s}] = m[\vec{p} \ \vec{r} \ \vec{s}] + n[\vec{q} \ \vec{r} \ \vec{s}] + l[\vec{p} \ \vec{q} \ \vec{s}], \] then the values of \( m, n, l \) respectively are:

A plane passes through the point \( (1, -2, 1) \) and is perpendicular to both the planes \[ 2x - 2y - 2z = 5 \quad \text{and} \quad x - y + 2z = 24 \] Then, the distance of the point \( (1, 2, 2) \) from this plane is: