mathematics Questions

It is given that there exist constants $A$ and $B$ such that

for all $x \in \mathbb{R}$. Find the values of $A$ and $B$. Hence, write down

in partial fractions.

Find the remainder when $2x^3 + 7x^2 + 5x + 9$ is divided by $(x - 1)(x - 2)$

Convert $2.31313131\ldots$ into a rational number

If $A(6, 7)$, $B(12, 9)$, and $C(15, 8)$, find the coordinates of the centroid and the area of triangle ABC

A quadratic polynomial $G(x)$ has remainders $\frac{1}{3}$, $\frac{1}{5}$, and $\frac{1}{8}$ when divided by $(x - 1)$, $(x - 3)$, and $(x - 6)$ respectively. Show that $(x - 1)$, $(x - 3)$, and $(x - 6)$ are factors of the polynomial

Hence, find $f(x)$

Let $f(x) = x^3 + 2x^2 + mx + n$. When $f(x)$ is divided by $(x - 2)$ and $(x - 3)$, the remainders are 33 and 70 respectively. Find the values of $m$ and $n$

Find the partial fraction decomposition of

If

then prove that

Find the partial fractions of

If $x = y \tan \theta - x \tan^2 \theta \quad \text{and} \quad 2x \sin \alpha = y \cos \alpha - 2x \cos \alpha \tan \theta,$ then show that $\cot \alpha = \tan 2\theta$

If $\tan \alpha = k \tan \beta,$ then prove that

When $x^3 + Ax^2 + Bx + 9$ is divided by $x^2 + 2$, the remainder is $6x - 1$. Then find the values of $A$ and $B$.

Let

and

Show that

Then, express

and find the values of $A$, $B$, $C$, and $D$. Hence, find the remainder when $f(x)$ is divided by $(x^2 + 1)(x + 1)$

Prove that:

If

then prove that

If $A(8, 9)$, $B(2, 2)$, and $C(6, 4)$ are the vertices of triangle $ABC$, find the coordinates of the centroid of triangle $ABC$.

Convert

into a rational number.

Given the binary number 10 (base 2), convert it to its decimal equivalent.

Given the binary number 01 (base 2), convert it to its decimal equivalent.

Given the binary number 11 (base 2), convert it to its decimal (base 10) equivalent.

Convert the binary number 1001 (base 2) into its decimal (base 10) equivalent.

Convert the binary number 0110 (base 2) into its decimal (base 10) equivalent.

If $x = \cos 3\theta + \sin 3\theta$ and $y = \cos \theta - \sin \theta$, then show that $x - y = 2y \sin 2\theta$

Convert the binary number 1111₂ into its decimal (base 10) equivalent.

Convert the binary number 1100₂ into its decimal (base 10) equivalent.

Convert the binary number 1101₂ into its decimal (base 10) equivalent.

Convert the binary number 1011₂ into its decimal (base 10) equivalent.

Convert the binary number 0111₂ into its decimal (base 10) equivalent.

Convert the binary number 0101₂ into its decimal (base 10) equivalent.

Convert the 8-bit binary number 1101 0010₂ into its decimal (base 10) equivalent.

Convert the 8-bit binary number 0110 0111₂ into its decimal (base 10) equivalent.

Convert the 8-bit binary number 10010010₂ into its decimal (base 10) equivalent.

Convert the 8-bit binary number 01110111₂ into its decimal (base 10) equivalent.

If $\cot \theta = \frac{4}{3}$, where $\pi < \theta < \frac{3\pi}{2}$, prove that: $\sin(2\theta) = \frac{24}{25}$, and $\tan(2\theta) = \frac{24}{7}$

Convert the 16-bit binary number:

into its decimal (base-10) equivalent.

Find the partial fraction decomposition of the function:

If $A + B + C = \pi$, prove that:

Convert 2.4141414141… into a rational number.

Find the remainder when $f(x) = x^3 + 5x^2 + 7x + 11$ is divided by $(x - 1)$

It is given that there exist constants $A$ and $B$ such that

for all $x \in \mathbb{R}$. Find the values of $A$, $B$, and $C$. Hence, write down

in partial fractions.