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Interact with our math tutor chatbot. Practice math by grade, topic and skill. Try our daily quiz below.

I can do calculations like 5+5, 2*(7+8), 3/9, sin(45), five times five, sqrt(16) or 2^10.

id: 960

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Glossary

Count : To count means to find out the number of objects by adding one for every object.

Hint

Count up by 8 starting at 2.

Solution

The correct answer is B. Counting up from 2, we reach the right answer after 8 steps: 2, 3, 4, 5, 6, 7, 8, 9, 10.

What is 2+8?

A 11
B 10
C 8
D 16

id: 3933

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Glossary

Division : Division is the mathematical operation that divides objects equally into groups. More generally $x\div y$ is defined as the number that if multiplied by $y$ equals $x.$

Remainder : In the division of a whole number $x$ by a natural number $y$ the remainder is the unique whole number $r$ with $0\leq r\lt y$ with $x=m\cdot y+r$ for some whole number $m.$ The remainder is the number of leftover wholes in the division. For example the remainder of the division of 14 by 3 is 2 as $14=4\cdot 3+2.$ Remainders are fundamental for the concept of congruence modulo $y$ in number theory.

Standard algorithm : By standard algorithm we refer to any one of the algorithms that are commonly taught at school to add, subtract, multiply or divide multi-digit numbers such as long division and partial products multiplication.

Hint

Use the standard algorithm for division.

Solution

The correct answer is 2. Using the standard algorithm for division we get 9692:5=1938 R 2: $$9692$$ $$5000$$ $$\overline{4692}$$ $$4500$$ $$\overline{192}$$ $$150$$ $$\overline{42}$$ $$40$$ $$\overline{2}$$

What is the remainder in the division 9,692:5?

id: 11622

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Glossary

Function : A function is a mapping in which every element in one set is mapped to exactly one element of a second set. Most often the mapping is described using a rule. For example the function $f(x)=x+1$ maps 2 to 3 and -1 to 0.

Graph : The graph of a function is the set of points $\{(x,f(x))\}$ that can be blotted in a coordinate system.

Monotonically decreasing : A function $f$ is called monotonically decreasing if we have $f(x)\geq f(y)$ for every $x\lt y.$ For example the function $f(x)=-x$ is monotonically decreasing and so is the function $f(x)=e^{-x}.$ A differentiable function $f$ is monotonically decreasing if and only if $f^{\prime}\leq 0.$

Monotonically increasing : A function $f$ is called monotonically increasing if we have $f(x)\leq f(y)$ for every $x\lt y.$ For example the function $f(x)=x$ is monotonically increasing and so is the function $f(x)=x^3.$ A differentiable function $f$ is monotonically increasing if and only if $f^{\prime}\geq 0.$

Point : A point is an element in a space. Shapes are made of sets of points.

Squared : $x$ squared refers to the number $x^2=x\cdot x.$ For example 3 squared equals 9.

Hint

Draw the graph of the function by calculating a few points including $f(9)$, $f(10)$, $f(11)$, $f(12)$ and $f(13).$

Solution

The correct answer is B. If $x_2>x_1>11,$ we can rewrite $x_1=11+h_1$ and $x_2=11+h_1+h_2$ with $h_1>0$ and $h_2>0.$ We have $f(x_2)-f(x_1)=(x_2-11)^2-(x_1-11)^2$ $=(h_1+h_2)^2-h_1^2$ $=h_1^2+2\cdot h_1h_2+h_2^2-h_1^2$ $=2\cdot h_1h_2+h_2^2>0,$ so $f$ is monotonically increasing for $x>11$. If $11>x_1>x_2$, we can rewrite $x_1=11-h_1$ and $x_1=11-h_1-h_2$ with $h_1>0$ and $h_2>0.$ We have $f(x_2)-f(x_1)=(x_2-11)^2-(x_1-11)^2$ $=(-h_1-h_2)^2-h_1^2$ $=h_1^2+2\cdot h_1h_2+h_2^2-h_1^2$ $=2\cdot h_1h_2+h_2^2>0,$ so $f$ is monotonically decreasing for $x<11$.

Which statement is true about the function $f(x)=(x-11)^2?$

A $f$ is monotonically decreasing.
B $f$ is monotonically increasing for $x>11$ and monotonically decreasing for $x<11.$
C $f$ is monotonically decreasing for $x>11$ and monotonically increasing for $x<11.$
D $f$ is monotonically increasing.

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