CBSE Class 9 Mathematics — NCERT Solutions

From the figure, the door segment $D_1R_1$ lies on the x-axis.

• So its distance from the y-axis is the x-coordinate of $D_1$.
• Its distance from the x-axis is 0, because it is on the x-axis.

Since the door lies on the x-axis and the room’s left wall is the y-axis, the left end of the door is at $x=8$ in the figure. Hence the coordinates of $D_1$ are $(8,0)$.

For $2x^2 - 5x + 3$, the highest power of the variable $x$ is $2$. So the degree is 2.

For $y^3 + 2y - 1$, the highest power of $y$ is $3$. So the degree is 3.

The numbers $11, 13, 17, 19$ are all prime numbers: each has exactly two factors, $1$ and itself.

The next three prime numbers after $19$ are 23, 29, 31.

No, natural numbers are not closed under subtraction.

Examples:

• $5 - 2 = 3$, which is a natural number.
• But $2 - 5 = -3$, which is not a natural number.
• Also, $1 - 1 = 0$, and $0$ is not included in the set of natural numbers given in the chapter.

So subtraction does not always give a natural number.

Using $(a+b)^2=a^2+2ab+b^2$ with $a=7x$ and $b=4y$:

$$(7x+4y)^2=(7x)^2+2(7x)(4y)+(4y)^2$$
$$=49x^2+56xy+16y^2$$

For a circle through two points A and B, the smallest possible circle is when AB itself is a diameter. Then the centre is the midpoint of AB, so the radius is half of AB. Any smaller radius cannot pass through both points.

For a circle, the circumference is $C=2\pi r$.

Given $C=44$ cm and using $\pi=\frac{22}{7}$:

$$44=2\times \frac{22}{7}\times r$$

$$44=\frac{44}{7}r$$

$$r=44\times \frac{7}{44}=7$$

So the radius is 7 cm.

This is certain because Monday always comes after Sunday in the weekly sequence.

• Natural numbers: each term is 1 more than the previous term, so the next few are 7, 8, 9, ...
• Odd numbers: each term increases by 2, so the next few are 13, 15, 17, ...
• Triangular numbers: the differences increase by 1 each time; the next few are 28, 36, 45, ...
• Square numbers: each term is the square of a natural number; the next few are 49, 64, 81, ...