Find five rational numbers between 1 and 2.
Question: Find five rational numbers between 1 and 2. Solution: We need to find 5 rational numbers between 1 and 2. Consider, $1=\frac{1}{1}$ $\Rightarrow 1=\frac{1}{1} \times \frac{6}{6}$ $\Rightarrow 1=\frac{6}{6}$ And $\Rightarrow 2=\frac{2}{1}$ $\Rightarrow 2=\frac{2}{1} \times \frac{6}{6}$ $\Rightarrow 2=\frac{12}{6}$...
Read More →The radius of an air bubble is increasing at the rate of
Question: The radius of an air bubble is increasing at the rate of $\frac{1}{2} \mathrm{~cm} / \mathrm{s}$. At what rate is the volume of the bubble increasing when the radius is 1 $\mathrm{cm} ?$ Solution: The air bubble is in the shape of a sphere. Now, the volume of an air bubble (V) with radius (r) is given by, $V=\frac{4}{3} \pi r^{3}$ The rate of change of volume (V)with respect to time (t)is given by, $\frac{d V}{d t}=\frac{4}{3} \pi \frac{d}{d r}\left(r^{3}\right) \cdot \frac{d r}{d t} \...
Read More →Is zero a rational number? Can you write it in the form
Question: Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ ? Solution: Yes, zero is a rational number because it is either terminating or non-terminating so we can write in the form of $\frac{p}{q}$, where $p$ and $q$ are natural numbers and $q$ is not equal to zero. So, $p=0, q=1,2,3 \ldots .$ Therefore, $\frac{p}{q}=\frac{0}{1}$ or $\frac{0}{2}$ or $\frac{0}{3} \ldots . .$...
Read More →Every real number is either
Question: Every real number is either ________ or _______ number. Solution: The real number includes all the rational as well as irrational numbers.Hence, everyreal number is eitherrationalorirrationalnumber....
Read More →The sum of a rational number and an irrational number is
Question: The sum of a rational number and an irrational number is ________ number. Solution: The sum of a rational number and an irrational numberalways results in an irrational number.Hence, the sum of a rational number and an irrational number is anirrational number....
Read More →0.3+0.4 is equal to _________.
Question: $0 . \overline{3}+0 . \overline{4}$ is equal to Solution:...
Read More →A particle moves along the curve
Question: A particle moves along the curve $6 y=x^{3}+2$. Find the points on the curve at which the $y$-coordinate is changing 8 times as fast as the $x$ coordinate. Solution: The equation of the curve is givenas: $6 y=x^{3}+2$ The rate of change of the position of the particle with respect to time (t)is given by, $6 \frac{d y}{d t}=3 x^{2} \frac{d x}{d t}+0$ $\Rightarrow 2 \frac{d y}{d t}=x^{2} \frac{d x}{d t}$ Whenthey-coordinate of the particle changes 8 times as fast as the $x$-coordinate i....
Read More →The simplest form of 1.6 is
Question: The simplest form of $1 . \overline{6}$ is Solution: Hence, the simplest form of $1 . \overline{6}$ is $\frac{5}{3}$....
Read More →The product of a non-zero rational number with an irrational number is always an
Question: The product of a non-zero rational number with an irrational number is always an ________ number. Solution: The product of a non-zero rational number with an irrational number always results in an irrational number.Hence, the product of a non-zero rational number with an irrational number is always anirrational number....
Read More →π is an _______ number.
Question: is an _______ number. Solution: The decimal expansion of neither terminates nor repeatsafter finitely many digits.Therefore, it is an irrational number.Hence, is anirrational number....
Read More →Find the derivative of x at x = 1.
Question: Find the derivative ofxatx= 1. Solution: Letf(x) =x. Accordingly, $f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$ $=\lim _{h \rightarrow 0} \frac{(1+h)-1}{h}$ $=\lim _{h \rightarrow 0} \frac{h}{h}$ $=\lim _{b \rightarrow 0}(1)$ $=1$ Thus, the derivative of $x$ at $x=1$ is 1 ....
Read More →A ladder 5 m long is leaning against a wall.
Question: A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall? Solution: Letym be the height of the wall at which the ladder touches. Also, let the foot of the ladder bexmaway from the wall. Then, by Pythagoras theorem, we have: $x^{2}+y^{2}=25$ [Length of the ladder $\left.=5 \mathrm{~m}\right]$ $\Rightarrow ...
Read More →Every recurring decimal is a
Question: Every recurring decimal is a _________ number. Solution: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.Hence, every recurring decimal is arationalnumber...
Read More →Find the derivative of 99x at x = 100.
Question: Find the derivative of 99xatx= 100. Solution: Letf(x) = 99x. Accordingly, $f^{\prime}(100)=\lim _{h \rightarrow 0} \frac{f(100+h)-f(100)}{h}$ $=\lim _{h \rightarrow 0} \frac{99(100+h)-99(100)}{h}$ $=\lim _{h \rightarrow 0} \frac{99 \times 100+99 h-99 \times 100}{h}$ $=\lim _{h \rightarrow 0} \frac{99 h}{h}$ $=\lim _{h \rightarrow 0}(99)=99$ Thus, the derivative of $99 x$ at $x=100$ is 99 ....
Read More →The value of 1.999. in the form of
Question: The value of $1.999$. in the form of $\frac{m}{n}$, where $m$ and $n$ are integers and $n \neq 0$, is Solution: Hence, the value of $1.999 \ldots$ in the form of $\frac{m}{n}$, where $m$ and $n$ are integers and $n \neq 0$, is $\frac{2}{1}$....
Read More →The decimal expansion of
Question: The decimal expansion of $\sqrt{2}$ is and Solution: $\sqrt{2}$ is an irrational number. The decimal expansion of an irrational number neither terminates nor repeats after finitely many digits. Hence, the decimal expansion of $\sqrt{2}$ is non-terminating and non-repeating....
Read More →A balloon, which always remains spherical has a variable radius.
Question: A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm. Solution: The volume of a sphere $(V)$ with radius $(r)$ is given by $V=\frac{4}{3} \pi r^{3}$. Rate of change of volume(V)with respect to its radius (r)is given by, $\frac{d V}{d r}=\frac{d}{d r}\left(\frac{4}{3} \pi r^{3}\right)=\frac{4}{3} \pi\left(3 r^{2}\right)=4 \pi r^{2}$ Therefore, when radius = 10 cm, $\frac{d V}{d r}=4 \pi(1...
Read More →The decimal expansion of an irrational number is non-terminating and
Question: The decimal expansion of an irrational number is non-terminating and ________. Solution: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.In case of irrational number, the decimal expansion neither terminates nor repeatsafter finitely many digits.Hence,the decimal expansion of an irrational number is non-terminating andnon-repeating....
Read More →The decimal expansion of a rational number is either
Question: The decimal expansion of a rational number is either ______ or _______. Solution: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.Hence, the decimal expansion of a rational number is eitherterminatingorrecurring....
Read More →Find the derivative of x2 – 2 at x = 10.
Question: Find the derivative of $x^{2}-2$ at $x=10$. Solution: Let $f(x)=x^{2}-2$. Accordingly, $f^{\prime}(10)=\lim _{h \rightarrow 0} \frac{f(10+h)-f(10)}{h}$ $=\lim _{h \rightarrow 0} \frac{\left[(10+h)^{2}-2\right]-\left(10^{2}-2\right)}{h}$ $=\lim _{h \rightarrow 0} \frac{10^{2}+2 \cdot 10 \cdot h+h^{2}-2-10^{2}+2}{h}$ $=\lim _{h \rightarrow 0} \frac{20 h+h^{2}}{h}$ $=\lim _{h \rightarrow 0}(20+h)=(20+0)=20$ Thus, the derivative of $x^{2}-2$ at $x=10$ is 20 ....
Read More →A balloon, which always remains spherical on inflation,
Question: A balloon, which always remains sphericalon inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm. Solution: The volume ofa sphere (V)with radius (r) is given by, $V=\frac{4}{3} \pi r^{3}$ Rate of change of volume (V)with respect to time (t)is given by, $\frac{d V}{d t}=\frac{d V}{d r} \cdot \frac{d r}{d t}$ [By chain rule] $=\frac{d}{d r}\left(\frac{4}{3} \pi r^{3}\right) ...
Read More →The smallest rational number by which
Question: The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is (a) $\frac{1}{10}$ (b) $\frac{3}{10}$ (c) 3 (d) 30 Solution: Give number is $\frac{1}{3}$. Now multiplying by $\frac{3}{10}$ in the given number, we have $\frac{1}{3}=\frac{1}{3} \times \frac{3}{10}$ $\Rightarrow \frac{1}{3}=\frac{1}{10}$ $\Rightarrow \frac{1}{3}=0.1$ Hence the correct option is...
Read More →For what integers m and n does
Question: If $f(x)= \begin{cases}m x^{2}+n, x0 \\ n x+m, 0 \leq x \leq 1 . \text { For what integers } m \text { and } n \text { does } \lim _{x \rightarrow 0} f(x) \text { and } \lim _{x \rightarrow 1} f(x) \text { exist? } \\ n x^{3}+m, x1\end{cases}$ Solution: The given function is $f(x)= \begin{cases}m x^{2}+n, x0 \\ n x+m, 0 \leq x \leq 1 \\ n x^{3}+m, x1\end{cases}$ $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0}\left(m x^{2}+n\right)$ $=m(0)^{2}+n$ $=n$ $\lim _{x \rightarrow 0^{...
Read More →The number of consecutive zeros in
Question: The number of consecutive zeros in $2^{3} \times 3^{4} \times 5^{4} \times 7$, is (a) 3(b) 2(c) 4(d) 5 Solution: We are given the following expression and asked to find out the number of consecutive zeros $2^{3} \times 3^{4} \times 5^{4} \times 7$ We basically, will focus on the powers of 2 and 5 because the multiplication of these two number gives one zero. So $2^{3} \times 3^{4} \times 5^{4} \times 7=2^{3} \times 5^{4} \times 3^{4} \times 7$ $=2^{3} \times 5^{3} \times 5 \times 3^{4}...
Read More →The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute.
Question: The lengthxof a rectangle is decreasing at the rate of 5 cm/minute and the widthyis increasing at the rate of 4 cm/minute. Whenx= 8 cm andy= 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle. Solution: Since the length(x)is decreasing at the rate of 5 cm/minute and the width (y)is increasing at the rate of 4 cm/minute, we have: $\frac{d x}{d t}=-5 \mathrm{~cm} / \mathrm{min}$ and $\frac{d y}{d t}=4 \mathrm{~cm} / \mathrm{min}$ (a) The perimeter(P)of...
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