Find of function.
Question: Find $\frac{d y}{d x}$ of function. $x^{y}+y^{x}=1$ Solution: The given function is $x^{y}+y^{x}=1$ Let $x^{y}=u$ and $y^{x}=v$ Then, the function becomesu+v= 1 $\therefore \frac{d u}{d x}+\frac{d v}{d x}=0$ ...(1) $u=x^{y}$ $\Rightarrow \log u=\log \left(x^{y}\right)$ $\Rightarrow \log u=y \log x$ Differentiating both sides with respect tox, we obtain $\frac{1}{u} \frac{d u}{d x}=\log x \frac{d y}{d x}+y \cdot \frac{d}{d x}(\log x)$ $\Rightarrow \frac{d u}{d x}=u\left[\log x \frac{d y...
Read More →Name important defence mechanisms in plants against herbivory.
Question: Name important defence mechanisms in plants against herbivory. Solution: Several plants have evolved various mechanisms both morphological and chemical to protect themselves against herbivory. (1)Morphological defence mechanisms: (a)Cactus leaves (Opuntia) are modified into sharp spines (thorns) to deter herbivores from feeding on them. (b)Sharp thorns along with leaves are present inAcaciato deter herbivores. (c)In some plants, the margins of their leaves are spiny or have sharp edges...
Read More →Find the coordinates of the focus, axis of the parabola,
Question: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for $y^{2}=10 x$ Solution: The given equation is $y^{2}=10 x$. Here, the coefficient of $x$ is positive. Hence, the parabola opens towards the right. On comparing this equation with $y^{2}=4 a x$, we obtain $4 a=10 \Rightarrow a=\frac{5}{2}$ $\therefore$ Coordinates of the focus $=(a, 0)=\left(\frac{5}{2}, 0\right)$ Since the given equation involves $y^{2}$, the axis of...
Read More →if p and q are two prime number,
Question: if pandqare two prime number, then what is their LCM? Solution: It is given thatpandqare two prime numbers; we have to find their LCM. We know that the factors of any prime number are 1 and the prime number itself. For example, let $p=2$ and $q=3$ Thus, the factors are as follows $p=2 \times 1$ And $q=3 \times 1$ Now, the LCM of 2 and 3 is $2 \times 3=6$. Thus the HCF of $p$ and $q$ is $p \times q$....
Read More →If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population?
Question: If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population? Solution: A population grows exponentially if sufficient amounts of food resources are available to the individual. Its exponential growth can be calculated by the following integral form of the exponential growth equation: $N_{t}=N_{0} e^{r t}$ Where, Nt= Population density after timet NO= Population density at time zero r= Intrinsic rate of natural increase e...
Read More →Find the coordinates of the focus, axis of the parabola,
Question: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for$x^{2}=-16 y$ Solution: The given equation is $x^{2}=-16 y$. Here, the coefficient ofyis negative. Hence, the parabola opens downwards. On comparing this equation with $x^{2}=-4$ ay, we obtain $-4 a=-16 \Rightarrow a=4$ $\therefore$ Coordinates of the focus $=(0,-a)=(0,-4)$ Since the given equation involves $x^{2}$, the axis of the parabola is the $y$-axis. Equation ...
Read More →If p and q are two prime number,
Question: Ifpandqare two prime number, then what is their HCF? Solution: It is given thatpandqare two prime numbers; we have to find their HCF. We know that the factors of any prime number are 1 and the prime number itself. For example, let $p=2$ and $q=3$ Thus, the factors are as follows $p=2 \times 1$ And $q=3 \times 1$ Now, the HCF of 2 and 3 is 1. Thus the HCF ofpandqis 1...
Read More →Find the coordinates of the focus, axis of the parabola,
Question: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for$y^{2}=-8 x$ Solution: The given equation is $y^{2}=-8 x$. Here, the coefficient of $x$ is negative. Hence, the parabola opens towards the left. On comparing this equation with $y^{2}=-4 a x$, we obtain $-4 a=-8 \Rightarrow a=2$ $\therefore$ Coordinates of the focus $=(-a, 0)=(-2,0)$ Since the given equation involves $y^{2}$, the axis of the parabola is the $x$-axis....
Read More →List the attributes that populations but not individuals possess.
Question: List the attributes that populations but not individuals possess. Solution: A population can be defined as a group of individuals of the same species residing in a particular geographical area at a particular time and functioning as a unit. For example, all human beings living at a particular place at a particular time constitute the population of humans. The main attributes or characteristics of a population residing in a given area are:- (a)Birth rate(Natality): It is the ratio of li...
Read More →What is a lemma?
Question: What is a lemma? Solution: A proven statement used as a stepping-stone toward the proof of another statement is called lemma. For example: Euclids Division Lemma: Letaandbbe any two positive integers. Then, there exist unique integersqandrsuch that $a=b q+r, 0 \leq rb$ If $b \mid a$ then $r=0$. Otherwise, $r$ satisfies the stronger inequality $0rb$....
Read More →Differentiate the function with respect to x.
Question: Differentiate the function with respect tox. $(x \cos x)^{x}+(x \sin x)^{\frac{1}{x}}$ Solution: Let $y=(x \cos x)^{x}+(x \sin x)^{\frac{1}{x}}$ Also, let $u=(x \cos x)^{x}$ and $v=(x \sin x)^{\frac{1}{x}}$ $\therefore y=u+v$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ ...(1) $u=(x \cos x)^{x}$ $\Rightarrow \log u=\log (x \cos x)^{x}$ $\Rightarrow \log u=x \log (x \cos x)$ $\Rightarrow \log u=x[\log x+\log \cos x]$ $\Rightarrow \log u=x \log x+x \log \cos x$ Different...
Read More →Most living organisms cannot survive at temperature above 45°C°.
Question: Most living organisms cannot survive at temperature above 45C. How are some microbes able to live in habitats with temperatures exceeding 100C? Solution: Archaebacteria (Thermophiles) are ancient forms of bacteria found in hot water springs and deep sea hydrothermal vents. They are able to survive in high temperatures (which far exceed 100C) because their bodies have adapted to such environmental conditions. These organisms contain specialized thermo-resistant enzymes, which carry out ...
Read More →Find the coordinates of the focus, axis of the parabola,
Question: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for$x^{2}=6 y$ Solution: The given equation is $x^{2}=6 y$. Here, the coefficient ofyis positive. Hence, the parabola opens upwards. On comparing this equation with $x^{2}=4 a y$, we obtain $4 a=6 \Rightarrow a=\frac{3}{2}$ $\therefore$ Coordinates of the focus $=(0, a)=\left(0, \frac{3}{2}\right)$ Since the given equation involves $x^{2}$, the axis of the parabola is t...
Read More →What is an algorithm?
Question: What is an algorithm? Solution: Algorithm is a step-by-step procedure for calculations. For example: Euclid's Division Algorithm: In order to compute the HCF of two positive integers say $a$ and $b$, with $ab$ by using Euclid's algorithm, we follow the following steps: $\underline{\text { STEP I }}$ : Apply Euclid's Division Lemma to $a$ and $b$ and obtain whole numbers $q_{1}$ and $r_{1}$, such that $a=b q_{1}+r_{1}, 0 \leq r_{1}b$ $\underline{\text { STEP II: If }} r_{1}=0, b$ is the...
Read More →Define phenotypic adaptation. Give one example.
Question: Define phenotypic adaptation. Give one example. Solution: Phenotypic adaptation involves changes in the body of an organism in response to genetic mutation or certain environmental changes. These responsive adjustments occur in an organism in order to cope with environmental conditions present in their natural habitats. For example, desert plants have thick cuticles and sunken stomata on the surface of their leaves to prevent transpiration. Similarly, elephants have long ears that act ...
Read More →If a marine fish is placed in a fresh water aquarium,
Question: If a marine fish is placed in a fresh water aquarium, will the fish be able to survive? Why or why not? Solution: If a marine fish is placed in a fresh water aquarium, then its chances of survival will diminish. This is because their bodies are adapted to high salt concentrations of the marine environment. In fresh water conditions, they are unable to regulate the water entering their body (through osmosis). Water enters their body due to the hypotonic environment outside. This results...
Read More →How is diapause different from hibernation?
Question: How is diapause different from hibernation? Solution: Diapause is a stage of suspended development to cope with unfavourable conditions. Many species of Zooplankton and insects exhibit diapause to tide over adverse climatic conditions during their development. Hibernation or winter sleep is a resting stage where in animals escape winters (cold) by hiding themselves in their shelters. They escape the winter season by entering a state of inactivity by slowing their metabolism. The phenom...
Read More →Find the coordinates of the focus, axis of the parabola, the equation of directrix
Question: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for $y^{2}=12 x$ Solution: The given equation is $y^{2}=12 x$. Here, the coefficient ofxis positive. Hence, the parabola opens towards the right. On comparing this equation with $y^{2}=4 a x$, we obtain $4 a=12 \Rightarrow a=3$ $\therefore$ Coordinates of the focus $=(a, 0)=(3,0)$ Since the given equation involves $y^{2}$, the axis of the parabola is the $x$-axis. Equat...
Read More →Write whether
Question: Write whether $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}$ on simplification gives a rational or an irrational number. Solution: Let us simplify $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}$ $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}=\frac{6 \sqrt{5}+6 \sqrt{5}}{2 \sqrt{5}}$ $=\frac{12 \sqrt{5}}{2 \sqrt{5}}$ =6 6is rational number...
Read More →Does the point (–2.5, 3.5) lie inside,
Question: Does the point $(-2.5,3.5)$ lie inside, outside or on the circle $x^{2}+y^{2}=25 ?$ Solution: The equation of the given circle is $x^{2}+y^{2}=25$. $x^{2}+y^{2}=25$ $\Rightarrow(x-0)^{2}+(y-0)^{2}=5^{2}$, which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=0, k=0$, and $r=5$ $\therefore$ Centre $=(0,0)$ and radius $=5$ Distance between point $(-2.5,3.5)$ and centre $(0,0)$ $=\sqrt{(-2.5-0)^{2}+(3.5-0)^{2}}$ $=\sqrt{6.25+12.25}$ $=\sqrt{18.5}$ $=4.3$ (approx.) $5$ Since the dista...
Read More →Has the rational number
Question: Has the rational number $\frac{441}{2^{2} \times 5^{7} \times 7^{2}}$ a terminating or a nonterminating decimal representation? Solution: We have, $\frac{441}{2^{2} \times 5^{7} \times 7^{2}}$ Theorem states: Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is not of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers. Then,xhas a decimal expression which is non-terminating repeating. This is clear that the prime factorization of ...
Read More →The decimal expansion of the rational number
Question: The decimal expansion of the rational number $\frac{43}{2^{4} \times 5^{3}}$ will terminate after how many places of decimals? Solution: We have, $\frac{43}{2^{4} \times 5^{3}}$ Theorem states: Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers. Then,xhas a decimal expression which terminates afterkplaces of decimals, wherekis the larger ofmandn. This is clear that the prime ...
Read More →Consult internet and find out how to make orally active protein pharmaceutical.
Question: Consult internet and find out how to make orally active protein pharmaceutical. What is the major problem to be encountered? Solution: Orally active protein pharmaceuticals contain biologically active materials such as peptides or proteins, antibodies, and polymeric beads. It is administrated orally into the body through various formulations. It involves the encapsulation of protein or peptide in liposomes or formulations using penetration enhancers. These proteins or peptides are used...
Read More →Find the equation of a circle with centre (2, 2)
Question: Find the equation of a circle with centre (2, 2) and passes through the point (4, 5). Solution: The centre of the circle is given as (h,k) = (2, 2). Since the circle passes through point (4, 5), the radius (r) of the circle is the distance between the points (2, 2) and (4, 5). $\therefore r=\sqrt{(2-4)^{2}+(2-5)^{2}}=\sqrt{(-2)^{2}+(-3)^{2}}=\sqrt{4+9}=\sqrt{13}$ Thus, the equation of the circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$ $(x-2)^{2}+(y-2)^{2}=(\sqrt{13})^{2}$ $x^{2}-4 x+4+y^{2}-4 y...
Read More →Complete the missing entries in the following factor tree.
Question: Complete the missing entries in the following factor tree. Solution: We need to fill the values foraandbin the following factor tree: It is clear from the factor tree above that $b=3 \times 7$ $b=21$ Also, $a=2 \times b$ $a=2 \times 21$ $a=42$ Thus, the missing entries are 21and 42....
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