If u, v and w are functions of x, then show that
Question: Ifu,vandware functions ofx, then show that $\frac{d}{d x}(u, v, w)=\frac{d u}{d x} v \cdot w+u \cdot \frac{d v}{d x} \cdot w+u \cdot v \cdot \frac{d w}{d x}$ in two ways-first by repeated application of product rule, second by logarithmic differentiation. Solution: Let $y=u \cdot v \cdot w=u \cdot(v \cdot w)$ By applying product rule, we obtain $\frac{d y}{d x}=\frac{d u}{d x} \cdot(v \cdot w)+u \cdot \frac{d}{d x}(v \cdot w)$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x} v \cdot w+u\le...
Read More →Find the coordinates of the foci, the vertices, the length of major axis,
Question: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse$\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$ Solution: The given equation is $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. Here, the denominator of $\frac{x^{2}}{36}$ is greater than the denominator of $\frac{y^{2}}{16}$. Therefore, the major axis is along thex-axis, while the minor axis is along they-axis. On comparing the given equation with $\...
Read More →In Q.No. 7, HCF (a, b) is
Question: In Q.No. 7, HCF (a,b) is (a) pq (b) $p^{3} q^{3}$ (c) $p^{3} q^{2}$ (d) $p^{2} q^{2}$ Solution: Two positive integers are expressed as follows: $a=p q^{2}$ $b=p^{3} q$ pandqare prime numbers. Then, taking the smallest powers ofpandqin the values foraandbwe get $\operatorname{HCF}(a, b)=p q$ Hence the correct choice is (a)....
Read More →In Q.No. 7, HCF (a, b) is
Question: In Q.No. 7, HCF (a,b) is (a) pq (b) $p^{3} q^{3}$ (c) $p^{3} q^{2}$ (d) $p^{2} q^{2}$ Solution: Two positive integers are expressed as follows: $a=p q^{2}$ $b=p^{3} q$ pandqare prime numbers. Then, taking the smallest powers ofpandqin the values foraandbwe get $\operatorname{HCF}(a, b)=p q$ Hence the correct choice is (a)....
Read More →List any three important characteristics of a population and explain
Question: List any three important characteristics of a population and explain 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. Three important characteristics of a population are: (a) Birth rate(Natality): It is the ratio of live births in an area to the ...
Read More →Select the statement which explains best parasitism.
Question: Select the statement which explains best parasitism. (a)One organism is benefited. (b)Both the organisms are benefited. (c)One organism is benefited, other is not affected. (d)One organism is benefited, other is affected. Solution: (d)One organism is benefited, other is affected. Parasitism is an interaction between two species in which one species (parasite) derives benefit while the other species (host) is harmed. For example, ticks and lice (parasites) present on the human body repr...
Read More →If two positive ingeters a and b are expressible in the form
Question: If two positive ingeters $a$ and $b$ are expressible in the form $a=p q^{2}$ and $b=p^{3} q ; p, q$ being prime number, then LCM ( $a$,b) is (a) $p q$ (b) $p^{3} q^{3}$ (c) $p^{3} q^{2}$ (d) $p^{2} q^{2}$ Solution: Two positive integers are expressed as follows: $a=p q^{2}$ $b=p^{3} q$ pandqare prime numbers. Then, taking the highest powers ofpandqin the values foraandbwe get: $\operatorname{LCM}(a, b)=p^{3} q^{2}$ Hence the correct choice is (c)....
Read More →Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0),
Question: Find the equation of the parabola that satisfiesthe following conditions:Vertex (0, 0), passing through (5, 2) and symmetric with respect toy-axis Solution: Since the vertex is $(0,0)$ and the parabola is symmetric about the $y$-axis, the equation of the parabola is either of the form $x^{2}=4 a y$ or $x^{2}=-4 a y$. The parabola passes through point (5, 2), which lies in the first quadrant. Therefore, the equation of the parabola is of the form $x^{2}=4 a y$, while point $(5,2)$ must ...
Read More →If p
Question: If $p_{1}$ and $p_{2}$ are two odd prime numbers such that $p_{1}p_{2}$, then $p_{1}^{2}-p_{2}^{2}$ is (a) an even number (b) an odd number (c) an odd prime number (d) a prime number Solution: Let the two odd prime numbers $p_{1}$ and $p_{2}$ be 5 and 3 . Then, $p_{1}^{2}=5^{2}$ $=25$ And $p_{2}^{2}=3^{2}$ = 9 Thus, p_{1}{ }^{2}-p_{2}{ }^{2}=25-9 = 16 16 is even number. Take another example, with $p_{1}$ and $p_{2}$ be 11 and 7 . Then, $p_{1}^{2}=11^{2}$ = 121 And $p_{2}{ }^{2}=7^{2}$ ...
Read More →Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along x-axis
Question: Find the equation of the parabola that satisfiesthe following conditions:Vertex (0, 0) passing through (2, 3) and axis is alongx-axis Solution: Since the vertex is $(0,0)$ and the axis of the parabola is the $x$-axis, the equation of the parabola is either of the form $y^{2}=4 a x$ or $y^{2}=-4 a x$. The parabola passes through point (2, 3), which lies in the first quadrant. Therefore, the equation of the parabola is of the form $y^{2}=4 a x$, while point $(2,3)$ must satisfy the equat...
Read More →Differentiate
Question: Differentiate $\left(x^{5}-5 x+8\right)\left(x^{3}+7 x+9\right)$ in three ways mentioned below (i) By using product rule. (ii) By expanding the product to obtain a single polynomial. (iii) By logarithmic differentiation. Do they all give the same answer? Solution: Let $y=\left(x^{5}-5 x+8\right)\left(x^{3}+7 x+9\right)$ (i) Let $x^{2}-5 x+8=u$ and $x^{3}+7 x+9=v$ $\therefore y=u v$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x} \cdot v+u \cdot \frac{d v}{d x} \quad$ (By using product rul...
Read More →With the help of suitable diagram describe the logistic population growth curve.
Question: With the help of suitable diagram describe the logistic population growth curve. Solution: The logistic population growth curve is commonly observed in yeast cells that are grown under laboratory conditions. It includes five phases: the lag phase, positive acceleration phase, exponential phase, negative acceleration phase, and stationary phase. (a)Lag phase: Initially, the population of the yeast cell is very small. This is because of the limited resource present in the habitat. (b)Pos...
Read More →Define the following terms and give one example for each:
Question: Define the following terms and give one example for each: (a)Commensalism (b)Parasitism (c)Camouflage (d)Mutualism (e)Interspecific competition Solution: (a)Commensalism:Commensalism is an interaction between two species in which one species gets benefited while the other remains unaffected. An orchid growing on the branches of a mango tree and barnacles attached to the body of whales are examples of commensalisms. (b)Parasitism: It is an interaction between two species in which one sp...
Read More →The number of decimal place after which the decimal expansion of the rational number
Question: The number of decimal place after which the decimal expansion of the rational number $\frac{23}{2^{2} \times 5}$ will terminate, is (a) 1 (b) 2 (c) 3 (d) 4 Solution: We have, $\frac{23}{2^{2} \times 5^{1}}$ 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. ...
Read More →Define population and community.
Question: Define population and community. Solution: Population: 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. Community: A community is defined as a group of individuals of different species, living within a certain geographical area. Such individuals can be simi...
Read More →Give an example for:
Question: Give an example for: (a)An endothermic animal (b)An ectothermic animal (c)An organism of benthic zone Solution: (a)Endothermic animal: Birds such as crows, sparrows, pigeons, cranes, etc. and mammals such as bears, cows, rats, rabbits, etc. are endothermic animals. (b)Ectothermic animal: Fishes such as sharks, amphibians such as frogs, and reptiles such as tortoise, snakes, and lizards are ectothermic animals. (c)Organism of benthic zone: Decomposing bacteria is an example of an organi...
Read More →The sum of the exponents of the prime factors in the prime factorisation of 196,
Question: The sum of the exponents of the prime factors in the prime factorisation of 196, is(a) 1 (b) 2 (c) 4 (d) 6 Solution: Using the factor tree for prime factorization, we have: Therefore, $196=2 \times 2 \times 7 \times 7$ $196=2^{2} \times 7^{2}$ The exponents of 2 and 7 are 2 and 2 respectively. Thus the sum of the exponents is 4 Hence the correct choice is (c)....
Read More →List the various abiotic environmental factors.
Question: List the various abiotic environmental factors. Solution: All non- living components of an ecosystem form abiotic components. It includes factors such as temperature, water, light, and soil....
Read More →Write a short note on
Question: Write a short note on (a)Adaptations of desert plants and animals (b)Adaptations of plants to water scarcity (c)Behavioural adaptations in animals (d)Importance of light to plants (e)Effect of temperature or water scarcity and the adaptations of animals. Solution: (a)Adaptations of desert plants and animals: (i)Adaptations of desert plants: Plants found in deserts are well adapted to cope with harsh desert conditions such as water scarcity and scorching heat. Plants have an extensive r...
Read More →If n
Question: Ifn= 23✕ 34✕ 54✕ 7, then the number of consecutive zeros inn, wherenis a natural number, is(a) 2 (b) 3 (c) 4 (d) 7 Solution: Since, it is given that $n=2^{3} \times 3^{4} \times 5^{4} \times 7$ $=2^{3} \times 5^{4} \times 3^{4} \times 7$ $=2^{3} \times 5^{3} \times 5 \times 3^{4} \times 7$ $=(2 \times 5)^{3} \times 5 \times 3^{4} \times 7$ $=5 \times 3^{4} \times 7 \times(10)^{3}$ So, this means the given numbernwill end with 3 consecutive zeroes....
Read More →Find the derivative of the function given by
Question: Find the derivative of the function given by $f(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)$ and hence find $f^{\prime}(1)$. Solution: The given relationship is $f(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)$ Taking logarithm on both the sides, we obtain $\log f(x)=\log (1+x)+\log \left(1+x^{2}\right)+\log \left(1+x^{4}\right)+\log \left(1+x^{8}\right)$ Differentiating both sides with respect tox, we obtain $\frac{1}{f(x)} \cdot \frac{d...
Read More →The LCM of two numbers is 1200. Which of the following cannot be their HCF?
Question: The LCM of two numbers is 1200. Which of the following cannot be their HCF?(a) 600 (b) 500 (c) 400 (d) 200 Solution: It is given that the LCM of two numbers is 1200. We know that the HCF of two numbers is always the factor of LCM Checking all the options: (a) 600 is the factor of 1200. So this can be the HCF. (b) 500 is not the factor of 1200. So this cannot be the HCF. (c) 400 is the factor of 1200. So this can be the HCF. (d) 200 is the factor of 1200. So this can be the HCF. Hence t...
Read More →Distinguish between the following:
Question: Distinguish between the following: (a)Hibernation and Aestivation (b)Ectotherms and Endotherms Solution: (a)Hibernation and Aestivation (b) Ectotherms and Endotherms...
Read More →The LCM of two numbers is 1200. Which of the following cannot be their HCF?
Question: The LCM of two numbers is 1200. Which of the following cannot be their HCF?(a) 600 (b) 500 (c) 400 (d) 200 Solution: It is given that the LCM of two numbers is 1200. We know that the HCF of two numbers is always the factor of LCM Checking all the options: (a) 600 is the factor of 1200. So this can be the HCF. (b) 500 is not the factor of 1200. So this cannot be the HCF. (c) 400 is the factor of 1200. So this can be the HCF. (d) 200 is the factor of 1200. So this can be the HCF. Hence t...
Read More →The exponent of 2 in the prime factorisation of 144, is
Question: The exponent of 2 in the prime factorisation of 144, is(a) 4 (b) 5 (c) 6 (d) 3 Solution: Using the factor tree for prime factorization, we have: Therefore, $144=2 \times 2 \times 2 \times 2 \times 3 \times 3$ $144=2^{4} \times 3^{2}$ Thus, the exponent of 2 in 144 is 4. Hence the correct choice is (a)....
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