Define ecological pyramids and describe with examples, pyramids of number and biomass.
Question: Define ecological pyramids and describe with examples, pyramids of number and biomass. Solution: An ecological pyramid is a graphical representation of various ecological parameters such as the number of individuals present at each trophic level, the amount of energy, or the biomass present at each trophic level. Ecological pyramids represent producers at the base, while the apex represents the top level consumers present in the ecosystem. There are three types of pyramids: (a)Pyramid ...
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=a(\theta-\sin \theta), y=a(1+\cos \theta)$ Solution: The given equations are $x=a(\theta-\sin \theta)$ and $y=a(1+\cos \theta)$ Then, $\frac{d x}{d \theta}=a\left[\frac{d}{d \theta}(\theta)-\frac{d}{d \theta}(\sin \theta)\right]=a(1-\cos \theta)$ $\frac{d y}{d \theta}=a\left[\frac{d}{d \theta}(1)+\frac{d}{d \theta}(\cos \theta)\right]=a[0+(-\sin \theta)]=-a \sin \t...
Read More →Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length
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$16 x^{2}+y^{2}=16$ Solution: The given equation is $16 x^{2}+y^{2}=16$. It can be written as $16 x^{2}+y^{2}=16$ Or, $\frac{x^{2}}{1}+\frac{y^{2}}{16}=1$ Or, $\frac{x^{2}}{1^{2}}+\frac{y^{2}}{4^{2}}=1$ $\ldots(1)$ Here, the denominator of $\frac{y^{2}}{4^{2}}$ is greater than the denominator of $\frac{x^{2}}{1^{2}}$. Therefore, the...
Read More →The decimal expansion of the rational number
Question: The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after (a) one decimal place (b) two decimal place (c) three decimal place (d) four decimal place Solution: We have, $\frac{14587}{1250}=\frac{14587}{2^{1} \times 5^{4}}$ 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 afterkplace...
Read More →If a
Question: Ifa= 23✕ 3,b= 2 ✕ 3 ✕ 5,c= 3n✕ 5 and LCM (a,b,c) = 23✕ 32✕ 5, thenn=(a) 1 (b) 2 (c) 3 (d) 4 Solution: LCM $(a, b, c)=2^{3} \times 3^{2} \times 5 \ldots \ldots$ (I) We have to find the value forn Also $a=2^{3} \times 3$ $b=2 \times 3 \times 5$ $c=3^{n} \times 5$ We know that the while evaluating LCM, we take greater exponent of the prime numbers in the factorization of the number. Therefore, by applying this rule and taking $n \geq 1$ we get the LCM as $\operatorname{LCM}(a, b, c)=2^{3}...
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=\cos \theta-\cos 2 \theta, y=\sin \theta-\sin 2 \theta$ Solution: The given equations are Then, $\frac{d x}{d \theta}=\frac{d}{d \theta}(\cos \theta-\cos 2 \theta)=\frac{d}{d \theta}(\cos \theta)-\frac{d}{d \theta}(\cos 2 \theta)$ $=-\sin \theta-(-2 \sin 2 \theta)=2 \sin 2 \theta-\sin \theta$ $\frac{d y}{d \theta}=\frac{d}{d \theta}(\sin \theta-\sin 2 \theta)=\frac...
Read More →Describe the components of an ecosystem.
Question: Describe the components of an ecosystem. Solution: An ecosystem is defined as an interacting unit that includes both the biological community as well as the non-living components of an area. The living and the non-living components of an ecosystem interact amongst themselves and function as a unit, which gets evident during the processes of nutrient cycling, energy flow, decomposition, and productivity. There are many ecosystems such as ponds, forests, grasslands, etc. The two componen...
Read More →What is the percentage of photosynthetically active radiation (PAR), in the incident solar radiation.
Question: What is the percentage of photosynthetically active radiation (PAR), in the incident solar radiation. (a)100% (b)50 % (c)1-5% (d)2-10% Solution: (b)50% Out of total incident solar radiation, about fifty percent of it forms photosynthetically active radiation or PAR....
Read More →Find the coordinates of the foci, the vertices, the length of major axis, the minor 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$36 x^{2}+4 y^{2}=144$ Solution: The given equation is $36 x^{2}+4 y^{2}=144$. It can be written as $36 x^{2}+4 y^{2}=144$ Or, $\frac{x^{2}}{4}+\frac{y^{2}}{36}=1$ $\mathrm{Or}, \frac{x^{2}}{2^{2}}+\frac{y^{2}}{6^{2}}=1$ $\ldots(1)$ Here, the denominator of $\frac{y^{2}}{6^{2}}$ is greater than the denominator of $\frac{x^{2}}{2^{2}...
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=4 t, y=\frac{4}{t}$ Solution: The given equations are $x=4 t$ and $y=\frac{4}{t}$ $\frac{d x}{d t}=\frac{d}{d t}(4 t)=4$ $\frac{d y}{d t}=\frac{d}{d t}\left(\frac{4}{t}\right)=4 \cdot \frac{d}{d t}\left(\frac{1}{t}\right)=4 \cdot\left(\frac{-1}{t^{2}}\right)=\frac{-4}{t^{2}}$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\righ...
Read More →Secondary producers are
Question: Secondary producers are (a)Herbivores (b)Producers (c)Carnivores (d)None of the above Solution: (d)None of the above Plants are the only producers. Thus, they are called primary producers. There are no other producers in a food chain....
Read More →The second trophic level in a lake is-
Question: The second trophic level in a lake is- (a)Phytoplankton (b)Zooplankton (c)Benthos (d)Fishes Solution: (b)Zooplankton Zooplankton are primary consumers in aquatic food chains that feed upon phytoplankton. Therefore, they are present at the second trophic level in a lake....
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=\sin t, y=\cos 2 t$ Solution: The given equations arex= sintandy= cos 2t Then, $\frac{d x}{d t}=\frac{d}{d t}(\sin t)=\cos t$ $\frac{d y}{d t}=\frac{d}{d t}(\cos 2 t)=-\sin 2 t \cdot \frac{d}{d t}(2 t)=-2 \sin 2 t$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\right)}=\frac{-2 \sin 2 t}{\cos t}=\frac{-2 \cdot 2 \sin t \cos t}...
Read More →Find the coordinates of the foci, the vertices, the length of major axis, the minor axis,
Question: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse$\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$ Solution: The given equation is $\frac{x^{2}}{25}+\frac{y^{2}}{100}=1$ or $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{10^{2}}=1$. Here, the denominator of $\frac{y^{2}}{100}$ is greater than the denominator of $\frac{x^{2}}{25}$. Therefore, the major axis is along the $y$-axis, while the minor axis is al...
Read More →The second trophic level in a lake is-
Question: The second trophic level in a lake is- (a)Phytoplankton (b)Zooplankton (c)Benthos (d)Fishes Solution: (b)Zooplankton Zooplankton are primary consumers in aquatic food chains that feed upon phytoplankton. Therefore, they are present at the second trophic level in a lake....
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=a \cos \theta, y=b \cos \theta$ Solution: The given equations are $x=a \cos \theta$ and $y=b \cos \theta$ Then, $\frac{d x}{d \theta}=\frac{d}{d \theta}(a \cos \theta)=a(-\sin \theta)=-a \sin \theta$ $\frac{d y}{d \theta}=\frac{d}{d \theta}(b \cos \theta)=b(-\sin \theta)=-b \sin \theta$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d \theta}\right)}{\left(\fra...
Read More →If HCF (26, 169) = 13, then LCM (26, 169) =
Question: If HCF (26, 169) = 13, then LCM (26, 169) =(a) 26 (b) 2 (c) 3 (d) 4 Solution: $\operatorname{HCF}(26,169)=13$ We have to find the value for $\operatorname{LCM}(26,169)$ We know that the product of numbers is equal to the product of their HCF and LCM. Therefore, $13(\mathrm{LCM})=26(169)$ $\mathrm{LCM}=\frac{26(169)}{13}$ $\mathrm{LCM}=338$ Hence the correct choice is (c)....
Read More →Find the coordinates of the foci, the vertices, the length of major axis, the minor 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$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ Solution: The given equation is $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ or $\frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{2}}=1$ Here, the denominator of $\frac{x^{2}}{16}$ is greater than the denominator of $\frac{y^{2}}{9}$. Therefore, the major axis is along thex-axis, while the minor axis is along they-ax...
Read More →Which one of the following has the largest population in a food chain?
Question: Which one of the following has the largest population in a food chain? (a)Producers (b)Primary consumers (c)Secondary consumers (d)Decomposers Solution: (d) Decomposers Decomposers include micro-organisms such as bacteria and fungi. They form the largest population in a food chain and obtain nutrients by breaking down the remains of dead plants and animals....
Read More →The HCF of 95 and 152, is
Question: The HCF of 95 and 152, is(a) 57 (b) 1 (c) 19 (d) 38 Solution: Using the factor tree for 95, we have: Using the factor tree for 152, we have: Therefore, $95=5 \times 19$ $152=2^{3} \times 19$ $\operatorname{HCF}(95,152)=19$ Hence the correct choice is (c)....
Read More →If x and y are connected parametrically by the equation,
Question: If $x$ and $y$ are connected parametrically by the equation, without eliminating the parameter, find $\frac{d y}{d x}$. $x=2 a t^{2}, y=a t^{4}$ Solution: The given equations are $x=2 a t^{2}$ and $y=a t^{4}$ Then, $\frac{d x}{d t}=\frac{d}{d t}\left(2 a t^{2}\right)=2 a \cdot \frac{d}{d t}\left(t^{2}\right)=2 a \cdot 2 t=4 a t$ $\frac{d y}{d t}=\frac{d}{d t}\left(a t^{4}\right)=a \cdot \frac{d}{d t}\left(t^{4}\right)=a \cdot 4 \cdot t^{3}=4 a t^{3}$ $\therefore \frac{d y}{d x}=\frac{\...
Read More →Find the coordinates of the foci, the vertices, the length of major axis,
Question: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse$\frac{x^{2}}{4}+\frac{y^{2}}{25}=1$ Solution: The given equation is $\frac{x^{2}}{4}+\frac{y^{2}}{25}=1$ or $\frac{x^{2}}{2^{2}}+\frac{y^{2}}{5^{2}}=1$. Here, the denominator of $\frac{y^{2}}{25}$ is greater than the denominator of $\frac{x^{2}}{4}$. Therefore, the major axis is along they-axis, while the minor axis is along thex-a...
Read More →If the LCM of a and 18 is 36 and the HCF of a and 18 is 2,
Question: If the LCM ofaand 18 is 36 and the HCF ofaand 18 is 2, thena=(a) 2 (b) 3 (c) 4 (d) 1 Solution: $\operatorname{LCM}(a, 18)=36$ $\operatorname{HCF}(a, 18)=2$ We know that the product of numbers is equal to the product of their HCF and LCM. Therefore, $18 a=2(36)$ $a=\frac{2(36)}{18}$ $a=4$ Hence the correct choice is (c)....
Read More →Fill in the blanks.
Question: Fill in the blanks. (a)Plants are called as_________ because they fix carbon dioxide. (b)In an ecosystem dominated by trees, the pyramid (of numbers) is _________ type. (c)In aquatic ecosystems, the limiting factor for the productivity is _________. (d)Common detritivores in our ecosystem are_________. (e)The major reservoir of carbon on earth is_________. Solution: (a)Plants are called asbecause they fix carbon dioxide. (b)In an ecosystem dominated by trees, the pyramid (of numbers) i...
Read More →If two positive integers m and n are expressible in the form m
Question: If two positive integers $m$ and $n$ are expressible in the form $m=p q^{3}$ and $n=p^{3} q^{2}$, where $p, q$ are prime numbers, then $\operatorname{HCF}(m, n)=$ (a) $p q$ (b) pq2 (c) $p^{3} q^{2}$ (d) $p^{2} q^{2}$ Solution: Two positive integers are expressed as follows: $m=p q^{3}$ $n=p^{3} q^{2}$ pandqare prime numbers. Then, taking the smallest powers ofpandqin the values formandnwe get $\operatorname{HCF}(m, n)=p q^{2}$ Hence the correct choice is (b)....
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