Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola$\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ Solution: The given equation is $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ or $\frac{x^{2}}{4^{2}}-\frac{y^{2}}{3^{2}}=1$. On comparing this equation with the standard equation of hyperbola i.e., $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, we obtain $a=4$ and $b=3 .$ We know that $a^{2}+b^{2}=c^{2}$. $\therefore c^{2}=4^{2}+3^{2}=25$ $\...
Read More →What is the significance of the slope of regression in a species − area relationship?
Question: What is the significance of the slope of regression in a species area relationship? Solution: The slope of regression (z) has a great significance in order to find a species-area relationship. It has been found that in smaller areas (where the species-area relationship is analyzed), the value of slopes of regression is similar regardless of the taxonomic group or the region. However, when a similar analysis is done in larger areas, then the slope of regression is much steeper....
Read More →Give three hypotheses for explaining why tropics show greatest levels of species richness.
Question: Give three hypotheses for explaining why tropics show greatest levels of species richness. Solution: There are three different hypotheses proposed by scientists for explaining species richness in the tropics. (1)Tropical latitudes receive more solar energy than temperate regions, which leads to high productivity and high species diversity. (2)Tropical regions have less seasonal variations and have a more or less constant environment. This promotes the niche specialization and thus, hig...
Read More →Find the equation for the ellipse that satisfies the given conditions:
Question: Find the equation for the ellipse that satisfies the given conditions:Major axis on thex-axis and passes through the points (4, 3) and (6, 2). Solution: Since the major axis is on thex-axis, the equation of the ellipse will be of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$\ldots(1)$ Where, $a$ is the semi-major axis The ellipse passes through points (4, 3) and (6, 2). Hence, $\frac{16}{a^{2}}+\frac{9}{b^{2}}=1$ $\ldots(2)$ $\frac{36}{a^{2}}+\frac{4}{b^{2}}=1$ $\ldots(3)$ On s...
Read More →How do ecologists estimate the total number of species present in the world?
Question: How do ecologists estimate the total number of species present in the world? Solution: The diversity of living organisms present on the Earth is very vast. According to an estimate by researchers, it is about seven millions. The total number of species present in the world is calculated by ecologists by statistical comparison between a species richness of a well studied group of insects of temperate and tropical regions. Then, these ratios are extrapolated with other groups of plants a...
Read More →If
Question: If $y=5 \cos x-3 \sin x$, prove that $\frac{d^{2} y}{d x^{2}}+y=0$ Solution: It is given that, $y=5 \cos x-3 \sin x$ Then $\frac{d y}{d x}=\frac{d}{d x}(5 \cos x)-\frac{d}{d x}(3 \sin x)=5 \frac{d}{d x}(\cos x)-3 \frac{d}{d x}(\sin x)$ $=5(-\sin x)-3 \cos x=-(5 \sin x+3 \cos x)$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}[-(5 \sin x+3 \cos x)]$ $=-\left[5 \cdot \frac{d}{d x}(\sin x)+3 \cdot \frac{d}{d x}(\cos x)\right]$ $=-[5 \cos x+3(-\sin x)]$ $=-[5 \cos x-3 \sin x]$ $=-y$ $\th...
Read More →Name the three important components of biodiversity.
Question: Name the three important components of biodiversity. Solution: Biodiversity is the variety of living forms present in various ecosystems. It includes variability among life forms from all sources including land, air, and water. Three important components of biodiversity are: (a)Genetic diversity (b)Species diversity (c)Ecosystem diversity...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $\sin (\log x)$ Solution: Let $y=\sin (\log x)$ Then, $\frac{d y}{d x}=\frac{d}{d x}[\sin (\log x)]=\cos (\log x) \cdot \frac{d}{d x}(\log x)=\frac{\cos (\log x)}{x}$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[\frac{\cos (\log x)}{x}\right]$ $=\frac{x \cdot \frac{d}{d x}[\cos (\log x)]-\cos (\log x) \cdot \frac{d}{d x}(x)}{x^{2}}$ $=\frac{x \cdot\left[-\sin (\log x) \cdot \frac{d}{d x}(\log x)\right]-\cos (\log x) \cdot 1}{...
Read More →Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0),
Question: Find the equation for the ellipse that satisfies the given conditions:Centre at (0, 0), major axis on they-axis and passes through the points (3, 2) and (1, 6). Solution: Since the centre is at (0, 0) and the major axis is on they-axis, the equation of the ellipse will be of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$ Where, $a$ is the semi-major axis The ellipse passes through points (3, 2) and (1, 6). Hence, $\frac{9}{b^{2}}+\frac{4}{a^{2}}=1$ $\frac{1}{b^{2}}+\frac{36}{a^{2...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $\log (\log x)$ Solution: Let $y=\log (\log x)$ Then, $\frac{d y}{d x}=\frac{d}{d x}[\log (\log x)]=\frac{1}{\log x} \cdot \frac{d}{d x}(\log x)=\frac{1}{x \log x}=(x \log x)^{-1}$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[(x \log x)^{-1}\right]=(-1) \cdot(x \log x)^{-2} \cdot \frac{d}{d x}(x \log x)$\ $=\frac{-1}{(x \log x)^{2}} \cdot\left[\log x \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}(\log x)\right]$ $=\frac{-1}{(x ...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $\tan ^{-1} x$ Solution: Let $y=\tan ^{-1} x$ Then, $\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^{2}}$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{1}{1+x^{2}}\right)=\frac{d}{d x}\left(1+x^{2}\right)^{-1}=(-1) \cdot\left(1+x^{2}\right)^{-2} \cdot \frac{d}{d x}\left(1+x^{2}\right)$ $=\frac{-1}{\left(1+x^{2}\right)^{2}} \times 2 x=\frac{-2 x}{\left(1+x^{2}\right)^{2}}$...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $e^{6 x} \cos 3 x$ Solution: Let $y=e^{6 x} \cos 3 x$ Then, $\frac{d y}{d x}=\frac{d}{d x}\left(e^{6 x} \cdot \cos 3 x\right)=\cos 3 x \cdot \frac{d}{d x}\left(e^{6 x}\right)+e^{6 x} \cdot \frac{d}{d x}(\cos 3 x)$ $=\cos 3 x \cdot e^{6 x} \cdot \frac{d}{d x}(6 x)+e^{6 x} \cdot(-\sin 3 x) \cdot \frac{d}{d x}(3 x)$ $=6 e^{6 x} \cos 3 x-3 e^{6 x} \sin 3 x$ ..(1) $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(6 e^{6 x} \cos 3 x-3 e...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $e^{x} \sin 5 x$ Solution: Let $y=e^{x} \sin 5 x$ $\frac{d y}{d x}=\frac{d}{d x}\left(e^{x} \sin 5 x\right)=\sin 5 x \cdot \frac{d}{d x}\left(e^{x}\right)+e^{x} \frac{d}{d x}(\sin 5 x)$ $=\sin 5 x \cdot e^{x}+e^{x} \cdot \cos 5 x \cdot \frac{d}{d x}(5 x)=e^{x} \sin 5 x+e^{x} \cos 5 x \cdot 5$ $=e^{x}(\sin 5 x+5 \cos 5 x)$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[e^{x}(\sin 5 x+5 \cos 5 x)\right]$ Then, $=(\sin 5 x+5 \cos ...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $x^{3} \log x$ Solution: Let $y=x^{3} \log x$ Then, $\frac{d y}{d x}=\frac{d}{d x}\left[x^{3} \log x\right]=\log x \cdot \frac{d}{d x}\left(x^{3}\right)+x^{3} \cdot \frac{d}{d x}(\log x)$ $=\log x \cdot 3 x^{2}+x^{3} \cdot \frac{1}{x}=\log x \cdot 3 x^{2}+x^{2}$ $=x^{2}(1+3 \log x)$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[x^{2}(1+3 \log x)\right]$ $=(1+3 \log x) \cdot \frac{d}{d x}\left(x^{2}\right)+x^{2} \frac{d}{d x}(1...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $\log x$ Solution: Let $y=\log x$ Then, $\frac{d y}{d x}=\frac{d}{d x}(\log x)=\frac{1}{x}$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{1}{x}\right)=\frac{-1}{x^{2}}$...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $x \cdot \cos x$ Solution: Let $y=x \cdot \cos x$ Then, $\frac{d y}{d x}=\frac{d}{d x}(x \cdot \cos x)=\cos x \cdot \frac{d}{d x}(x)+x \frac{d}{d x}(\cos x)=\cos x \cdot 1+x(-\sin x)=\cos x-x \sin x$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}[\cos x-x \sin x]=\frac{d}{d x}(\cos x)-\frac{d}{d x}(x \sin x)$ $=-\sin x-\left[\sin x \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}(\sin x)\right]$ $=-\sin x-(\sin x+x \cos x)$ $=-(x \cos x...
Read More →The remainder when the square of any prime number greater than 3 is divided by 6,
Question: The remainder when the square of any prime number greater than 3 is divided by 6, is(a) 1 (b) 3 (c) 2 (d) 4 [Hint: Any prime number greater than 3 is of the from $6 k \pm 1$, where $k$ is a natural number and $(6 k \pm 1)^{2}=36 k^{2} \pm 12 k+$ $1=6 k(6 k \pm 2)+1]$ Solution: Any prime number greater than 3 is of the form $6 k \pm 1$, where $k$ is a natural number. Thus, $(6 k \pm 1)^{2}=36 k^{2} \pm 12 k+1$ $=6 k(6 k \pm 2)+1$ When, $6 k(6 k \pm 2)+1$ is divided by 6 , we get, $k(6 k...
Read More →Find the equation for the ellipse that satisfies the given conditions:
Question: Find the equation for the ellipse that satisfies the given conditions:b= 3,c= 4, centre at the origin; foci on thexaxis. Solution: It is given thatb= 3,c= 4, centre at the origin; foci on thexaxis. Since the foci are on thex-axis, themajor axisis along thex-axis. Therefore, the equation of the ellipse will be of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where a is the semi-major axis. Accordingly, $b=3, c=4$. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore a^{2}=3^{2}+4^{...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $x^{20}$ Solution: Let $y=x^{20}$ Then $\frac{d y}{d x}=\frac{d}{d x}\left(x^{20}\right)=20 x^{19}$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(20 x^{19}\right)=20 \frac{d}{d x}\left(x^{19}\right)=20 \cdot 19 \cdot x^{18}=380 x^{18}$...
Read More →If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF,
Question: If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then the product of two numbers is(a) 203400 (b) 194400 (c) 198400 (d) 205400 Solution: Let the HCF bexand the LCM of the two numbers bey. It is given that the sum of the HCF and LCM is 1260 $x+y=1260 \ldots \ldots(i)$ And, LCM is 900 more than HCF. $y=x+900 \ldots \ldots(i i)$ Substituting (ii) in (i), we get: $x+x+900=1260$ $2 x+900=1260$ $2 x=1260-900$ $2 x=360$ $x=180$ Substituting $x=180$ in...
Read More →Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4
Question: Find the equation for the ellipse that satisfies the given conditions:Foci (3, 0),a= 4 Solution: Foci (3, 0),a= 4 Since the foci are on thex-axis, themajor axisis along thex-axis. Therefore, the equation of the ellipse will be of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where $a$ is the semi-major axis. Accordingly,c= 3 anda= 4. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore 4^{2}=b^{2}+3^{2}$ $\Rightarrow 16=b^{2}+9$ $\Rightarrow b^{2}=16-9=7$ Thus, the equation of the...
Read More →Find the second order derivatives of the function.
Question: Find the second order derivatives of the function. $x^{2}+3 x+2$ Solution: Let $y=x^{2}+3 x+2$ Then, $\frac{d y}{d x}=\frac{d}{d x}\left(x^{2}\right)+\frac{d}{d x}(3 x)+\frac{d}{d x}(2)=2 x+3+0=2 x+3$ $\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(2 x+3)=\frac{d}{d x}(2 x)+\frac{d}{d x}(3)=2+0=2$...
Read More →The LCM and HCF of two rational numbers are equal,
Question: The LCM and HCF of two rational numbers are equal, then the numbers must be (a) prime (b) co-prime (c) composite (d) equal Solution: Let the two numbers beaandb. (a) If we assume that theaandbare prime. Then, $\operatorname{HCF}(a, b)=1$ $\operatorname{LCM}(a, b)=a b$ (b) If we assume thataandbare co-prime. Then, $\operatorname{HCF}(a, b)=1$ $\operatorname{LCM}(a, b)=a b$ (c) If we assume thataandbare composite. Then, $\operatorname{HCF}(a, b)=1$ or any other highest common integer' $\...
Read More →If
Question: If $x=\sqrt{a^{\sin ^{-1} t}}, y=\sqrt{a^{\cos ^{-1} t}}$, show that $\frac{d y}{d x}=-\frac{y}{x}$ Solution: The given equations are $x=\sqrt{a^{\sin ^{-1} t}}$ and $y=\sqrt{a^{\cos ^{-1} t}}$ $x=\sqrt{a^{\sin ^{-1}},}$ and $y=\sqrt{a^{\cos ^{-1}},}$ $\Rightarrow x=\left(a^{\sin ^{-1} t}\right)^{\frac{1}{2}}$ and $y=\left(a^{\cos ^{-1} t}\right)^{\frac{1}{2}}$ $\Rightarrow x=a^{\frac{1}{2} \sin ^{-1} t}$ and $y=a^{\frac{1}{2} \cos ^{-1} t}$ Consider $x=a^{\frac{1}{2} \sin ^{-1} 1}$ Ta...
Read More →Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)
Question: Find the equation for the ellipse that satisfies the given conditions:Length of minor axis 16,foci (0, 6) Solution: Length of minor axis = 16; foci = (0, 6). Since the foci are on they-axis, themajor axisis along they-axis. Therefore, the equation of the ellipse will be of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$, where $a$ is the semi-major axis. Accordingly, 2b= 16⇒b= 8 andc= 6. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore a^{2}=8^{2}+6^{2}=64+36=100$ $\Rightarrow a=...
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