Find the equation for the ellipse that satisfies the given conditions: Length of major
Question: Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci $(\pm 5,0)$ Solution: Length of major axis $=26 ;$ foci $=(\pm 5,0)$. 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, 2a= 26⇒a= 13 andc= 5. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore 13^{2}=b^{2}+5^{2}$ $\Rightarr...
Read More →If $n$ is any natural number,
Question: If $n$ is any natural number, then $6^{n}-5^{n}$ always ends with (a) 1 (b) 3 (c) 5 (d) 7 [Hint: For any $n \in N, 6^{n}$ and $5^{n}$ end with 6 and 5 respectively. Therefore, $6^{n}-5^{n}$ always ends with $6-5=1$.] Solution: We know that $6^{n}$ will end in 6 And $5^{n}$ will end in 5 . Now, $6^{n}-5^{n}$ always end with $6-5=1$ Hence the correct choice in (a)....
Read More →If $n$ is a natural number,
Question: If $n$ is a natural number, then $9^{2 n}-4^{2 n}$ is always divisible by (a) 5 (b) 13 (c) both 5 and 13 (d) None of these [Hint : $9^{2 n}-4^{2 n}$ is of the form $a^{2 n}-b^{2 n}$ which is divisible by both $a-b$ and $a+b$. So, $9^{2 n}-4^{2 n}$ is divisible by both $9-$ $4=5$ and $9+4=13$.] Solution: We know that $a^{2 n}-b^{2 n}$ is always divisible by both $a-b$ and $a+b$. So, $9^{2 n}-4^{2 n}$ is always divisible by both $9-4=5$ and $9+5=13$. Hence, the correct choice is $(c)$....
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: Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis $(\pm 1,0)$ Solution: Ends of major axis $(0, \pm \sqrt{5})$, ends of minor axis $(\pm 1,0)$ Here, 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, $a=\sqrt{5}$ and $b=1$. Thus, the equation of the ellipse is $\frac{x^{2}}{1^...
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 $\frac{d y}{d x}$. $x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$ Solution: The given equations are $x=a(\cos \theta+\theta \sin \theta)$ and $y=a(\sin \theta-\theta \cos \theta)$ Then, $\frac{d x}{d \theta}=a\left[\frac{d}{d \theta} \cos \theta+\frac{d}{d \theta}(\theta \sin \theta)\right]=a\left[-\sin \theta+\theta \frac{d}{d \theta}(\sin \theta)+\sin \thet...
Read More →Find the equation for the ellipse that satisfies the given conditions: Ends of major axis
Question: Find the equation for the ellipse that satisfies the given conditions: Ends of major axis$(\pm 3,0)$, ends of minor axis $(0, \pm 2)$ Solution: Ends of major axis $(\pm 3,0)$, ends of minor axis $(0, \pm 2)$ Here, 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, $a=3$ and $b=2$. Thus, the equation of the ellipse is $\frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2...
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) $\frac{3}{10}$ (b) $\frac{1}{10}$ (c) 3 (d) $\frac{3}{100}$ Solution: For terminating the decimal expansion after one place of decimal, the highest power of $m$ and $n$ in $2^{m} \times 5^{n}$ should be 1 . Let $m=n=1$ We will get: $\frac{1}{3} \times \frac{3}{10}=\frac{1}{10}$ $=0.1$ Thus, it is evident that we multiplied it by $\frac{3}{...
Read More →Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)
Question: Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 6,0)$, foci $(\pm 4,0)$ Solution: Vertices $(\pm 6,0)$, foci $(\pm 4,0)$ Here, the vertices are on the $x$-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,a= 6,c= 4. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore 6^{2}=b^{2}+4^{2}$ $\Rightarrow 36=b^{2}+16$ $\Rightarrow b^{2}=36-16$ $\Rightarro...
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 $\frac{d y}{d x}$. $x=a \sec \theta, y=b \tan \theta$ Solution: The given equations are $x=a \sec \theta$ and $y=b \tan \theta$ Then, $\frac{d x}{d \theta}=a \cdot \frac{d}{d \theta}(\sec \theta)=a \sec \theta \tan \theta$ $\frac{d y}{d \theta}=b \cdot \frac{d}{d \theta}(\tan \theta)=b \sec ^{2} \theta$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d \theta}\right)}{\left(\frac{d x...
Read More →The smallest number by which
Question: The smallest number by which $\sqrt{27}$ should be multiplied so as to get a rational number is (a) $\sqrt{27}$ (b) $3 \sqrt{3}$ (c) $\sqrt{3}$ (d) 3 Solution: $\sqrt{27}=\sqrt{3 \times 3 \times 3}$ $=3 \sqrt{3}$ Out of the given choices $\sqrt{3}$ is the only smallest number by which if we multiply $\sqrt{27}$ we get a rational number. Hence, the correct choice is (c)....
Read More →(a) an integer (b) a rational number (c) a natural number (d) an irrational number
Question: 3. $\overline{27}$ is (a) an integer (b) a rational number (c) a natural number (d) an irrational number Solution: We have, $3 . \overline{27}=3.27272727 \ldots$ Let $x=3.27272727 \ldots$ Then, $100 x=327.272727 \ldots$ $10000 x=32727.2727 \ldots$ Subtract these to get $9900 x=32400$ $x=\frac{32400}{9900}$ $x=\frac{324}{99}$ Thus, we can also conclude that all infinite repeating decimals are rational numbers. Hence, the correct choice is (b)....
Read More →Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)
Question: Find the equation for the ellipse that satisfies the given conditions: Vertices $(0, \pm 13)$, foci $(0, \pm 5)$ Solution: Vertices $(0, \pm 13)$, foci $(0, \pm 5)$ Here, the vertices are on 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,a= 13 andc= 5. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore 13^{2}=b^{2}+5^{2}$ $\Rightarrow 169=b^{2}+25$ $\Rightarrow b^{2}=169-25$...
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 $\frac{d y}{d x}$. $x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t$ Solution: The given equations are $x=a\left(\cos t+\log \tan \frac{t}{2}\right)$ and $y=a \sin t$ Then, $\frac{d x}{d t}=a \cdot\left[\frac{d}{d t}(\cos t)+\frac{d}{d t}\left(\log \tan \frac{t}{2}\right)\right]$ $=a\left[-\sin t+\frac{1}{\tan \frac{t}{2}} \cdot \frac{d}{d t}\left(\tan \frac{t}{2}\right)\right]...
Read More →Outline salient features of carbon cycling in an ecosystem
Question: Outline salient features of carbon cycling in an ecosystem Solution: The carbon cycle is an important gaseous cycle which has its reservoir pool in the atmosphere. All living organisms contain carbon as a major body constituent. Carbon is a fundamental element found in all living forms. All biomolecules such as carbohydrates, lipids, and proteins required for life processes are made of carbon. Carbon is incorporated into living forms through a fundamental process called photosynthesis....
Read More →If 3 is the least prime factor of number a and 7 is the least prime factor of number b,
Question: If 3 is the least prime factor of numberaand 7 is the least prime factor of numberb, then the least prime factor ofa+b, is(a) 2 (b) 3 (c) 5 (d) 10 Solution: Since $7+3=10$ The least prime factor of $a+b$ has to be $2 ;$ unless $a+b$ is a prime number greater than 2 . Suppose $a+b$ is a prime number greater than 2 . Then $a+b$ must be an odd number o one ofaorbmust be an even number. Suppose then that $a$ is even. Then the least prime factor of $a$ is 2 ; which is not 3 or 7 . So $a$ ca...
Read More →Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)
Question: Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5,0)$, foci $(\pm 4,0)$, Solution: Vertices $(\pm 5,0)$, foci $(\pm 4,0)$ Here, the vertices are on 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, $a=5$ and $c=4$. It is known that $a^{2}=b^{2}+c^{2}$. $\therefore 5^{2}=b^{2}+4^{2}$ $\Rightarrow 25=b^{2}+16$ $\Rightarrow b^{2}=25-16$ $\Ri...
Read More →Write important features of a sedimentary cycle in an ecosystem.
Question: Write important features of a sedimentary cycle in an ecosystem. Solution: Sedimentary cycles have their reservoirs in the Earths crust or rocks. Nutrient elements are found in the sediments of the Earth. Elements such as sulphur, phosphorus, potassium, and calcium have sedimentary cycles. Sedimentary cycles are very slow. They take a long time to complete their circulation and are considered as less perfect cycles. This is because during recycling, nutrient elements may get locked in ...
Read More →Which of the following rational numbers have terminating decimal?
Question: Which of the following rational numbers have terminating decimal? (i) $\frac{16}{225}$ (ii) $\frac{5}{18}$ (iii) $\frac{2}{21}$ (iv) $\frac{7}{250}$ (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (i) and (iv) Solution: (i) We have, $\frac{16}{225}=\frac{16}{3^{2} \times 5^{2}}$ 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 deci...
Read More →Give an account of energy flow in an ecosystem.
Question: Give an account of energy flow in an ecosystem. Solution: Energy enters an ecosystem from the Sun. Solar radiations pass through the atmosphere and are absorbed by the Earths surface. These radiations help plants in carrying out the process of photosynthesis. Also, they help maintain the Earths temperature for the survival of living organisms. Some solar radiations are reflected by the Earths surface. Only 2-10 percent of solar energy is captured by green plants (producers) during phot...
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$4 x^{2}+9 y^{2}=36$ Solution: The given equation is $4 x^{2}+9 y^{2}=36$. It can be written as $4 x^{2}+9 y^{2}=36$ Or, $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ $\mathrm{Or}, \frac{x^{2}}{3^{2}}+\frac{y^{2}}{2^{2}}=1$ $\ldots(1)$ Here, the denominator of $\frac{x^{2}}{3^{2}}$ is greater than the denominator of $\frac{y^{2}}{2^{2}}$. The...
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 $\frac{d y}{d x}$. $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$ Solution: The given equations are $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}$ and $y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$ Then, $\frac{d x}{d t}=\frac{d}{d t}\left[\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}\right]$ $=\frac{\sqrt{\cos 2 t} \cdot \frac{d}{d t}\left(\sin ^{3} t\right)-\sin ^{3} t \cdot \...
Read More →Define decomposition and describe the processes and products of decomposition.
Question: Define decomposition and describe the processes and products of decomposition. Solution: Decomposition is the process that involves the breakdown of complex organic matter or biomass from the body of dead plants and animals with the help of decomposers into inorganic raw materials such as carbon dioxide, water, and other nutrients. The various processes involved in decomposition are as follows: (1)Fragmentation: It is the first step in the process of decomposition. It involves the brea...
Read More →Which of the following rational numbers have terminating decimal?
Question: Which of the following rational numbers have terminating decimal? (i) $\frac{16}{225}$ (ii) $\frac{5}{18}$ (iii) $\frac{2}{21}$ (iv) $\frac{7}{250}$ Solution: (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (i) and (iv)...
Read More →What is primary productivity?
Question: What is primary productivity? Give brief description of factors that affect primary productivity. Solution: It is defined as the amount of organic matter or biomass produced by producers per unit area over a period of time. Primary productivity of an ecosystem depends on the variety of environmental factors such as light, temperature, water, precipitation, etc. It also depends on the availability of nutrients and the availability of plants to carry out photosynthesis....
Read More →If $p$ and $q$ are co-prime numbers,
Question: If $p$ and $q$ are co-prime numbers, then $p^{2}$ and $q^{2}$ are (a) coprime (b) not coprime (c) even (d) odd Solution: We know that the co-prime numbers have no factor in common, or, their HCF is 1. Thus, $p^{2}$ and $q^{2}$ have the same factors with twice of the exponents of $p$ and $q$ respectively, which again will not have any common factor. Thus we can conclude that $p^{2}$ and $q^{2}$ are co-prime numbers. Hence, the correct choice is (a)....
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