What is the ecological principle behind the biological control method of managing with pest insects?
Question: What is the ecological principle behind the biological control method of managing with pest insects? Solution: The basis of various biological control methods is on the concept of predation. Predation is a biological interaction between the predator and the prey, whereby the predator feeds on the prey. Hence, the predators regulate the population of preys in a habitat, thereby helping in the management of pest insects....
Read More →Two numbers have 12 as their HCF and 350 as their LCM (True/False).
Question: Two numbers have 12 as their HCF and 350 as their LCM (True/False). Solution: Two numbers have 12 as their HCF and 350 as their LCM (True/False). False. Reason: We know that HCF should divide LCM. But, the HCF 12 does not divide the LCM 350....
Read More →If a and b are relatively prime numbers, then what is their LCM?
Question: Ifaandbare relatively prime numbers, then what is their LCM? Solution: It is given thataandbare two relatively prime numbers; we have to find their LCM. We know that two numbers are relatively prime if they dont have any common divisor. Also, the factors of any prime number are 1 and the prime number itself. For example, leta= 7 andb= 20 Thus, the factors are as follows a= 7 1 And b = 22 5 1 Now, the LCM of 7 and 20 is 140. Thus the HCF ofaandbisab....
Read More →If a and b are relatively prime numbers,
Question: Ifaandbare relatively prime numbers, then what is their HCF? Solution: It is given thataandbare two relatively prime numbers; we have to find their HCF. We know that two numbers are relatively prime if they dont have any common divisor. Also, the factors of any prime number are 1 and the prime number itself. For example, leta = 7andb= 20 Thus, the factors are as follows a= 7 1 And b= 22 5 1 Now, the HCF of 7 and 20 is 1. Thus the HCF ofaandbis 1...
Read More →Find of function.
Question: Find $\frac{d y}{d x}$ of function. $x y=e^{(x-y)}$ Solution: The given function is $x y=e^{(x-y)}$ Taking logarithm on both the sides, we obtain $\log (x y)=\log \left(e^{x-y}\right)$ $\Rightarrow \log x+\log y=(x-y) \log e$ $\Rightarrow \log x+\log y=(x-y) \times 1$ $\Rightarrow \log x+\log y=x-y$ Differentiating both sides with respect tox, we obtain $\frac{d}{d x}(\log x)+\frac{d}{d x}(\log y)=\frac{d}{d x}(x)-\frac{d y}{d x}$ $\Rightarrow \frac{1}{x}+\frac{1}{y} \frac{d y}{d x}=1-...
Read More →For what value of n,
Question: For what value ofn, 2n✕ 5nends in 5. Solution: We need to find the value of $n$, for which $2^{n} \times 5^{n}$ ends in 5 . Clearly, $2^{n} \times 5^{n}=(2 \times 5)^{n}$ $=10^{n}$ Also, all the values of $n$ will make $10^{n}$ end in 0 . Thus, there is no value of $n$ for which $2^{n} \times 5^{n}$ ends in 5 ....
Read More →The sum of two irrational number is an irrational number (True/False).
Question: The sum of two irrational number is an irrational number (True/False). Solution: The sum of two irrational numbers is an irrational number (True/False) False Reason: However, $\sqrt{2}$ is not rational because there is no fraction, no ratio of integers that will equal $\sqrt{2}$. It calculates to be a decimal that never repeats and never ends. The same can be said for $\sqrt{3}$. Also, there is no way to write $\sqrt{2}+\sqrt{3}$ as a fraction. In fact, the representation is already in...
Read More →The product of two irrational numbers is an irrational number (True/False).
Question: The product of two irrational numbers is an irrational number (True/False). Solution: The product of two irrational numbers is an irrational number (True/False) False Reason: Let us assume the two irrational numbers be $\sqrt{2}$ and $\sqrt{3}$ Sometimes, it is and sometimes it isn't. $\sqrt{2}$ And $\sqrt{3}$ are both irrational as their product is $\sqrt{6}$ Now $\sqrt{2}$ and $\sqrt{8}$ are both irrational but their product, $\sqrt{16}$ is rational (in fact, it equals 4 )...
Read More →Every odd integer is of the form 2m − 1,
Question: Every odd integer is of the form 2m 1, wheremis an integer (True/False). Solution: Every odd integer is of the form $2 m-1$, where $m$ is an integer (True/False) True Reason: Let the various values ofmas -1, 0 and 9. Thus, the values for $2 m-1$ become $-3,-1$ and 17 respectively. These are odd integers....
Read More →Find of function.
Question: Find $\frac{d y}{d x}$ of function. $(\cos x)^{y}=(\cos y)^{x}$ Solution: The given function is $(\cos x)^{y}=(\cos y)^{x}$ Taking logarithm on both the sides, we obtain $y \log \cos x=x \log \cos y$ Differentiating both sides, we obtain $\log \cos x \cdot \frac{d y}{d x}+y \cdot \frac{d}{d x}(\log \cos x)=\log \cos y \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}(\log \cos y)$ $\Rightarrow \log \cos x \frac{d y}{d x}+y \cdot \frac{1}{\cos x} \cdot \frac{d}{d x}(\cos x)=\log \cos y \cdot...
Read More →Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) focus (–2, 0)
Question: Find the equation of the parabola that satisfiesthe following conditions: Vertex (0, 0) focus (2, 0) Solution: Vertex (0, 0) focus (2, 0) Since the vertex of the parabola is $(0,0)$ and the focus lies on the negative $x$-axis, $x$-axis is the axis of the parabola, while the equation of the parabola is of the form $y^{2}=-$ $4 a x$. Since the focus is $(-2,0), a=2$. Thus, the equation of the parabola is $y^{2}=-4(2) x$, i.e., $y^{2}=-8 x$...
Read More →Every even integer is of the form 2m,
Question: Every even integer is of the form 2m, wheremis an integer (True/False). Solution: Every even integer is of the form 2m, wheremis an integer (True/False) True Reason: Let the various values ofmas -1, 0 and 9. Thus, the values for 2m become -2, 0 and 18 respectively....
Read More →The product of any three consecutive natural number is divisible by 6 (True/False).
Question: The product of any three consecutive natural number is divisible by 6 (True/False). Solution: The product of any three natural numbers is divisible by 6. True Reason: Let the three consecutive natural numbers be 1,2 and 3. Their product is 6, which is divisible by 6 Let the other set of three consecutive natural numbers be 3, 4 and 5. Their product is 60, which is divisible by 6...
Read More →Find the equation of the parabola that satisfies the following conditions:
Question: Find the equation of the parabola that satisfiesthe following conditions:Vertex (0, 0); focus (3, 0) Solution: Vertex (0, 0); focus (3, 0) Since the vertex of the parabola is $(0,0)$ and the focus lies on the positive $x$-axis, $x$-axis is the axis of the parabola, while the equation of the parabola is of the form $y^{2}=4 a x$. Since the focus is (3, 0),a= 3. Thus, the equation of the parabola is $y^{2}=4 \times 3 \times x$, i.e., $y^{2}=12 x$...
Read More →The sum of two prime number is always a prime number (True/ False).
Question: The sum of two prime number is always a prime number (True/ False). Solution: The sum of two prime numbers is always a prime number. False Reason: Let us prove the above by taking an example. Let the two given prime numbers be 2 and 7. Thus, their sum, i.e; 9 is not a prime number. Hence the above statement is false...
Read More →Find the equation of the parabola that satisfies the following conditions: Focus (0, –3); directrix y = 3
Question: Find the equation of the parabola that satisfies the following conditions: Focus $(0,-3)$; directrix $y=3$ Solution: Focus $=(0,-3) ;$ directrix $y=3$ Since the focus lies on the $y$-axis, the $y$-axis is the axis of the parabola. Therefore, the equation of the parabola is either of the form $x^{2}=4 a y$ or $x^{2}=-4 a y$ It is also seen that the directrix, $y=3$ is above the $x$-axis, while the focus $(0,-3)$ is below the $x$-axis. Hence, the parabola is of the form $x^{2}=-4$ ay. He...
Read More →π is an irrational number (True/False).
Question: is an irrational number (True/False). Solution: Here $\pi$ is an irrational number True Reason: Rational number is one that can be expressed as the fraction of two integers. Rational numbers converted into decimal notation always repeat themselves somewhere in their digits. For example, 3 is a rational number as it can be written as 3/1 and in decimal notation it is expressed with an infinite amount of zeros to the right of the decimal point. 1/7 is also a rational number. Its decimal ...
Read More →Find of function.
Question: Find $\frac{d y}{d x}$ of function. $y^{x}=x^{y}$ Solution: The given function is $y^{x}=x^{y}$' Taking logarithm on both the sides, we obtain $x \log y=y \log x$ Differentiating both sides with respect tox, we obtain $\log y \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}(\log y)=\log x \cdot \frac{d}{d x}(y)+y \cdot \frac{d}{d x}(\log x)$ $\Rightarrow \log y \cdot 1+x \cdot \frac{1}{y} \cdot \frac{d y}{d x}=\log x \cdot \frac{d y}{d x}+y \cdot \frac{1}{x}$ $\Rightarrow \log y+\frac{x}{y...
Read More →Find the equation of the parabola that satisfies the following conditions:
Question: Find the equation of the parabola that satisfies the following condifions: Focus $(6,0)$; directrix $x=-6$ Solution: Focus $(6,0)$; directrix, $x=-6$ Since the focus lies on thex-axis, thex-axis is the axis of the parabola. Therefore, the equation of the parabola is either of the form $y^{2}=4 a x$ or $y^{2}=-4 a x$ It is also seen that the directrix, $x=-6$ is to the left of the $y$-axis, while the focus $(6,0)$ is to the right of the $y$-axis. Hence, the parabola is of the form $y^{2...
Read More →HCF of two numbers is always a factor of their LCM (True/False).
Question: HCF of two numbers is always a factor of their LCM (True/False). Solution: HCF of two numbers is always a factor of their LCM True Reason: The HCF is a factor of both the numbers which are factors of their LCM.Thus the HCF is also a factor of the LCM of the two numbers....
Read More →What is the HCF of the smallest composite number and the smallest prime number?
Question: What is the HCF of the smallest composite number and the smallest prime number? Solution: The smallest composite number is 4 The smallest prime number is 2 Thus, the HCF of and is 2....
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 $x^{2}=-9 y$ Solution: The given equation is $x^{2}=-9 y$. Here, the coefficient ofyis negative. Hence, the parabola opens downwards. On comparing this equation with $x^{2}=-4 a y$, we obtain $-4 a=-9 \Rightarrow b=\frac{9}{4}$ $\therefore$ Coordinates of the focus $=(0,-a)=\left(0,-\frac{9}{4}\right)$ Since the given equation involves $x^{2}$, the axis of the parab...
Read More →What is a composite number?
Question: What is a composite number? Solution: A composite number is a positive integer which has a divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
Read More →An orchid plant is growing on the branch of mango tree.
Question: An orchid plant is growing on the branch of mango tree. How do you describe this interaction between the orchid and the mango tree? Solution: An orchid growing on the branch of a mango tree is an epiphyte. Epiphytes are plants growing on other plants which however, do not derive nutrition from them. Therefore, the relationship between a mango tree and an orchid is an example of commensalisms, where one species gets benefited while the other remains unaffected. In the above interaction,...
Read More →What is the total number of factors of a prime number?
Question: What is the total number of factors of a prime number? Solution: We know that the factors of any prime number are 1 and the prime number itself. For example, let p = 2 Thus, the factors are as follows $p=2 \times 1$ Hence, the total number of factors of a prime number is 2...
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