Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? B = {x : x ϵ N, 2x + 3 = 4}. Solution: Natural numbers = 1, 2, 3, 4, 5, 6, If x = 1, then 2x + 3 = 2(1) + 3 = 2 + 3 = 5 4 no elements in the set B because the equation 2x + 3 = 4 is not satisfied by any natural number of x. It is a : set....
Read More →Find the values of a so that the function
Question: Find the values ofaso that the function $f(x)=\left\{\begin{array}{cl}a x+5, \text { if } x \leq 2 \\ x-1, \text { if } x2\end{array}\right.$ is continuous at $x=2$ Solution: Given: $f(x)=\left\{\begin{array}{l}a x+5, \text { if } x \leq 2 \\ x-1, \text { if } x2\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=2)=\lim _{\mathrm{x} \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} a(2-h)+5=2 a+5$ $(\mathrm{RHL}$ at $x=2)=\lim _{\mathrm{x} \rightarrow 2^{...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $A=\{x: x \in N, 1x \leq 2\}$ Solution: Natural numbers = 1, 2, 3, 4, 5, 6, 7, Natural number greater than 1 (1 x) = 2, 3, 4, 5, .. Natural number less than or equal to 2 (x 2) = 1 A number cannot be simultaneously greater than 1 and less than equal to 2 no elements in this set It is a : set....
Read More →Find the sum
Question: Find the sum (i) $1+(-2)+(-5)+(-8)+\ldots+(-236)$ (ii) $\left(4-\frac{1}{n}\right)+\left(4-\frac{2}{n}\right)+\left(4-\frac{3}{n}\right)+\ldots$ upto $n$ terms. (iii) $\frac{a-b}{a+b}+\frac{3 a-2 b}{a+b}+\frac{5 a-3 b}{a+b}+\ldots$ to 11 terms. Solution: (i) Here. first term $(a)=1$ and common difference $(d)=(-2)-1=-3$ $\because$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $\Rightarrow$ $S_{n}=\frac{n}{2}[2 \times 1+(n-1) \times(-3)]$ $\Rightarrow$ $S_{n}=\frac{n}{2}(2...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? Set of even prime numbers. Solution: Prime numbers = Those numbers which are divisible by 1 and number itself. Prime numbers = 2, 3, 5, 7, 11, 13, Even Prime number = 2 set is not empty. It is not a : set...
Read More →Determine the value of the constant k so that the function
Question: Determine the value of the constantkso that the function $f(x)=\left\{\begin{array}{rr}\frac{\sin 2 x}{5 x}, \text { if } \quad x \neq 0 \\ k , \text { if } x=0\end{array}\right.$is continuous at $x=0$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\sin 2 x}{5 x}, \text { if } x \neq 0 \\ k, \text { if } x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 2 x}{5 x}=k$ $\Rightarrow \lim _...
Read More →A cylindrical vessel, without lid, has to be tin-coated on its both sides.
Question: A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs 3.50 per 1000 cm2. Solution: Let $r \mathrm{~cm}$ and $h \mathrm{~cm}$ be the radius of the cylindrical vessel. Given : Radius, $r=70 \mathrm{~cm}$ Height, $h=1.4 \mathrm{~m}=140 \mathrm{~cm}$ Rate of tin $-$ plating $=$ Rs $3.50$ per 1000 square centimetre Cost of tin - plating the cylindrical vessel ...
Read More →Determine the value of the constant k so that the function
Question: Determine the value of the constantkso that the function $f(x)=\left\{\begin{array}{ll}k x^{2}, \text { if } x \leq 2 \\ 3, \text { if } x2\end{array}\right.$ is continuous at $x=2$ Solution: Given: $f(x)=\left\{\begin{array}{l}k x^{2}, \text { if } x \leq 2 \\ 3, \text { if } x2\end{array}\right.$ If $f(x)$ is continuous at $x=2$, then $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=f(2)$ .....(1) Now, $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)...
Read More →Find the ratio between the total surface area of a cylinder to its curved surface area,
Question: Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm. Solution: Let $S_{1}$ and $S_{2}$ be the total surface area and curved surface area, respectively. Given : Height, $h=7.5 \mathrm{~cm}$ Radius, $r=3.5 \mathrm{~cm}$ $\mathrm{~S}_{1}=2 \pi r(r+h)$ $\mathrm{S}_{2}=2 \pi r h$ According to the question: $\frac{S_{1}}{S_{2}}=\frac{2 \pi r(r+h)}{2 \pi r h}$ $\frac{S_{1}}{S_{2}}=\frac{\mathrm{r}+\math...
Read More →The first term of an AP is – 5 and the last
Question: The first term of an AP is 5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms and the common difference. Solution: Let the first term, common difference and the number of terms of an AP are a, d and n respectively. Given that, first term (a) = 5 and last term (l) = 45 Sum of the terms of the AP = 120 ⇒ Sn= 120 We know that, if last term of an AP is known, then sum of n terms of an AP is, $S_{n}=\frac{n}{2}(a+l)$ $\Rightarrow$$120=\frac{n}...
Read More →The sum of the radius of the base and height of a solid cylinder is 37 m.
Question: The sum of the radius of the base and height of a solid cylinder is 37 m. If the total surface area of the solid cylinder is 1628 m2, find the circumference of its base. Solution: Let $r$ and $h$ be the radius and height of the solid cylinder. Given : $r+h=37 \mathrm{~m}$ Total surface area, $S=2 \pi r(r+h)$ $1628=2 \pi \times r \times 37$ $\mathbf{r}=\frac{1628}{2 \pi \times 37}$ $=\frac{1628}{232.477}$ $=7 \mathrm{~m}$ Circumference of its base, $S_{1}=2 \pi \mathrm{r}$ $=\left(2 \ti...
Read More →For what value of k is the function
Question: For what value ofkis the function $f(x)=\left\{\begin{aligned} \frac{\sin 5 x}{3 x}, \text { if } \quad x \neq 0 \\ k \text {, if } x=0 \end{aligned}\right.$ is continuous at $x=0 ?$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\sin 5 x}{3 x}, \text { if } x \neq 0 \\ k, \text { if } x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{\mathrm{x} \rightarrow 0} \frac{\sin 5 x}{3 x}=k$ $\Rightarrow \lim _{\mathrm{x}...
Read More →Find the sum of the two middle most terms
Question: Find the sum of the two middle most terms of an AP $\frac{-4}{3},-1, \frac{-2}{3}, \ldots, \ldots, 4 \frac{1}{3}$ Solution: Here, first term $(a)=-\frac{4}{3}$, common difference $(d)=-1+\frac{4}{3}=\frac{1}{3}$ and the last term $(l)=4 \frac{1}{3}=\frac{13}{3}$ $\because n$th term of an AP, $l=a_{n}=a+(n-1) d$ $\Rightarrow \quad \frac{13}{3}=-\frac{4}{3}+(n-1) \frac{1}{3}$ $\Rightarrow \quad 13=-4+(n-1)$ $\Rightarrow \quad n-1=17$ $\Rightarrow \quad n=18$ [even] So, the two middle mos...
Read More →The total surface area of a hollow cylinder which is open from both sides is 4620 sq.
Question: The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm, area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder. Solution: Given: Total surface area of the cylinder $=4620 \mathrm{~cm}^{2}$ Area of the base ring $=115.5 \mathrm{~cm}^{2}$ Height, $h=7 \mathrm{~cm}$ Let $R$ be the radius of the outer ring and $r$ be the radius of the inner ring. Area of the base ring $=\pi R^{2}-\pi r^{2}$ $115.5=\pi\left(R^{2}-r^{2}\right)$...
Read More →Determine the value of the constant k so that the function
Question: Determine the value of the constantkso that the function $f(x)=\left\{\begin{array}{rlr}\frac{x^{2}-3 x+2}{x-1}, \text { if } x \neq 1 \\ k , \text { if } x=1\end{array}\right.$ is continuous at $x=1$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-3 x+2}{x-1}, \text { if } x \neq 1 \\ k, \text { if } x=1\end{array}\right.$ If $f(x)$ is continuous at $x=1$, then, $\lim _{x \rightarrow 1} f(x)=f(1)$ $\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{x^{2}-3 x+2}{x-1}=k$ $\Rig...
Read More →Determine the value of the constant k so that the function
Question: Determine the value of the constantkso that the function $f(x)=\left\{\begin{array}{rlr}\frac{x^{2}-3 x+2}{x-1}, \text { if } x \neq 1 \\ k , \text { if } x=1\end{array}\right.$ is continuous at $x=1$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-3 x+2}{x-1}, \text { if } x \neq 1 \\ k, \text { if } x=1\end{array}\right.$ If $f(x)$ is continuous at $x=1$, then, $\lim _{x \rightarrow 1} f(x)=f(1)$ $\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{x^{2}-3 x+2}{x-1}=k$ $\Rig...
Read More →How many numbers lie between 10 and 300,
Question: How many numbers lie between 10 and 300, which divided by 4 leave a remainder 3? Solution: Here, the first number is 11, which divided by 4 leave remainder 3 between 10 and 300. Last term before 300 is 299, which divided by 4 leave remainder 3. 11,15,19,23. 299 Here, first term (a) = 11, common difference d = 15 -11 = 4 $\because \quad$ nth term, $a_{n}=a+(n-1) d=l \quad$ [last term] $\Rightarrow \quad 299=11+(n-1) 4$ $\Rightarrow \quad 4(n-1)=288$ $\Rightarrow \quad(n-1)=72$ $\therefo...
Read More →For what value of k is the following function continuous at x = 1?
Question: For what value ofkis the following function continuous atx= 1? $f(x)=\left\{\begin{array}{rr}\frac{x^{2}-1}{x-1}, x \neq 1 \\ k , x=1\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x-1}, \quad x \neq 1 \\ k, \quad x=1\end{array}\right.$ If $f(x)$ is continuous at $x=1$, then $\lim _{x \rightarrow 1} f(x)=f(1)$ $\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{x^{2}-1}{x-1}=k$ $\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{(x-1)(x+1)}{x-1}=k$ $\Ri...
Read More →Twenty one cylindrical pillars of the Parliament House are to be cleaned.
Question: Twenty one cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m, what will be the cost of cleaning them at the rate of Rs 2.50 per square metre? Solution: Given: Diameter of the pillars $=0.5 \mathrm{~m}$ Radius of the pillars, $r=0.25 \mathrm{~m}$ Height of the pillars, $h=4 \mathrm{~m}$ Number of pillars $=21$ Rate of cleaning $=$ Rs $2.50$ per square metre Curved surface area of one pillar $=2 \pi r h$ $=2 \times \...
Read More →The diameter of a roller is 84 cm and its length is 120 cm.
Question: The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions moving once over to level a playground. What is the area of the playground? Solution: Given: Diameter of the roller $=84 \mathrm{~cm}$ $\therefore$ Radius, $r=\frac{\text { Diameter }}{2}=42 \mathrm{~cm}$ In 1 revolution, it covers the distance of its lateral surface area. Roller is a cylinder of height, $h=120 \mathrm{~cm}$ Radius $=42 \mathrm{~cm}$ Lateral surface area of the cylinder $=2 \p...
Read More →Discuss the continuity
Question: Discuss the continuity of $f(x)=\left\{\begin{array}{ll}2 x-1, x0 \\ 2 x+1 , x \geq 0\end{array}\right.$ at $x=0$ Solution: $f(x)= \begin{cases}2 x-1, x0 \\ 2 x+1, x \geq 0\end{cases}$ $($ LHL at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=2(0)-1=-1$ $(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=2(0)+1=1$ $\Rightarrow \lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0+} f(x)$ Hence,f(x) is discontinuous atx = 0....
Read More →The inner diameter of a circular well is 3.5 m. It is 10 m deep.
Question: The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find the cost of plastering its inner curved surface at Rs 4 per square metre. Solution: Given: Inner diameter of the circular well $=3.5 \mathrm{~m}$ $\therefore$ Inner radius of the circular well, $r=1.75 \mathrm{~m}$ Depth of the circular well, $h=10 \mathrm{~m}$ Inner curved surface area, $S=2 \pi r h$ $\mathrm{S}=2 \pi \times 1.75 \times 10 \mathrm{~m}^{2}=2 \times \frac{22}{7} \times 1.75 \times 10 \mathrm{~m}^{2}$ ...
Read More →Discuss the continuity of the function f(x) at the point x = 1/2, where
Question: Discuss the continuity of the functionf(x) at the pointx= 1/2, where $f(x)=\left\{\begin{array}{cc}x, \quad 0 \leq x1 / 2 \\ 1 / 2, \quad x=1 / 2 \\ 1-x, 1 / 2x \leq 1\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}x, 0 \leq x\frac{1}{2} \\ \frac{1}{2}, x=\frac{1}{2} \\ 1-x, \frac{1}{2}x \leq 1\end{array}\right.$ We observe $\left(\mathrm{LHL}\right.$ at $\left.x=\frac{1}{2}\right)=\lim _{x \rightarrow \frac{1}{2}^{-}} f(x)=\lim _{h \rightarrow 0} f\left(\frac{1}{2}-h...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? Set of odd natural numbers divisible by 2. Solution: Natural numbers = 1, 2, 3, 4, 5, Odd Natural numbers = 1, 3, 5, 7, 9, 11, No odd natural number is divisible by 2. no elements in this set It is a : set....
Read More →Match each of the sets on the left described in the roster from with the same
Question: Match each of the sets on the left described in the roster from with the same set on the right described in the set-builder from: Solution: (i) {-5, 5} It can be seen that if we take the square of -5 and 5, the result will be 25 If $x=-5$, then $(-5)^{2}=25$ If $x=5$, then $(5)^{2}=25$ and -5, 5 both are integers So, $\left\{x: x \in Z\right.$ and $\left.x^{2}=25\right\}$ $\therefore$ (i) matches (c) (ii) $\{1,2,3,6,9,18\}$ Divisor of 18 are $18=18 \times 1$ $18=9 \times 2$ $18=6 \time...
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