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Question: If $f(x)=\left\{\begin{array}{cl}\frac{1-\cos k x}{x \sin x}, x \neq 0 \\ \frac{1}{2} , x=0\end{array}\right.$ is continuous at $x=0$, find $k$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos k x}{x \sin x}, x \neq 0 \\ \frac{1}{2}, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f^{\prime}(x)=f(0)$ ....(1) Consider: $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{1-\cos k x}{x \sin x}\right)=\lim _{x \rightarrow 0}\le...
Read More →Find the sum of first 17 terms
Question: Find the sum of first 17 terms of an AP whose 4th and 9th terms are 15 and 30, respectively. Solution: Let the first term, common difference and the number of terms in an AP are a, d and n,respectively We know that, the $n$th term of an AP, $T_{n}=a+(n-1) d$ $\ldots$ (i) $\therefore \quad$ 4th term of an AP, $T_{4}=a+(4-1) d=-15$ [given] $\Rightarrow \quad a+3 d=-15 \quad \ldots$ (ii) and 9th term of an AP, $T_{a}=a+(9-1) d=-30$ [given] $\Rightarrow \quad a+8 d=-30$ .....(iii) Now, sub...
Read More →Which of the following are pairs of equal sets?
Question: Which of the following are pairs of equal sets? A = set of letters in the word, ALLOY. B = set of letters in the word, LOYAL. Solution: Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements we write A = B We have, A = set of letters in the word, ALLOY A = {A, L, O, Y} and B = set of letters in the word, LOYAL B = {L, O, Y, A} Here, A = B because the elements in both the sets are equal. The repetition of elements in a set does not change a set. Thus, ...
Read More →Find the (i) curved surface area
Question: Find the (i) curved surface area (ii) total surface area and (iii) volume of a right circular cylinder whose height is 15 cm and the radius of the base is 7 cm. Solution: Given : Height, $h=15 \mathrm{~cm}$ Radius, $r=7 \mathrm{~cm}$ (i) Curved surface area, $S_{1}=2 \pi r h$ $=2 \times \frac{22}{7} \times 7 \times 15$ $=660 \mathrm{~cm}^{2}$ (ii) Total surface area, $S_{2}=2 \pi r(r+h)$ $=2 \times \frac{22}{7} \times 7 \times(7+15)$ $=44 \times 22$ $=968 \mathrm{~cm}^{2}$ (iii) Volume...
Read More →Which of the following are examples of the singleton set?
Question: Which of the following are examples of the singleton set? (i) $\left\{x: x \in Z, x^{2}=4\right\}$. (ii) $\{x: x \in Z, x+5=0\}$. (iii) $\{x: x \in Z,|x|=1\}$. (iv) $\left\{x: x \in N, x^{2}=16\right\}$. (v) $\{x: x$ is an even prime number $\}$ Solution: (i) Integers $=\ldots-3,-2,-1,0,1,2,3, \ldots$ Given equation: $x^{2}=4$ $\Rightarrow x=\sqrt{4}$ $\Rightarrow x=\pm 2$ If $x=-2$, then $x^{2}=(-2)^{2}=4$ If $x=2$, then $x^{2}=(2)^{2}=4$ So, there are two elements in a set. $\therefo...
Read More →Determine the values of a, b, c for which the function
Question: Determine the values ofa,b,cfor which the function $f(x)=\left\{\begin{array}{cl}\frac{\sin (a+1) x+\sin x}{x}, \text { for } x0 \\ c , \quad \text { for } x=0 \text { is continuous at } x=0 . \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}}, \text { for } x0\end{array}\right.$ Solution: The given function can be rewritten as: $f(x)= \begin{cases}\frac{\sin (a+1) x+\sin x}{x}, \text { for } x0 \\ c , \text { for } x=0 \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b^{\frac{3}{2}}} , \text { for }...
Read More →A hollow cylindrical pipe is 21 dm long.
Question: A hollow cylindrical pipe is 21 dm long. Its outer and inner diameters are 10 cm and 6 cm respectively. Find the volume of the copper used in making the pipe. Solution: Let the length of the cylinder pipe be $h=21, \mathrm{dm}=210 \mathrm{~cm}$. Let the outer and the inner radius of the pipe be $R \mathrm{~cm}$ and $r \mathrm{~cm}$, re $s$ pectively. $\therefore 2 R=10$ and $2 r=6$ $R=5 \mathrm{~cm}$ and $r=3 \mathrm{~cm}$ Volume of the copper used in making the pipe, $V=\pi\left(R^{2}...
Read More →If sn denotes the sum of first
Question: If sndenotes the sum of first n terms of an AP, then prove thats12=3(s8-s4) Solution: $\because$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ ...(i) $\therefore$ $S_{8}=\frac{8}{2}[2 a+(8-1) d]=4(2 a+7 d)=8 a+28 d$ and $S_{4}=\frac{4}{2}[2 a+(4-1) d]=2(2 a+3 d)=4 a+6 d$ Now, $S_{8}-S_{4}=8 a+28 d-4 a-6 d=4 a+22 d$ ... (ii) and $S_{12}=\frac{12}{2}[2 a+(12-1) d]=6(2 a+11 d)$ $=3(4 a+22 d)=3\left(S_{8}-S_{4}\right) \quad$ [from Eq. (ii)] $\therefore$ $S_{12}=3\left(S_{8}-S...
Read More →Determine the values of a, b, c for which the function
Question: Determine the values ofa,b,cfor which the function $f(x)=\left\{\begin{array}{cl}\frac{\sin (a+1) x+\sin x}{x}, \text { for } x0 \\ c , \quad \text { for } x=0 \text { is continuous at } x=0 . \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}}, \text { for } x0\end{array}\right.$ Solution: The given function can be rewritten as: $f(x)= \begin{cases}\frac{\sin (a+1) x+\sin x}{x}, \text { for } x0 \\ c , \text { for } x=0 \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b^{\frac{3}{2}}} , \text { for }...
Read More →The circumference of the base of a cylinder is 88 cm and its height is 15 cm.
Question: The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find the volume of the cylinder. Solution: Let $r \mathrm{~cm}$ be the radius of a cylinder. Circumference of the cylinder, $S=2 \pi r$ Given : Height, $h=15 \mathrm{~cm}$ Circumference, $S=88 \mathrm{~cm}$ $\mathrm{S}=2 \pi \mathrm{r}$ $88=2 \times \frac{22}{7} \times \mathrm{r}$ $r=\frac{88 \times 7}{44}$ r = 14cm Volume of cylinder, $V=\pi r^{2} h$ $=\frac{22}{7} \times 14^{2} \times 15$ $=9240 \mathrm{~cm...
Read More →In an AP, if sn = 3n2 + 5n and
Question: In an AP, if sn= 3n2+ 5n and ak= 164, then find the value of k. Solution: $\because n$th term of an AP, $a_{n}=S_{n}-S_{n-1}$ $=3 n^{2}+5 n-3(n-1)^{2}-5(n-1)$ $\left[\because S_{n}=3 n^{2}+5 n\right.$ (given) $]$ $=3 n^{2}+5 n-3 n^{2}-3+6 n-5 n+5$ $a_{n}=6 n+2$ $\ldots($ i) or $a_{k}=6 k+2=164$ $\left[\because a_{k}=164\right.$ (given)) $\Rightarrow \quad 6 k=164-2=162$ $\therefore \quad k=27$...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $G=\{0\}$ Solution: Since, 0 G the set is not empty It is not a : set...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $F=\{x: x \in Q, 1x2\}$ Solution: Here, $x \in Q$ i.e. x is a rational number We know that If $\mathrm{a}$ and $\mathrm{b}$ are two rational numbers, then $\frac{a+b}{2}$ is a rational number between $\mathrm{a}$ and $\mathrm{b}$ such that $a\frac{a+b}{2}b$ So, the rational number 1 and 2 is $\frac{1+2}{2}=\frac{3}{2}$ the set is not empty It is not a : set....
Read More →Find the value of k if f(x) is continuous
Question: Find the value of $k$ if $f(x)$ is continuous at $x=\pi / 2$, where $f(x)=\left\{\begin{array}{rr}\frac{k \cos x}{\pi-2 x}, x \neq \pi / 2 \\ 3 , x=\pi / 2\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{k \cos x}{\pi-2 x}, x \neq \frac{\pi}{2} \\ 3, x=\frac{\pi}{2}\end{array}\right.$ If $f(x)$ is continuous at $x=\frac{\pi}{2}$, then $\lim _{x \rightarrow \frac{\pi}{2}} f(x)=f\left(\frac{\pi}{2}\right)$ $\Rightarrow \lim _{x \rightarrow \frac{\pi}{2}} \frac{k \c...
Read More →The area of the base of a right circular cylinder is 616 cm
Question: The area of the base of a right circular cylinder is 616 cm2and its height is 25 cm. Find the volume of the cylinder. Solution: Let the area of the base of a right circular cylinder be $S \mathrm{~cm}^{2}$. Given : $S=616 \mathrm{~cm}^{2}$ Height, $h=25 \mathrm{~cm}$ Let the radius of a right circular cylinder be $r \mathrm{~cm}$. $S=\pi r^{2}$ $616=\frac{22}{7} \times \mathrm{r}^{2}$ $\mathbf{r}^{2}=\frac{616 \times 7}{22}$ $\mathrm{r}^{2}=196$ r = 14 cm Volume of the cylinder, $\math...
Read More →In an AP, if sn =n (4n + 1),
Question: In an AP, if sn=n (4n + 1), then find the AP. Solution: We know that, the n th term of an AP is $a_{n}=S_{n}-S_{n-1}$ $a_{n}=n(4 n+1)-(n-1)\{4(n-1)+1\} \quad\left[\because S_{n}=n(4 n+1)\right]$ $\Rightarrow \quad a_{n}=4 n^{2}+n-(n-1)(4 n-3)$ $=4 n^{2}+n-4 n^{2}+3 n+4 n-3=8 n-3$ Put $n=1, \quad a_{1}=8(1)-3=5$ Put $n=2, \quad a_{2}=8(2)-3=16-3=13$ Put $n=3, \quad a_{3}=8(3)-3=24-3=21$ Hence, the required AP is 5,13, 21,...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $E=\{x: x \in W, x+3 \leq 3\}$ Solution: Whole numbers = 0, 1, 2, 3, If we take $x=0$ then $x+3=0+3=3$ If we take $x=1$ then $x+3=1+3=43$ So, 0 is the element of set $E$ because it satisfies the given equation. $\therefore$ It is not a : set....
Read More →If an = 3 — 4n, then show
Question: If an= 3 4n, then show thata1,a2, a3, form an AP. Also, findS20. Solution: Given that, nth term of the series is an = 3 4n (i) Put $n=1, \quad a_{1}=3-4(1)=3-4=-1$ Put $n=2, \quad a_{2}=3-4(2)=3-8=-5$ Put $n=3, \quad a_{3}=3-4(3)=3-12=-9$ Put $n=4, \quad a_{4}=3-4(4)=3-16=-13$ So, the series becomes $-1,-5,-9,-13, \ldots$ We see that, $a_{2}-a_{1}=-5-(-1)=-5+1=-4$ $a_{3}-a_{2}=-9-(-5)=-9+5=-4$ $a_{4}-a_{2}=-13-(-9)=-13+9=-4$ i.e. $\quad a_{2}-a_{1}=a_{3}-a_{2}=a_{4}-a_{3}=\ldots=-4$ Si...
Read More →Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are:
Question: Find the volume of a cylinder, if the diameter (d) of its base and its altitude (h) are: (i)d= 21 cm,h= 10 cm (ii)d= 7 m,h= 24 m Solution: (i) Given : $d=21 \mathrm{~cm}$, radius, $r=\frac{d}{2}=10.5 \mathrm{~cm}$ height, $h=10 \mathrm{~cm}$ Volume of the cylinder, $V=\pi r^{2} h$ $=\frac{22}{7} \times(10.5)^{2} \times 10$ $=3465 \mathrm{~cm}^{3}$ (ii) Given : $d=7 \mathrm{~m}$, radius, $r=\frac{d}{2}=3.5 \mathrm{~m}$ height $h=24 \mathrm{~m}$ Volume of the cylinder, $\mathrm{V}=\pi r^...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $D=\left\{x: x \in N, x^{2}+1=0\right\}$ Solution: Natural numbers = 1, 2, 3, 4, 5, 6, If $x=1$, then $x^{2}+1=(1)^{2}+1=1+1=2 \neq 0$ 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 →Prove that the function
Question: Prove that the function $f(x)=\left\{\begin{array}{cc}\frac{x}{|x|+2 x^{2}}, x \neq 0 \\ k , x=0\end{array}\right.$ remains discontinuous at $x=0$, regardless the choice of $k$. Solution: The given function can be rewritten as: $f(x)=\left\{\begin{array}{l}\frac{x}{x+2 x^{2}}, x0 \\ \frac{-x}{x-2 x^{2}}, x0 \\ k, x=0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{c}\frac{1}{2 x+1}, x0 \\ \frac{1}{2 x-1}, x0 \\ k, x=0\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=0)=\li...
Read More →Find the volume of a cylinder whose
Question: Find the volume of a cylinder whose (i)r= 3.5 cm,h= 40 cm (ii)r= 2.8 m,h= 15 m Solution: (i) Given : $r=3.5 \mathrm{~cm}, h=40 \mathrm{~cm}$ Volume of cylinder, $\mathrm{V}=\pi r^{2} h$ $=\frac{22}{7} \times(3.5)^{2} \times 40$ $=1540 \mathrm{~cm}^{3}$ (ii) Given : $r=2.8 \mathrm{~m}, h=15 \mathrm{~m}$ Volume of cylinder, $\mathrm{V}=\pi r^{2} h$ $=\frac{22}{7} \times(2.8)^{2} \times 15$ $=369.6 \mathrm{~m}^{3}$...
Read More →Which of the following are examples of the : set?
Question: Which of the following are examples of the : set? $C=\{x: x$ is prime, $90x96\}$ Solution: Prime numbers = Those numbers which are divisible by 1 and number itself. Prime numbers $=2,3,5,7,11,13, \ldots, 83,89,97, \ldots$ Prime number greater than $90=97$ Prime number less than $96=89$ Prime number less than 96 but greater than $90=\phi$ $\therefore$ The set is empty $\therefore$ It is a : set...
Read More →Which term of the AP – 2, – 7, – 12,…
Question: Which term of the AP 2, 7, 12, will be 77 ? Find the sum of this AP upto the term 77. Solution: Given, AP -2,-7,-12, Let the nth term of an AP is 77. Then, first term (a) = 2 and common difference (d) = 7 (- 2) = 7 + 2 = 5. nth term of an AP, Tn= a + (n 1)d $\Rightarrow \quad-77=-2+(n-1)(-5)$ $\Rightarrow \quad-75=-(n-1) \times 5$ $\Rightarrow \quad(n-1)=15 \Rightarrow n=16$ So, the 16th term of the given AP will be $-77$. Now the sum of $n$ terms of an $A P$ is $S_{n}=\frac{n}{2}[2 a+...
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^{...
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