Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}$ Solution: Let $y=\frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}$ $\Rightarrow y=\frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}} \times \frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}$ $\Rightarrow y=\frac{\left(\sqrt{x^{2}+1}+\sqrt{x^{2}-1}\right)^{2}}{\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)\lef...
Read More →A die has its six faces marked 0, 1, 1, 1, 6, 6.
Question: A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded. (i) How many different scores are possible? (ii) What is the probability of getting a total of 7? Solution: Given, a die has its six faces marked {0,1,1,1,6, 6} Total sample space, n(S) = 62= 36 (i) The different score which are possible are 6 scores e., 0,1,2,6,7 and12. (ii) Let E = Event of getting a sum 7 $=\{(1,6),(1,6),(1,6),(1,6),(1,6),(1,6),(6,1),(6,1),(6,1),(6,1),...
Read More →Factorise:
Question: Factorise:p2+ 6p+ 8 Solution: The given expression is $p^{2}+6 p+8$. Find two numbers that follow the conditions given below: Sum $=6$ Product $=8$ Clearly, the numbers are 4 and 2 . $p^{2}+6 p+8=p^{2}+4 p+2 p+8$ $=p(p+4)+2(p+4)$ $=(p+4)(p+2)$...
Read More →In a game, the entry fee is of ₹ 5.
Question: In a game, the entry fee is of ₹ 5. The game consists of a tossing a coin 3 times. If one or two heads show, Sweta gets her entry fee back. If she throws 3 heads, she receives double the entry fees. Otherwise she will lose. For tossing a coin three times, find the probability that she (i) loses the entry fee. (ii) gets double entry fee. (iii) just gets her entry fee. Solution: Total possible outcomes of tossing a coin 3 times, S = {(HHH), (TTT), (HTT), (THT), (TTH), (THH), (HTH), (HHT)...
Read More →Factorise:
Question: Factorise:z2+ 12z+ 27 Solution: The given expression is $z^{2}+12 z+27$. Find two numbers that follow the conditions given below: Sum $=12$ Product $=27$ Clearly, the numbers are 9 and $3 .$ $z^{2}+12 z+27=z^{2}+9 z+3 z+27$ $=z(z+9)+3(z+9)$ $=(z+9)(z+3)$...
Read More →Factorise:
Question: Factorise:y2+ 10y+ 24 Solution: The given expression is $y^{2}+10 y+24$. Find two numbers that follow the conditions given below: Sum $=10$ Product $=24$ Clearly, the numbers are 6 and 4 . $y^{2}+10 y+24=y^{2}+6 y+4 y+24$ $=y(y+6)+4(y+6)$ $=(y+6)(y+4)$...
Read More →A child’s game has 8 triangles of
Question: A childs game has 8 triangles ofwhich 3 are blue and rest are red, and 10 squares of which 6 are blue and rest are red. One piece is lost at random. Find the probability that it is a (i) triangle (ii) square (iii)square of blue colour (iv) triangle of red colour Solution: Total number of figures $n(S)=8$ triangles $+10$ squares $=18$ (i) $P$ (lost piece is a triangle) $=\frac{8}{18}=\frac{4}{9}$ (ii) $P$ (lost piece is a square) $=\frac{10}{18}=\frac{5}{9}$ (iii) $P$ (square of blue co...
Read More →Factorise:
Question: Factorise:x2+ 5x+ 6 Solution: The given expression is $x^{2}+5 x+6$. Find two numbers that follow the conditions given below : Sum $=5$ Product $=6$ Clearly, the numbers are 3 and 2 . $x^{2}+5 x+6=x^{2}+3 x+2 x+6$ $=x(x+3)+2(x+3)$ $=(x+3)(x+2)$...
Read More →A carton of 24 bulbs contain 6 defective bulbs.
Question: A carton of 24 bulbs contain 6 defective bulbs. One bulb is drawn at random. What is the probability that the bulb is not defective? If the bulb selected is defective and it is not replaced and a second bulb is selected at random from the rest, what is the probability that the second bulb is defective? Solution: Total number of bulbs, n (S) = 24 Let $E_{1}=$ Event of selecting not defective bulb $=$ Event of selecting good bulbs $n\left(E_{1}\right)=18$ $\therefore$$P\left(E_{1}\right)...
Read More →Factorise:
Question: Factorise:(l+m)2 4lm Solution: We have: $(l+m)^{2}-4 l m=\left(l^{2}+m^{2}+2 l m\right)-4 l m$ $=l^{2}+m^{2}+2 l m-4 l m$ $=l^{2}+m^{2}-2 l m$ $=(l)^{2}+(m)^{2}-2 \times l \times m$ $=(l-m)^{2}$ $\therefore(l+m)^{2}-4 l m=(l-m)^{2}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{x} \log (\sin 2 x)$ Solution: Let $y=e^{x} \log (\sin 2 x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[e^{x} \log (\sin 2 x)\right]$ We have $(\text { uv })^{\prime}=$ vu' $+$ uv' (product rule) $\Rightarrow \frac{d y}{d x}=\log (\sin 2 x) \frac{d}{d x}\left(e^{x}\right)+e^{x} \frac{d}{d x}[\log (\sin 2 x)]$ We know $\frac{d}{d x}\left(e^{x}\right)=e^{x}$ and $\frac{d}{d x}(\log ...
Read More →Factorise:
Question: Factorise:m4+ 2m2n2+n4 Solution: We have: $m^{4}+2 m^{2} n^{2}+n^{4}=\left(m^{2}\right)^{2}+2 \times m^{2} \times n^{2}+\left(n^{2}\right)^{2}$ $=\left(m^{2}+n^{2}\right)^{2}$ $\therefore m^{4}+2 m^{2} n^{2}+n^{4}=\left(m^{2}+n^{2}\right)^{2}$...
Read More →Box A contains 25 slips of which 19 are marked
Question: Box A contains 25 slips of which 19 are marked ₹ 1 and other are marked ₹ 5 each. Box B contains 50 slips of which 45 are marked ₹ 1 each and others are marked ₹ 13 each. Slips of both boxes are poured into a third box and resuffled. A slip is drawn at random. What is the probability that it is marked other than ₹ 1? Solution: Total number of slips in a box, n(S) = 25 + 50 = 75 Erom the chart it is clear that, there are 11 slips which are marked other than $₹ 1$. $\therefore \quad$ Req...
Read More →There are 1000 sealed envelopes in a box,
Question: There are 1000 sealed envelopes in a box, 10 of them contain a cash prize of ₹ 100 each, 100 of them contain a cash prize of ₹ 50 each and 200 of them contain a cash prize of ₹ 10 each and rest do not contain any cash prize. If they are well shuffled and an envelope is picked up out, what is the probability that it contains no cash prize? Solution: Total number of sealed envelopes in a box, n (S) = 1000 Number of envelopes containing cash prize = 10 + 100 + 200 = 310 Number of envelope...
Read More →A letter of english alphabets is chosen at random.
Question: A letter of english alphabets is chosen at random. Determine the probability that the letter is a consonant Solution: We know that, in english alphabets, there are (5 vowels + 21 consonants)=26 letters. So, total number of outcomes in english alphabets are, n(S) = 26 Let $E=$ Event of choosing a english alphabet, which is a consonent $=\{b, c, d, t, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z\}$ $\therefore \quad n(E)=21$ Hence, required probability $=\frac{n(E)}{n(S)}=\frac{21}{...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{2}\{\log (2 x+3)\}$ Solution: Let $y=\sin ^{2}\{\log (2 x+3)\}$ On differentiating y with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\sin ^{2}\{\log (2 \mathrm{x}+3)\}\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2 \sin ^{2-1}\{\log (2 \mathrm{x}+3)\}...
Read More →Factorise:
Question: Factorise:a2b2 6abc+ 9c2 Solution: We have: $a^{2} b^{2}-6 a b c+9 c^{2}=(a b)^{2}-2 \times a b \times 3 c+(3 c)^{2}$ $=(a b-3 c)^{2}$...
Read More →Cards with numbers 2 to 101 are placed in a box.
Question: Cards with numbers 2 to 101 are placed in a box. A card is selected at random. Find the probability that the card has (i) an even number (ii) a square number Solution: Total number of out comes with numbers 2 to 101, n(s) =100 (i) Let $E_{1}=$ Event of selectina a card which is an even number $=\{2,4,6, \ldots 100\}$ [in an AP, $l=a+(n-1) d$, here $l=100, a=2$ and $d=2 \Rightarrow \begin{aligned} 100=2+(n-1) 2 \Rightarrow(n-1)=49 \Rightarrow n=50] \end{aligned}$ $\therefore$$n\left(E_{...
Read More →Solve the following
Question: Factorise:m2 4mn+ 4n2 Solution: We have: $m^{2}-4 m n+4 n^{2}=m^{2}-2 \times m \times 2 n+(2 n)^{2}$ $=(m-2 n)^{2}$ $\therefore m^{2}-4 m n+4 n^{2}=(m-2 n)^{2}$...
Read More →Factorise:
Question: Factorise:16x2 24x+ 9 Solution: We have: $16 x^{2}-24 x+9=(4 x)^{2}-2 \times 4 x \times 3+(3)^{2}$ $=(4 x-3)^{2}$ $\therefore 16 x^{2}-24 x+9=(4 x-3)^{2}$...
Read More →Factorise:
Question: Factorise:9y2 12y+ 4 Solution: We have: $9 y^{2}-12 y+4=(3 y)^{2}-2 \times 3 y \times 2+(2)^{2}$ $=(3 y-2)^{2}$ $\therefore 9 y^{2}-12 y+4=(3 y-2)^{2}$...
Read More →An integer is chosen between 0 and 100.
Question: An integer is chosen between 0 and 100. What is the probability that it is (i) divisible by 7? (ii) not divisible by 7? Solution: The number of integers between 0 and 100 is n(S)= 99 (i) Let $E_{1}=$ Event of choosing an integer which is divisible by 7 $=$ Event of choosing an integer which is multiple of 7 $=\{7,14,21,28,35,42,49,56,63,70,77,84,91,98\}$ $\therefore \quad n\left(E_{1}\right)=14$ $\therefore \quad P\left(E_{1}\right)=\frac{n\left(E_{1}\right)}{n(S)}=\frac{14}{99}$ (ii) ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{3 x^{2} \sin x}{\sqrt{7-x^{2}}}$ Solution: Let $y=\frac{3 x^{2} \sin x}{\sqrt{7-x^{2}}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\frac{3 x^{2} \sin x}{\sqrt{7-x^{2}}}\right)$ $\Rightarrow \frac{d y}{d x}=3 \frac{d}{d x}\left(\frac{x^{2} \sin x}{\sqrt{7-x^{2}}}\right)$ Recall that $\left(\frac{\mathrm{u}}{\mathrm{v}}\right)^{\prime}=\frac{\mathrm{vu}^{\prime}-\mathrm{uv}^{\p...
Read More →Factorise:
Question: Factorise:1 6x+ 9x2 Solution: We have: $1-6 x+9 x^{2}=9 x^{2}-6 x+1$ $=(3 x)^{2}-2 \times 3 x \times 1+(1)^{2}$ $=(3 x-1)^{2}$ $\therefore 1-6 x+9 x^{2}=(3 x-1)^{2}$...
Read More →All the jacks, queensapd kings are removed
Question: All the jacks, queensapd kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value. (i) 7 (ii) greater than 7 (iii) Less than 7 Solution: In out of 52 playing cards, 4 jacks, 4 queens and 4 kings are removed, then the remaining cards are left, n(S) = 52 3 x 4 = 40. (i) Let $E_{1}=$ Event of getting a card whose value is ...
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