Factorise:
Question: Factorise:121a2 88ab+ 16b2 Solution: We have: $121 a^{2}-88 a b+16 b^{2}=(11 a)^{2}-2 \times 11 a \times 4 b+(4 b)^{2}$ $=(11 a-4 b)^{2}$ $\therefore 121 a^{2}-88 a b+16 b^{2}=(11 a-4 b)^{2}$...
Read More →Factorise:
Question: Factorise:p2 10p+ 25 Solution: We have: $p^{2}-10 p+25=p^{2}-2 \times p \times 5+(5)^{2}$ $=(p-5)^{2}$ $\therefore p^{2}-10 p+25=(p-5)^{2}$...
Read More →Refer to Q.28. What is the probability that the card is
Question: Refer to Q.28. What is the probability that the card is (i) a club (ii) 10 of hearts Solution: (i) Let $E_{3}=$ Event of getting a club $n\left(E_{3}\right)=(13-3)=10$ $\therefore$ Required probability $=\frac{n\left(E_{3}\right)}{n(S)}=\frac{10}{49}$ (ii) Let $E_{4}=$ Event of getting 10 of hearts $n\left(E_{4}\right)=1$ [because in 52 playing cards only 13 are the heart cards and only one10 in 13 heart cards] $\therefore \quad$ Required probability $=\frac{n\left(E_{4}\right)}{n(S)}=...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log (3 x+2)-x^{2} \log (2 x-1)$ Solution: Let $y=\log (3 x+2)-x^{2} \log (2 x-1)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\log (3 x+2)-x^{2} \log (2 x-1)\right]$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}[\log (3 \mathrm{x}+2)]-\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{x}^{2} \log (2 \mathrm{x}-1)\right]$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm...
Read More →The king, queen and jack of clubs are removed
Question: The king, queen and jack of clubs are removed from a deck of 52 playing cards and then well shuffled. Now, one card is drawn at fandom from the remaining cards. Determine the probability that the card is (i) a heart (ii) a king Solution: If we remove one king, one queen and one jack of clubs from 52 cards, then the remaining cards left, n(S) = 49 (I) Let $E_{1}=$ Event of getting a heart $n\left(E_{1}\right)=13$ $\therefore$$P\left(E_{1}\right)=\frac{n\left(E_{1}\right)}{n(S)}=\frac{13...
Read More →Factorise:
Question: Factorise:49a2+ 84ab+ 36b2 Solution: We have: $49 a^{2}+84 a b+36 b^{2}=(7 a)^{2}+2 \times 7 a \times 6 b+(6 b)^{2}$ $=(7 a+6 b)^{2}$ $\therefore 49 a^{2}+84 a b+36 b^{2}=(7 a+6 b)^{2}$...
Read More →A bag contains 10 red, 5 blue and 7 green balls.
Question: A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a (i) red ball (ii) green ball (iii) not a blue ball Solution: if a ball is drawn out of 22 balls (5 blue + 7 green + 10 red), then the total number of outcomes are $n(S)=22$ (i) Let $E_{1}=$ Event of getting a red ball $n\left(E_{1}\right)=10$ $\therefore \quad$ Required probability $=\frac{n\left(E_{1}\right)}{n(S)}=\frac{10}{22}=\frac{5}{11}$ (ii) Let $E_{2}=$ Event of...
Read More →Factorise:
Question: Factorise: $z^{2}+z+\frac{1}{4}$ Solution: We have: $z^{2}+z+\frac{1}{4}=z^{2}+2 \times z \times \frac{1}{2} \times\left(\frac{1}{2}\right)^{2}$ $=\left(z+\frac{1}{2}\right)^{2}$ $\therefore z^{2}+z+\frac{1}{4}=\left(z+\frac{1}{2}\right)^{2}$...
Read More →Two dice are thrown at the same time.
Question: Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2. Solution: The total number of sample space in two dice, n (S) = 6 x 6 = 36 Let $E=$ Event of getting the numbers whose difference is 2 $=\{(1,3),(2,4),(3,5),(4,6),(3,1),(4,2),(5,3),(6,4)\}$ $\therefore \quad n(E)=8$ $\therefore \quad P(E)=\frac{n(E)}{n(S)}=\frac{8}{36}=\frac{2}{9}$...
Read More →A coin is tossed 3 times.
Question: A coin is tossed 3 times. List the possible outcomes. Find the probability of getting (i) all heads (ii) atleast 2 heads Solution: The possible outcomes if a coin is tossed 3 times is S = {(HHH), (TTT), (HTT), (THT), (TEH), (THH), (HTH), (HHT)} (i) Let $E_{1}=$ Event of getting all heads $=\{(H H H)\}$ $\therefore \quad n\left(E_{1}\right)=1$ $\therefore \quad P\left(E_{1}\right)=\frac{n\left(E_{1}\right)}{n(S)}=\frac{1}{8}$ (ii) Let $E_{2}=$ Event of getting atleast 2 heads $=\{(H H T...
Read More →Factorise:
Question: Factorise:9m2+ 24m+ 16 Solution: We have: $9 m^{2}+24 m+16=(3 m)^{2}+2 \times 3 m \times 4+(4)^{2}$ $=(3 m+4)^{2}$ $\therefore 9 m^{2}+24 m+16=(3 m+4)^{2}$...
Read More →Factorise:
Question: Factorise:36a2+ 36a+ 9 Solution: We have: $36 a^{2}+36 a+9=(6 a)^{2}+2 \times 6 a \times 3+(3)^{2}$ $=(6 a+3)^{2}$ $\therefore 36 a^{2}+36 a+9=(6 a+3)^{2}$...
Read More →A coin is tossed two times.
Question: A coin is tossed two times. Find the probability of getting atmost one head. Solution: The possible outcomes,if a coin is tossed 2 times is $S=\{(H H),(T T),(H T),(T H)\}$ $\therefore$$n(\mathrm{~S})=4$ I et $F=$ Fvent of chetting atmost one head $=\{(T T),(H T),(T H)\}$ $\therefore$ $n(E)=3$ Hence, required probability $=\frac{n(E)}{n(S)}=\frac{3}{4}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $x \sin (2 x)+5^{x}+k^{k}+\left(\tan ^{2} x\right)^{3}$ Solution: Let $y=x \sin (2 x)+5^{x}+k^{k}+\left(\tan ^{2} x\right)^{3}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[x \sin 2 x+5^{x}+k^{k}+\left(\tan ^{2} x\right)^{3}\right]$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x} \sin 2 \mathrm{x})+\frac{\mathrm{d}}{\mathrm{dx}}\left(5^{\mathrm{x}...
Read More →Factorise:
Question: Factorise:4y2+ 20y+ 25 Solution: We have: $4 y^{2}+20 y+25=(2 y)^{2}+2 \times 2 y \times 5+(5)^{2}$ $=(2 y+5)^{2}$ $\therefore 4 y^{2}+20 y+25=(2 y+5)^{2}$...
Read More →Two dice are numbered 1, 2, 3, 4, 5, 6
Question: Two dice are numbered 1, 2, 3, 4, 5, 6 and 1, 1, 2, 2, 3, 3, respectively. They are thrown and the sum of the numbers on them is noted. Find the probability of getting each sum from 2 to 9, separately. Solution: Number of total outcomes = 36 (i) Let $E_{1}=$ Event of getting sum $2=\{(1,1),(1,1)\}$ $n\left(E_{1}\right)=2$ $\therefore$$P\left(E_{1}\right)=\frac{n\left(E_{1}\right)}{n(S)}=\frac{2}{36}=\frac{1}{18}$ (ii) Let $E_{2}=$ Event of getting sum $3=\{(1,2),(1,2),(2,1),(2,1)\}$ $n...
Read More →Factorise:
Question: Factorise:x2+ 6ax+ 9a2 Solution: We have: $x^{2}+6 a x+9 a^{2}=x^{2}+2 \times x \times 3 a+(3 a)^{2}$ $=(x+3 a)^{2}$ $\therefore x^{2}+6 a x+9 a^{2}=(x+3 a)^{2}$...
Read More →Factorise:
Question: Factorise:9 + 6z+z2 Solution: We have: $9+6 z+z^{2}=z^{2}+6 z+9$ $=z^{2}+2 \times x \times 3+(3)^{2}$ $=(z+3)^{2}$ $\therefore 9+6 z+z^{2}=(z+3)^{2}$...
Read More →Factorise:
Question: Factorise:1 + 2x+x2 Solution: We have: $1+2 x+x^{2}=x^{2}+2 x+1$ $=x^{2}+2 \times x \times 1+(1)^{2}$ $=(x+1)^{2}$ $\therefore 1+2 x+x^{2}=(x+1)^{2}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{2^{x} \cos x}{\left(x^{2}+3\right)^{2}}$ Solution: Let $y=\frac{2^{x} \cos x}{\left(x^{2}+3\right)^{2}}$ On differentiating y with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\frac{2^{x} \cos x}{\left(x^{2}+3\right)^{2}}\right]$ Recall that $\left(\frac{\mathrm{u}}{\mathrm{v}}\right)^{\prime}=\frac{\mathrm{vu}^{\prime}-\mathrm{uv}^{\prime}}{\mathrm{v}^{2}}$ (quotient rule) $\Rightarrow \frac{d y}{d x}=\fr...
Read More →Two dice are thrown at the same time
Question: Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is less than 9. Solution: Number of total outcomes = 36 When product of numbers appearing on them is less than 9, then possible ways are (1,6), (1,5) (1,4), (1,3), (1,2), (1,1), (2, 2), (2, 3), (2, 4), (3, 2), (4, 2), (4,1), (3,1), (5,1), (6,1) and (2,1). Number of possible ways = 16 Required probability $=\frac{16}{36}=\frac{4}{9}$...
Read More →Factorise:
Question: Factorise:x2+ 14x+ 49 Solution: We have: $x^{2}+14 x+49=x^{2}+2 \times x \times 7+(7)^{2}$ $=(x+7)^{2}$ $\therefore x^{2}+14 x+49=(x+7)^{2}$...
Read More →Two dice are thrown together.
Question: Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is (i) 6 (ii) 12 (iii) 7 Solution: Number of total outcomes = 36 (i) When product of the numbers on the top of the dice is $6 .$ So, the possible ways are $(1,6),(2,3),(3,2)$ and $(6,1)$. Number of possible ways $=4$ $\therefore \quad$ Required probability $=\frac{4}{36}=\frac{1}{9}$ (ii) When product of the numbers on the top of the dice is $12 .$ So, the possible ways are $(2,6),...
Read More →Factorise:
Question: Factorise:x2+ 8x+ 16 Solution: We have: $x^{2}+8 x+16=x^{2}+2 \times x \times 4+(4)^{2}$ $=(x+4)^{2}$ $\therefore x^{2}+8 x+16=(x+4)^{2}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log \left(\tan ^{-1} x\right)$ Solution: Let $y=\log \left(\tan ^{-1} x\right)$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left(\tan ^{-1} \mathrm{x}\right)\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=\frac{1}{\tan ^{-1} x} \frac{d}{d x}\left(\tan ^{-1} x\right)$ [using ch...
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