A card is drawn from a deck of 52 cards.
Question: A card is drawn from a deck of 52 cards. The event E is that card is not an ace of hearts. The number of outcomes favourable to E is (a) 4 (b) 13 (c) 48 (d) 51 Solution: (d) In a deck of 52 cards, there are 13 cards of heart and 1 is ace of heart. Hence, the number of outcomes favourable to $E=51$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\sqrt{\cot x}}$ Solution: Let $\mathrm{y}=\mathrm{e}^{\sqrt{\cot \mathrm{x}}}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sqrt{\cot \mathrm{x}}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\sqrt{\cot x}} \frac{\mat...
Read More →Factorise: 16(2p − 3q)2 − 4(2p − 3q)
Question: Factorise:16(2p 3q)2 4(2p 3q) Solution: We have: $16(2 p-3 q)^{2}-4(2 p-3 q)=(2 p-3 q)\{16(2 p-3 q)-4\}$ $=(2 p-3 q)(32 p-48 q-4)$ $\therefore 16(2 p-3 q)^{2}-4(2 p-3 q)=(2 p-3 q)(32 p-48 q-4)$...
Read More →When a die is thrown,
Question: When a die is thrown, the probability of getting an odd number less than 3 is , (a) $\frac{1}{6}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) 0 Solution: (a) When a die-is thrown, then total number of outcomes $=6$ Odd number less than 3 is 1 only. Number of possible outcomes $=1$ Required probability $=\frac{1}{6}$...
Read More →Factorise:
Question: Factorise:2a+ 6b 3(a+ 3b)2 Solution: We have: $2 a+6 b-3(a+3 b)^{2}=2(a+3 b)-3(a+3 b)^{2}$ $=(a+3 b)\{2-3(a+3 b)\}$ $=(a+3 b)(2-3 a-9 b)$ $\therefore 2 a+6 b-3(a+3 b)^{2}=(a+3 b)(2-3 a-9 b)$...
Read More →The probability that a non-leap your selected
Question: The probability that a non-leap your selected at random will contains 53 Sunday is (a) $\frac{1}{7}$ (b) $\frac{2}{7}$ (c) $\frac{3}{7}$ (d) $\frac{5}{7}$ Solution: (a) A non-leap year has 365 days and therefore 52 weeks and 1 day. This 1 day may be Sunday or Monday or Tuesday or Wednesday or Thursday or Friday or Saturday. Thus, out of 7 possibilities, 1 favourable event is the event that the one day is Sunday....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\sqrt{\cot x}}$ Solution: Let $\mathrm{y}=\mathrm{e}^{\sqrt{\cot \mathrm{x}}}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\sqrt{\cot \mathrm{x}}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\sqrt{\cot x}} \frac{\mat...
Read More →If a card is selected from a deck of 52 cards,
Question: If a card is selected from a deck of 52 cards, then the probability of its being a red face card is (a) $\frac{3}{26}$ (b) $\frac{3}{13}$ (C) $\frac{2}{13}$ (d) $\frac{1}{2}$ Solution: (c) In a deck of 52 cards, there are 12 face cards i.e. 6 red and 6 black cards. So, probability of getting a red face $c a r d=\frac{6}{52}=\frac{3}{26}$...
Read More →Factorise:
Question: Factorise:3(a 2b)2 5(a 2b) Solution: $3(a-2 b)^{2}-5(a-2 b)=(a-2 b)\{3(a-2 b)-5\}$ $=(a-2 b)(3 a-6 b-5)$ $\therefore 3(a-2 b)^{2}-5(a-2 b)=(a-2 b)(3 a-6 b-5)$...
Read More →If P (A) denotes the probability of an event A,
Question: If P (A) denotes the probability of an event A, then (a) P(A) 0 (b) P(A) 1 (c) 0 P(A) 1 (d) -1 P(A) 1 Solution: (c)Since, probability of an event always lies between 0 and 1....
Read More →The probability expressed as a percentage
Question: The probability expressed as a percentage of a particular occurrence can never be (a) less than 100 (b) less than 0 (c) greater than 1 (d) anything but a whole number Solution: (b)We know that, the probability expressed as a percentage always lie between 0 and 100. So, it cannot be less than 0....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\tan 3 x}$ Solution: Let $y=e^{\tan 3 x}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\tan 3 x}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=e^{\tan 3 x} \frac{d}{d x}(\tan 3 x)$ [using chain rule] We have $\frac{d}{d x}(\tan x)=\sec ^{2} x$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin (\log \sin x)$ Solution: Let $y=\sin (\log \sin x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}[\sin (\log (\sin x))]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\sin \mathrm{x})=\cos \mathrm{x}$ $\Rightarrow \frac{d y}{d x}=\cos (\log (\sin x)) \frac{d}{d x}[\log (\sin x)]$ [using chain rule] We have $\frac{d}{d x}(\log x)=\frac{1}{x}$ $\Rightarrow \frac{d y}{d x}=\cos (\log (\sin x))\...
Read More →If the probability of an event is P,
Question: If the probability of an event is P, then the probability of its completmentry event will be (a) P-1 (b) $P$ (c) $1-P$ (d) $1-\frac{1}{P}$ Solution: (c)Since, probability of an event + probability of its complementry event = 1 So, probability of its complementry event = 1 Probability of an event = 1 P...
Read More →Factorise:
Question: Factorise:(x+ 5)2 4(x+ 5) Solution: We have: $(x+5)^{2}-4(x+5)=(x+5)\{(x+5)-4\}$ $=(x+5)(x+5-4)$ $=(x+5)(x+1)$ $\therefore(x+5)^{2}-4(x+5)=(x+5)(x+1)$...
Read More →An event is very unlikely to happen.
Question: An event is very unlikely to happen. Its probability is closest to (a) 0.0001 (b) 0.001 (c) 0.01 (d) 0.1 Solution: (a)The probability of an event which is very unlikely to happen is closest to zero and from the given options 0.0001 is closest to zero....
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{3 x} \cos (2 x)$ Solution: Let $y=e^{3 x} \cos (2 x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{3 x} \cos 2 x\right)$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}\left(e^{3 x} \times \cos 2 x\right)$ Recall that (uv) $^{\prime}=v u^{\prime}+u v^{\prime}$ (product rule) $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\cos 2 \mathrm{x} \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^...
Read More →Which of the following cannot be
Question: Which of the following cannot be the probability of an event? (a) $\frac{1}{2}$ (b) $0.1$ (c) 3 (d) $\frac{17}{16}$ Solution: (d)Since, probability of an event always lies between 0 and 1....
Read More →Factorise:
Question: Factorise:9a(3a 5b) 12a2(3a 5b) Solution: We have: $9 a(3 a-5 b)-12 a^{2}(3 a-5 b)=(3 a-5 b)\left(9 a-12 a^{2}\right)=3 a(3 a-5 b)(3-4 a)$...
Read More →Factorise:
Question: Factorise:x3(2ab) +x2(2ab) Solution: We have: $x^{3}(2 a-b)+x^{2}(2 a-b)=(2 a-b)\left(x^{3}+x^{2}\right)=x^{2}(x+1)(2 a-b)$...
Read More →If an event cannot occur,
Question: If an event cannot occur,then its probability is (a) 1 (b) $\frac{3}{4}$ (c) $\frac{1}{2}$ (d) 0 Solution: (d)The event which cannot occur is said to be impossible event and probability of impossible event is zero....
Read More →Factorise: 6a(a − 2b) + 5b(a − 2b)
Question: Factorise:6a(a 2b) + 5b(a 2b) Solution: We have: $6 a(a-2 b)+5 b(a-2 b)=(a-2 b)(6 a+5 b)$...
Read More →Consider the following distribution
Question: Consider the following distribution the frequency of the class 30-40 is (a) 3 (b) 4 (c) 3 (d) 4 Solution: Hence,frequency in the class interval 30-40 is 3...
Read More →Factorise:
Question: Factorise: (i)x(x+ 3) + 5(x+ 3) (ii) 5x(x 4) 7(x 4) (iii) 2m(1 n) + 3(1 n) Solution: (i) $x(x+3)+5(x+3)=(x+3)(x+5)$ (ii) $5 x(x-4)-7(x-4)=(x-4)(5 x-7)$ (iii) $2 m(1-n)+3(1-n)=(1-n)(2 m+3)$...
Read More →Factorise:
Question: Factorise: (i)x(x+ 3) + 5(x+ 3) (ii) 5x(x 4) 7(x 4) (iii) 2m(1 n) + 3(1 n) Solution: (i) $x(x+3)+5(x+3)=(x+3)(x+5)$ (ii) $5 x(x-4)-7(x-4)=(x-4)(5 x-7)$ (iii) $2 m(1-n)+3(1-n)=(1-n)(2 m+3)$...
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