Prove
Question: $\int x^{2}\left(1-\frac{1}{x^{2}}\right) d x$ Solution: $\int x^{2}\left(1-\frac{1}{x^{2}}\right) d x$ $=\int\left(x^{2}-1\right) d x$ $=\int x^{2} d x-\int 1 d x$ $=\frac{x^{3}}{3}-x+\mathrm{C}$...
Read More →In a group of 950 persons, 750 can speak Hindi and 460 can speak English.
Question: In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: (i) how many can speak both Hindi and English: (ii) how many can speak Hindi only; (iii) how many can speak English only. Solution: LetABdenote the sets of the persons who like Hindi English, respectively. Given: $n(A)=750$ $n(B)=460$ $n(A \cup B)=950$ (i) We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 950=750+460-n(A \cap B)$ $\Rightarrow n(A \cap B)=260$ Thus, 260 persons can speak both Hindi a...
Read More →In a group of 950 persons, 750 can speak Hindi and 460 can speak English.
Question: In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: (i) how many can speak both Hindi and English: (ii) how many can speak Hindi only; (iii) how many can speak English only. Solution: LetABdenote the sets of the persons who like Hindi English, respectively. Given: $n(A)=750$ $n(B)=460$ $n(A \cup B)=950$ (i) We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 950=750+460-n(A \cap B)$ $\Rightarrow n(A \cap B)=260$ Thus, 260 persons can speak both Hindi a...
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Question: $\int\left(4 e^{3 x}+1\right) d x$ Solution: $\int\left(4 e^{3 x}+1\right) d x$ $=4 \int e^{3 x} d x+\int 1 d x$ $=4\left(\frac{e^{3 x}}{3}\right)+x+\mathrm{C}$ $=\frac{4}{3} e^{3 x}+x+\mathrm{C}$...
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=x^{3}-6 x^{2}+11 x-6, g(x)=x^{2}-3 x+2$ Solution: Here, $f(x)=x^{3}-6 x^{2}+11 x-6$ $g(x)=x^{2}-3 x+2$ First we need to find the factors of $x^{2}-3 x+2$ $\Rightarrow x^{2}-2 x-x+2$ ⟹ x(x - 2) -1(x - 2) ⟹ (x - 1) and (x - 2) are the factors To prove that g(x) is the factor of f(x), The results of f(1) and f(2) should be zero Let, x 1 = 0 x = 1 substitute the value of x in f(x) $f(1)=1^{3}-6...
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Question: $\sin 2 x-4 e^{3 x}$ Solution: The anti derivative of $\left(\sin 2 x-4 e^{3 x}\right)$ is the function of $x$ whose derivative is $\left(\sin 2 x-4 e^{3 x}\right)$. It is known that, $\frac{d}{d x}\left(-\frac{1}{2} \cos 2 x-\frac{4}{3} e^{3 x}\right)=\sin 2 x-4 e^{3 x}$ Therefore, the anti derivative of $\left(\sin 2 x-4 e^{3 x}\right)$ is $\left(-\frac{1}{2} \cos 2 x-\frac{4}{3} e^{3 x}\right)$....
Read More →Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:
Question: Solve the following system of linear equation graphically and shade the region between the two lines andx-axis: (i) $2 x+3 y=12$, $x-y=1$ (ii) $3 x+2 y-4=0$, $2 x-3 y-7=0$ (iii) $3 x+2 y-11=0$ $2 x-3 y+10=0$ Solution: (i) The given equations are: $2 x+3 y=12$ ....(i) $x-y=1$....(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0+3 y=12$ $\Rightarrow y=4$ $x=0, \quad y=4$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow 2 x+3 \times 0=12$ $\Rightarrow x=6$ $x=6, \q...
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=2 x^{3}-9 x^{2}+x+13, g(x)=3-2 x$ Solution: Here, $f(x)=2 x^{3}-9 x^{2}+x+13$ g(x) = 3 - 2x To prove that g(x) is the factor of f(x), To prove that g(x) is the factor of f(x), we should show ⟹ f(3/2) = 0 here, 3 - 2x = 0 ⟹ -2x = -3 ⟹ 2x = 3 ⟹ x =3/2 Substitute the value of x in f(x) $f(3 / 2)=2(3 / 2)^{3}-9(3 / 2)^{2}+(3 / 2)+13$ =2(27/8) 9(9/4) + 3/2 + 12 =(27/4) (81/4) + 3/2 + 12 Taking L...
Read More →A survey shows that 76% of the Indians like oranges,
Question: A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas? Solution: LetABdenote the sets of the Indians who like oranges bananas, respectively. Given : $n(A)=76 \%$ $n(B)=62 \%$ $n(A \cup B)=100 \%$ $n(A \cap B)=?$ We know : $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 100=76+62-n(A \cap B)$ $\Rightarrow n(A \cap B)=38$ Therefore, $38 \%$ of the Indians like both oranges \ bananas....
Read More →A survey shows that 76% of the Indians like oranges,
Question: A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas? Solution: LetABdenote the sets of the Indians who like oranges bananas, respectively. Given : $n(A)=76 \%$ $n(B)=62 \%$ $n(A \cup B)=100 \%$ $n(A \cap B)=?$ We know : $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 100=76+62-n(A \cap B)$ $\Rightarrow n(A \cap B)=38$ Therefore, $38 \%$ of the Indians like both oranges \ bananas....
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Question: Cos 3x Solution: The anti derivative of cos 3xis a function ofxwhose derivative is cos 3x. It is known that, $\frac{d}{d x}(\sin 3 x)=3 \cos 3 x$ $\Rightarrow \cos 3 x=\frac{1}{3} \frac{d}{d x}(\sin 3 x)$ $\therefore \cos 3 x=\frac{d}{d x}\left(\frac{1}{3} \sin 3 x\right)$ Therefore, the anti derivative of $\cos 3 x$ is $\frac{1}{3} \sin 3 x$....
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=3 x^{3}+x^{2}-20 x+12, g(x)=3 x-2$ Solution: Here, $f(x)=3 x^{3}+x^{2}-20 x+12$ g(x) = 3x - 2 To prove that g(x) is the factor of f(x), we should show ⟹ f(2/3) = 0 here, 3x - 2 = 0 ⟹ 3x = 2 ⟹ x =2/3 Substitute the value of x in f(x) $f(2 / 3)=3(2 / 3)^{3}+(2 / 3)^{2}-20(2 / 3)+12$ =3(8/27) + 4/9 40/3 + 12 =8/9 + 4/9 40/3 + 12 =12/9 40/3 + 12 Taking L.C.M $=\frac{12-120+108}{9}$ $=\frac{120-...
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Question: $(a x+b)^{2}$ Solution: The anti derivative of $(a x+b)^{2}$ is the function of $x$ whose derivative is $(a x+b)^{2}$. It is known that, $\frac{d}{d x}(a x+b)^{3}=3 a(a x+b)^{2}$ $\Rightarrow(a x+b)^{2}=\frac{1}{3 a} \frac{d}{d x}(a x+b)^{3}$ $\therefore(a x+b)^{2}=\frac{d}{d x}\left(\frac{1}{3 a}(a x+b)^{3}\right)$ Therefore, the anti derivative of $(a x+b)^{2}$ is $\frac{1}{3 a}(a x+b)^{3}$....
Read More →Let A and B be two sets such that :
Question: Let $A$ and $B$ be two sets such that : $n(A)=20, n(A \cup B)=42$ and $n(A \cap B)=4$. Find (i) $n(B)$ (ii) $n(A-B)$ (iii) $n(B-A)$ Solution: Given: $n(A)=20, n(A \cup B)=42$ and $n(A \cap B)=4$ (i) We know : $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 42=20+n(B)-4$ $\Rightarrow n(B)=26$ (ii) $n(A-B)=n(A)-n(A \cap B)$ $\Rightarrow n(A-B)=20-4=16$ (iii) We know that sets follow the commutative property. $\therefore n(A \cap B)=n(B \cap A)$ $n(B-A)=n(B)-n(B \cap A)$ $\Rightarrow n(B...
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=x^{3}-6 x^{2}-19 x+84, g(x)=x-7$ Solution: Here, $f(x)=x^{3}-6 x^{2}-19 x+84$ g(x) = x - 7 To prove that g(x) is the factor of f(x), we should show ⟹ f(7) = 0 here, x - 7 = 0 ⟹ x = 7 Substitute the value of x in f(x) $f(7)=7^{3}-6(7)^{2}-19(7)+84$ = 343 - (6 * 49) - (19 * 7) + 84 = 342 - 294 - 133 + 84 = 427 - 427 = 0 Since, the result is 0 g(x) is the factor of f(x)...
Read More →Given that
Question: e2x Solution: The anti derivative of $e^{2 x}$ is the function of $x$ whose derivative is $e^{2 x}$. It is known that, $\frac{d}{d x}\left(e^{2 x}\right)=2 e^{2 x}$ $\Rightarrow e^{2 x}=\frac{1}{2} \frac{d}{d x}\left(e^{2 x}\right)$ $\therefore e^{2 x}=\frac{d}{d x}\left(\frac{1}{2} e^{2 x}\right)$ Therefore, the anti derivative of $e^{2 x}$ is $\frac{1}{2} e^{2 x}$....
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=x^{5}+3 x^{4}-x^{3}-3 x^{2}+5 x+15, g(x)=x+3$ Solution: Here, $f(x)=x^{5}+3 x^{4}-x^{3}-3 x^{2}+5 x+15$ g(x) = x + 3 To prove that g(x) is the factor of f(x), we should show ⟹ f(-3) = 0 here, x + 3 = 0 ⟹ x = -3 Substitute the value of x in f(x) $f(-3)=(-3)^{5}+3(-3)^{4}-(-3)^{3}-3(-3)^{2}+5(-3)+15$ = 243 + 243 + 27 - 27 - 15 + 15 = 0 Since, the result is 0 g(x) is the factor of f(x)...
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=3 x^{4}+17 x^{3}+9 x^{2}-7 x-10, g(x)=x+5$ Solution: Here, $f(x)=3 x^{4}+17 x^{3}+9 x^{2}-7 x-10$ g(x) = x + 5 To prove that g(x) is the factor of f(x), we should show ⟹ f(-5) = 0 here, x + 5 = 0 ⟹ x = - 5 Substitute the value of x in f(x) $f(-5)=3(-5)^{4}+17(-5)^{3}+9(-5)^{2}-7(-5)-10$ = (3 * 625) + (12 * (-125)) + (9*25) + 35 - 10 = 1875 - 2125 + 225 + 35 - 10 = 2135 - 2135 = 0 Since, the...
Read More →given value
Question: sin 2x Solution: The anti derivative of sin 2xis a function ofxwhose derivative is sin 2x. It is known that, $\frac{d}{d x}(\cos 2 x)=-2 \sin 2 x$ $\Rightarrow \sin 2 x=-\frac{1}{2} \frac{d}{d x}(\cos 2 x)$ $\therefore \sin 2 x=\frac{d}{d x}\left(-\frac{1}{2} \cos 2 x\right)$ Therefore, the anti derivative of $\sin 2 x$ is $-\frac{1}{2} \cos 2 x$...
Read More →Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not:
Question: Use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x), or not: $f(x)=x^{3}-6 x^{2}+11 x-6, g(x)=x-3$ Solution: Here, $f(x)=x^{3}-6 x^{2}+11 x-6$ g(x) = x - 3 To prove that g(x) is the factor of f(x), we should show ⟹ f(3) = 0 here, x - 3 = 0 ⟹ x = 3 Substitute the value of x in f(x) $f(3)=3^{3}-6 *(3)^{2}+11(3)-6$ = 27 - (6*9) + 33 - 6 = 27 - 54 + 33 - 6 = 60 - 60 = 0 Since, the result is 0 g(x) is the factor of f(x)...
Read More →Find the remainder when
Question: Find the remainder when $x^{3}+3 x^{3}+3 x+1$ is divided by, 1. $x+1$ 2. $x-1 / 2$ 3. $x$ 4. $x+\pi$ 5. $5+2 \mathrm{x}$ Solution: Here, $f(x)=x^{3}+3 x^{2}+3 x+1$ by remainder theorem 1 ⟹ x + 1 = 0 ⟹ x = -1 substitute the value of x in f(x) $f(-1)=(-1)^{3}+3(-1)^{2}+3(-1)+1$ = -1 + 3 - 3 + 1 = 0 2. x - 1/2 Here, $f(x)=x^{3}+3 x^{2}+3 x+1$ By remainder theorem ⟹x -1/2= 0 ⟹ x =1/2 substitute the value of x in f(x) $f(1 / 2)=(1 / 2)^{3}+3(1 / 2)^{2}+3(1 / 2)+1$ $=(1 / 2)^{3}+3(1 / 2)^{2}...
Read More →In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks.
Question: In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea? Solution: Let A denote the set of the people who like tea B denote the set of the people who like coffee. Given : $n(A \cup B)=70$ $n(A)=52$ $n(B)=37$ To find: $n(A \cap B)$ We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 70=52+37-n(A \cap B)$ $\Rightarrow n(A \cap B)=19$ Therefore, 19 people like both tea \ coffee.iven...
Read More →In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks.
Question: In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea? Solution: Let A denote the set of the people who like tea B denote the set of the people who like coffee. Given : $n(A \cup B)=70$ $n(A)=52$ $n(B)=37$ To find: $n(A \cap B)$ We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 70=52+37-n(A \cap B)$ $\Rightarrow n(A \cap B)=19$ Therefore, 19 people like both tea \ coffee.iven...
Read More →In a school there are 20 teachers who teach mathematics or physics.
Question: In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics? Solution: LetAbe the number of teachers who teach mathematics Bbe the number of teachers who teach physics. Given : $n(A)=12$ $n(A \cup B)=20$ $n(A \cap B)=4$ To find: $n(B)$ We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 20=12+n(B)-4$ $\Rightarrow n(B)=20-8=12$ Therefore, 12 teachers teach physics....
Read More →If the polynomials ax3 + 3x2 − 13
Question: If the polynomials $a x^{3}+3 x^{2}-13$ and $2 x^{3}-5 x+a$ when divided by $(x-2)$ leave the same remainder, Find the value of a Solution: Here, the polynomials are $f(x)=a x^{3}+3 x^{2}-13$ $p(x)=2 x^{3}-5 x+a$ equate, x - 2 = 0 x = 2 substitute the value of x in f(x) and p(x) $f(2)=(2)^{3}+3(2)^{2}-13$ = 8a + 12 - 13 = 8a - 1 ..... 1 $p(2)=2(2)^{3}-5(2)+a$ = 16 - 10 + a = 6 + a .... 2 f(2) = p(2) ⟹ 8a - 1 = 6 + a ⟹ 8a - a = 6 + 1 ⟹7a = 7 ⟹a = 1 The value of a = 1...
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