For any set A, (A')' is equal to
Question: For any setA, (A')' is equal to (a) $A^{\prime}$ (b) $A$ (c) $\phi$ (d) none of these. Solution: (b)AThe complement of the complement of a set is the set itself....
Read More →Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.
Question: Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis ofxin each system. (i) $2 x+y=6$ $x-2 y=-2$ (ii) $2 x-y=2$ $4 x-y=8$ (iii) $x+2 y=5$ $2 x-3 y=-4$ (iv) $2 x+3 y=8$ $x-2 y=-3$ Solution: (i) The given equations are $2 x+y=6$...(i) $x-2 y=-2$....(ii) The two points satisfying (i) can be listed in a table as, It is seen that the solution of the given system of equations is given byx= 2,y= 2. Also, it...
Read More →For any set A, (A')' is equal to
Question: For any setA, (A')' is equal to (a) $A^{\prime}$ (b) $A$ (c) $\phi$ (d) none of these. Solution: (b)AThe complement of the complement of a set is the set itself....
Read More →In a survey it was found that 21 persons liked product
Question: In a survey it was found that 21 persons liked productP1, 26 liked productP2and 29 liked productP3. If 14 persons liked productsP1andP2; 12 persons liked productP3andP1; 14 persons liked productsP2andP3and 8 liked all the three products. Find how many liked productP3only. Solution: Let $P_{1}, P_{2}$ and $P_{3}$ denote the sets of persons liking products $P_{1}, P_{2}$ and $P_{3}$, respectively. Also, let $U$ be the universal set. Thus, we have: $n\left(P_{1}\right)=21, n\left(P_{2}\ri...
Read More →If x3 + ax2 − bx + 10 is divisible by x3
Question: If $x^{3}+a x^{2}-b x+10$ is divisible by $x^{3}-3 x+2$, find the values of $a$ and $b$ Solution: Here, $f(x)=x^{3}+a x^{2}-b x+10$ $g(x)=x^{3}-3 x+2$ first, we need to find the factors of g(x) $g(x)=x^{3}-3 x+2$ $=x^{3}-2 x-x+2$ = x(x - 2) -1(x - 2) = (x - 1) and (x - 2) are the factors From factor theorem, if x = 1, 2 are the factors of f(x) then f(1) = 0 and f(2) = 0 Let, us take x - 1 ⟹ x - 1 = 0 ⟹ x = 1 Substitute the value of x in f(x) $f(1)=1^{3}+a(1)^{2}-b(1)+10$ = 1 + a - b + ...
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Question: $\frac{2 x}{1+x^{2}}$ Solution: Let $1+x^{2}=t$ $\therefore 2 x d x=d t$ $\Rightarrow \int \frac{2 x}{1+x^{2}} d x=\int \frac{1}{t} d t$ $=\log |t|+\mathrm{C}$ $=\log \left|1+x^{2}\right|+\mathrm{C}$ $=\log \left(1+x^{2}\right)+\mathrm{C}$...
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Question: If $\frac{d}{d x} f(x)=4 x^{3}-\frac{3}{x^{4}}$ such that $f(2)=0$, then $f(x)$ is (A) $x^{4}+\frac{1}{x^{3}}-\frac{129}{8}$ (B) $x^{3}+\frac{1}{x^{4}}+\frac{129}{8}$ (C) $x^{4}+\frac{1}{x^{3}}+\frac{129}{8}$ (D) $x^{3}+\frac{1}{x^{4}}-\frac{129}{8}$ Solution: It is given that, $\frac{d}{d x} f(x)=4 x^{3}-\frac{3}{x^{4}}$ $\therefore$ Anti derivative of $4 x^{3}-\frac{3}{x^{4}}=f(x)$ $\therefore f(x)=\int 4 x^{3}-\frac{3}{x^{4}} d x$ $f(x)=4 \int x^{3} d x-3 \int\left(x^{-4}\right) d x...
Read More →In a survey of 100 students, the number of students studying the various languages were found to be :
Question: In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find: (i) How many students were studying Hindi? (ii) How many students were studying English and Hindi? Solution: Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively. Also, let U be the universal set. Now, we have: $n(...
Read More →In a survey of 100 students, the number of students studying the various languages were found to be :
Question: In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find: (i) How many students were studying Hindi? (ii) How many students were studying English and Hindi? Solution: Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively. Also, let U be the universal set. Now, we have: $n(...
Read More →Find the values of a and b so that (x + 1) and (x - 1) are the factors of
Question: Find the values of $a$ and $b$ so that $(x+1)$ and $(x-1)$ are the factors of $x^{4}+a x^{3}-3 x^{2}+2 x+b$ Solution: Here, $f(x)=x^{4}+a x^{3}-3 x^{2}+2 x+b$ The factors are (x + 1) and (x - 1) From factor theorem, if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0 Let, us take x + 1 ⟹ x + 1 = 0 ⟹ x = -1 Substitute value of x in f(x) $f(-1)=(-1)^{4}+a(-1)^{3}-3(-1)^{2}+2(-1)+b$ = 1 - a - 3 - 2 + b = -a + b - 4 ... 1 Let, us take x - 1 ⟹ x - 1 = 0 ⟹ x = 1 Substitute value...
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Question: The anti derivative of $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$ equals (A) $\frac{1}{3} x^{\frac{1}{3}}+2 x^{\frac{1}{2}}+\mathrm{C}$ (B) $\frac{2}{3} x^{\frac{2}{3}}+\frac{1}{2} x^{2}+\mathrm{C}$ (C) $\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+\mathrm{C}$ (D) $\frac{3}{2} x^{\frac{3}{2}}+\frac{1}{2} x^{\frac{1}{2}}+\mathrm{C}$ Solution: $\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x$ $=\int x^{\frac{1}{2}} d x+\int x^{-\frac{1}{2}} d x$ $=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{...
Read More →Solve the following system of linear equations graphically :
Question: Solve the following system of linear equations graphically : 3x+y 11 = 0,xy 1 = 0.Shade the region bounded by these lines andy-axis. Also, find the area of the region bounded by the these lines andy-axis. Solution: The given equations are $3 x+y-11=0$$\ldots \ldots($ i $)$ $x-y-1=0$$\ldots \ldots($ ii $)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 3 \times 0+y=11$ $\Rightarrow y=11$ $x=0, \quad y=11$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 3 x+0=11$ $\Rightarro...
Read More →Find the values of p and q so that x4
Question: Find the values of $p$ and $q$ so that $x^{4}+p x^{3}+2 x^{2}-3 x+q$ is divisible by $\left(x^{2}-1\right)$ Solution: Here, $f(x)=x^{4}+p x^{3}+2 x^{2}-3 x+q$ $g(x)=x^{2}-1$ first, we need to find the factors of $x^{2}-1$ $\Rightarrow x^{2}-1=0$ $\Rightarrow x^{2}=1$ $\Rightarrow x=\pm 1$ $\Rightarrow(x+1)$ and $(x-1)$ From factor theorem, if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0 Let us take, x + 1 ⟹ x + 1 = 0 ⟹ x = -1 Substitute the value of x in f(x) $f(-1)=(-...
Read More →In a survey of 100 persons it was found that 28 read magazine A,
Question: In a survey of 100 persons it was found that 28 read magazineA, 30 read magazineB, 42 read magazineC, 8 read magazinesAandB, 10 read magazinesAandC, 5 read magazinesBandCand 3 read all the three magazines. Find: (i) How many read none of three magazines? (ii) How many read magazineConly? Solution: Let A, B C be the sets of the persons who read magazines A, B and C, respectively. Also, let U denote the universal set. We have:n(U) = 100 $n(\mathrm{~A})=28, n(\mathrm{~B})=30, n(\mathrm{C}...
Read More →In a survey of 100 persons it was found that 28 read magazine A,
Question: In a survey of 100 persons it was found that 28 read magazineA, 30 read magazineB, 42 read magazineC, 8 read magazinesAandB, 10 read magazinesAandC, 5 read magazinesBandCand 3 read all the three magazines. Find: (i) How many read none of three magazines? (ii) How many read magazineConly? Solution: Let A, B C be the sets of the persons who read magazines A, B and C, respectively. Also, let U denote the universal set. We have:n(U) = 100 $n(\mathrm{~A})=28, n(\mathrm{~B})=30, n(\mathrm{C}...
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Question: $\int \frac{2-3 \sin x}{\cos ^{2} x} d x$ Solution: $\int \frac{2-3 \sin x}{\cos ^{2} x} d x$ $=\int\left(\frac{2}{\cos ^{2} x}-\frac{3 \sin x}{\cos ^{2} x}\right) d x$ $=\int 2 \sec ^{2} x d x-3 \int \tan x \sec x d x$ $=2 \tan x-3 \sec x+\mathrm{C}$...
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Question: $\int \frac{\sec ^{2} x}{\operatorname{cosec}^{2} x} d x$ Solution: $\int \frac{\sec ^{2} x}{\operatorname{cosec}^{2} x} d x$ $=\int \frac{1}{\frac{\cos ^{2} x}{1} d x}$ $=\int \frac{\sin ^{2} x}{\cos ^{2} x} d x$ $=\int \tan ^{2} x d x$ $=\int\left(\sec ^{2} x-1\right) d x$ $=\int \sec ^{2} x d x-\int 1 d x$ $=\tan x-x+C$...
Read More →Find α, β if (x + 1) and (x + 2) are the factors
Question: Find $\alpha, \beta$ if $(x+1)$ and $(x+2)$ are the factors of $x^{3}+3 x^{2}-2 a x+\beta$ Solution: Given, $f(x)=x^{3}+3 x^{2}-2 \alpha x+\beta$ and the factors are $(x+1)$ and $(x+2)$ From factor theorem, if they are the factors of f(x) then results of f(-2) and f(-1) should be zero Let, x + 1 = 0 ⟹ x = -1 Substitute value of x in f(x) $f(-1)=(-1)^{3}+3(-1)^{2}-2 \alpha(-1)+\beta$ = 1 + 3 + 2 + = 2 + + 2 ... 1 Let, x + 2 = 0 ⟹ x = -2 Substitute value of x in f(x) $f(-2)=(-2)^{3}+3(-2...
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Question: $\int \sec x(\sec x+\tan x) d x$ Solution: $\int \sec x(\sec x+\tan x) d x$ $=\int\left(\sec ^{2} x+\sec x \tan x\right) d x$ $=\int \sec ^{2} x d x+\int \sec x \tan x d x$ $=\tan x+\sec x+\mathrm{C}$...
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Question: $\int\left(2 x^{2}-3 \sin x+5 \sqrt{x}\right) d x$ Solution: $\int\left(2 x^{2}-3 \sin x+5 \sqrt{x}\right) d x$ $=2 \int x^{2} d x-3 \int \sin x d x+5 \int x^{\frac{1}{2}} d x$ $=\frac{2 x^{3}}{3}-3(-\cos x)+5\left(\frac{x^{\frac{3}{2}}}{3}\right)+\mathrm{C}$ $=\frac{2}{3} x^{3}+3 \cos x+\frac{10}{3} x^{\frac{3}{2}}+\mathrm{C}$...
Read More →A survey of 500 television viewers produced the following information;
Question: $\cap$ A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games? Solution: Let F, H B denote the sets of students who watch football, hockey and basketball, respectively. Also, let U be the universa...
Read More →A survey of 500 television viewers produced the following information;
Question: $\cap$ A survey of 500 television viewers produced the following information; 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games? Solution: Let F, H B denote the sets of students who watch football, hockey and basketball, respectively. Also, let U be the universa...
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Question: $\int\left(2 x-3 \cos x+e^{x}\right) d x$ Solution: $\int\left(2 x-3 \cos x+e^{x}\right) d x$ $=2 \int x d x-3 \int \cos x d x+\int e^{x} d x$ $=\frac{2 x^{2}}{2}-3(\sin x)+e^{x}+\mathrm{C}$ $=x^{2}-3 \sin x+e^{x}+\mathrm{C}$...
Read More →Find the values of a and b, if x2 - 4
Question: Find the values of $a$ and $b$, if $x^{2}-4$ is a factor of $a x^{4}+2 x^{3}-3 x^{2}+b x-4$ Solution: Given, $f(x)=a x^{4}+2 x^{3}-3 x^{2}+b x-4$ $g(x)=x^{2}-4$ first we need to find the factors of g(x) $\Rightarrow x^{2}-4$ $\Rightarrow x^{2}=4$ $\Rightarrow x=\sqrt{4}$ $\Rightarrow x=\pm 2$ (x - 2) and (x + 2) are the factors By factor therorem if (x - 2) and (x + 2) are the factors of f(x) the result of f(2) and f(-2) should be zero Let, x - 2 = 0 ⟹ x = 2 Substitute the value of x i...
Read More →Solve graphically the system of linear equations:
Question: Solve graphically the system of linear equations: $4 x-3 y+4=0$ $4 x+3 y-20=0$ Find the area bounded by these lines andx-axis. Solution: The given equations are $4 x-3 y+4=0$$. .(i)$ $4 x+3 y-20=0$$. .(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 4 \times 0-3 y=-4$ $\Rightarrow y=4 / 3$ $x=0, \quad y=4 / 3$ Putting $y=0$ in equation $(i,$, we get: $\Rightarrow 4 x-3 \times 0=-4$ $\Rightarrow x=-1$ $x=-1, \quad y=0$ Use the following table to draw the graph. The graph of ...
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