If A and B are two sets such that n
Question: If $A$ and $B$ are two sets such that $n(A)=70, n(B)=60, n(A \cup B)=110$, then $n(A \cap B)$ is equal to (a) 240 (b) 50 (c) 40 (d) 20 Solution: (d) 20 We have: $n(\mathrm{~A} \cap \mathrm{B})=n(\mathrm{~A})+n(\mathrm{~B})-n(\mathrm{~A} \cup \mathrm{B})$ $=70+60-110$ $=20$...
Read More →If A and B are two sets such that n
Question: If $A$ and $B$ are two sets such that $n(A)=70, n(B)=60, n(A \cup B)=110$, then $n(A \cap B)$ is equal to (a) 240 (b) 50 (c) 40 (d) 20 Solution: (d) 20 We have: $n(\mathrm{~A} \cap \mathrm{B})=n(\mathrm{~A})+n(\mathrm{~B})-n(\mathrm{~A} \cup \mathrm{B})$ $=70+60-110$ $=20$...
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Question: For two sets $A \cup B=A$ iff (a) $B \subseteq A$ (b) $A \subseteq B$ (c) $A \neq B$ (d) $A=B$ Solution: (a) $B \subseteq A$ The union of two sets is a set of all those elements that belong to A or to B or to both A and B. If $A \cup B=A$, then $B \subseteq A$...
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Question: For two sets $A \cup B=A$ iff (a) $B \subseteq A$ (b) $A \subseteq B$ (c) $A \neq B$ (d) $A=B$ Solution: (a) $B \subseteq A$ The union of two sets is a set of all those elements that belong to A or to B or to both A and B. If $A \cup B=A$, then $B \subseteq A$...
Read More →If A and B are two disjoint sets,
Question: $\cap$ If $A$ and $B$ are two disjoint sets, then $n(A \cup B)$ is equal to (a) $n(A)+n(B)$ (b) $n(A)+n(B)-n(A \cap B)$ (c) $n(A)+n(B)+n(A \cap B)$ (d) $n(A) n(B)$ (e) $n(A)-n(B)$ Solution: (a) $n(A)+n(B)$ Two sets are disjoint if they do not have a common element in them, i.e., $A \cap B=\emptyset$. $\therefore n(\mathrm{~A} \cup \mathrm{B})=n(\mathrm{~A})+n(\mathrm{~B})$...
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Question: $\frac{\sin x}{(1+\cos x)^{2}}$ Solution: Let $1+\cos x=t$ $\therefore-\sin x d x=d t$ $\Rightarrow \int \frac{\sin x}{(1+\cos x)^{2}} d x=\int-\frac{d t}{t^{2}}$ $=-\int t^{-2} d t$ $=\frac{1}{t}+\mathrm{C}$ $=\frac{1}{1+\cos x}+\mathrm{C}$...
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Question: $\frac{\sin x}{1+\cos x}$ Solution: Let $1+\cos x=t$ $\therefore-\sin x d x=d t$ $\Rightarrow \int \frac{\sin x}{1+\cos x} d x=\int-\frac{d t}{t}$ $=-\log |t|+\mathrm{C}$ $=-\log |1+\cos x|+\mathrm{C}$...
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Question: $\cot x \log \sin x$ Solution: Let $\log \sin x=t$ $\Rightarrow \frac{1}{\sin x} \cdot \cos x d x=d t$ $\therefore \cot x d x=d t$ $\Rightarrow \int \cot x \log \sin x d x=\int t d t$ $=\frac{t^{2}}{2}+C$ $=\frac{1}{2}(\log \sin x)^{2}+C$...
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Question: $\frac{\cos x}{\sqrt{1+\sin x}}$ Solution: Let $1+\sin x=t$ $\therefore \cos x d x=d t$ $\Rightarrow \int \frac{\cos x}{\sqrt{1+\sin x}} d x=\int \frac{d t}{\sqrt{t}}$ $=\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+\mathrm{C}$ $=2 \sqrt{t}+\mathrm{C}$ $=2 \sqrt{1+\sin x}+\mathrm{C}$...
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Question: $\sqrt{\sin 2 x} \cos 2 x$ Solution: Let $\sin 2 x=t$ $\therefore 2 \cos 2 x d x=d t$ $\Rightarrow \int \sqrt{\sin 2 x} \cos 2 x d x=\frac{1}{2} \int \sqrt{t} d t$ $=\frac{1}{2}\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+\mathrm{C}$ $=\frac{1}{3} t^{\frac{3}{2}}+\mathrm{C}$ $=\frac{1}{3}(\sin 2 x)^{\frac{3}{2}}+\mathrm{C}$...
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Question: $\frac{\cos \sqrt{x}}{\sqrt{x}}$ Solution: Let $\sqrt{x}=t$ $\therefore \frac{1}{2 \sqrt{x}} d x=d t$ $\Rightarrow \int \frac{\cos \sqrt{x}}{\sqrt{x}} d x=2 \int \cos t d t$ $=2 \sin t+\mathrm{C}$ $=2 \sin \sqrt{x}+\mathrm{C}$...
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Question: $\frac{1}{\cos ^{2} x(1-\tan x)^{2}}$ Solution: $\frac{1}{\cos ^{2} x(1-\tan x)^{2}}=\frac{\sec ^{2} x}{(1-\tan x)^{2}}$ Let $(1-\tan x)=t$ $\therefore-\sec ^{2} x d x=d t$ $\Rightarrow \int \frac{\sec ^{2} x}{(1-\tan x)^{2}} d x=\int \frac{-d t}{t^{2}}$ $=-\int t^{-2} d t$ $=+\frac{1}{t}+\mathrm{C}$ $=\frac{1}{(1-\tan x)}+\mathrm{C}$...
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Question: $\frac{2 \cos x-3 \sin x}{6 \cos x+4 \sin x}$ Solution: $\frac{2 \cos x-3 \sin x}{6 \cos x+4 \sin x}=\frac{2 \cos x-3 \sin x}{2(3 \cos x+2 \sin x)}$ Let $3 \cos x+2 \sin x=t$ $\therefore(-3 \sin x+2 \cos x) d x=d t$ $\int \frac{2 \cos x-3 \sin x}{6 \cos x+4 \sin x} d x=\int \frac{d t}{2 t}$ $=\frac{1}{2} \int \frac{1}{t} d t$ $=\frac{1}{2} \log |t|+\mathrm{C}$ $=\frac{1}{2} \log |2 \sin x+3 \cos x|+\mathrm{C}$...
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Question: $\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}$ Solution: Let $\sin ^{-1} x=t$ $\therefore \frac{1}{\sqrt{1-x^{2}}} d x=d t$ $\Rightarrow \int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x=\int t d t$ $=\frac{t^{2}}{2}+\mathrm{C}$ $=\frac{\left(\sin ^{-1} x\right)^{2}}{2}+\mathrm{C}$...
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Question: $\sec ^{2}(7-4 x)$ Solution: Let $7-4 x=t$ $\therefore-4 d x=d t$ $\therefore \int \sec ^{2}(7-4 x) d x=\frac{-1}{4} \int \sec ^{2} t d t$ $=\frac{-1}{4}(\tan t)+\mathrm{C}$ $=\frac{-1}{4} \tan (7-4 x)+\mathrm{C}$...
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Question: $\tan ^{2}(2 x-3)$ Solution: $\tan ^{2}(2 x-3)=\sec ^{2}(2 x-3)-1$ Let $2 x-3=t$ $\therefore 2 d x=d t$ $\Rightarrow \int \tan ^{2}(2 x-3) d x$ $=\int\left[\sec ^{2}(2 x-3)-1\right] d x$ $=\int\left[\sec ^{2} t-1\right] \frac{d t}{2}$ $=\frac{1}{2}\left[\int \sec ^{2} t d t-\int 1 d t\right]$ $=\frac{1}{2}[\tan t-t+\mathrm{C}]$ $=\frac{1}{2}[\tan (2 x-3)-(2 x-3)+\mathrm{C}]$...
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Question: $\frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}}$ Solution: Let $e^{2 x}+e^{-2 x}=t$ $\therefore\left(2 e^{2 x}-2 e^{-2 x}\right) d x=d t$ $\Rightarrow 2\left(e^{2 x}-e^{-2 x}\right) d x=d t$ $\Rightarrow \int\left(\frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}}\right) d x=\int \frac{d t}{2 t}$ $=\frac{1}{2} \int \frac{1}{t} d t$ $=\frac{1}{2} \log |t|+\mathrm{C}$ $=\frac{1}{2} \log \left|e^{2 x}+e^{-2 x}\right|+\mathrm{C}$...
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Question: $\frac{e^{2 x}-1}{e^{2 x}+1}$ Solution: $\frac{e^{2 x}-1}{e^{2 x}+1}$ Dividing numerator and denominator byex, we obtain $\frac{\frac{\left(e^{2 x}-1\right)}{e^{x}}}{\frac{\left(e^{2 x}+1\right)}{e^{x}}}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$ Let $e^{x}+e^{-x}=t$ $\therefore\left(e^{x}-e^{-x}\right) d x=d t$ $\Rightarrow \int \frac{e^{2 x}-1}{e^{2 x}+1} d x=\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$ $=\int \frac{d t}{t}$ $=\log |t|+\mathrm{C}$ $=\log \left|e^{x}+e^{-x}\right|+\mathrm{C}$...
Read More →In set-builder method the null set is represented by
Question: In set-builder method the null set is represented by (a) { } (b) Φ (c) $|x: x \neq x|$ (d) $|x: x=x|$ Solution: (c) $\{x: x \neq x\}$...
Read More →In set-builder method the null set is represented by
Question: In set-builder method the null set is represented by (a) { } (b) Φ (c) $|x: x \neq x|$ (d) $|x: x=x|$ Solution: (c) $\{x: x \neq x\}$...
Read More →Form the pair of linear equations in the following problems, and find their solution graphically:
Question: Form the pair of linear equations in the following problems, and find their solution graphically:(i) 10 students of class X took part in Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and a pen.(iii) Champa went to a 'sale' to purchase some pants and skirts. When her friends ...
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Question: $\frac{e^{\tan ^{-1} x}}{1+x^{2}}$ Solution: Let $\tan ^{-1} x=t$ $\therefore \frac{1}{1+x^{2}} d x=d t$ $\Rightarrow \int \frac{e^{\tan ^{-1} x}}{1+x^{2}} d x=\int e^{t} d t$ $=e^{\prime}+\mathrm{C}$ $=e^{\tan ^{-1} x}+\mathrm{C}$...
Read More →If A = |1, 2, 3, 4, 5|,
Question: IfA= |1, 2, 3, 4, 5|, then the number of proper subsets ofAis (a) 120 (b) 30 (c) 31 (d) 32 Solution: (c) 31 The number of proper subsets of any set is given by the formula $2^{n}-1$, where $n$ is the number of elements in the set. Here, n= 5 $\therefore$ Number of proper subsets of $A=2^{5}-1=31$...
Read More →If A = |1, 2, 3, 4, 5|,
Question: IfA= |1, 2, 3, 4, 5|, then the number of proper subsets ofAis (a) 120 (b) 30 (c) 31 (d) 32 Solution: (c) 31 The number of proper subsets of any set is given by the formula2n12n-1, wherenis the number of elements in the set. Here, n= 5 $\therefore$ Number of proper subsets of $A=2^{5}-1=31$...
Read More →If A = |1, 2, 3, 4, 5|,
Question: IfA= |1, 2, 3, 4, 5|, then the number of proper subsets ofAis (a) 120 (b) 30 (c) 31 (d) 32 Solution: (c) 31 The number of proper subsets of any set is given by the formula2n12n-1, wherenis the number of elements in the set. Here, n= 5 $\therefore$ Number of proper subsets of $A=2^{5}-1=31$...
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