Prove that:
Question: Prove that: $\cos ^{-1}\left(1-2 x^{2}\right)=2 \sin ^{-1} x$ Solution: To Prove: $\cos ^{-1}\left(1-2 x^{2}\right)=2 \sin ^{-1} x$ Formula Used: $\cos 2 A=1-2 \sin ^{2} A$ Proof: $\mathrm{LHS}=\cos ^{-1}\left(1-2 \mathrm{x}^{2}\right) \ldots(1)$ Let $x=\sin A \ldots$ (2) Substituting $(2)$ in $(1)$, LHS $=\cos ^{-1}\left(1-2 \sin ^{2} \mathrm{~A}\right)$ $=\cos ^{-1}(\cos 2 \mathrm{~A})$ $=2 \mathrm{~A}$ From $(2), A=\sin ^{-1} x$ $2 A=2 \sin ^{-1} x$ $=\mathrm{RHS}$ Therefore, $\math...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}+1}{(x-2)^{2}(x+3)} d x$ Solution: $I=\int \frac{x^{2}+2}{(x-2)^{2}(x+3)} d x$ $\frac{\mathrm{x}^{2}+2}{(\mathrm{x}-2)^{2}(\mathrm{x}+3)}=\frac{\mathrm{A}}{\mathrm{x}-2}+\frac{\mathrm{B}}{(\mathrm{x}-2)^{2}}+\frac{\mathrm{c}}{\mathrm{x}+3}$ $x^{2}+1=A(x-2)(x+3)+B(x+3)+C(x-2)^{2}$ Put $x=2$ $4+1=B \times 5$ $5=5 B$ $B=\frac{5}{5}=1$ Put $x=-3$ $10=C \times 25$ $C=\frac{10}{25}=\frac{2}{5}$ Equating coefficients of constants $1=-6 A+3 B+4...
Read More →Prove that:
Question: Prove that: $\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ Solution: To Prove: $\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ Formula Used: $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ Proof: $\mathrm{LHS}=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) \ldots$ (1) $=\tan ^{-1}\left(\frac{x+\left(\frac{2 x}{1-x^{2}}\right)}{1-\left(...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{2 x+1}{(x+2)(x-3)^{2}}$ Solution: $I=\int \frac{2 x+1}{(x+2)(x-3)^{2}} d x$ $\frac{2 x+1}{(x+2)(x-3)^{2}}=\frac{A}{x+2}+\frac{B}{x-3}+\frac{c}{(x-3)^{2}}$ $2 x+1=A(x-3)^{2}+B(x+2)(x-3)+C(x+2)$ $2 x+1=A x^{2}-3 A x+9 A+B x^{2}-5 B x-6 B+C x+2 C$ Put $x=3$ $7=5 C$ $C=\frac{7}{5}$ Put $x=-2$ $-3=0 A$ $-11=4 C$ $C=-\frac{11}{4}$ Equating coefficients of constants $-2=3 A+3 B+C$ $-2=3 A+3 \times \frac{-5}{2}-\frac{11}{4}$ $A=\frac{11}{4}$ Thus, $...
Read More →Prove that:
Question: Prove that: $\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)=3 \tan ^{-1} x,|x|\frac{1}{\sqrt{3}}$ Solution: To Prove: $\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)=3 \tan ^{-1} x$ Formula Used: $\tan 3 A=\frac{3 \tan A-\tan ^{3} A}{1-3 \tan ^{2} A}$ Proof: $\mathrm{LHS}=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right) \cdots$ Let $x=\tan A \ldots(2)$ Substituting (2) in (1), $\mathrm{LHS}=\tan ^{-1}\left(\frac{3 \tan A-\tan ^{3} A}{1-3 \tan ^{2} A}\right)$ $=\tan ^{-1}(\tan...
Read More →Prove that:
Question: Prove that: $\cos ^{-1}\left(4 x^{3}-3 x\right)=3 \cos ^{-1} x, \frac{1}{2} \leq x \leq 1$ Solution: To Prove: $\cos ^{-1}\left(4 x^{3}-3 x\right)=3 \cos ^{-1} x$ Formula Used: $\cos 3 A=4 \cos ^{3} A-3 \cos A$ Proof: LHS $=\cos ^{-1}\left(4 x^{3}-3 x\right) \ldots$ (1) Let $x=\cos A \ldots(2)$ Substituting (2) in (1), $\mathrm{LHS}=\cos ^{-1}\left(4 \cos ^{3} \mathrm{~A}-3 \cos \mathrm{A}\right)$ $=\cos ^{-1}(\cos 3 A)$ $=3 \mathrm{~A}$ From $(2), A=\cos ^{-1} x$ $3 A=3 \cos ^{-1} x$ ...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{3 x-2}{(x+1)^{2}(x+3)}$ Solution: $I=\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x$ $\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{c}{x+3}$ $3 x-2=A(x+1)(x+3)+B(x+3)+C(x+1)^{2}$ Put $x=-1$ $-3-2=A \times 0+B \times(-1+3)+C \times 0$ $-5=2 B$ $B=-\frac{5}{2}$ Put $x=-3$ $-9-2=C \times(-2)(-2)$ $-11=4 C$ $C=-\frac{11}{4}$ Equating coefficients of constants $-2=3 A+3 B+C$ $-2=3 A+3 \times \frac{-5}{2}-\frac{11}{4}$ $A=\frac{11}{4...
Read More →Between sodium hydrogen carbonate
Question: Between sodium hydrogen carbonate and magnesium hydroxide which is a better antacid and why? Solution: Magnesium hydroxide is a better antacid because it maintains a pH level in the stomach and insoluble in the stomach. It does not make stomach alkaline whereas sodium hydrogen carbonate makes stomach alkaline....
Read More →prove that
Question: Prove that $\sin ^{-1}\left(3 x-4 x^{3}\right)=3 \sin ^{-1} x,|x| \leq \frac{1}{2}$ Solution: To Prove: $\sin ^{-1}\left(3 x-4 x^{3}\right)=3 \sin ^{-1} x$ Formula Used: $\sin 3 A=3 \sin A-4 \sin ^{3} A$ Proof: LHS $=\sin ^{-1}\left(3 x-4 x^{3}\right) \ldots$ (1) Let $x=\sin A \ldots(2)$ Substituting (2) in (1), $\mathrm{LHS}=\sin ^{-1}\left(3 \sin \mathrm{A}-4 \sin ^{3} \mathrm{~A}\right)$ $=\sin ^{-1}(\sin 3 \mathrm{~A})$ $=3 \mathrm{~A}$ From $(2), A=\sin ^{-1} x$ $3 A=3 \sin ^{-1} ...
Read More →What is the basic difference between
Question: What is the basic difference between antiseptics and disinfectants? Solution: 1. Antiseptics are applied to living tissues whereas disinfectants are applied on non-living substances 2. Antiseptics are antimicrobial whereas disinfectants are anti-microbial too 3. Antiseptics applied on cuts or wounds whereas disinfectants are used in floors, toilets etc....
Read More →What is the scientific explanation
Question: What is the scientific explanation for the feeling of depression? Solution: A hormone called Noradrenaline controls the mood swings. Depression can be caused due to the low level of noradrenaline which hampers the signal activities in the brain....
Read More →What are analgesics?
Question: What are analgesics? Solution: The analgesics drug is a neurologically active drug which is used to reduce pain. They dont have any side effects....
Read More →Why is it safer to use soap from
Question: Why is it safer to use soap from the environmental point of view? Solution: When compared to the detergents soaps are safer to use which is biodegradable. It doesnt have a polluting nature....
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{3}+x+1}{x^{2}-1}$ Solution: Let $I=\int \frac{x^{3}+x+1}{x^{2}-1} d x=\int\left(x+\frac{2 x+1}{x^{2}-1}\right) d x$ Now, Let $\frac{2 x+1}{x^{2}-1}=\frac{A}{x+1}+\frac{B}{x-1}$ $2 x+1=A(x-1)+B(x+1)$ Put $x=1$ $2+1=A \times 0+B \times 2$ $3=2 B$ $B=\frac{3}{2}$ Put $x=-1$ $-2+1=-2 A+B \times 0$ $-1=-2 \mathrm{~A}$ $A=\frac{1}{2}$ $I=\int x d x+\frac{1}{2} \int \frac{d x}{x+1}+\frac{3}{2} \int \frac{d x}{x-1}$ $\int \frac{\mathrm{dx}}{\math...
Read More →How does the branching of the hydrocarbon
Question: How does the branching of the hydrocarbon chain of synthetic detergents affect their biodegradability? Solution: Lesser the branching lesser is the non-polluting nature of the detergent and increased in branching causes the polluting nature of the detergent to increase....
Read More →Prove that:
Question: Prove that: $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=2 \sin ^{-1} x,|x| \leq \frac{1}{\sqrt{2}}$ Solution: To Prove: $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)=2 \sin ^{-1} x$ Formula Used: $\sin 2 A=2 \times \sin A \times \cos A$ Proof: $\mathrm{LHS}=\sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right) \cdots$(1) Let $x=\sin A \ldots$ (2) Substituting $(2)$ in $(1)$, LHS $=\sin ^{-1}\left(2 \sin A \sqrt{1-\sin ^{2} A}\right)$ $=\sin ^{-1}(2 \times \sin A \times \cos A)$ $=\si...
Read More →Dishwashing soaps are synthetic detergents.
Question: Dishwashing soaps are synthetic detergents. What is their chemical nature? Solution: Detergents are synthetic detergents which contain non-ionic detergents that have a cleansing property. They combine with the dirt and impurities and make them soluble....
Read More →Hair shampoos belong to
Question: Hair shampoos belong to which class of synthetic detergent? Solution: Cationic detergents are used in hair shampoos. eg: cetyltrimethylammonium bromide. Cationic detergents are quarternary ammonium salts of acetates, chlorides or bromides....
Read More →Which category of the synthetic
Question: Which category of the synthetic detergents is used in toothpaste? Solution: Anionic detergents are used in toothpaste to clean teeth and form suitable foam. Eg: sodium or ammonium lauryl sulphate...
Read More →Explain why some times foaming is
Question: Explain why some times foaming is seen in river water near the place where sewage water is poured after treatment? Solution: The foam is due to the non-biodegradable detergents which are present in water after sewage treatment. A detergent is a water-soluble cleansing agent which combines with impurities and dirt to make them more soluble....
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}+1}{\left(x^{2}+4\right)\left(x^{2}+25\right)} d x$ Solution: Denominator is factorized, so let separate the fraction through partial fraction, hence let $\frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+4\right)\left(\mathrm{x}^{2}+25\right)}=\frac{\mathrm{Ax}+\mathrm{B}}{\left(\mathrm{x}^{2}+4\right)}+\frac{\mathrm{Cx}+\mathrm{D}}{\mathrm{x}^{2}+25} \ldots \ldots$ (i) $\Rightarrow \frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+4\right)\left...
Read More →If the soap has high alkali content it irritates the skin.
Question: If the soap has high alkali content it irritates the skin. How can the amount of excess alkali be determined? What can be the source of excess alkali? Solution: Excess of alkali can be found out using acid-base titration. The alkali that is formed during the hydrolyses of oil during soap preparation may be a cause of excess alkali in the soap....
Read More →What is a soft soap?
Question: What is a soft soap? Solution: They are easily soluble which contains potassium salts of fatty acids as a major component....
Read More →Both antacids and antiallergic drugs
Question: Both antacids and antiallergic drugs are antihistamines but they cannot replace each other. Explain why? Solution: Antacids are used for the treatment of acid in the stomach and antihistamines inhibit the action of histamine in the body. Both antacids and antiallergic drugs work on different receptors in the body and hence they cannot replace each other....
Read More →Aspirin is a pain relieving antipyretic
Question: Aspirin is a pain relieving antipyretic drug but can be used to prevent heart attack. Explain. Solution: Aspirin prevents blood clotting in the heart as it has anti-blood-clotting action. This action helps in preventing heart attack....
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