Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \cos ^{3} x d x$ Solution: Let $\mathrm{I}=\int \mathrm{x} \cos ^{3} \mathrm{x} \mathrm{dx}$ we know that, $\cos ^{3} t d t=\frac{3 \cos t+\cos 3 t}{4}$ $=\int x\left(\frac{3 \cos x+\cos 3 x}{4}\right) d x$ $=\frac{1}{4} \int x(3 \cos x+\cos 3 x) d x$ Using integration by parts, $I=\frac{1}{4}\left[x \int(3 \cos x+\cos 3 x) d x-\int 1 \int(3 \cos x+\cos 3 x) d x\right]$ $=\frac{1}{4}\left[x\left(3 \sin x+\frac{\sin 3 x}{3}\right)-\int\left(3 \s...
Read More →Which of the following are peroxoacids
Question: Which of the following are peroxoacids of sulphur? (i) H2SO5 and H2S2O8 (ii) H2SO5 and H2S2O7 (iii) H2S2O7 and H2S2O8 (iv) H2S2O6 and H2S2O7 Solution: Option (i) H2SO5 and H2S2O8 is the answer....
Read More →Which of the following is not tetrahedral
Question: Which of the following is not tetrahedral in shape? (i) NH4+ (ii) SiCl4 (iii) SF4 (iv) SO42 Solution: Option (iii)SF4 is the answer....
Read More →The oxidation state of a central atom
Question: The oxidation state of a central atom in the anion of compound NaH2PO2 will be ______. (i) +3 (ii) +5 (iii) +1 (iv) 3 Solution: Option (iii)+1 is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cos ^{3} \sqrt{x} d x$ Solution: Let $\sqrt{x}=t$ $x=t^{2}$ $\mathrm{d} \mathrm{x}=2 \mathrm{tdt}$ let $\mathrm{I}=2 \int \mathrm{t} \cos ^{3} \mathrm{tdt}$ we know that, $\cos ^{3} t d t=\frac{3 \cos t+\cos 3 t}{4}$ $=2 \int \mathrm{t}\left(\frac{3 \cos t+\cos 3 t}{4}\right) \mathrm{dt}$ $=\frac{1}{2} \int \mathrm{t}(3 \cos t-\cos 3 \mathrm{t}) \mathrm{dt}$ Using integration by parts, $=\frac{1}{2}\left[t\left(3 \sin t+\frac{1}{3} \sin 3 t\righ...
Read More →In the preparation of HNO3,
Question: In the preparation of HNO3, we get NO gas by catalytic oxidation of ammonia. The moles of NO produced by the oxidation of two moles of NH3 will be ______. (i) 2 (ii) 3 (iii) 4 (iv) 6 Solution: Option (i)2is the answer....
Read More →On heating ammonium dichromate
Question: On heating ammonium dichromate and barium azide separately we get (i) N2 in both cases (ii) N2 with ammonium dichromate and NO with barium azide (iii) N2O with ammonium dichromate and N2 with barium azide (iv) N2O with ammonium dichromate and NO2 with barium azide Solution: Option (i)N2 in both cases is the answer....
Read More →Find the value
Question: Let $A=\{1,2,3,4\} .$ Let $f: A \rightarrow A$ and $g: A \rightarrow A$, defined by $f=\{(1,4),(2,1),(3,3),(4,2)\}$ and $g=\{(1,3),(2,1),(3,2),(4,4)\}$ Find (i) g of (ii) f o g (iii) f o f. Solution: (i) $\mathrm{g}$ o $\mathrm{f}$ To find: $g$ o $f$ Formula used: $g$ o $f=g(f(x))$ Given: $f=\{(1,4),(2,1),(3,3),(4,2)\}$ and $g=\{(1,3),(2,1)$ $(3,2),(4,4)\}$ Solution: We have, $g \circ f(1)=g(f(1))=g(4)=4$ $g \circ f(2)=g(f(2))=g(1)=3$ $g \circ f(3)=g(f(3))=g(3)=2$ $g \circ f(4)=g(f(4))...
Read More →Elements of group-15 form compounds in +5 oxidation state.
Question: Elements of group-15 form compounds in +5 oxidation state. However, bismuth forms only one well-characterised compound in +5 oxidation state. The compound is (i) Bi2O5 (ii) BiF5 (iii) BiCl5 (iv) Bi2S5 Solution: Option (ii)BiF5 is the answer....
Read More →A brown ring is formed in the ring test for NO3
Question: A brown ring is formed in the ring test for NO3 ion. It is due to the formation of (i) [Fe(H2O)5 (NO)]2+ (ii) FeSO4.NO2 (iii) [Fe(H2O)4 (NO)]2+ (iv) FeSO4.HNO3 Solution: Option (i)[Fe(H2O)5 (NO)]2+ is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \sin ^{3} x d x$ Solution: Let $\mathrm{I}=\int \mathrm{x} \sin ^{3} \mathrm{x} \mathrm{dx}$ We know that, $\sin ^{3} x=\frac{3 \sin x-\sin 3 x}{4}$ $=\int x\left(\frac{3 \sin x-\sin 3 x}{4}\right) d x$ $=\frac{1}{4} \int x(3 \sin x-\sin 3 x) d x$ Using integration by parts, $I=\frac{1}{4}\left[x \int(3 \sin x-\sin 3 x) d x-\int 1 \int(3 \sin x-\sin 3 x) d x\right]$ $=\frac{1}{4}\left[x\left(-3 \cos x+\frac{\cos 3 x}{3}\right)-\int\left(-3 \cos...
Read More →Which of the following statements
Question: Which of the following statements is wrong? (i) Single NN bond is stronger than the single PP bond. (ii) PH3 can act as a ligand in the formation of a coordination compound with transition elements. (iii) NO2 is paramagnetic. (iv) Covalency of nitrogen in N2O5 is four Solution: Option (i)Single NN bond is stronger than the single PP bond is the answer....
Read More →Maximum covalency of nitrogen
Question: Maximum covalency of nitrogen is ______________. (i) 3 (ii) 5 (iii) 4 (iv) 6 Solution: Option (iii) 4 is the answer....
Read More →Which of the following elements does
Question: Which of the following elements does not show allotropy? (i) Nitrogen (ii) Bismuth (iii) Antimony (iv) Arsenic Solution: Option (i)Nitrogen is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{3} \sqrt{x} d x$ Solution: Let $\sqrt{X}=t$ $\mathrm{x}=\mathrm{t}^{2}$ $\mathrm{dx}=2 \mathrm{tdt}$ $I=2 \int t \sin ^{3} t d t$ $=2 \int \mathrm{t}\left(\frac{3 \sin t-\sin 3 t}{4}\right) \mathrm{dt}$ $=\frac{1}{2} \int \mathrm{t}(3 \sin t-\sin 3 \mathrm{t}) \mathrm{dt}$ Using integration by parts, $=\frac{1}{2}\left[t\left(-3 \cos t+\frac{1}{3} \cos 3 t\right)-\int\left(-3 \cos t+\frac{\cos 3 t}{3}\right) d t\right]$ $=\frac{1}{2}\left[...
Read More →On heating lead, nitrate forms oxides of nitrogen
Question: On heating lead, nitrate forms oxides of nitrogen and lead. The oxides formed are ______. (i) N2O, PbO (ii) NO2, PbO (iii) NO, PbO (iv) NO, PbO2 Solution: Option (ii)NO2, PbO is the answer....
Read More →Strong reducing behaviour of H3PO2
Question: Strong reducing behaviour of H3PO2 is due to (i) The low oxidation state of phosphorus (ii) Presence of two OH groups and one PH bond (iii) Presence of one OH group and two PH bonds (iv) High electron gain enthalpy of phosphorus Solution: Option (iii)Presence of one OH group and two PH bondsis the answer....
Read More →Which of the following acids forms
Question: Which of the following acids forms three series of salts? (i) H3PO2 (ii) H3BO3 (iii) H3PO4 (iv) H3PO3 Solution: Option (iii)H3PO4 is the answer....
Read More →On heating with concentrated NaOH
Question: On heating with concentrated NaOH solution in an inert atmosphere of CO2, white phosphorus gives a gas. Which of the following statement is incorrect about the gas? (i) It is highly poisonous and has smelled like rotten fish. (ii) Its a solution in water decomposes in the presence of light. (iii) It is more basic than NH3 (iv) It is less basic than NH3 Solution: Option (iii)It is more basic than NH3 is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$ Solution: Let $I=\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$ We are splitting this in to two functions First we find the integral of: $\int \frac{\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}$ Put $1-x^{2}=t$ $-2 x d x=d t$ $\int \frac{\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}=-\frac{1}{2} \int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}}}=-\sqrt{\mathrm{t}}=-\sqrt{1-\mathrm{x}^{2}}$ $I=\int \frac{...
Read More →Bond dissociation enthalpy of E—H (E = element)
Question: Bond dissociation enthalpy of EH (E = element) bonds are given below. Which of the compounds will act as the strongest reducing agent? (i) NH3 (ii) PH3 (iii) AsH3 (iv) SbH3 Solution: Option (iv)SbH3 is the answer....
Read More →Affinity for hydrogen decreases in the group
Question: Affinity for hydrogen decreases in the group from fluorine to iodine. Which of the halogen acids should have the highest bond dissociation enthalpy? (i) HF (ii) HCl (iii) HBr (iv) HI Solution: Option (i)HF is the answer....
Read More →Which of the following pairs of ions
Question: Which of the following pairs of ions are isoelectronic and isostructural? (i) CO32-, NO3- (ii) ClO3-, CO32- (iii) SO32-, NO3- (iv) ClO3-, SO32 Solution: Option (i) is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left(\tan ^{-1} x^{2}\right) x d x$ Solution: Let $I=\int\left(\tan ^{-1} x^{2}\right) x d x$ $x^{2}=t$ $2 x d x=d t$ $I=\frac{1}{2} \int\left(\tan ^{-1} t\right) d t$ Using integration by parts, $=\frac{1}{2}\left(\tan ^{-1} \mathrm{t} \int \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \tan ^{-1} \mathrm{t} \int \mathrm{dt}\right)$ We know that, $\frac{d}{d t} \tan ^{-1} t=\frac{1}{2\left(1+t^{2}\right)}$ $=\frac{1}{2}\left[\mathrm{t} \tan ^{-...
Read More →Which of the following elements can
Question: Which of the following elements can be involved in pd bonding? (i) Carbon (ii) Nitrogen (iii) Phosphorus (iv) Boron Solution: Option (iii)Phosphorus is the answer....
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