Evaluate the following integrals:
Question: Evaluate the following integrals: $\int e^{x}(\cos x-\sin x) d x$ Solution: Let $\mathrm{I}=\int \mathrm{e}^{\mathrm{x}}(\cos \mathrm{x}-\sin \mathrm{x}) \mathrm{dx}$ Using integration by parts, $=\int e^{x} \cos x d x-\int e^{x} \sin x d x$ We know that, $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=-\sin \mathrm{x}$ $=\cos x \int e^{x}-\int \frac{d}{d x} \cos x \int e^{x}-\int e^{x} \sin x d x$ $=e^{x} \cos x+\int e^{x} \sin x d x-\int e^{x} \sin x d x$ $=e^{x} \cos x+c$...
Read More →In the preparation of H2SO4 by Contact Process,
Question: In the preparation of H2SO4 by Contact Process, why is SO3 not absorbed directly in water to form H2SO4? Solution: Dissolution of SO3 in water is highly exothermic. This leads to the formation of a mist of tiny droplets which are highly corrosive and they can even attack the lead pipelines (lead pipelines are used to cover the tower in contact process)....
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Question: Let $f: R \rightarrow R: f(x)=(2 x+1)$ and $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$ Write down the formulae for (i) (g o f) (ii) (f o g) (iii) (f o f) (iv) (g o g) Solution: (i) g o f To find: g o f Formula used: g o f = g(f(x)) Given: (i) f : R R : f(x) = (2x + 1) (ii) $g: R \rightarrow R: g(x)=\left(x^{2}-2\right)$ Solution: We have, $g \circ f=g(f(x))=g(2 x+1)=\left[(2 x+1)^{2}-2\right]$ $\Rightarrow 4 x^{2}+4 x+1-2$ $\Rightarrow 4 x^{2}+4 x-1$ Ans). $g \circ f(x)=4 x^{2}+4 x...
Read More →Which of the following statements are true?
Question: Which of the following statements are true? (i) The only type of interactions between particles of noble gases is due to weak dispersion forces. (ii) Ionisation enthalpy of molecular oxygen is very close to that of xenon. (iii) Hydrolysis of XeF6 is a redox reaction. (iv) Xenon fluorides are not reactive. Solution: Option (i) and (ii) are the answers....
Read More →In which of the following reactions conc.
Question: In which of the following reactions conc. H2SO4 is used as an oxidising reagent? (i) CaF2+ H2SO4 CaSO4+ 2HF (ii) 2HI + H2SO4 I2 + SO2+ 2H2O (iii) Cu + 2H2SO4 CuSO4 + SO2 + 2H2O (iv) NaCl + H2SO4 NaHSO4 + HCl Solution: Option (ii) and (iii) are the answers....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$ Solution: Let I $=\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$ $\sin ^{-1} x=t$ $\frac{1}{\sqrt{1-x^{2}}} d x=d t$ $I=\int \frac{\sin ^{2} t \times t d t}{1-\sin ^{2} t}$ $=\int \frac{\operatorname{tsin}^{2} t}{\cos ^{2} t} d t$ $=\int t \tan ^{2} t d t$ $=\int t\left(\sec ^{2} t-1\right) d t$ Using integration by parts, $=\int \operatorname{tsec}^{2} t d t-\int t d...
Read More →Which of the following statements are correct?
Question: Which of the following statements are correct? (i) SS bond is present in H2S2O6 (ii) In peroxosulphuric acid (H2SO5) sulphur is in +6 oxidation state. (iii) Iron powder along with Al2O3 and K2O is used as a catalyst in the preparation of NH3 by Habers process. (iv) Change in enthalpy is positive for the preparation of SO3 by catalytic oxidation of SO2 Solution: Option (i) and (ii) are the answers....
Read More →Which of the following orders are correct as
Question: Which of the following orders are correct as per the properties mentioned against each? (i) As2O3 SiO2 P2O3 SO2 Acid strength. (ii) AsH3 PH3 NH3 Enthalpy of vaporization. (iii) S O Cl F More negative electron gain enthalpy (iv) H2O H2S H2Se H2Te Thermal stability. Solution: Option (i) and (iv) are the answers....
Read More →Which of the following statements are correct?
Question: Which of the following statements are correct? (i) All three NO bond lengths in HNO3 are equal. (ii) All PCl bond lengths in PCl5 molecule in the gaseous state are equal. (iii) P4 molecule in white phosphorus have angular strain therefore white phosphorus is very reactive. (iv) PCl is ionic in the solid-state in which cation is tetrahedral and the anion is octahedral. Solution: Option (iii) and (iv) are the answers....
Read More →Which of the following statements are correct
Question: Which of the following statements are correct for SO2 gas? (i) It acts as a bleaching agent in moist conditions. (ii) Its molecule has linear geometry. (iii) Its dilute solution is used as a disinfectant. (iv) It can be prepared by the reaction of dilute H2SO4 with metal sulphide. Solution: Option (i) and (iii) are the answers....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{3} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} d x$ Solution: Let $I=\int \frac{x^{3} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} d x$ $\sin ^{-1} x^{2}=t$ $\frac{1}{\sqrt{1-x^{4}}} 2 x d x=d t$ $I=\int \frac{x^{2} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} x d x$ $=\int(\sin t) t \frac{d t}{2}$ Using integration by parts, $=\frac{1}{2}\left[\mathrm{t} \int \sin t \mathrm{~d} t-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \sin t \mathrm{dt}\right]$ $=\frac{1}...
Read More →Which of the following statements are correct?
Question: Which of the following statements are correct? (i) Among halogens, radius ratio between iodine and fluorine is maximum. (ii) Leaving FF bond, all halogens have weaker XX bond than XX bond in interhalogens. (iii) Among interhalogen compounds, the maximum number of atoms are present in iodine fluoride. (iv) Interhalogen compounds are more reactive than halogen compounds. Solution: Option (i), (iii) and (iv) are the answers....
Read More →Which of the following is correct for P4
Question: Which of the following is correct for P4 the molecule of white phosphorus? (i) It has 6 lone pairs of electrons. (ii) It has six PP single bonds. (iii) It has three PP single bonds. (iv) It has four lone pairs of electrons. Solution: Option (ii)It has six PP single bonds. and (iv)It has four lone pairs of electronsare the answers....
Read More →Which of the following options are not in accordance
Question: Which of the following options are not in accordance with the property mentioned against them? (i) F2 Cl2 Br2 I2 Oxidising power. (ii) MI MBr MCl MF Ionic character of metal halide. (iii) F2 Cl2 Br2 I2 Bond dissociation enthalpy. (iv) HI HBr HCl HF Hydrogen-halogen bond strength. Solution: Option (ii) and (iii) are the answers...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x$ Solution: Let $I=\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x$ Let $\mathrm{x}=\mathrm{a} \tan ^{2} \theta$ $\mathrm{dx}=2 \mathrm{a} \tan ^{2} \theta \sec ^{2} \theta$ $I=\int\left(\sin ^{-1} \sqrt{\frac{\operatorname{atan}^{2} \theta}{a+\operatorname{atan}^{2} \theta}}\right) 2 a \tan ^{2} \theta \sec ^{2} \theta d \theta$ $=\int \sin ^{-1}(\sin \theta) 2 a \tan ^{2} \theta \sec ^{2} \theta d \theta$ $=\int 2 \t...
Read More →If chlorine gas is passed through a hot NaOH solution,
Question: If chlorine gas is passed through a hot NaOH solution, two changes are observed in the oxidation number of chlorine during the reaction. These are ________ and _________. (i) 0 to +5 (ii) 0 to +3 (iii) 0 to 1 (iv) 0 to +1 Solution: Option (i) and (iii) are the answers....
Read More →Which of the following is isoelectronic pair?
Question: Which of the following is isoelectronic pair? (i) ICl2, ClO2 (ii) BrO2, BrF2+ (iii) ClO2, BrF (iv) CN, O3 Solution: Option (ii)BrO2, BrF2+is the answer....
Read More →Find the value
Question: Let $f: R \rightarrow R: f(x)=x^{2}$ and $g: R \rightarrow R: g(x)=(x+1)$. Show that $(g \circ f) \neq(f \circ g) .$ Solution: To prove: $(g \circ f) \neq(f \circ g)$ Formula used: (i) g o f = g(f(x)) (ii) f o g = f(g(x)) Given: (i) $f: R \rightarrow R: f(x)=x^{2}$ (ii) $g: R \rightarrow R: g(x)=(x+1)$ Proof: We have, $g \circ f=g(f(x))=g\left(x^{2}\right)=\left(x^{2}+1\right)$ fo $g=f(g(x))=g(x+1)=\left[(x+1)^{2}+1\right]=x^{2}+2 x+2$ From the above two equation we can say that (g o f...
Read More →Reduction potentials of some ions are given below.
Question: Reduction potentials of some ions are given below. Arrange them in decreasing order of oxidising power. (i) ClO4 IO4 BrO4 (ii) IO4 BrO4 ClO4 (iii) BrO4 IO4 ClO4 (iv) BrO4 ClO4 IO4 Solution: Option (iii)BrO4 IO4 ClO4is the answer....
Read More →In solid state PCl5 is a _________.
Question: In solid state PCl5 is a _________. (i) covalent solid (ii) octahedral structure (iii) ionic solid with [PCl6]+ octahedral and [PCl4] tetrahedra (iv) ionic solid with [PCl4]+tetrahedral and [PCl6] octahedral Solution: Option (iv) ionic solid with [PCl4]+tetrahedral and [PCl6] octahedralis the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}} d x$ Solution: Let $\mathrm{I}=\int \tan ^{-1} \sqrt{\frac{1-\mathrm{x}}{1+\mathrm{x}}} \mathrm{dx}$ $\mathrm{x}=\cos \theta ; \mathrm{dx}=-\sin \theta \mathrm{d} \theta$ $I=\int \tan ^{-1}\left(\tan \frac{\theta}{2}\right)-\sin \theta d \theta$ $=-\frac{1}{2} \int \theta \sin \theta d \theta$ Using integration by parts, $=-\frac{1}{2}\left[\theta \int \sin \theta d \theta-\int \frac{d}{d \theta} \theta \int \sin...
Read More →In the preparation of compounds of Xe,
Question: In the preparation of compounds of Xe, Bartlett had taken O2+Pt F6 as a base compound. This is because (i) both O2 and Xe has the same size. (ii) both O2 and Xe has the same electron gain enthalpy. (iii) both O2 and Xe has almost same ionisation enthalpy. (iv) both Xe and O2 are gases. Solution: Option (iii)both O2 and Xe has almost same ionisation enthalpy is the answer....
Read More →A black compound of manganese reacts
Question: A black compound of manganese reacts with a halogen acid to give greenish yellow gas. When an excess of this gas reacts with NH3 an unstable trihalide is formed. In this process the oxidation state of nitrogen changes from _________. (i) 3 to +3 (ii) 3 to 0 (iii) 3 to +5 (iv) 0 to 3 Solution: Option (i) 3 to +3 is the answer....
Read More →Solve this
Question: Let $f:\{3,9,12\} \rightarrow\{1,3,4\}$ and $g:\{1,3,4,5\} \rightarrow\{3,9\}$ be defined as $f=\{(3,1),(9,3),(12,4)\}$ and $g=\{(1,3),(3,3),(4,9),(5,9)\}$ Find (i) (g off) (ii) (f o g). Solution: (i) g o f To find: g o f Formula used: g o f = g(f(x)) Given: f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3),(4, 9), (5, 9)} Solution: We have, $g \circ f(3)=g(f(3))=g(1)=3$ $g \circ f(9)=g(f(9))=g(3)=3$ $g \circ f(12)=g(f(12))=g(4)=9$ Ans) g o f = {(3, 3), (9, 3), (12, 9)} (ii) f o g ...
Read More →Hot conc. H2SO4 acts as a moderately strong
Question: Hot conc. H2SO4 acts as a moderately strong oxidising agent. It oxidises both metals and nonmetals. Which of the following element is oxidised by conc H2SO4 into two gaseous products? (i) Cu (ii) S (iii) C (iv) Zn Solution: Option (iii)Cis the answer....
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