, defined by 
is both one – one and onto.Solution: $\begin{array}{l} f(n)=\left\{\begin{array}{l} \frac{1}{2}(n-1), \text { when } n \text { is odd } \\ -\frac{1}{2} n, \text { when } n \text { is even } \end{array}\right. \\ f(1)=0 \\...
and exceeds twice the length of the altitude by 
. Find the length of each side of the triangle.Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...
Let one side of the right-angled triangle be $x \mathrm{~m}$ and the other side be $(x+4) \mathrm{m}$. On applying Pythagoras theorem, we have: $\begin{array}{l} 20^{2}=(x+4)^{2}+x^{2} \\...
Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) m$ $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text { Altitude } \\ \therefore...
Let the altitude of the triangle be $x \mathrm{~m}$. Therefore, the base will be $3 x \mathrm{~m}$. $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text {...
is one – one but not onto.Solution: In the range of $\mathrm{N} \mathrm{f}(\mathrm{x})$ is monotonically increasing. $\therefore f(n)=n^{2}+n+1$ is one one. But Range of $f(n)=[0.75, \infty) \neq N($ codomain $)$ Thus,...
. If the base of the triangle exceeds the altitude by 
, find the dimensions of the triangle.Let the altitude of the triangle be $x \mathrm{~cm}$ Therefore, the base of the triangle will be $(x+10) \mathrm{cm}$ $\begin{array}{l} \text { Area of triangle }=\frac{1}{2} x(x+10)=600 \\...
is one – one into 
is one – one intoSolution: (i) $f: N \rightarrow N: f(x)=x^{3}$ is one - one into. $f(x)=x^{3}$ As the function $f(x)$ is monotonically increasing from the domain $N \rightarrow N$ $\therefore f(x)$ is one -one...
Let the length and breadth of the rectangular garden be $x$ and $y$ meter, respectively. Given: $x y=180 \mathrm{sq} \mathrm{m}$$\ldots(i)$ and $\begin{array}{l} 2 y+x=39 \\ \Rightarrow x=39-2 y...
, more than the width of the rectangle. Their areas being equal, find the dimensions.Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...
. If the difference in their perimeter be 
, find the sides of the two squareLet the length of the side of the first and the second square be $x$ and $y \cdot$ respectively. According to the question: $x^{2}+y^{2}=640$ Also, $\begin{array}{l} 4 x-4 y=64 \\ \Rightarrow x-y=16...
is one – one into. 
is many – one intoSolution: (i) $f: N \rightarrow N: f(x)=x^{2}$ is one - one into. As the function $f(x)$ is monotonically increasing from the domain $N \rightarrow N$ $\therefore f(x)$ is one -one Range of...
long and 
wide. There is a path of uniform width all around it, having an area of 
. Find the width of the pathLet the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...
and its area is 288 sq meters. Find the dimension of the plotLet the length and breadth of the rectangular plot be $x$ and $y$ meter, respectively. Therefore, we have: $\begin{array}{l} \text { Perimeter }=2(x+y)=62 \quad \ldots . .(i) \text { and } \\ \text...
![f:\left[0, \frac{\pi}{2}\right] \rightarrow R: f(x)=\sin x](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-902634cb75410b73f743099ea343b78d_l3.png "Rendered by QuickLaTeX.com")
and ![g:\left[0, \frac{\pi}{2}\right] \rightarrow R: g(x)=\cos x](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5d08fed556b082ab80b726e70e9cca27_l3.png "Rendered by QuickLaTeX.com")
. Show that each one of 
and 
is one one but 
is not one – one.Solution: $f: \left[0, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ for given function $\mathrm{f}(\mathrm{x})=\sin$ Recalling the graph for $\sin \mathrm{x}$, we realise that for any two values on...
Let the breath of the rectangular hall be $x$ meter. Therefore, the length of the rectangular hall will be $(x+3)$ meter. According to the question: $\begin{array}{l} x(x+3)=238 \\ \Rightarrow...
Let the length and breadth of the rectangle be $3 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 3 x \times x=147 \\ \Rightarrow 3 x^{2}=147 \\...
. Find the dimension of the rectangle.Let the length and breadth of the rectangle be $2 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 2 x \times x=288 \\ \Rightarrow 2 x^{2}=288 \\...
Let the tap of smaller diameter fill the tank in $x$ hours. $\therefore$ Time taken by the tap of larger diameter to fill the tank $=(x-9) h$ Suppose the volume of the tank be $V$. Volume of the...
is one – one and onto.Solution: We need to show that $f: R \rightarrow R$ given by $f(x)=x s$ is one-one and onto. A function which is onto has every element of co-domain mapped to the at least one element of Domain....
minutes. If on pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.Let the time taken by one pipe to fill the tank be $x$ minutes. $\therefore$ Time taken by the other pipe to fill the tank $=(x+5) \min$ Suppose the volume of the tank be $V$. Volume of the tank...
is many – one and into.Solution: We need to show that f: $\mathrm{R} \rightarrow \mathrm{R}$ given by $\mathrm{f}(\mathrm{x})=\mathrm{x} 4$ is many-one into. A function which is not onto is into. A function where more...
minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.Let one pipe fills the cistern in $x$ mins. Therefore, the other pipe will fill the cistern in $(x+3)$ mins. Time taken by both, running together, to fill the cistern $=3 \frac{1}{13} \min...
to finish a piece of work. If both 
and together can finish the work in 12 days, find the time taken by B to finish the work.Let B takes $x$ days to complete the work. Therefore, A will take $(x-10)$ days. $\begin{array}{l} \therefore \frac{1}{x}+\frac{1}{(x-10)}=\frac{1}{12} \\ \Rightarrow...
in still water, goes 
downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Downstream speed $=(9+x) \mathrm{km} / \mathrm{hr}$ Upstream speed $=(9-x) \mathrm{km} / \mathrm{hr}$ Distance covered...
. It can go upstream and 
downstream is 5 hours. Fid the speed of the stream.Speed of the boat in still water $=8 \mathrm{~km} / \mathrm{hr}$. Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Speed...
is many – one into.Solution: We need to show that $f: R \rightarrow R$ given by $f(x) = 1 + x^2$ is many-one into. A function which is not onto is into. A function where more than one element in Set A maps to one...
be defined by 
(ii) 
.Solution: (i) $\mathrm{f}(-1)$ $x=-1$, it is satisfying the condition $-2 \leq x \leq 3$ Therefore, $f(x)=x_{2}-2$ $\begin{aligned} \begin{aligned} \therefore \mathrm{f}(-1) =(-1)_{2}-2 \\ =1-2 \\...
, takes 1 hour more to go 
upstream than to return to the same spot. Find the speed of the stream.Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given: Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / h r$ Speed upstream...
. Travelling by the Deccan Queen, it takes 48 minutes less than another train. Calculate the speed of the Deccan Queen if the speeds of the two train differ by 
Let the speed of the Deccan Queen be $x \mathrm{~km} / \mathrm{hr}$. According to the question: Speed of another train $=(x-20) \mathrm{km} / \mathrm{hr}$ $\begin{array}{l} \therefore...
if its speed is increased by 
from its usual speed. Find its usual speed.Let the usual speed $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{(x+5)}=2 \\ \Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2 \\ \Rightarrow...
be defined by 
(ii) 
Solution: (i) $\mathrm{f}(2)$ $x=2$, it is satisfying the condition $-2 \leq x \leq 3$ Therefore, $f(x)=x_{2}-2$ $\begin{aligned} \therefore \mathrm{f}(2) =22-2 \\ =4-2 \\ =2 \\ \therefore...
at a uniform speed. Had the speed been 
more, it would have taken 30 minutes less for the journey. Find the original speed of the train.Let the original speed of the train be $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\frac{90}{x}-\frac{90}{(x+15)}=\frac{1}{2}$ $\begin{array}{l} \Rightarrow \frac{90(x+15)-90...
Solution: (i) Neither one-one nor onto $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ given by $\mathrm{f}(\mathrm{x})=|\mathrm{x}|=\left\{\begin{array}{l}\mathrm{x}, \text { if } \mathrm{x} \geq 0...
at a uniform speed. If the speed had been more, it would have taken 1 hour less for the same journey. Find the speed of the train.Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...
Solution: (i) One-One but not Onto $\mathrm{f}: \mathrm{N} \rightarrow \mathrm{N}$ be a mapping given by $\mathrm{f}(\mathrm{x})=\mathrm{x} 2$ For one-one $\begin{array}{l} f(x)=f(y) \\ x_{2}=y z \\...
and then travels a distance of 63 
at an average speed of 
more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?Let the first speed of the train be $x \mathrm{~km} / \mathrm{h}$. Time taken to cover $54 \mathrm{~km}=\frac{54}{x} h .$ New speed of the train $=(x+6) \mathrm{km} / \mathrm{h}$ $\therefore$ Time...
at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...
Solution: (i)Bijective function: It is, also known as one-one onto function and is a function where for every element of set A, there is exactly one image in set B, such that no element is set B is...
Solution: A function is stated as the relation between the two sets, where there is exactly one element in set B, for every element of set A. A function is represented as f: A → B, which means ‘f’...
Solution: $\begin{array}{l} \mathrm{A}=\{1,2,3,4\} \text { and } \mathrm{R}=\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3) \\ (3,2)\} \text { (Given) } \end{array}$ $\mathrm{R}$ is reflexive if $\mathrm{a}...
Solution: $A=\{1,2,3\}$ and $\bar{R}=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}$ (Given) $\mathrm{R}$ is reflexive if $\mathrm{a} \in \mathrm{A}$ and $(\mathrm{a}, \mathrm{a}) \in \mathrm{R}$ Here,...
for all 
Show that R satisfies none of reflexivity, symmetry and transitivity.Solution: $\mathrm{R}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}=\mathrm{b}_{2}\right\}$ for all $\mathrm{a}, \mathrm{b} \in \mathrm{N}$ (As given) Non-Reflexivity: Assume $a$ be an arbitrary...
Solution: $\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{d}(\mathrm{A}, \mathrm{B})<2$ units $\}$, where $\mathrm{d}(\mathrm{A}, \mathrm{B})$ is the distance between the points $\mathrm{A}$ and...
on 
, defined by 
is an equivalent relation.Solution: If $R$ is Reflexive, Symmetric and Transitive, then $R$ is an equivalence relation. Reflexivity: Suppose $a$ and $\mathrm{b}$ be an arbitrary element of $\mathrm{N} \times \mathrm{N}$...
and 
is divisible by 5
Show that is an equivalence relation on 
.Solution: $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $(\mathrm{a}-\mathrm{b})$ is divisible by 5$\}$ (As given) If $R$ is Reflexive, Symmetric and Transitive,...
be the set of all triangles in a plane. Show that the relation 
is an equivalence relation on 
.Solution: Suppose $\mathrm{R}=\left\{\left(\Delta 1, \Delta_{2}\right): \Delta_{1} \sim \Delta_{2}\right\}$ be a relation defined on A. (As given) If $\mathrm{R}$ is Reflexive, Symmetric and...
, the work done during combustion of 
of ethane (molar mass 
) at 
is A) 
B) 
C) 
D) 
Correct option is B $18.7 \mathrm{~kJ}$ $ \begin{array}{l} \mathrm{C}_{2} \mathrm{H}_{6(\mathrm{~g})}+\frac{7}{2} \mathrm{O}_{2(\mathrm{~g})} \longrightarrow...
Solution: (i) Reflexivity: Suppose $\mathrm{p}$ is an arbitrary element of $\mathrm{S}$. So now, $\mathrm{p} \leq \mathrm{p}$ $\Rightarrow(p, p) \in R$ Therefore, $\mathrm{R}$ is reflexive. (ii)...
Solution: (i) $A \subset B$ and $(A, B) \in R$ (As given) i.e. $\mathrm{A}$ is a proper subset of $\mathrm{B}$ But, B cannot be a proper subset of A Since, B can contain atleast one element that is...
B) 
C) 
D) 
Correct option is B Explanation: Because -CN has the highest atomic number, the atom with the highest atomic number that is immediately linked to carbon will be seen first, followed by the remaining...
Solution: Since a< 4, $a = 1, 2, 3$ Therefore, R $=$ {(1, 6), (2, 7), (3, 8)} As a result, Domain(R) = {1, 2, 3} and Range(R) = {6, 7, 8}
and 
. Find the domain and range of 
.Solution: Since 1<a< 5, $a = 2, 3, 4$ Therefore, R $= {\{(2,{\frac12}), (3,{\frac13}), (4,{\frac14})}\}$ As a result, Domain(R) = {2, 3, 4} and Range(R) =...
Solution: As $|a| < 3$, $a = −2$, $−1$, $0$, $1$, $2$ Therefore, R = {(−2, 3), (−1, 2), (0, 1), (1, 0), (2, 1)} As a result, Domain(R) = {-2, -1, 0, 1, 2} and Range(R) = {3, 2, 1, 0}
Solution: $x + 2y = 8$ (given) $x = 8 – 2y$ Putting $y = 1$ $x = 8 – 2(1) = 6$ Putting $y = 2$ $x = 8 – 2(2) = 4$ Putting $y = 3$ $x = 8 – 2(3) = 2$ Putting $y = 4$ $x = 8 – 2(4) = 0$; which is not...
: a is a prime number less than 5}. Find the range of R.Solution: R = {(2, 8), (3, 27) Therefore, Range of R = {8 27}
Solution: Set of all the first elements or $x$-coordinates of the ordered pairs is called Domain. Set of all the second elements or $y$-coordinates of the ordered pairs is called Range. Therefore,...
Correct option is A Explanation: Because of the availability of d-electrons, phosphorus is a 15th group element that forms the most stable pentavalent compound. The stability of the +5 oxidation...
Correct option is A Explanation: Passivation is the process of rendering a metal surface inert. Metal is treated with high oxidising chemicals, such as concentrated nitric acid, in this procedure....
of sucrose? (Given 
molar masses of sucrose 
, water 
, density of water 
) A) 
B) 
C) 
D) 
Correct option is A ($0.072 \mathrm{dm}^{3}$) $ \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}+\mathrm{H}_{2} \mathrm{O} \stackrel{\mathrm{H}^{+}}{\longrightarrow} \mathrm{C}_{6} \mathrm{H}_{12}...
B) 
C) 
D) 
Correct option is D) $\left.\mathrm{HCHO}\rangle \mathrm{CH}_{3} \mathrm{CHO}\right\rangle \mathrm{CH}_{3} \mathrm{COCH}_{3}$ The following is the right order of reactivity of aldehydes and ketones...
B is the correct answer The basicity of orthophosphorous acid is two due to the presence of two - OH groups in its structure.
D is the correct answer (Nylon-6, 6) A heteropolymer is one that is made up of two or more distinct (though often related) monomers. Hexamethylenediamine and adipic acid, which give nylon 66 its...
B) 
C) 
D) 
Correct option is C Explanation: A Nicol prism is a type of polarizer, which is an optical device that produces a polarised light beam. It is constructed in such a way that total internal reflection...
Correct option is D Explanation: Tellurium is a silvery-white metalloid that possesses both metal and non-metal characteristics. Sodium is a metal, while carbon and neon are non-metals. As a result,...
Correct option is B Explanation: Chromium (IV) dioxide is a ferromagnetic substance. These compounds' ferromagnetic properties allow them to be employed in data tapes, which are used to store data....
this reaction is known as A) Etard reaction B) Stephen reaction C) Hell-Vohlard-Zelinsky reaction D) Balz-Schiemann reactionB is the correct answer. Explanation: Using tin chloride and dilution, aldehydes (R-CHO) are synthesised from nitriles (R-CN). Stephen's reaction is the reaction of HCl in an acidic media. As an...
if the rate law expression is, rate ![=\mathrm{K}\left[\mathrm{O}_{3}\right][\mathrm{O}]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e69a8d7ed909b9b87bef5e92873c3d1c_l3.png "Rendered by QuickLaTeX.com")
the molecularity and order of the reaction are respectively A) 2 and 2 B) 2 and 
C) 2 and 1 D) 1 and 2Correct option is A (2 and 2) Rate $=\mathrm{K}\left[\mathrm{O}_{3}\right][\mathrm{O}]$ Order $=1+1=2$ Hence, the overall order of the reaction is 2 . Now, $ \mathrm{O}_{3(9)}+\mathrm{O}_{(9)}...
![\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]\left[\mathrm{Cr}(\mathrm{SCN})_{6}\right] \text { and }\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{2}(\mathrm{SCN})_{4}\right]\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{4}(\mathrm{SCN})_{2}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1c51c7d9e0fc5c8f37f7a46ee940d288_l3.png "Rendered by QuickLaTeX.com")
are the examples of what type of isomerism ? A) Ionisation isomerism B) Linkage isomerism C) Coordination isomerism D) Solvate isomerismCorrect option is C (Coordination isomerism) Explanation: The above examples are of coordination isomerism. It is because there are two coordination spheres in a compound.
Correct option is B Explanation: The change in entropy during hydrogen dissociation is less than zero, which indicates it is negative. Hydrogen dissociation is a non-spontaneous process. As a...
Correct option is C Explanation: Promethium (Pm) is a lanthanoid element with the atomic number 61 and the symbol Pm. Its isotopes are all radioactive. Lanthanoids such as gadolinium, holmium, and...
Correct option is A During the electrolysis of fused sodium chloride with a suitable electrode, the chloride ion is oxidised at the anode. Chloride ions are oxidised to chlorine gas by giving up...
Correct option is C Explanation: Silicate minerals make up more than 90% of the earth's crust. Feldspars and alkali feldspar are the most common silicates. Other common silicate minerals include...
Correct option is C) N-nitroso diethyl amine Explanation: N-nitroso diethylamine is formed by treating diethylamine with nitrous acid. It's a strong carcinogen.
Correct option is B Explanation: Dichromate oxidizes to $\mathrm{Cr}^{+3}$ from $\mathrm{Cr}_{2} \mathrm{O}_{7}^{-2}$ in acidic medium according to the reaction $ \mathrm{Cr}_{2}...
Correct option is C Explanation: The packing efficiency is the fraction of volume occupied by constituent particles in a crystal structure (volume of space occupied by the spheres/total volume). The...
Correct option is A) Propan-1-ol Explanation: Hydrogen bonds have the potential to occur in propan-1-ol. Because of the strongly E.N oxygen atom, intermolecular hydrogen bonding develop in...
B is the correct answer. Explanation: The leaching method concentrates aluminium ore. Impurities in aluminium ore include SiO2, iron oxides, and titanium oxide. The powdered ore is digested with a...
The correct answer is C(2-chlorobutanoic acid) In nature, 2-chlorobutanoic acid is the most acidic. Because there is an electron withdrawing group (Cl) next to the carboxylic acid group (COOH). As a...
![\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a99adc5c7be224b196c9e2507b50069e_l3.png "Rendered by QuickLaTeX.com")
is 
Calculate the percent dissociation of A) 
B) 
C) 78 D) 
Correct option is C Explanation: $ \mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right] \rightarrow 3 \mathrm{~K}^{+}+\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-} $ $ \mathrm{i}=\alpha...
Correct option is C Balz-Schiemann reaction Explanation: The Balz- Schiemann reaction is a chemical reaction in which anilines are transformed to aryl fluorides via diazonium fluoroborates....
and 
represent molar mass of solute, mass of solute and volume of solution in litres respectively, which among following equations is true? A) 
B) 
C) 
D) 
Correct option is C $\pi=\frac{\mathrm{TWR}}{\mathrm{VM}}$ $\pi=$ CRT where $\mathrm{C}$ is the concentration $\mathrm{C}=\frac{\text { Moles of solute }}{\text { volume }}$...
![\left[\mathrm{CO}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{NO}_{2}\right)_{3}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a87a614704284e544285de047b62719a_l3.png "Rendered by QuickLaTeX.com")
A) Triammine trinitrito – N cobalt (III) B) Triammine trinitrito 
cobalt (II) C) Triammine cobalt (III) nitrite D) Triammine trinitrito – N cobaltate (III)Correct option is A) Triammine trinitrito - N cobalt Triammine trinitro-N-cobalt is the correct IUPAC name (III). In this scenario, the groups are ordered alphabetically, with the ammine group...
The best option is A Explanation: An only p-block element's electrical arrangement is uncommon. There is only one atom of that element in each molecule. Because helium has the electrical...
B) 
C) 
D) 
Correct option is B Due to resonance stabilisation between the benzene nucleus and the nitrogen atom, dizonium salts with an aryl group immediately connected to the nitrogen atom are the most...
The correct option is A Explanation: Thermoplastic is a plastic material that can be melted and moulded repeatedly by just heating it. When heated, thermoplastics soften and become moldable. They...
![\mathrm{t}_{\frac{1}{2}}=[\mathrm{A}]_{0} 2 \mathrm{k}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4790aa54d3511e4a28ff21e7043d219a_l3.png "Rendered by QuickLaTeX.com")
B) 
C) ![t_{\frac{1}{2}}=\frac{[\mathrm{A}]_{0}}{2 \mathrm{k}}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3dde05612d23f52ee0597c3f471cfe53_l3.png "Rendered by QuickLaTeX.com")
D) ![\mathrm{t}_{\frac{1}{2}}=\frac{2[\mathrm{~A}]_{0}}{\mathrm{k}}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a9a16937140e0a1a6dbc47fd5be5c9a5_l3.png "Rendered by QuickLaTeX.com")
Correct option is C $\mathrm{t}_{1 / 2}=\frac{\left[\mathrm{A}_{0}\right]}{2 \mathrm{k}}$ The relationship between rate constant and half-life period of zero-order reaction is given by: $...
D is the correct answer. Nb−Ta Chemical twins are those in which some elements' characteristics are identical. Zr and Hf have comparable atomic sizes, as do Nb and Ta. This is related to the...
Correct option is C) ultraviolet light Chlorination of ethane is carried out in the presence of Ultraviolet light to form 1,2-hexachloro ethane.
Correct option is B) Valium A tranquillizer is a medicine that causes people and animals to become calm. Anxiety disorders, alcohol withdrawal symptoms, and muscular spasms are all treated with...
Correct option is B) Manganin Manganin foil and wire are employed in the production of resistors, notably ammeter shunts, due to its low temperature coefficient of resistance and long-term...
Correct option is A $99 \mathrm{~kJ}$ $ \mathrm{V}_{1}=1 \mathrm{~m}^{3} $ Pressure $=100 \mathrm{kPa}=100000 \mathrm{~Pa}$ $\mathrm{V}_{2}=10 \mathrm{dm}^{3}=(1 / 100) \mathrm{m}^{3}$ So, $W=P...
Correct option is B Only three isomers of mono hydroxy derivatives of a hydrocarbon with five carbon atoms and one methyl group as a branch are feasible.
Correct option is (A) The number of moles of solute dissolved in 1dm3 of the solution is called molarity. Molarity = (moles of solute dissoved) / (volume of solvent in dm3) So, answer is...
Correct option is D) ΔT = 0 T=0 for adiabatic expansion. An ideal gas is enclosed in an insulated container and then allowed to expand in a vacuum for adiabatic free expansion. The work done by or...
Correct option is B ($0.1 \mathrm{~h}^{-1}$) We know, $ \mathrm{t}_{\frac{1}{2}}=\frac{0.693}{\mathrm{k}} $ where $t_{\frac{1}{2}}$ is the half life and $\mathrm{k}$ is rate constant. $...
Correct option is B Aldehydic group Glucose reacts with hydrogen cyanide to form glucose cyanohydrin. In this reaction functional group is replaced that is −CHO group.
Correct option is D) tert-butyl methyl ether
B is the correct answer. Pentaerythrityl stearate is a kind of stearic acid. The uncharged, hydrophilic headgroups of non-ionic detergents distinguish them from ionic detergents. Polyoxyethylene or...
Correct option is (A)Disulphurous acid Disulphurous acid(H2S2O5) contain S-S bond in its structure.
on the set 
is defined as ![\[a * b=\left\{\begin{array}{ll} a+b ; & \text { if } a+b<6 \\ a+b-6 ; & \text { if } a+b \geq 6 \end{array}\right.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c98db70d6373f329a8c6803f36eeb4ad_l3.png "Rendered by QuickLaTeX.com")
To find: identity and inverse elementFor a binary operation if a*e = a, then e s called the right identity If $\mathrm{e}^{*} \mathrm{a}=\mathrm{a}$ then $\mathrm{e}$ is called the left identity For the given binary operation,...
, we define 
Show that is commutative but not associative.$\text { let } a=1, b=2 \in N$ $a * b=1^{3}+2^{3}=9$ And $b^{*} a=2^{3}+1^{3}=9$ => ${ }^{*}$ is commutative. Let $c=3$ $\begin{array}{l} (a * b)^{*} c=9 * c=9^{3}+3^{3} \\ a *(b * c)=a...
on 
, defined by 
is neither commutative nor associative.let $\mathrm{a}=1, \mathrm{~b}=0 \in \mathrm{R}-\{-1\}$ $a * b=\frac{1}{0+1}=1$ And $b * a=\frac{0}{1+1}=0$ => $^{*}$ is not commutative. Let $c=3$ $\begin{array}{l} (a * b) * c=1^{*}...
be the set of all positive rational numbers. (iii) Show that * is not associative.(iii) let $c=3$. $(a * h) * c=1.5 *^{*} c=\frac{1}{-}(15+2)=275$ $a *(b * c)=a * \frac{1}{2}(2+3)=1 * 2.5=\frac{1}{2}(1+2.5)=1.75$ hence * is not associative.
for all values ofa, 
(iii) Find the identity element in N. (iv) Find all invertible elements in 
.(iii)let $\mathrm{x} \in \mathrm{N}$ and $\mathrm{x} * 1=\operatorname{lcm}(x, 1)=\mathrm{x}=\operatorname{lcm}(1, \mathrm{x})$ 1 is the identity element. (iv) let there exist $y$ in $n$ such that...
be a binary operation on 
, defined by 
for all a. . Show that * is neither commutative nor associative.To prove: $*$ is neither commutative nor associative Let us assume that * is commutative $\Rightarrow a^{b}=b^{a}$ for all $a, b \in N$ This is valid only for $a=b$ For example take $a=1, b=2$...
on 
defined by 
is not a binary operation.To prove: $*$ is not a binary operation Since, a and $b$ are defined on positive integer set And $\mathrm{a}^{*} \mathrm{~b}=|\mathrm{a}-\mathrm{b}|$ $\Rightarrow a^{*} b=(a-b)$, when $a>b$...
of all integers defined by a 
, find 
$2 * 4$ since, $a^{*} b=a+3 b^{2}$ $\Rightarrow 2 * 4=\left(2+3 \times 4^{2}\right)=2+48=50$
of all rational numbers given as 
for all 
Q. Find 
and 
Is 
$3 * 5$ and $5 * 3$ $a * b=(2 a-b)^{2}$ $\Rightarrow 3 * 5=(6-5)^{2}=1$ $5 * 3=(10-3)^{2}=49$ $\Rightarrow 3 * 5$ is not equal to $5 * 3$
. Find the value of 
given that 
To find: value of $x$ Since, $a * b=\frac{a b}{5}$ $\Rightarrow \mathrm{x} * 5=\frac{5 \mathrm{x}}{5}=\mathrm{x}$ Now $(2 * x)=\frac{2 x}{5}$ $\Rightarrow \frac{2 \mathrm{x}}{5}=10 \Rightarrow...
be a binary operation on the set 
of all integers, defined by a 
Find the value of 
$4 * 5$ $a * b=3 a+4 b-2$ Since, $a=4$ and $b=5$ $\begin{array}{l} \Rightarrow 4 * 5=3 \times 4+4 \times 5-2=12+20-2=30 \\ \Rightarrow 4 * 5=30 \end{array}$
$\begin{array}{l} \sin \left(\frac{\pi}{2}+\frac{\pi}{3}\right) \\ =\sin \left(\frac{5 \pi}{6}\right) \\ =\sin \left(\pi-\frac{\pi}{6}\right) \\ =\sin \frac{\pi}{6} \\ =\frac{1}{2} \end{array}$
$\cos \left\{\pi-\frac{\pi}{6}+\frac{\pi}{6}\right\}$ $\begin{array}{l} =\cos \{\pi\} \\ =\cos \left(\frac{\pi}{2}+\frac{\pi}{2}\right) \\ =-1 \end{array}$
(iv) 
(iii) Let $\tan ^{-1}(-\sqrt{3})=x$ $\Rightarrow-\tan ^{-1}(\sqrt{3})=x\left[\right.$ Formula: $\left.\tan ^{-1}(-x)=-\tan ^{-1}(x)\right]$ $\Rightarrow \sqrt{3}=-\tan x$ $\therefore...
(vii) Let $\operatorname{cosec}^{-1}(\sqrt{2})=x$ $\Rightarrow \sqrt{2}=\operatorname{cosec} x$ $\therefore \mathrm{x}=\frac{\pi}{4}$
(vi) 
(v) Let $\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)=x$ $\Rightarrow \frac{1}{\sqrt{3}}=\tan \mathrm{x}$ [We know which value of $\mathrm{x}$ $\therefore \mathrm{x}=\frac{\pi}{6}$ (vi) Let $\sec...
(iv) 
(iii) Let $\cos ^{-1}\left(\frac{1}{2}\right)=x$ $\Rightarrow \frac{1}{2}=\cos x$ [We know which value of $x$ when put in this expression will give us this result] $\therefore...
(ii) 
(i) Let $\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=x$ $\Rightarrow \frac{\sqrt{3}}{2}=\sin x$ => $\therefore \mathrm{x}=\frac{\pi}{3}$ (ii) Let $\sin ^{-1}\left(\frac{1}{2}\right)=x$ $\Rightarrow...
Given sin-1 (2a/ 1+ a2) + sin-1 (2b/ 1+ b2) = 2 tan-1 x
(ii) 
$(i)$ $(ii)$
(x) 
$(ix)$ $(x)$
(a) 4 (b) 5 (c) 3 (d) 2 Answer: (c) 3 We have: a5 = 20 and a7 + a11 = 64. Let a be the first term and d be the common difference of the AP. Then, Thus, the common difference of the AP is...
(viii) 
$(vii)$ $(viii)$
(vi) 
$(v)$ $(vi)$
(iv) 
$(iii)$ $(iv)$
(ii) 
$(i)$ $(ii)$
(iv) 
$(iii)$ $(iv)$
(a) 50 (b) -50 (c) 30 (d) -30 Answer: (b) -50
(ii) 
$(i)$ $(ii)$
(a) (3n + 8) (b) (4n – 7) (c) (15 – 2n) (d) (2n – 15) Answer: (d) (2n – 15)
Since, cos-1 (a/x) – cos-1 (b/x) = cos-1 (1/b) – cos-1 (1/a)
(a) (6n - 2) (b) (7n – 3) (c) (8n – 2) (d) (8n + 2) Answer: (c) (8n – 2)
Given Cos (sin -1 3/5 + sin-1 5/13)
(a) ( 5 - 2n) (b) ( 6 – 2n) (c) (2n – 5) (d) (2n – 6) Answer: (b) ( 6 – 2n)
(a) 6 (b) 9 (c) 15 (d) -3 Solution: (a) b
$(iii)$
(a) 6n+3 (b) 15 (c) 12 (d) 21 Solution: (b) 15
(ii) 
$(i)$ $(ii)$ LHS
(a) 19 (b) 23 (c) 22 (d) cannot be determined Solution: (c) 22
, find Since, $\cos ^{-1} x+\sin ^{-1} x=\pi / 2$ => $\cos ^{-1} x=\pi / 2-\sin ^{-1} x$ Substituting this in $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}=17 \pi^{2} / 36$ $\left(\sin...
is$ (a)\,\sqrt{84} $ $ (b)\,\sqrt{98} $ $ (c)\,\sqrt{70} $ $ (d)\,\sqrt{112} $ Solution: $ (d)\,\sqrt{112} $ The AP is $ \sqrt{7},\sqrt{28},\sqrt{63},...... $ $ =\sqrt{7},\sqrt{4\times...
$\cot \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=0$ => $\begin{array}{l} \left(\cos ^{-1} 3 / 5+\sin ^{-1} x\right)=\cot ^{-1}(0) \\ \left(\operatorname{Cos}^{-1} 3 / 5+\sin ^{-1} x\right)=\pi /...
Given sin-1 x + sin-1 y = π/3 ……. (i) And cos-1 x – cos-1 y = π/6 ……… (ii)
Since, cos-1 x + cos-1 y = π/4
$(v)$ $=>0$
(a)1/3 (b)-1/3 (c) b (d) –b Solution: (d) -b
for 
(iv) Cot 
$(iii)$ $(iv)$
(ii) for 
$(i)$ $(ii)$
(a) p (b) –p (c) -1 (d) 1 Solution: (c) -1
$(iii)$
(ii) Sec 
$(i)$ $(ii)$
$(ix)$ .
(viii) 
$(vii)$ $(viii)$
(vi) 
(v) \[\begin{array}{*{35}{l}} {} \\ Let\text{ }co{{s}^{-1}}\left( 8/17 \right)\text{ }=\text{ }y \\ cos\text{ }y\text{ }=\text{ }8/17\text{ }where\text{ }y\text{ }\in \text{ }\left[ 0,\text{ }\pi...
(iv) 
(iii) (iv)
(ii) 
(i) \[\begin{array}{*{35}{l}} Given\text{ }sin\text{ }\left( si{{n}^{-1}}~7/25 \right) \\ let\text{ }y\text{ }=\text{ }si{{n}^{-1}}~7/25 \\ sin\text{ }y\text{ }=\text{ }7/25\text{ }where\text{...
f(x) = mx + c, Checking the differentiability at x = 0 This is the derivative of a function at x = 0, and also this is the derivative of this function at every value of x.
, find f’ (4).f(x) = x3 + 7x2 + 8x – 9, => Checking the differentiability at x = 4
, Ø’ (5) = 97, find λ.Finding the value of λ given in the real function and we are given with the differentiability of the function f(x) = λx2 + 7x – 4 at x = 5 which is f ‘(5) = 97 =>
![\[~\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{9}{{\mathbf{x}}^{\mathbf{2}~}}+\text{ }\mathbf{12}\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{9}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c388eec2224700cda456bd12406ee5c9_l3.png "Rendered by QuickLaTeX.com")
We are given with a polynomial function f(x) = 2x3 – 9x2 + 12x + 9, and we have
– 4x + 7, show that f’ (5) = 2 f’ (7/2)Since, a polynomial and a constant function is continuous and differentiable everywhere => f(x) is continuous and differentiable for x ∈ (-1, 0) and x ∈ (0, 1) and (1, 2). Checking continuity...
is defined as follows 
Is continuous at 
, but not differentiable thereat.Since, LHL = RHL = f (2) Hence, F(x) is continuous at x = 2 Checking the differentiability at x = 2 $=> 5$ Since, (RHD at x = 2) ≠ (LHD at x = 2) Hence, f (2) is not differentiable at x =...
checking differentiability of given function at x = 3 => LHD (at x = 3) = RHD (at x = 3) = 12 Since, (LHD at x = 3) = (RHD at x = 3) Hence, f(x) is differentiable at x = 3.
is not differentiable at x = 0.Since, LHD and RHD does not exist at x = 0 Hence, f(x) is not differentiable at x = 0
Correct answer is C.
is connected across the terminal of a cell of e.m.f. ‘ 
‘ the current is ‘ 
‘. When the resistance is changed to 
‘ the current is ‘ 
‘. The internal resistance of the cell is
Correct answer is C. $\begin{array}{l} I_{1}=\frac{E}{r+R_{1}} \\ I_{2}=\frac{E}{r+R_{2}} \\ \frac{I_{1}}{I_{2}}=\frac{r+R_{2}}{r+R_{1}} \\ I_{1} r+I_{1} R_{1}=I_{2} r+I_{2} R_{2} \\...
![\left[L^{2} M^{1} T^{-1}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e3ae8151d351e21d072dc99a517680a5_l3.png "Rendered by QuickLaTeX.com")
![\left[L^{2} M^{2} T^{-2}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-570bf7fd6b73030ec948ef99c8b75448_l3.png "Rendered by QuickLaTeX.com")
![\left[L^{2} M^{1} T^{1}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8a2e9f8d186a5b9fceddc341575a2dfb_l3.png "Rendered by QuickLaTeX.com")
![\left[L^{2} M^{1} T^{-2}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6d349a07b4098874e03d0238509c1b23_l3.png "Rendered by QuickLaTeX.com")
Correct answer is A.
Correct answer is D.
, when mass of 
is attached to the lower end. Energy gained by the wire is (Take 
)
Correct answer is B
‘. The respective forces exerted by water on the bases are ‘ 
and respective weights are 
. Then
Correct answer is B.
Correct answer is A.
and 
. The resonating length of the air column in its fundamental mode will be (velocity of sound in air 
Correct answer is C. $\begin{array}{l} 2 n=n_{2}-n_{1} \\ =350-250=100 \\ n=\frac{100}{2}=50 H z \\ n=\frac{v}{4 l_{0}}=\frac{340}{4 \times l_{0}} \\ \therefore l=\frac{340}{4 \times...
Correct answer is C.
from the centre of magnet is
Correct answer is B. Magnetic induction at axial point $=\overrightarrow{\mathrm{B}}$ axial $=\frac{\mu \circ}{4 \pi} \frac{2 \overrightarrow{\mathrm{m}}}{\mathrm{r}^{3}}$ Magnetic induction at...
Correct answer is B.
‘. The magnetic induction at the midway between them is 
permeability of free space)
Correct answer is A. $\begin{array}{l} \mathrm{x}=\frac{\mathrm{r}}{2} \\ \mathrm{~B}_{1}=-\frac{\mu_{0} \mathrm{i}}{2 \pi \mathrm{x}} \\ \mathrm{B}_{2}=\frac{\mu_{0} \mathrm{i}}{2...
along the direction of 
is
Correct answer is C.
‘ at potential 
‘. The charge on capacitor is increased to ‘ 
‘ so that its potential increases to 
, then the difference between final and initial energy stored in capacitor is
Correct answer is B.
refractive index of oil and angle made by the object at the aperture = 
)
Correct answer is D.
Correct answer is A.
weight is suspended from a weightless spring and it has period ‘ 
. If now ‘ 
weight is suspended from the same spring, the new period will be
Correct answer is B. $\begin{array}{l} \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \\ \mathrm{T}_{1}=\mathrm{T} \\ \mathrm{m}_{1}=1 \mathrm{~kg} \\ \mathrm{~m}_{2}=4 \mathrm{~kg} \\...
and a length of 
. If the horizontal circle traced by the bob has a diameter of 
, then the tension in the string would be nearly 
Correct answer is D.
and 
respectively. The distance between two coherent sources is
Correct answer is C. When, 1) magnified, distance between images, $\mathrm{d}_{1}=2.4$ 2) dimnished, distance between images, $\mathrm{d}_{2}=0.6$ $\therefore$ Distance between virtual sources...
Correct answer is B. $\begin{array}{l} \mathrm{n}=\mathrm{n}_{0} \mathrm{e}^{-\alpha t} \\ \alpha_{\mathrm{A}}=\ln 2 / 20 \\ \alpha_{\mathrm{B}}=\ln 2 / 40 \end{array}$ After 80 min, remaining...
Correct answer is C.
![\left[L^{-1} M^{0} T^{0} I^{1}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-61bc00b014420d451f210b1b6d369577_l3.png "Rendered by QuickLaTeX.com")
![\left[L^{-1} M^{0} T^{0} I^{-1}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c460b6b0a8d363b4b9c32706e423f306_l3.png "Rendered by QuickLaTeX.com")
![\left[L^{-1} M^{1} T^{1} I^{-1}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9ab8d27e17f1706b0fe994dd5197b794_l3.png "Rendered by QuickLaTeX.com")
Correct answer is A.
and 
. The number of beats produced per minute by these two waves isCorrect answer is B. $\begin{array}{l} \mathrm{Y}_{1}=4 \sin (500 \pi \mathrm{t}) \\ \mathrm{Y}_{2}=2 \sin (506 \pi \mathrm{t}) \\ \mathrm{f}_{1}=\frac{\omega_{1}}{2 \pi}=\frac{500 \pi}{2 \pi}=250...
, the diameter of the tenth orbit is
Correct answer is C. Radius of an atom is directly proportional to $n^{2}$ and radius of $n$th orbit which is equal to radius of first orbit multiplied by $n^{2}$ So that will give you. Diameter of...
‘ is in unison with a tuning fork of frequency’n’. When the vibrating length of the wire is reduced to ‘ 
‘, it produces ‘ 
‘ beats per second with the fork. The frequency of the fork is
Correct answer is D. For the length $L_{1}, n_{1}=\frac{1}{2 L_{1}} \sqrt{\frac{T}{m}}=n \ldots(1)$ where $m=$ mass per unit length For the length $L_{2}, n_{2}=\frac{1}{2 L_{2}} \sqrt{\frac{T}{m}}...
Correct answer is A.
. The distance travelled by it between 1 st and 2 nd second of its motion is (A = amplitude of S. H. M.)
Correct answer is C. In $2 \mathrm{~s}$ (which is equal to $\frac{T}{4}$ ), one amplitude will be convered. In $1 \mathrm{st}$ second $x=a \sin \left(\frac{\pi}{4}\right)=\frac{a}{\sqrt{2}}$...
‘ is the radius of bore of tube then
Correct answer is B. We have, $\mathrm{h}=\frac{2 \sigma \cos \theta}{\mathrm{r} \rho \mathrm{g}}$ $\Rightarrow \mathrm{h} \cdot \mathrm{r}=$ constant
Correct answer is A.
‘ is enclosed by a Gaussian spherical surface of radius ‘R’. If, now, radius of the spherical surface is doubled the outward electric flux willCorrect answer is B. Method 1: Using Number of Field Lines - We know that, Electric flux is proportional to number of field lines. - Form figure, we see that the number of field lines crossing both...
when the body is at 
above the temperature of surroundings. When the body is at 
above the temperature of surroundings, its rate of cooling will be
Correct answer is C. Rate of cooling $\frac{\mathrm{dT}}{\mathrm{dt}}=-\frac{\mathrm{k}}{\mathrm{ms}}\left(\mathrm{T}-\mathrm{T}_{\mathrm{s}}\right)$ Or $2=\frac{-\mathrm{k}}{\mathrm{ms}}(50)$ Or...