Class 10

### Prove that the function 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,...

### The length of a rectangle is thrice as long as the side of a square. The side of the square is , 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...

### Let and . 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...

### A motor boat whose speed in still water is , 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...

### A train travels at a certain average speed for a distanced of 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...

### A train covers a distance of 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...

### Define each of the following: (i) bijective function (ii) many – one function Give an example of each type of functions.

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...

### Define a function. What do you mean by the domain and range of a function? Give examples.

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’...

### On the set S of all real numbers, define a relation R = {(a, b) : a ≤ b}. Show that R is (i) reflexive (ii) transitive

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)...

### Let S be the set of all sets and let R = {(A, B) : A ⊂ B)}, i.e., A is a proper subset of B. Show that R is (i) not symmetric.

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...

### Which among the following functional groups has been given the highest priority while assigning R-S configuration? A) 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...

### Let R = {(a, b) : a, b ∈ N and b = a + 5, a < 4}. Find the domain and range of R.

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}

### Let 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) =...

### Let R = {(a, b) : b = |a – 1|, a ∈ Z and la| < 3}. Find the domain and range of 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}

### Let R = (x, y) : x + 2y = be are relation on N. Write the range of R.

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...

### Let R = { : 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}

### Find the domain and range of the relation R = {(-1, 1), (1, 1), (-2, 4), (2, 4)}.

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,...

### Which among the following group 15 element forms most stable pentavalent compound? A) Phosphorus B) Antimony C) Bismuth D) Arsenic

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...

### The process in which metal surface is made inactive is called A) Passivation B) Galvanizing C) Corrosion D) Pickling

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....

### are the examples of what type of isomerism ? A) Ionisation isomerism B) Linkage isomerism C) Coordination isomerism D) Solvate isomerism

Correct option is C (Coordination isomerism) Explanation: The above examples are of coordination isomerism. It is because there are two coordination spheres in a compound.

### For which among the following reactions, change in entropy is less than zero ? A) Sublimation of Iodine B) Dissociation of Hydrogen C) Formation of water D) Thermal decomposition of Calcium Carbonate

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...

### The only radioactive element among the lanthanoids is A) Gadolinium B) Holmium C) Promethium D) Neodynium

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...

### The overall reaction taking place at anode during electrolysis of fused sodium chloride using suitable electrode is A) Oxidation of chloride B) Reduction of sodium ions C) Reduction of chlorine D) Oxidation of Sodium atoms

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...

### What is the most abundant element on earth ? A) Hydrogen B) Nitrogen C) Oxygen D) Silicon

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...

### Diethyl amine when treated with nitrous acid yields A) Diethyl ammonium nitrite B) Ethyl alcohol C) N-nitroso diethyl amine D) Triethyl ammonium nitrite

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.

### Replacement of diazonium group by fluorine is known as A) Gattermann reaction B) Sandmeyer reaction C) Balz-Schiemann reaction D) Etard reaction

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....

### If 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 $\mathrm{C}=\frac{\text { Moles of solute }}{\text { volume }}$...

### The correct IUPAC name of 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...

### Electronic configuration of only one P block element is exceptional. One molecule of that element consists of how many atoms of it ? A) One B) Two C) Three D) Four

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...

### Which of the following is the most stable diazonium salt ? A) 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...

### Which polymer among the following polymers does NOT soften on heating ? A) Bakelite B) Polythene C) Polystyrene D) PVC

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...

### What is the possible number of monohydroxy derivatives of a hydrocarbon consisting of five carbon atoms with one methyl group as a branch ? A) 2 B) 3 C) 4 D) 5

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.

### Molarity is defined as A) the number of moles of solute dissolved in one dm3 of the solution B) the number of moles of solute dissolved in 1 kg of solvent C) the number of moles of solute dissolved in 1 dm3 of the solvent D) the number of moles of solute dissolved in 100 ml of the solvent

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...

### Which among the following is a feature of adiabatic expansion ? A) ΔV < 0 B) ΔU < 0 C) ΔU > 0 D) ΔT = 0

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...

### Let 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 $b=5$ $\begin{array}{l} \Rightarrow 4 * 5=3 \times 4+4 \times 5-2=12+20-2=30 \\ \Rightarrow 4 * 5=30 \end{array}$

### Evaluate

$\begin{array}{l} \sin \left(\frac{\pi}{2}+\frac{\pi}{3}\right) \\ =\sin \left(\frac{5 \pi}{6}\right) \\ =\sin \left(\pi-\frac{\pi}{6}\right) \\ =\sin \frac{\pi}{6} \\ =\frac{1}{2} \end{array}$

### Evaluate

$\cos \left\{\pi-\frac{\pi}{6}+\frac{\pi}{6}\right\}$ $\begin{array}{l} =\cos \{\pi\} \\ =\cos \left(\frac{\pi}{2}+\frac{\pi}{2}\right) \\ =-1 \end{array}$

(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] $\therefore... read more ### Find the principal value of : (i) (ii) (i) Let$\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

### Prove that: (i) (ii)

$(i)$   $(ii)$

### Prove the following results: (ix) (x)

$(ix)$ $(x)$

### The 5th term of an AP is 20 and the sum of its 7th and 11th terms is 64. The common difference of the AP is

(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...

### Prove the following results: (vii) (viii)

$(vii)$ $(viii)$

### Prove the following results: (v) (vi)

$(v)$ $(vi)$

### Prove the following results: (iii) (iv)

$(iii)$ $(iv)$

### Prove the following results: (i) (ii)

$(i)$ $(ii)$

### Evaluate the following: (iii) (iv)

$(iii)$ $(iv)$

### The 5th term of an AP is -3 and its common difference is -4. The sum of the first 10 terms is

(a) 50 (b) -50 (c) 30 (d) -30 Answer: (b) -50

### Evaluate the following: (i) (ii)

$(i)$ $(ii)$

### The 7th term of an AP is -1 and its 16th term is 17. The nth term of the AP is

(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)

### The sum of the first n terms of an AP is 4n2 + 2n.The nth term of this AP is

(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)

### The sum of first n terms of an AP is (5n-n2) The nth term of the AP is

(a) ( 5 - 2n) (b) ( 6 – 2n) (c) (2n – 5) (d) (2n – 6) Answer: (b) ( 6 – 2n)

### The sum of the first n terms of an AP is (3n + 6n). The common difference of the AP is

(a) 6                  (b) 9               (c) 15                  (d) -3 Solution: (a) b

### Prove the following results: (iii)

$(iii)$

### If the nth term of an AP is (2n + 1) then the sum of its first three terms is

(a) 6n+3 (b) 15 (c) 12 (d) 21 Solution: (b) 15

### Prove the following results: (i) (ii)

$(i)$ $(ii)$ LHS

### If 4, x1,x2, x3, 28 are in AP then x3 =?

(a) 19 (b) 23 (c) 22 (d) cannot be determined Solution: (c) 22

Correct answer is B. $\begin{array}{l} \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \\ \mathrm{T}_{1}=\mathrm{T} \\ \mathrm{m}_{1}=1 \mathrm{~kg} \\ \mathrm{~m}_{2}=4 \mathrm{~kg} \\... read more ### A conical pendulum has bob of mass of and a length of . If the horizontal circle traced by the bob has a diameter of , then the tension in the string would be nearly A. B. C. D. Correct answer is D. read more ### In Biprism experiment the distance between the two virtual images of the slits in the magnified and diminished position are and respectively. The distance between two coherent sources isA. B. C. D. Correct answer is C. When, 1) magnified, distance between images,$\mathrm{d}_{1}=2.4$2) dimnished, distance between images,$\mathrm{d}_{2}=0.6\therefore$Distance between virtual sources... read more ### Half-lives of two radioactive elements A and B are 20 minute and 40 minute respectively. Initially the samples have equal nuclei. After 80 minute, the ratio of decayed number of A and B nuclei will beA. B. C. D. Correct answer is B.$\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}$After 80 min, remaining... read more ### The S.I. unit of impulse is the same as that ofA. linear velocity.B. angular velocity.C. linear momentum.D. angular momentum. Correct answer is C. read more ### Dimensions of magnetic intensity areA. B. C. D. Correct answer is A. read more ### Two vibrating tuning forks produce waves which are represented by and . The number of beats produced per minute by these two waves isA. 60B. 180C. 360D. 3 Correct answer is B.$\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...

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...