Solution: The distance between the two points (x1,y1) ( x2,y2) : $d=\surd ({{x}_{2}}-{{x}_{1}}){}^\text{2}+({{y}_{2}}-{{y}_{1}}){}^\text{2}$ The distance between A (–3, –14) and B (a, –5): $=\surd...
An ideal gas is permitted to grow against a consistent strain of 2 bar from 10 L to 50 L in one stage. Compute the measure of work done by the gas. In the event that a similar development were done reversibly, will the work is done be higher or lower than the prior case?
solution: \[\begin{array}{*{35}{l}} Measure\text{ }of\text{ }work\text{ }done\text{ }=\text{ }-\text{ }pext\text{ }V \\ ~ \\ =\text{ }\text{ }2\text{ }bar\text{ }\times \text{ }\left(...
What type of a quadrilateral do the points A (2, –2), B (7, 3), C (11, –1) and D (6, –6) taken in that order, form?
Solution: A (2, –2), B (7, 3), C (11, –1) and D (6, –6) are the given points. Now using the distance formula, $d~=\text{ }\surd \text{ }({{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{...
The enthalpy of response for the response :
What will be standard enthalpy of development of H2O (l)?
solution: For the given response : \[2H2\left( g \right)\text{ }+\text{ }O2\left( g \right)\text{ }\to 2H2O\left( l \right)\] the standard enthalpy of response is \[Hr\Theta \text{ }=\text{...
The enthalpy of vapourisation of CCl4 is 30.5 kJ mol–1. Compute the warmth needed for the vapourisation of 284 g of CCl4 at steady tension.
solution: The enthalpy of vapourisation is given \[\begin{array}{*{35}{l}} for\text{ }1\text{ }mole\text{ }of\text{ }CCl4\text{ }=\text{ }30.5\text{ }kJ\text{ }mol1 \\ ~ \\...
The net enthalpy change of a response is the measure of energy needed to break every one of the bonds in reactant atoms less the measure of energy needed to shape every one of the bonds in the item particles. What will be the enthalpy change for the accompanying response?
Considering that Bond energy of H2, Br2 and HBr is 435 kJ mol–1, 192 kJ mol–1 and 368 kJ mol–1 separately.
solution: For the response \[H2\left( g \right)\text{ }+\text{ }Br2\left( g \right)\text{ }\to 2HBr\left( g \right)\] \[\begin{array}{*{35}{l}} Enthalpy\text{ }change \\ ~ \\ =\text{...
On the off chance that the burning of 1g of graphite produces 20.7 kJ of warmth, what will be molar enthalpy change? Offer the meaning of the hint too.
solution: The warmth of ignition ∆Hc of graphite (for example carbon) is given as \[=\text{ }20.7\text{ }kJ\] for 1g of graphite (C). \[1\text{ }mole\text{ }of\text{ }Carbon\text{...
Find the points on the x–axis which are at a distance of 2√5 from the point (7, –4). How many such points are there?
Solution: (x, 0) = Let coordinates of the point (given that the point lies on x axis) ${{x}_{1}}=7.\text{ }{{y}_{1}}=-4$ ${{x}_{2}}=x.\text{ }{{y}_{2}}=0$ Distance $=\surd...
The contrast among CP and CV can be inferred utilizing the experimental connection
. Work out the distinction among CP and CV for 10 moles of an optimal gas.
solution: For an optimal gas, the distinction between these two \[is\text{ }CP\text{ }\text{ }CV\text{ }=\text{ }nR,\] the all inclusive gas consistent and \[where\text{ }n=\text{ }no.\text{...
Name the type of triangle formed by the points A (–5, 6), B (–4, –2) and C (7, 5).
Solution: A (–5, 6), B (–4, –2) and C (7, 5) are the given points. Now, using the distance formula, $d~=\text{ }\surd \text{ }({{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{...
Warmth limit (Cp) is a broad property yet explicit warmth (c) is an escalated property. What will be the connection between Cpand c for 1 mol of water?
solution: \[\begin{array}{*{35}{l}} For\text{ }water,\text{ }molar\text{ }warmth\text{ }limit\text{ }=\text{ }18\text{ }x\text{ }Specific\text{ }warmth\text{ }or \\ ~ \\ Cp~=\text{...
Development of gas in a vacuum is called free extension. Ascertain the work is done and the adjustment of inner energy when 1 liter of an ideal gas grows isothermally into a vacuum until its complete volume is 5 liter?
solution: Work done in vacuum is determined by : \[\begin{array}{*{35}{l}} -\text{ }w\text{ }=\text{ }Text\text{ }\left( Vinitial\text{ }\text{ }Vfinal\text{ } \right) \\ ~ \\...
State whether the following statements are true or false. Justify your answer. Points A (4, 3), B (6, 4), C (5, –6) and D (–3, 5) are the vertices of a parallelogram.
Solution: The statement given in the question is false. Justification: A (4, 3), B (6, 4), C (5, –6) and D (–3, 5) are the points given. We need to find the distance between A and B...
Despite the fact that warmth is a way work yet warms consumed by the framework under certain particular conditions is free of way. What are those conditions? Clarify.
solution: Warmth is free of the way under 2 conditions: At the point when the volume of the framework is kept steady By first law of thermodynamics: ...
State whether the following statements are true or false. Justify your answer. Points A (3, 1), B (12, –2) and C (0, 2) cannot be the vertices of a triangle.
Solution: The statement given in the question is true. Justification: Coordinates of A $=\text{ }({{x}_{1}},\text{ }{{y}_{1}})\text{ }=\text{ }\left( 3,\text{ }1 \right)$ Coordinates of B $=\text{...
State whether the following statements are true or false. Justify your answer. Point P (0, 2) is the point of intersection of y–axis and perpendicular bisector of line segment joining the points A (–1, 1) and B (3, 3).
Solution: The statement given in the question is false. Justification: We know that the points on the perpendicular bisector of the line segment joining two points are equidistant from the two...
Foresee the adjustment of inward energy for a secluded framework at steady volume.
solution: For a disengaged framework \[q=0\text{ }and\text{ }w=0\] What's more, as indicated by first law of thermodynamics: \[U=\text{ }q\text{ }+\text{ }w\text{ }\left(...
State whether the following statements are true or false. Justify your answer. The points (0, 5), (0, –9) and (3, 6) are collinear.
Solution: The statement given in the question is false. Justification: If the area of a triangle formed by its points equals 0, then the points are collinear. Provided, ${{x}_{1}}~=\text{ }0,\text{...
State whether the following statements are true or false. Justify your answer. Point P (– 4, 2) lies on the line segment joining the points A (– 4, 6) and B (– 4, – 6).
Solution: The statement given in the question is true. Justification: Equation of the line conatining the points A and B using the two-point form is, $\frac{y-6}{x+4}=\frac{-6-6}{-4+4}$...
State whether the following statements are true or false. Justify your answer. △ABC with vertices A (–2, 0), B (2, 0) and C (0, 2) is similar to △DEF with vertices D (–4, 0) E (4, 0) and F (0, 4).
Solution: The statement given in the question is true. Justification: Distance formula, $d~=\text{ }\surd \text{ }({{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{...
Which amount out of ΔrG and ΔrGΘwill be zero at harmony?
solution: Gibbs energy for a response in which all reactants and items are in standard state. ΔrG° is identified with the harmony consistent of the response as follows ...
The molar enthalpy of vapourisation of CH3)2CO is not exactly that of water. Why?
solution: Water has solid hydrogen bonds and the high extremity likewise accumulates in coming about it to bubble at higher temperatures. Thus water has a higher molar enthalpy than CH3)2CO....
Distinguish the state capacities and way works out of the accompanying : enthalpy, entropy, heat, temperature, work, free energy.
solution: State capacities: enthalpy, entropy, temperature and free energy. Way works: Heat and work
The standard molar entropy of H2O (l ) is 70 J K–1 mol–1. Will the standard molar entropy of H2O(s) be more, or under 70 J K–1 mol–1?
solution: The standard molar entropy of H20 (1) is 70 J K-1 mol-1. The strong type of H20 is ice. In ice, atoms of H20 are less irregular than in fluid water. In this way, molar entropy of...
As warm harmony complies with the zeroth law of thermodynamics, temperature of framework and environmental elements will be a similar when they are in warm balance.
solution: For the given response \[N2O4\text{ }\left( g \right)\leftrightharpoons 2NO2\text{ }\left( g \right)\text{ }the\text{ }worth\text{ }of\text{ }Kp\text{ }=\text{ }0.98.\]...
Expansion in enthalpy of the environmental elements is equivalent to the lessening in enthalpy of the framework. Will the temperature of the framework and environmental elements be a similar when they are in warm balance?
solution: As warm balance submits to the zeroth law of thermodynamics, temperature of framework and environmental factors will be a similar when they are in warm balance.
The warmth affects a framework and temperature is the proportion of normal tumultuous movement of particles in the framework. Compose the numerical connection which relates these three boundaries.
solution: The numerical connection which relates these three boundaries is \[\Delta S\text{ }=\text{ }qrev/T\] where ΔS is the adjustment of entropy and T represents...
. Utilize the accompanying information to ascertain Δlattice Hθfor NaBr.
solution: Sublimation of the metal(ΔsubHΘ) →Ionization of the metal (ΔiHΘ) →Dissociation of the non-metal (ΔdissHΘ) →Gain of electrons by the non-metal(ΔegHΘ) \[\Delta f\text{ }H\theta...
The enthalpy of atomisation for the response
. What is the bond energy of the C–H bond?
solution: For 1 C-H bond, the bond energy will be equivalent to 1/4 that of the enthalpy of atomisation \[=\text{ }\left( 1665/4 \right)\text{ }=\text{ }416.25\text{ }kJ\text{...
. Enthalpy is a broad property. As a rule, if the enthalpy of a general response A→B along one course is Δr H and Δr H1, ΔrH2, ΔrH3 … .. address enthalpies of middle responses prompting item B. What will be the connection between ΔrH for generally speaking response and ΔrH1, ΔrH2… .. and so forth for moderate responses.
solution: For the response, A→B the development of B goes through a few middle of the road responses with various enthalpy esteems Δr H1, ΔrH2, ΔrH3… .., and the general enthalpy change is Δr...
. The worth of
. Compute the enthalpy change for the accompanying response :
solution: Enthalpy change of a response is determined as : Σbond enthalpy of reactants-Σbond enthalpy of items for the deterioration \[\begin{array}{*{35}{l}} 2NH3\left( g...
Standard molar enthalpy of arrangement, Δf Hθis simply a unique instance of enthalpy of response, Δr Hθ. Is the Δr Hθfor the accompanying response same as Δf Hθ? Offer the justification behind your response.
solution: The given response \[CaO\left( s \right)\text{ }+\text{ }CO2\left( g \right)\text{ }\to CaCO3\left( s \right)\] is demonstrating that it is happening in the standard type of 1 mole...
. One mole of CH3)2CO requires less warmth to disintegrate than 1 mol of water. Which of the two fluids has a higher enthalpy of vapourisation?
solution: Among the two fluids, water has a higher enthalpy of vapourisation (burning-through higher warmth energy). Thusly, ∆Hvapourisation (water) > ∆Hvapourisation...
18.0 g of water totally vapourises at 100°C and 1 bar pressure and the enthalpy change in the process is 40.79 kJ mol–1. What will be the enthalpy change for vapourising two moles of water under similar conditions? What is the standard enthalpy of vapourisation for water?
solution: Enthalpy change of vapourisation for \[1\text{ }mole\text{ }=\text{ }40.79\text{ }kJ\text{ }mol1\] enthalpy change of vapourisation for \[2\text{ }moles\text{ }of\text{ }water\text{...
. Think about the accompanying response among zinc and oxygen and pick the right alternatives out of the choices given underneath :
(i) The enthalpy of two moles of ZnO is not exactly the absolute enthalpy of two moles of Zn and one mole of oxygen by 693.8 kJ. (ii) The enthalpy of two moles of ZnO is more than the absolute enthalpy of two moles of Zn and one mole of oxygen by 693.8 kJ. (iii) 693.8 kJ mol–1 energy is advanced in the response. (iv) 693.8 kJ mol–1 energy is caught up in the response.
solution: Choice (I) and (iii) are the appropriate responses
. For an optimal gas, crafted by reversible extension under isothermal condition can be determined by utilizing the articulation
An example containing 1.0 mol of an ideal gas is extended isothermally and reversibly to multiple times of its unique volume, in two separate tests. The extension is completed at 300 K and 600 K separately. Pick the right alternative. (I) Work done at 600 K is multiple times the work done at 300 K. (ii) Work done at 300 K is double the work done at 600 K. (iii) Work done at 600 K is double the work done at 300 K. (iv) ∆U = 0 in the two cases.
solution: Alternative (iii) and (iv) are the appropriate responses. work done at 600 K is double the work done at 300 K. Since each case includes isothermal extension of an optimal gas, there...
The immediacy implies, having the capacity to continue without the help of an outer organization. The cycles which happen immediately are (I) stream of warmth from colder to hotter body. (ii) gas in a compartment contracting into one corner. (iii) gas extending to fill the accessible volume. (iv) consuming carbon in oxygen to give carbon dioxide.
solution: Alternative (iii) and (iv) are the appropriate responses. Gas grows or diffuses in accessible space suddenly, e.g., spillage of cooking gas gives smell of ethyl mercaptan...
In an exothermic response, heat is advanced, and the framework loses warmth to the encompassing. For such a framework (I) qp will be negative (ii) ∆rH will be negative (iii) qp will be positive (iv) ∆rH will be positive
solution: Choice (I) and (ii) are the appropriate responses. For an exothermic response\[,\text{ }qp~=\text{ }-\text{ }ve,\text{ }\gamma H\text{ }=\text{ }-\text{ }ve\]
. Thermodynamics essentially manages (I) interrelation of different types of energy and their change from one structure to another. (ii) energy changes in the cycles which rely just upon starting and last conditions of the minute frameworks containing a couple of particles. (iii) how and at what rate these energy changes are done. (iv) the framework in harmony state or moving from one balance state to another harmony state.
solution: Alternative (I) and (iv) are the appropriate responses. Thermodynamics manages interrelation of different types of energy and their change into one another. It additionally manages...
. Which of coming up next isn’t right? (I) ∆G is zero for a reversible response (ii) ∆G is positive for an unconstrained response (iii) ∆G is negative for an unconstrained response (iv) ∆G is positive for a non-unconstrained response
solution: Alternative (ii) is the appropriate response. ∆G gives a basis for suddenness at consistent strain and temperature. (I) If ∆G is negative (< 0). the cycle is...
. Enthalpy of sublimation of a substance is equivalent to (I) enthalpy of combination + enthalpy of vapourisation (ii) enthalpy of combination (iii) enthalpy of vapourisation (iv) double the enthalpy of vapourisation
solution: Choice (I) is the appropriate response. Enthalpy of sublimation of a substance is equivalent to enthalpy of combination + enthalpy of vapourisation. Sublimation is immediate...
The enthalpies of components in their standard states are taken as nothing. The enthalpy of arrangement of a compound (I) is consistently negative (ii) is consistently sure (iii) possibly certain or negative (iv) is rarely negative
solution: Choice (iii) is the appropriate response. Warmth of arrangement of a compound might be positive or negative.
. Consider the responses given beneath. Based on these responses discover which of the arithmetical relations given in choices (I) to (iv) is right?
solution: Choice (iii) is the appropriate response. x > y because same bonds are formed in reactions (i) and (ii) but bonds between reactant molecules are broken only in reaction...
Based on thermochemical conditions (a), (b) and (c), discover which of the logarithmic connections given in alternatives (I) to (iv) is right.
solution: Choice (iii) is the appropriate response. \[\begin{array}{*{35}{l}} \left( a \right)\text{ }C\text{ }\left( graphite \right)\text{ }+\text{ }O2\text{ }\left( g \right)\text{ }\to...
The entropy change can be determined by utilizing the articulation
At the point when water freezes in a glass container, pick the right assertion among the accompanying : (I) ∆S (framework) diminishes however ∆S (environmental factors) stays as before. (ii) ∆S (framework) increments yet ∆S (environmental elements) diminishes. (iii) ∆S (framework) diminishes yet ∆S (environmental elements) increments. (iv) ∆S (framework) diminishes and ∆S (environmental factors) likewise diminishes.
solution: Alternative (iii) is the appropriate response. During the method involved with freezing energy is released,which is consumed by the environmental factors. Therefore,the entropy off...
Hydrogen bonds are formed in many compounds e.g., H2O, HF, NH3. The boiling point of such compounds depends to a large extent on the strength of hydrogen bond and the number of hydrogen bonds. The correct decreasing order of the boiling points of the above compounds is : (i) HF > H2O > NH3 (ii) H2O > HF > NH3 (iii) NH3 > HF > H2O (iv) NH3 > H2O > HF
Solution: Option (ii) is the answer.
The types of hybrid orbitals of nitrogen in NO2+, NO3- and NH4+respectively are expected to be
(i) sp, sp3 and sp2
(ii) sp, sp2 and sp3
(iii) sp2, sp and sp3
(iv) sp2, sp3 and sp
Solution: Option (ii) is the answer. The hybridisation of each molecule gives us an idea about the hybrid orbitals.
In an adiabatic interaction, no exchange of warmth happens among framework and environmental elements. Pick the right choice with the expectation of complimentary extension of an optimal gas under adiabatic condition from the accompanying.
solution: Choice (iii) is the appropriate response. With the expectation of complimentary extension w = 0 For adiabatic cycle q = 0 From first law of thermodynamics, ...
. ∆fUᶱ of arrangement of CH4 (g) at certain temperature is – 393 kJ mol–1. The worth of ∆ fHᶱ is (I) zero (ii) < ∆f Uᶱ (iii) > ∆f Uᶱ (iv) equivalent to ∆f Uᶱ
solution: Choice (ii) is the appropriate response.
During complete burning of one mole of butane, 2658 kJ of warmth is delivered. The thermochemical response for above change is
solution: Choice (iii) is the appropriate response. Exothermic reaction for combustion of one mole of butane is represented as \[\left( iii \right)\text{ }C4H10\left( g \right)\text{...
. The volume of gas is decreased to half from its unique volume. The particular warmth will be ______. (I) decrease to half (ii) be multiplied (iii) stay consistent (iv) increment multiple times
solution: Alternative (iii) is the appropriate response. The particular warmth of a substance is the warmth needed to raise the temperature of 1 gram of a substance by one degree (1 K or 1...
The condition of a gas can be portrayed by citing the relationship between___. (I) pressure, volume, temperature (ii) temperature, sum, pressure (iii) the sum, volume, temperature (iv) pressure, volume, temperature, sum
solution: Alternative (iv) is the appropriate response. Condition of a framework can be portrayed by state capacities or state factors which are pressure, volume, temperature and measure of...
Which of the accompanying assertions is right? (I) The presence of responding species in a covered measuring utencil is an illustration of an open framework. (ii) There is a trade of energy just as a matter between the framework also, the environmental elements in a shut framework. (iii) The presence of reactants in a shut vessel made down of copper is an illustration of a shut framework. (iv) The presence of reactants in a canteen jar or some other shut protected vessel is an illustration of a shut framework.
solution: Alternative (iii) is the appropriate response. For a shut vessel made down of copper, regardless of can be traded between the framework and the environmental elements however energy trade...
Form the pair of linear equations for the following problems and find their solution by substitution method.(i) The difference between two numbers is 26 and one number is three times the other. Find them.(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
Arrangement (i): Leave the two numbers alone x and y individually, to such an extent that y > x. As indicated by the inquiry, \[y\text{ }=\text{ }3x\text{ }\ldots \text{ }\ldots \text{ }\ldots...
Choose the correct answer from the given four options in the following questions: The distance of the point P (–6, 8) from the origin is (A) 8 (B) 2√7 (C) 10 (D) 6
Solution: Option (C) 10 is the correct answer. The formula for distance: ${{d}^{2}}~=\text{ }{{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{ }{{y}_{1}})}^{2}}$ According to the...
Choose the correct answer from the given four options in the following questions: The area of a triangle with vertices A (3, 0), B (7, 0) and C (8, 4) is (A) 14 (B) 28 (C) 8 (D) 6
Solution: Option (C) 8 is the correct answer. The vertices of the triangle are, $A\text{ }({{x}_{1}},\text{ }{{y}_{1}})=\left( 3,\text{ }0 \right)$ $B\text{ }({{x}_{2}},\text{ }{{y}_{2}})=\left(...
Choose the correct answer from the given four options in the following questions: The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is (A) 5 (B) 12 (C) 11 (D) 7+ √5
Solution: Option (B) 12 is the correct answer. (0, 4), (0, 0) and (3, 0) are the vertices of a triangle. The perimeter of triangle AOB = Sum of the length of all its sides: = distance between...
Choose the correct answer from the given four options in the following questions: AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is (A) 5 (B) 3 (C) √34 (D) 4
Solution: Option (C) √34 is the correct answer. The three vertices are: $A\text{ }=\text{ }\left( 0,\text{ }3 \right)$, $O\text{ }=\text{ }\left( 0,\text{ }0 \right)$ , $B\text{ }=\text{ }\left(...
Choose the correct answer from the given four options in the following questions: The distance between the points (0, 5) and (–5, 0) is (A) 5 (B) 5√2 (C) 2√5 (D) 10
Solution: Option (B) 5√ 2 is the correct answer. Distance formula: ${{d}^{2}}~=\text{ }{{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{ }{{y}_{1}})}^{2}}$ According to the given...
Choose the correct answer from the given four options in the following questions: The distance between the points A (0, 6) and B (0, –2) is (A) 6 (B) 8 (C) 4 (D) 2
Solution: Option (B) 8 is the correct answer. The formula for distance : ${{d}^{2}}~=\text{ }{{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{ }{{y}_{1}})}^{2}}$ According to the...
Choose the correct answer from the given four options in the following questions: The distance of the point P (2, 3) from the x-axis is (A) 2 (B) 3 (C) 1 (D) 5
Solution: Option (B) 3 is the correct answer. We all know that, On the Cartesian plane (x, y) is a point in first quadrant. Then, Perpendicular distance from Y–axis = x, and Perpendicular distance...
A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
Solution: Let MN the flag pole = 18 m and its shadow LM = 9.6 m. The distance of the top of the pole be LN, N from the far end, L of the shadow. By Pythagoras theorem in right angled ∆LMN,...
For going to a city B from city A, there is a route via city C such that AC⊥CB, AC = 2 x km and CB = 2 (x + 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
Solution: According to the given question, AC⊥CB, $AC\text{ }=\text{ }2x\text{ }km$, $CB=2\left( x+7 \right)km$ and $AB=26\text{ }km$ As a result, we get triangle ACB right angled at C. Now, using...
A 5 m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.
Solution: Let 5 m be the length of the ladder AC. Let 4m be the height of the wall on which ladder is placed is BC From right angled triangle EBD, Now using the Pythagoras Theorem,...
In Fig 6.17, if PQRS is a parallelogram and AB||PS, then prove that OC||SR.
Solution: According to the given question, The given figure PQRS is a parallelogram, As a result, PQ || SR and PS || QR. It is also given that, AB || PS. To prove: OC || SR From triangles OPS and...
Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.
Solution: Let's assume a triangle ABC in which a line DE parallel to BC intersects AB at D and AC at E. To prove: The two sides are divided by DE in the same ratio. $AD/DB\text{ }=\text{ }AE/EC$...
It is given that ∆ ABC ~ ∆ EDF such that AB = 5 cm, AC = 7 cm, DF= 15 cm and DE = 12 cm. Find the lengths of the remaining sides of the triangles.
Solution: According to the given question, ∆ABC ∼ ∆EDF By the property of similar triangle, we all know that, corresponding sides of ∆ABC and ∆EDF are in the same ratio. $AB/ED\text{ }=\text{...
In Fig. 6.16, if ∠A = ∠C, AB = 6 cm, BP = 15 cm, AP = 12 cm and CP = 4 cm, then find the lengths of PD and CD.
Solution: According to the given question, $\angle A\text{ }=~\angle C$, $AB\text{ }=\text{ }6\text{ }cm$, $BP\text{ }=\text{ }15\text{ }cm$, $AP\text{ }=\text{ }12\text{ }cm$ $CP\text{ }=\text{...
If ∆ABC ∼ ∆DEF, AB = 4 cm, DE = 6, EF = 9 cm and FD = 12 cm, then find the perimeter of ∆ABC.
Solution: According to the given question, $AB\text{ }=\text{ }4\text{ }cm$, $DE\text{ }=\text{ }6\text{ }cm$ $EF\text{ }=\text{ }9\text{ }cm$ $FD\text{ }=\text{ }12\text{ }cm$ Also, ∆ABC ∼ ∆DEF We...
Find the altitude of an equilateral triangle of side 8 cm.
Solution: Let an equilateral triangle of side 8 cm be ABC. $AB\text{ }=\text{ }BC\text{ }=\text{ }CA\text{ }=\text{ }8\text{ }cm$. (all sides of an equilateral triangle is equal) Construct an...
In figure, if AB || DC and AC, PQ intersect each other at the point O. Prove that OA.CQ = 0C.AP.
Solution: According to the given question, At point O, AC and PQ intersect each other and AB||DC. From triangles AOP and COQ, $\angle AOP\text{ }=~\angle COQ$[As they are vertically opposite angles]...
Diagonals of a trapezium PQRS intersect each other at the point 0, PQ || RS and PQ = 3 RS. Find the ratio of the areas of Δ POQ and Δ ROS.
Solution: According to the given question, The given figure, PQRS is a trapezium in which PQ || RS and PQ = 3RS $PQ/RS\text{ }=\text{ }3/1\text{ }=\text{ }3$…(i) In triangles POQ and ROS, $\angle...
In figure, if ∠1 =∠2 and ΔNSQ = ΔMTR, then prove that ΔPTS ~ ΔPRQ.
Solution: According to the given question, $\text{ }\Delta NSQ~\cong ~\Delta MTR$ $\angle 1\text{ }=~\angle 2$ Since, $\Delta NSQ\text{ }=\text{ }\Delta MTR$ As a result, $SQ\text{ }=\text{...
In a ΔPQR, PR2 – PQ2 = QR2 and M is a point on side PR such that QM ⊥ PR. Prove that QM2 =PM × MR.
Solution: According to the given question, In triangle PQR, $P{{R}^{2}}~=\text{ }Q{{R}^{2}}$ and QM⊥PR Using the Pythagoras theorem, we obtain, $P{{R}^{2}}~=\text{ }P{{Q}^{2}}~+\text{ }Q{{R}^{2}}$...
Thermodynamics isn’t worried about______. (I) energy changes associated with a substance response. (ii) the degree to which a substance response continues. (iii) the rate at which a response continues. (iv) the practicality of a synthetic response.
solution: Choice (iii) is the appropriate response. This is because Thermodynamics informs us concerning the practicality, energy changes and degree of compound response. It doesn't informs us...
Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is (i) 6 (ii) 12 (iii) 7
Number of absolute results = 36 (I) When result of the numbers on the highest point of the dice = 6. The potential results = (1, 6), (2,3), (3, 2), (6, 1). Subsequently, number of conceivable ways =...
Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on the dice is (i) 7? (ii) a prime number? (iii) 1?
As indicated by the inquiry, Two dice are tossed all the while. In this way, that number of potential results = 36 (I) Sum of the numbers showing up on the dice is 7. Thus, the potential...
Two dice are thrown at the same time. Find the probability of getting A Same number on both dice. Different numbers on both dice.
Two dice are tossed simultaneously. Along these lines, absolute number of potential results = 36 (I) Same number on both dice. Potential results = (1,1), (2,2), (3, 3), (4, 4), (5, 5), (6,...
The weight of coffee in 70 packets are shown in the following table : Weight (in g) Number of packets 200-201 12 201-202 26 202-203 20 203-204 9 204-205 2 205-206 1 Determine the modal weight.
In the given information, the most noteworthy recurrence is 26, which lies in the stretch 201 – 202 Here, l = 201,fm = 26,f1 = 12,f2 = 20 and (class width) h = 1 Subsequently, the modular weight =...
The monthly income of 100 families are given as below : Income (in Rs) Number of families 0-5000 8 5000-10000 26 10000-15000 41 15000-20000 16 20000-25000 3 25000-30000 3 30000-35000 2 35000-40000 1 Calculate the modal income.
As per the information given, The most elevated recurrence = 41, 41 lies in the stretch 10000 – 15000. Here, l = 10000, fm = 41,f1 = 26,f2 = 16 and h = 5000 \[=\text{ }10000\text{ }+\text{...
The maximum bowling speeds, in km per hour, of 33 players at a cricket coaching centre are given as follows: Speed (km/h) 85-100 100-115 115-130 130-145 Number of players 11 9 8 5 Calculate the median bowling speed.
First we develop the combined recurrence table Speed ( in km/h) Number of players Cumulative recurrence 85 – 100 11 11 100 – 115 ...
Weekly income of 600 families is tabulated below : Weekly income Number of families (in Rs) 0-1000 250 1000-2000 190 2000-3000 100 3000-4000 40 4000-5000 15 5000-6000 5 Total 600 Compute the median income.
Week by week Income Number of families (fi) Cumulative recurrence (cf) 0-1000 250 250 1000-2000 190 250 + 190 = 400 2000-3000 100 440 + 100 = 540...
Given below is a cumulative frequency distribution showing the marks secured by 50 students of a class: Marks Below 20 Below 40 Below 60 Below 80 Below 100 Number of students 17 22 29 37 50 Form the frequency distribution table for the data.
The recurrence circulation table for given information. Marks Number of understudies 0 – 20 12 20 – 40 22 – 17 = 5 40 – 60 29 – 22 = 7 60 – 80 37 – 29 = 8 80 – 100 50 – 37 =...
The following are the ages of 300 patients getting medical treatment in a hospital on a particular day: Age (in years) 10-20 20-30 30-40 40-50 50-60 60-70 Number of patients 60 42 55 70 53 20 Form: ALess than type cumulative frequency distribution. More than type cumulative frequency distribution
(I) Less than type Age (in year) Number of patients Under 10 0 Under 20 60 + 0 = 60 Under 30 60 + 42 = 102 Under 40 102 + 55 = 157 Under...
Find the unknown entries a, b, c, d, e, f in the following distribution of heights of students in a class: Height Frequency Cumulative frequency (in cm) 150-155 12 a 155-160 b 25 160-165 10 c 165-170 d 43 170-175 e 48 175-180 2 f Total 50
Tallness (in cm) Frequency Cumulative recurrence given Cumulative recurrence 150 – 155 12 a 12 155 – 160 b ...
Form the frequency distribution table from the following data : Marks (out of 90) Number of candidates More than or equal to 80 4 More than or equal to 70 6 More than or equal to 60 11 More than or equal to 50 17 More than or equal to 40 23 More than or equal to 30 27 More than or equal to 20 30 More than or equal to 10 32 More than or equal to 0 34
The recurrence dissemination table for the given information is: Class Interval Number of understudies 0-10 34 – 32 = 2 10-20 32 – 30 = 2 20-30 30 – 27 = 3 30-40 27 – 23 = 4...
The following table shows the cumulative frequency distribution of marks of 800 students in an examination: Marks Number of students Below 10 10 Below 20 50 Below 30 130 Below 40 270 Below 50 440 Below 60 570 Below 70 670 Below 80 740 Below 90 780 Below 100 800 Construct a frequency distribution table for the data above.
The recurrence circulation table for the given information is: Class Interval Number of understudies 0-10 10 10-20 50 – 10 = 40 20-30 130 – 50 = 80 30-40 270 – 130 = 140...
The following is the distribution of weights (in kg) of 40 persons : Weight (in kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 Number of persons 4 4 13 5 6 5 2 1 Construct a cumulative frequency distribution (of the less than type) table for the data above.
Weight (in kg) Cumulative recurrence Under 45 4 Under 50 4 + 4 = 8 Under 55 8 + 13 = 21 Under 60 21 + 5 = 26 Under 65 26 + 6 = 32 Under...
The mileage (km per litre) of 50 cars of the same model was tested by a manufacturer and details are tabulated as given below : Mileage (km/l) 10-12 12-14 14-16 16-18 Number of cars 7 12 18 13 Find the mean mileage. The manufacturer claimed that the mileage of the model was 16 km/litre. Do you agree with this claim?
Mileage (km L-1) Class – Marks (xi) Number of vehicles (fi) fixi 10 – 12 11 7 77 12 – 14 13 12 156 14 – 16 15 ...
The weights (in kg) of 50 wrestlers are recorded in the following table : Weight (in kg) 100-110 110-120 120-130 130-140 140-150 Number of wrestlers 4 14 21 8 3 Find the mean weight of the wrestlers.
Weight (in kg) Number of Wrestlers (fi) Class Marks (xi) Deviation (di = xi – a) fidi 100 – 110 4 105 –20 –80 110 – 120 14 ...
An aircraft has 120 passenger seats. The number of seats occupied during 100 flights is given in the following table : Number of seats 100-104 104-108 108-112 112-116 116-120 Frequency 15 20 32 18 15 Determine the mean number of seats occupied over the flights.
Class Interval Class Marks (xi) Frequency (fi) Deviation (di = xi – a) fidi 100 – 104 102 15 –8 –120 104 – 108 106 ...
The daily income of a sample of 50 employees are tabulated as follows : Income (in Rs) 1-200 201-400 401-600 601-800 Number of employees 14 15 14 7 Find the mean daily income of employees.
C.I xi di = (xi – a) Fi fidi 1 – 200 100.5 –200 14 –2800 201 – 400 300.5 0 15 0 401 – 600 ...
The following tabe gives the number of pages written by Sarika for completing her own book for 30 days : Number of pages written per day 16-18 19-21 22-24 25-27 28-30 Number of days 1 3 4 9 13 Find the mean number of pages written per day.
Class Marks Mid – Value (xi) Number of days (fi) fixi 15.5 – 18.5 17 1 17 18.5 – 21.5 20 3 60 21.5 – 24.5 ...
Calculate the mean of the following data : Class 4 – 7 8 –11 12– 15 16 –19 Frequency 5 4 9 10
The given information isn't constant. Thus, we deduct 0.5 from as far as possible and add 0.5 in the maximum furthest reaches of each class. Class Class Marks (xi) Frequency (fi) fixi 3.5...
The elements in which electrons are progressively filled in 4f-orbital are called
(i) actinoids
(ii) transition elements
(iii) lanthanoids
(iv) halogens
Option (iii) is the answer. In lanthanoids, the 4f orbital is gradually filled with electrons. Lanthanoids have a broad electrical configuration. [Xe]4f 1-145d0-16s2.
Calculate the mean of the scores of 20 students in a mathematics test : Marks 10-20 20-30 30-40 40-50 50-60 Number of students 2 4 7 6 1
We first, discover the class mark xi of each class and afterward continue as follows Class Class Marks (xi) Frequency (fi) fixi 10-20 15 2 30 20-30 ...
Find the mean of the distribution : Class 1-3 3-5 5-7 7-10 Frequency 9 22 27 17
We first, discover the class mark xi of each class and afterward continue as follows. Class Class Marks (xi) Frequency (fi) fixi 1-3 2 9 18 3-5 ...
In calculating the mean of grouped data, grouped in classes of equal width, we may use the formula where a is the assumed mean. a must be one of the mid-points of the classes. Is the last statement correct? Justify your answer.
No, the assertion isn't right. It isn't required that expected mean ought to be the mid – mark of the class span. a can be considered as any worth which is not difficult to work on it.
The median of an ungrouped data and the median calculated when the same data is grouped are always the same. Do you think that this is a correct statement? Give reason.
To ascertain the middle of an assembled information, the recipe utilized depends with the understanding that the perceptions in the classes are consistently disseminated or similarly divided....
Is it true to say that the mean, mode and median of grouped data will always be different? Justify your answer.
No, the upsides of mean, mode and middle of gathered information can be equivalent to well, it relies upon the sort of information given.
Will the median class and modal class of grouped data always be different? Justify your answer.
The middle class and modular class of assembled information isn't generally unique, it relies upon the information given.
In a family having three children, there may be no girl, one girl, two girls or three girls. So, the probability of each is ¼. Is this correct? Justify your answer.
No it isn't right that in a family having three youngsters, there might be no young lady, one young lady, two young ladies or three young ladies, the likelihood of each is ¼. . Let young men be B...
A game consists of spinning an arrow which comes to rest pointing at one of the regions (1, 2 or 3) (Fig. 13.1). Are the outcomes 1, 2 and 3 equally likely to occur? Give reasons.
All out no. of result = 360 \[p\left( 1 \right)=\text{ }90/360\text{ }=1/4\] \[p\left( 2 \right)\text{ }=\text{ }90/360\text{ }=\text{ }1/4\] \[p\left( 3 \right)\text{ }=\text{ }180/360\text{...
Apoorv throws two dice once and computes the product of the numbers appearing on the dice. Peehu throws one die and squares the number that appears on it. Who has the better chance of getting the number 36? Why?
Apoorv toss two dice on the double. Thus, the all out number of results = 36 Number of results for getting item 36 = 1(6×6) ∴ Probability for Apoorv = 1/36 Peehu tosses one kick the bucket, Thus,...
Which of the following cannot be the probability of an event? (A)1/3 (B) 0.1 (C) 3% (D)17/16
(D)17/16 Clarification: Likelihood of an occasion consistently lies somewhere in the range of 0 and 1. Likelihood of any occasion can't be mutiple or negative as (17/16) > 1 Consequently, choice...
If an event cannot occur, then its probability is (A)1 (B) ¾ (C) ½ (D) 0
(D) 0 Clarification: The occasion which can't happen is supposed to be incomprehensible occasion. The likelihood of incomprehensible occasion = zero. Thus, choice (D) is right
Consider the following distribution : Marks obtained Number of students More than or equal to 0 63 More than or equal to 10 58 More than or equal to 20 55 More than or equal to 30 51 More than or equal to 40 48 More than or equal to 50 42 The frequency of the class 30-40 is (A) 3 (B) 4 (C) 48 (D) 51
(A) 3 Clarification: Imprints Obtained Number of students Cumulative Frequency 0-10 (63 – 58) = 5 5 10-20 (58 – 55) = 3 3 20-30 (55 – 51) =...
The times, in seconds, taken by 150 athletes to run a 110 m hurdle race are tabulated below Class 13.8-14 14-14.2 14.2-14.4 14.4-14.6 14.6-14.8 14.8-15 Frequency 2 4 5 71 48 20 The number of athletes who completed the race in less than 14.6 seconds is : A11 (B) 71 (C) 82 (D) 130
(C) 82 Clarification: The quantity of competitors who finished the race in under 14.6 second= 2 + 4 + 5 + 71 = 82 Subsequently, choice (C) is right
Consider the data : Class 65-85 85-105 105-125 125-145 145-165 165-185 185-205 Frequency 4 5 13 20 14 7 4 The difference of the upper limit of the median class and the lower limit of the modal class is A0 (B) 19 (C) 20 (D) 38
(C) 20 Clarification: Class Frequency Cumulative Frequency 65-85 4 4 85-105 5 9 105-125 13 22 ...
For the following distribution: Marks Number of students Below 10 3 Below 20 12 Below 30 27 Belo w 40 57 Below 50 75 Below 60 80 The modal class is (A)10-20 (B) 20-30 (C) 30-40 (D) 50-60
(C) 30-40 Clarification: Marks Number of students Cumulative Frequency Underneath 10 3=3 3 10-20 (12 – 3) = 9 12 20-30 (27 – 12) = 15 27...
Consider the following frequency distribution: Class 0-05 6-11 12-17 18-23 24-29 Frequency 13 10 15 8 11 The upper limit of the median class is (A)17 (B) 17.5 (C) 18 (D) 18.5
(B) 17.5 Clarification: As per the inquiry, Classes are not constant, henceforth, we make the information nonstop by taking away 0.5 from lower limit and adding 0.5 to furthest reaches of each...
For the following distribution : Class 0-05 5-10 10-15 15-20 20-25 Frequency 10 15 12 20 9 the sum of lower limits of the median class and modal class is (A)15 (B) 25 (C) 30 (D) 35
(B) 25 Clarification: Class Frequency Cumulative Frequency 0-5 10 10 5-10 15 25 10-15 12 37 15-20 ...
The abscissa of the point of intersection of the less than type and of the more than type cumulative frequency curves of a grouped data gives its (A) mean (B) median (C) mode (D) all the three above
(B) Median Clarification: Since, the convergence point of not as much as ogive and more than ogive gives the middle on the abscissa, the abscissa of the mark of convergence of the not as much...
In the formula x = a + h(fiui/fi), for finding the mean of grouped frequency distribution, ui = (A) (xi+a)/h (B) h (xi – a) (C) (xi –a)/h (D) (a – xi)/h
(C) (xi – a)/h Clarification: As indicated by the inquiry, \[x\text{ }=\text{ }a\text{ }+\text{ }h\left( fiui/fi \right),\] Above equation is a stage deviation recipe. In the above recipe, xi is...
If xi’s are the mid points of the class intervals of grouped data, fi’s are the corresponding frequencies and x is the mean, then (fixi – ¯¯¯ x ) is equal to (A)0 (B) –1 (C) 1 (D) 2
(A) 0 Clarification: Mean (x) = Sum of the relative multitude of perceptions/Number of perceptions \[x\text{ }=\text{ }\left( f1x1\text{ }+\text{ }f2x2\text{ }+\text{ }\ldots \text{ }..+\text{ }fnxn...
While computing mean of grouped data, we assume that the frequencies are (A) Evenly distributed over all the classes (B) Centred at the class marks of the classes (C) Centred at the upper limits of the classes (D) Centred at the lower limits of the classes
(B) Centered at the class characteristics of the classes Clarification: In figuring the mean of assembled information, the frequencies are focused at the class signs of the classes. Subsequently,...
Choose the correct answer from the given four options: 1. In the formula For finding the mean of grouped data di’s are deviations from a of (A) Lower limits of the classes (B) Upper limits of the classes (C) Mid points of the classes (D) Frequencies of the class marks
(C) Mid marks of the classes Clarification: We know, \[di\text{ }=\text{ }xi\text{ }\text{ }a\] Where, xi are information and 'a' is the expected to be mean In this way, di are the deviations from...
The electronic configuration of gadolinium (Atomic number 64) is
Option (iii) is the answer. The electronic configuration of gandolium is [Xe] 4f7 5d1 6s2
Why should a magnesium ribbon be cleaned before burning in air?
Answer: Because Magnesium metal combines with ambient oxygen to generate Magnesium Oxide (MgO) layer, which is a very stable chemical, it is recommended that magnesium ribbon be cleaned prior to...
500 persons are taking a dip into a cuboidal pond which is 80 m long and 50 m broad. What is the rise of water level in the pond, if the average displacement of the water by a person is 0.04m3?
As indicated by the inquiry, Normal dislodging by an individual = 0.04 m3 Normal uprooting by 500 people = 500 × 0.04 = 20 m3 Subsequently, the volume of water brought up in lake = 20 m3 It is...
A solid iron cuboidal block of dimensions 4.4 m × 2.6 m × 1m is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
Thinking about cuboidal square Length, l = 4 m Expansiveness, b = 2.6 m Stature, h = 1 m We realize that, Volume of tank = lbh Volume of cuboid = 4.4(2.6)(1) = 11.44 m3 We realize that, The volume...
Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm?
Let the time taken by line to fill lake = t hours Water streams 15 km in 60 minutes, in this way, it will stream 15t meters in t hours. We realize that, Volume of cuboidal lake up to tallness 21 cm...
A factory manufactures 120000 pencils daily. The pencils are cylindrical in shape each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at Rs 0.05 per dm2.
The state of pencil = chamber. Let the sweep of base = r cm Outline of base = 1.5 cm Boundary of circle is 2πr = 1.5 cm r = 1.5/2π cm As per the inquiry, Tallness, h = 25 cm We realize that, Bended...
A heap of rice is in the form of a cone of diameter 9 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?
As indicated by the inquiry, Think about tapered load, Base Diameter = 9 cm Thus, base range, r = 4.5 cm Tallness, h = 3.5 cm We realize that, Inclination tallness, The condition of volume of cone =...
Water flows at the rate of 10m/minute through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm?
Let the time taken by line to fill vessel = t minutes Since water streams 10 m in 1 moment, it will stream 10t meters in t minutes. As indicated by the inquiry, Volume of funnel shaped vessel =...
The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen is used up on writing 3300 words on an average. How many words can be written in a bottle of ink containing one fifth of a litre?
Allow us first to work out the volume of barrel of pen that is of round and hollow shape Think about barrel, Since 1cm = 10 mm Base width = 5 mm = 0.5 cm Base span, r = 0.25 cm Stature, h = 7 cm We...
How many cubic centimetres of iron is required to construct an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm provided the thickness of the iron is 1.5 cm. If one cubic cm of iron weighs 7.5 g, find the weight of the box.
Let the length (l), breath (b), and stature (h) be the outside element of an open box and thickness be x. The volume of metal utilized in box = Volume of outer box – Volume of inner box Think about...
A rectangular water tank of base 11 m × 6 m contains water upto a height of 5 m. If the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank.
Volume of water in tank = volume of cuboidal tank up to a stature of 5 m As per the inquiry, For cuboidal tank Length, l = 11 m Broadness, b = 6 m Tallness, h = 5m We realize that the condition to...
A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of base radius 6 cm. Determine the height of the cone.
For side of the equator, Span, r = 8 cm We realize that, volume of side of the equator = 2/3 πr3, where, r = sweep of half of the globe In this way, we get, Volume of given side of the equator...
Two cones with same base radius 8 cm and height 15 cm are joined together along their bases. Find the surface area of the shape so formed.
As per the inquiry, We get the figure given underneath, We realize that, Complete surface space of shape framed = Curved space of first cone + Curved surface space of second cone Since, the two...
From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of the remaining solid.
From the figure, we get, Volume of staying strong = volume of shape – volume of cone For Cube Side, a = 7 cm We realize that, Volume of 3D shape = a3, where a = side of block Volume of 3D shape =...
Two identical cubes each of volume 64 cm3 are joined together end to end. What is the surface area of the resulting cuboid?
Let the side of one block = a Surfaces space of coming about cuboid = 2(Total surface space of a block) – 2(area of single surface) We realize that, All out surface space of solid shape = 6a2 ,...
A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts.
As per the inquiry, Tallness of cone = OM = 12 cm The cone is separated from mid-point. Consequently, let the mid-point of cone = P Operation = PM = 6 cm From △OPD and △OMN ∠POD = ∠POD [Common] ∠OPD...
A bucket is in the form of a frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are 28 cm and 21 cm, respectively. Find the height of the bucket.
As per the inquiry, The pail is as frustum of a cone. We realize that, Volume of frustum of a cone\[=\text{ }1/3\text{ }\pi h\left( r12\text{ }+\text{ }r22\text{ }+\text{ }r1r2 \right)\] , where, h...
How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9cm × 11cm × 12cm?
Volume of cuboid = lbh, where, l = length, b = expansiveness and h = stature Cuboidal lead: Length, l = 9 cm Expansiveness, b = 11 cm Stature, h = 12 cm Volume of lead \[=\text{ }9\left( 11...
Three metallic solid cubes whose edges are 3 cm, 4 cm and 5 cm are melted and formed into a single cube. Find the edge of the cube so formed.
We realize that, Volume of 3D shape = a3, where a = side of 3D square As per the inquiry, Side of first 3D shape, a1 = 3 cm Side of second 3D square, a2 = 4 cm Side of third 3D square, a3 = 5 cm...
A strong ball is actually fitted inside the cubical box of side a. The volume of the ball is 4/3πa3.
False, Clarification: Let the sweep of circle = r At the point when a strong ball is by and large fitted inside the cubical box of side a, We get, Width of ball = Edge length of 3D shape \[2r\text{...
A strong cone of sweep r and tallness h is set over a strong chamber having same base range and stature as that of a cone. The complete surface space of the consolidated strong is πr[√(r2 + h2 +3r + 2h].
False Clarification: At the point when a strong cone is put over a strong chamber of same base range, the foundation of cone and top of the chamber won't be shrouded in absolute surface region....
A solid cylinder of radius r and height h is placed over other cylinder of same height and radius. The total surface area of the shape so formed is 4πrh + 4πr2.
False, Clarification: As indicated by the inquiry, At the point when one chamber is set over another, the foundation of first chamber and top of other chamber won't be canvassed in all out surface...
Two indistinguishable strong sides of the equator of equivalent base range r cm are remained together along their bases. The all out surface space of the blend is 6πr2.
False, Clarification: At the point when two sides of the equator are consolidated along their bases, a circle of same base range is framed. Bended Surface Area of a circle = 4πr2.
A solid piece of iron in the form of a cuboid of dimensions 49cm × 33cm × 24cm, is moulded to form a solid sphere. The radius of the sphere is (A) 21cm (B) 23cm (C) 25cm (D) 19cm
(A) 21cm As we probably are aware, Volume of cuboid = lbh Where, l = length, b = broadness and h = tallness For given cuboid, Length, l = 49 cm Broadness, b = 33 cm Tallness, h = 24 cm Volume of...
A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8cm. The height of the cone is (A) 12cm (B) 14cm (C) 15cm (D) 18cm
(B) 14cm Volume of circular shell = Volume of cone recast by liquefying For Spherical Shell, Inside measurement, d1 = 4 cm Inside range, r1 = 2 cm [ as range = 1/2 diameter] Outer measurement, d2 =...
A hollow cube of internal edge 22cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that 1/8 space of the cube remains unfilled. Then the number of marbles that the cube can accommodate is (A) 142296 (B) 142396 (C) 142496 (D) 142596
(A) 142296 As indicated by the inquiry, Volume of block =223=10648cm3 Volume of block that stays unfilled \[=1/8\times 10648=1331cm3\] volume involved by circular marbles...
A cone is cut through a plane parallel to its base and then the cone that is formed on one side of that plane is removed. The new part that is left over on the other side of the plane is called (A) a frustum of a cone (B) cone (C) cylinder (D) sphere
(A) a frustum of a cone At the point when a cone is isolated into two sections by a plane through any point on its pivot corresponding to its base, the upper and lower parts got are cone and a...
A shuttle cock used for playing badminton has the shape of the combination of (A) a cylinder and a sphere (B) a cylinder and a hemisphere (C) a sphere and a cone (D) frustum of a cone and a hemisphere
(D) frustum of a cone and a side of the equator The plug of a van = hemispherical shapes The upper piece of a bus = state of frustum of a cone. Subsequently, it is a mix of frustum of a cone and a...
The shape of a gilli, in the gilli-danda game (see Fig. 12.4), is a combination of (A) two cylinders (B) a cone and a cylinder (C) two cones and a cylinder (D) two cylinders and a cone
(C) two cones and a chamber The left and right piece of a gilli = funnel shaped The focal piece of a gilli = round and hollow Thusly, it is a mix of a chamber and two cones.
The shape of a glass (tumbler) (see Fig. 12.3) is usually in the form of (A) a cone (B) frustum of a cone (C) a cylinder (D) a sphere
The correct answer is option(B) frustum of a cone
A plumbline (sahul) is the combination of (see Fig. 12.2) (A) a cone and a cylinder (B) a hemisphere and a cone (C) frustum of a cone and a cylinder (D) sphere and cylinder
(B) a half of the globe and a cone The upper piece of plumbline = hemispherical, The base piece of plumbline = cone shaped Accordingly, it is a blend of half of the globe and cone.
A surahi is the combination of (A) a sphere and a cylinder (B) a hemisphere and a cylinder (C) two hemispheres (D) a cylinder and a cone.
(A) a circle and a chamber The top piece of surahi = round and hollow shape Base piece of surahi = circular shape Subsequently, surahi is a mix of Sphere and a chamber.
Choose the correct answer from the given four options: A cylindrical pencil sharpened at one edge is the combination of (A) a cone and a cylinder (B) frustum of a cone and a cylinder (C) a hemisphere and a cylinder (D) two cylinders.
(A) a cone and a cylinder The Nib of a sharpened pencil = conical shape The rest of the part of a sharpened pencil = cylindrical Therefore, a pencil is a combination of cylinder and a cone.
Is the following statement true? Why? “Two quadrilaterals are similar, if their corresponding angles are equal”.
Solution: False If only the two corresponding angles of a quadrilateral are equal, then the given two quadrilaterals cannot be similar.
In ΔPQR and ΔMST, ∠P = 55°, ∠Q =25°, ∠M = 100° and ∠S = 25°. Is ΔQPR ~ ΔTSM? Why?
Solution: We all know that, When the three angles of a triangle are added then their sum equals to 180°. Then, from triangle PQR, $\angle P\text{ }+~\angle Q\text{ }+~\angle R\text{ }=\text{...
In figure, BD and CE intersect each other at the point P. Is ΔPBC ~ ΔPDE? Why?
Solution: True In triangles PBC and PDE, ∠EPD = ∠BPC [ as vertically opposite angles] $PB/PD\text{ }=\text{ }5/10\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$… (i) $PC/PE\text{ }=\text{...
A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reason for your answer.
Solution: True According to the given question, $PQ\text{ }=\text{ }12.5\text{ }cm$ $PA\text{ }=\text{ }5\text{ }cm$ $BR\text{ }=\text{ }6\text{ }cm$ $PB\text{ }=\text{ }4\text{ }cm$ Then, $QA\text{...
It is given that ΔDEF ~ ΔRPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P ? Why?
False We all know that, The corresponding angles of similar triangles are equal. As a result, we obtain, $\angle D\text{ }=~\angle R$ $\angle E\text{ }=~\angle P$ $\angle F\text{ }=\angle...
In figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30°. Then, ∠PBA is equal to (a) 50° (b) 30° (c) 60° (d) 100°
Solution: (d) 100° Explanation: From triangles APB and CPD, $\angle APB\text{ }=~\angle CPD\text{ }=\text{ }50{}^\circ $ (as they are vertically opposite angles) $AP/PD\text{ }=\text{ }6/5$ … (i)...
If in two Δ PQR, AB/QR = BC/PR = CA/PQ, then (a)Δ PQR~Δ CAB (b) Δ PQR ~ Δ ABC (c)Δ CBA ~ Δ PQR (d) Δ BCA ~ Δ PQR
Solution: (a)Δ PQR~Δ CAB Explanation: From triangles ABC and PQR, we have, AB/QR = BC/PR = CA/PQ When the sides of one triangle are proportional to the sides of the other given triangle, and even...
If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true? (a) BC · EF = AC · FD (b) AB · EF = AC · DE (c) BC · DE = AB · EF (d) BC · DE = AB · FD
Solution: (c) BC · DE = AB · EF Explanation: We all know that, If the sides of one triangle are proportionate to the sides of the other triangle, and the corresponding angles are all equal, the...
If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, the length of the sides of the rhombus is (a) 9 cm (b) 10 cm (c) 8 cm (d) 20 cm
Solution: (b) 10 cm Explanation: We all know that, A rhombus is a simple quadrilateral with four equal-length sides and diagonals that are perpendicular bisector of each other. Now according to the...
In figure, if ∠BAC =90° and AD⊥BC. Then, (a) BD.CD = BC² (b) AB.AC = BC² (c) BD.CD=AD² (d) AB.AC =AD²
Solution: c) BD.CD=AD² Explanation: From triangles ADB and ADC, Now according to the question, we have, ∠ADB = ∠ADC = 90° (Since AD ⊥ BC) ∠DBA = ∠DAC [As each angle = 90°- ∠C] Using AAA criterion...
1. Write the first terms of each of the following sequences whose term are: (iii) (iv)
An arithmetic progressions or arithmetic sequence is a number’s sequence such that the difference between the consecutive terms is constant. Solutions: (iii) ${{a}_{n}}={{3}^{n}}$ Given sequence...
In Fig. 9.18, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.
[Hint: Draw a line through Q and perpendicular to QP.] As per the inquiry, Digressions PQ and PR are attracted to a circle to such an extent that ∠RPQ = 30°. A harmony RS is attracted corresponding...
In a right triangle ABC in which ∠B = 90°, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Prove that the tangent to the circle at P bisects BC.
As per the inquiry, In a right point ΔABC is which ∠B = 90°, a circle is drawn with AB as distance across meeting the hypotenuse AC at P. Likewise PQ is a digression at P To Prove: PQ separates BC...
Two circles with centres O and O‘ of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O‘P are tangents to the two circles. Find the length of the common chord PQ.
As per the inquiry, Two circles with focuses O and O' of radii 3 cm and 4 cm, individually meet at two focuses P and Q, to such an extent that OP and O'P are digressions to the two circles and PQ is...
If AB is a chord of a circle with centre O, AOC is a diameter and AT is the tangent at A as shown in Fig. 9.17. Prove that ∠BAT = ∠ACB
As per the inquiry, A circle with focus O and AC as a measurement and AB and BC as two harmonies additionally AT is a digression at point A To Prove : ∠BAT = ∠ACB Verification : ∠ABC = 90° [Angle in...
From an external point P, two tangents, PA and PB are drawn to a circle with centre O. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the triangle PCD.
As per the inquiry, From an outside point P, two digressions, PA and PB are attracted to a circle with focus O. At a point E on the circle digression is drawn which crosses PA and PB at C and D,...
Let s denote the semi-perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that BD = s – b.
As indicated by the inquiry, A triangle ABC with BC = a , CA = b and AB = c . Likewise, a circle is engraved which contacts the sides BC, CA and AB at D, E and F individually and s is semi-border of...
If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA.
As per the inquiry, A Hexagon ABCDEF encompass a circle. To demonstrate: Stomach muscle + CD + EF = BC + DE + FA Verification: Digressions drawn from an outside highlight a circle are equivalent....
In Fig. 9.13, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.
As indicated by the inquiry, Abdominal muscle = CD Development: Produce AB and CD, to converge at P. Confirmation: Think about the circle with more noteworthy span. Digressions drawn from an outside...
Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Leave the lines alone l1 and l2. Expect that O contacts l₁ and l₂ at M and N, We get, OM = ON (Radius of the circle) In this way, From the middle "O" of the circle, it has equivalent separation from...
If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that angle DBC = 120°, prove that BC + BD = BO, i.e., BO = 2BC.
As per the inquiry, By RHS rule, ΔOBC and ΔOBD are harmonious By CPCT ∠OBC and ∠ OBD are equivalent In this way, \[\angle OBC\text{ }=\angle OBD\text{ }=60{}^\circ \] In triangle OBC, \[cos\text{...
Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.
We realize that, Sweep ⊥ Tangent = OR ⊥ PR i.e., ∠ORP = 90° Similarly, Sweep ⊥ Tangent = OQ ⊥PQ ∠OQP = 90° In quadrilateral ORPQ, Amount of every single inside point = 360º \[\angle ORP\text{...
Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle
From the figure, Harmony AB = 8 cm OC is opposite to the harmony AB AC = CB = 4 cm In right triangle OCA \[OC2\text{ }+\text{ }CA2\text{ }=\text{ }OA2\] \[OC2\text{ }=\text{ }52\text{ }\text{...
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a√2.
True, Digression is consistently opposite to the span at the resource. Subsequently, ∠RPT = 90 Assuming 2 digressions are drawn from an outer point, they are similarly disposed to the line fragment...
The angle between two tangents to a circle may be 0°.
True, Defense: The point between two digressions to a circle might be 0°only when both digression lines concur or are corresponding to one another.
The length of tangent from an external point P on a circle with centre O is always less than OP.
True, Defense: Think about the figure of a circle with focus O. Leave PT alone a digression drawn from outer point P. Presently, Joint OT. OT ⏊ PT We realize that, Digression anytime on the circle...
The length of tangent from an external point on a circle is always greater than the radius of the circle.
False, Support: Length of digression from an outside point P on a circle might possibly be more prominent than the sweep of the circle.
If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.
False, Support: For instance, Think about the given figure. In which we have a circle with focus O and AB a harmony with ∠AOB = 60° Since, digression to any point on the circle is opposite to the...
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is
(A) 4 cm (B) 5 cm (C) 6 cm (D) 8 cm As indicated by the inquiry, Span of circle, AO=OC = 5cm AM=8CM AM=OM+AO OM =AM-AO Subbing these qualities in the situation, OM= (8-5) =3CM OM is opposite to the...
From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is
(A) 60 cm2 (B) 65 cm2 (C) 30 cm2 (D) 32.5 cm2 Development: Draw a circle of span 5 cm with focus O. Leave P alone a point a ways off of 13 cm from O. Draw a couple of digressions, PQ and PR. OQ...
In Fig. 9.4, AB is a chord of the circle and AOC is its diameter such that ACB = 50°. If AT is the tangent to the circle at the point A, then BAT is equal to
(A) 65° (B) 60° (C) 50° (D) 40° As per the inquiry, A circle with focus O, measurement AC and ∠ACB = 50° AT is a digression to the circle at point A Since, point in a half circle is a right point...
In Fig. 9.3, if AOB = 125°, then COD is equal to
(A) 62.5° (B) 45° (C) 35° (D) 55° ABCD is a quadrilateral delineating the circle We realize that, the contrary sides of a quadrilateral d elineating a circle subtend strengthening points at the...
If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is
(A) 3 cm (B) 6 cm (C) 9 cm (D) 1 cm As per the inquiry, OA = 4cm, OB = 5cm What's more, OA ⊥ BC Consequently, \[OB2\text{ }=\text{ }OA2\text{ }+\text{ }AB2\] \[\Rightarrow 52\text{ }=\text{...
If tanθ + secθ = l, then prove that secθ = (l2 + 1)/2l.
Given: tan θ+ sec θ = l … eq. 1 Duplicating and isolating by (sec θ – tan θ) on numerator and denominator of L.H.S, Along these lines, sec θ – tan θ = 1 … eq.2 Adding eq. 1and eq. 2, we get \[\left(...
A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan α/(tan β – tan α)].
Considering that an upward banner staff of stature h is overcomed on an upward pinnacle of tallness H(say), with the end goal that FP = h and FO = H. The point of height of the base and top of the...
A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan α/(tan β – tan α)].
Considering that an upward banner staff of stature h is overcomed on an upward pinnacle of tallness H(say), with the end goal that FP = h and FO = H. The point of height of the base and top of the...
The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.
Let SQ = h be the pinnacle. ∠SPQ = 30° and ∠SRQ = 60° As per the inquiry, the length of shadow is 50 m long hen point of rise of the sun is 30° than when it was 60°. Thus, PR = 50 m and RQ = x m So...
The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is √st .
Let BC = s; PC = t Leave tallness of the pinnacle alone AB = h. ∠ABC = θ and ∠APC = 90° – θ (∵ the point of height of the highest point of the pinnacle from two focuses P and B are corresponding) ⇒...
Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.
Given: sin θ +2 cos θ = 1 Squaring on the two sides, \[\begin{array}{*{35}{l}} \left( sin\text{ }\theta \text{ }+2\text{ }cos\text{ }\theta \right)2\text{ }=\text{ }1 \\...
If 1 + sin2θ = 3sinθ cosθ , then prove that tanθ = 1 or ½.
Given: \[1+sin2\text{ }\theta \text{ }=\text{ }3\text{ }sin\text{ }\theta \text{ }cos\text{ }\theta \] Isolating L.H.S and R.H.S conditions with sin2 θ, We get, NCERT Exemplar Class 10 Maths Chapter...
The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.
Let PR = h meter, be the tallness of the pinnacle. The onlooker is remaining at point Q to such an extent that, the distance between the spectator and pinnacle is QR = (20+x) m, where \[QR\text{...
Prove that √(sec2 θ + cosec2 θ) = tan θ + cot θ
L.H.S= Since, NCERT Exemplar Class 10 Maths Chapter 8 Ex. 8.4 Question 2 = R.H.S Henceforth, demonstrated.
If cosecθ + cotθ = p, then prove that cosθ = (p2 – 1)/ (p2 + 1).
As per the inquiry, Since, NCERT Exemplar Class 10 Maths Chapter 8 Ex. 8.4 Question 1 Henceforth, demonstrated.
tan θ + tan (90° – θ) = sec θ sec (90° – θ)
using the formulae: tan (90° – θ) = cot θ tan θ + tan (90° – θ)= tan θ + cot θ we get,
Prove: 1 + (cot2 α/1+cosec α) = cosec α
proved
(√3+1) (3 – cot 30°) = tan3 60° – 2 sin 60°
L.H.S: (√3 + 1) (3 – Cot30°) \[=\text{ }\left( \surd 3\text{ }+\text{ }1 \right)\text{ }\left( 3\text{ }\text{ }\surd 3 \right)\text{ }\left[ \because cos\text{ }30{}^\circ =\surd 3 \right]\]...
(sin α + cos α) (tan α + cot α) = sec α + cosec α
=> LHS = RHS (PROVED)
If tan A = ¾, then sinA cosA = 12/25
Solution: As per the inquiry, tan A = ¾ We know, tan A = opposite/base Along these lines, tan A = 3k/4k Where, Opposite = 3k Base = 4k Utilizing Pythagoras Theorem, (hypotenuse)2 = (perpendicular)2...
Prove: tan A/(1+secA) – tan A/(1-secA) = 2cosec A
= LHS Using the formulae: sec2A – tan2A = 1 sec2A – 1 = tan2A we get, = RHS (PROVED)
Prove: sin θ/(1+cos θ) + (1+ cos θ)/sin θ = 2cosec θ
=> LHS = RHS (Proved)