A vessel in the form of a hollow hemisphere mounted by a hollow cylinder.
Question: A vessel in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. Solution: We have to find the inner surface area of a vessel which is in the form of a hemisphere mounted by a hollow cylinder. Radius of hemisphere and cylinder $(r)=7 \mathrm{~cm}$ Total height of vessel $(r+h)=13 \mathrm{~cm}$ So, the inner surface area of a vessel, $=2 \pi r(r+h)$ $=...
Read More →If the bisector of an angle of a triangle bisects the opposite side, then the triangle is
Question: If the bisector of an angle of a triangle bisects the opposite side, then the triangle is(a) scalene(b) equilateral(c) isosceles(d) right-angled Solution: (c) isosceles Let AD be the angle bisector of angle A in triangle ABC.Applying angle bisector theorem, we get: $\frac{A B}{A C}=\frac{B D}{D C}$ It is given that AD bisects BC.Therefore, BD = DC $\Rightarrow \frac{A B}{A C}=1$ $\Rightarrow A B=A C$ Therefore, the triangle is isosceles....
Read More →A cylindrical road roller made of iron is 1 m long,
Question: A cylindrical road roller made of iron is $1 \mathrm{~m}$ long, Its internal diameter is $54 \mathrm{~cm}$ and the thickness of the iron sheet used in making the roller is $9 \mathrm{~cm}$. Find the mass of the roller, if $1 \mathrm{~cm}^{3}$ of iron has $7.8 \mathrm{gm}$ mass. (Use $\pi=3.14$ ) Solution: We have to find the mass of the roller. Radius of inner cylinder $\left(r_{1}\right)=27 \mathrm{~cm}$ Radius of outer cylinder $\left(r_{2}\right)=(27+9) \mathrm{cm}$ $=36 \mathrm{~cm...
Read More →In the following pairs of halogen compounds,
Question: In the following pairs of halogen compounds, which compound undergoes faster SN1 reaction? (i) (ii) Solution: (i) SN1 reaction proceeds via the formation of carbocation. The alkyl halide (I) is 3 while (II) is 2. Therefore, (I) forms 3 carbocation while (II) forms 2 carbocation. Greater the stability of the carbocation, faster is the rate of SN1 reaction. Since 3 carbocation is more stable than 2 carbocation. (I), i.e. 2chloro-2-methylpropane, undergoes faster SN1 reaction than (II) i....
Read More →The line segments joining the midpoints of the adjacent sides of a quadrilateral form
Question: The line segments joining the midpoints of the adjacent sides of a quadrilateral form(a) a parallelogram(b) a rectangle(c) a square(d) a rhombus Solution: (a) a parallelogramThe line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram....
Read More →With increasing biasing voltage of a photodiode, the photocurrent magnitude :
Question: With increasing biasing voltage of a photodiode, the photocurrent magnitude :(1) remains constant(2) increases initially and after attaining certain value, it decreases(3) Increases linearly(4) increases initially and saturates finallyCorrect Option: , 4 Solution: (4) I-V characteristic of a photodiode is as follows : On increasing the biasing voltage of a photodiode, the magnitude of photocurrent first increases and then attains a saturation....
Read More →In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA = (3x −1)
Question: In the given figure,ABCDis a trapezium whose diagonalsACandBDintersect atOsuch thatOA= (3x1) cm,OB= (2x+ 1) cm,OC= (5x 3) cm andOD= (6x 5) cm. Then,x= ? (a) 2(b) 3(c) 2.5(d) 4 Solution: (a) 2We know that the diagonals of a trapezium are proportional.Therefore, $\frac{O A}{O C}=\frac{O B}{O D}$ $\Rightarrow \frac{3 X-1}{5 X-3}=\frac{2 X+1}{6 X-5}$ $\Rightarrow(3 X-1)(6 X-5)=(2 X+1)(5 X-3)$ $\Rightarrow 18 X^{2}-15 X-6 X+5=10 X^{2}-6 X+5 X-3$ $\Rightarrow 18 X^{2}-21 X+5=10 X^{2}-X-3$ $\...
Read More →Which alkyl halide from the following pairs would you expect to react more rapidly by
Question: Which alkyl halide from the following pairs would you expect to react more rapidly by an SN2 mechanism? Explain your answer. (i) (ii) (iii) Solution: (i) 2-bromobutane is a 2 alkylhalide whereas 1-bromobutane is a 1 alkyl halide.The approaching of nucleophile is more hindered in 2-bromobutane than in 1-bromobutane. Therefore, 1-bromobutane reacts more rapidly than 2-bromobutane by an SN2 mechanism. (ii) 2-Bromobutane is 2 alkylhalide whereas 2-bromo-2-methylpropane is 3 alkyl halide. T...
Read More →A cylindrical vessel of diameter 14 cm and height 42 cm
Question: A cylindrical vessel of diameter 14 cm and height 42 cm is fixed symmetrically inside a similar vessel of diameter 16 cm and height 42 cm. The total space between the two vessels is filled with cork dust for heat insulation purposes. How many cubic centimeters of cork dust will be required? Solution: We have to find the volume of cork dust filled between the two vessels. Radius of outer vessel $\left(r_{2}\right)=8 \mathrm{~cm}$ Radius of inner vessel $\left(r_{1}\right)=7 \mathrm{~cm}...
Read More →Arrange each set of compounds in order of increasing boiling points.
Question: Arrange each set of compounds in order of increasing boiling points. (i)Bromomethane, Bromoform, Chloromethane, Dibromomethane. (ii)1-Chloropropane, Isopropyl chloride, 1-Chlorobutane. Solution: (i) For alkyl halides containing the same alkyl group, the boiling point increases with an increase in the atomic mass of the halogen atom. Since the atomic mass of Br is greater than that of Cl, the boiling point of bromomethane is higher than that of chloromethane. Further, for alkyl halides ...
Read More →An wooden toy is made by scooping out a hemisphere of same
Question: An wooden toy is made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy. (Use = 22/7). Solution: Volume of wood in the toy = Volume of cylinder 2(Volume of hemisphere) $=\pi r^{2} h-2 \times \frac{2}{3} \pi r^{3}$ $=\frac{22}{7} \times(3.5)^{2} \times 10-2 \times \frac{2}{3} \times \frac{22}{7} \times(3.5)^{3}$ $=385-179.67$ $=205.33 \mathrm{~cm}^{3}...
Read More →If the diagonals of a quadrilateral divide each other proportionally, then it is a
Question: If the diagonals of a quadrilateral divide each other proportionally, then it is a(a) parallelogram(b) trapezium(c) rectangle(d) square Solution: (b) trapeziumDiagonals of a trapezium divide each other proportionally....
Read More →The lengths of the diagonals of a rhombus are 24 cm and 10 cm.
Question: The lengths of the diagonals of a rhombus are 24 cm and 10 cm. The length of each side of the rhombus is(a) 12 cm(b) 13 cm(c) 14 cm(d) 17 cm Solution: (b) 13 cm LetABCDbe the rhombus with diagonalsACandBDintersecting each other atO.We have:AC= 24 cm andBD= 10 cmWe know that diagonals of a rhombus bisect each other at right angles.Therefore applying Pythagoras theorem in right-angled triangleAOB,we get: $A B^{2}=A O^{2}+B O^{2}=12^{2}+5^{2}$ $=144+25=169$ $A B=\sqrt{169}=13$ Hence, the ...
Read More →In a rhombus of side 10 cm, one of the diagonals is 12 cm long. The length of the second diagonal is
Question: In a rhombus of side 10 cm, one of the diagonals is 12 cm long. The length of the second diagonal is(a) 20 cm(b) 18 cm(c) 16 cm(d) 22 cm Solution: (c) 16 cm LetABCDbe the rhombus with diagonalsACandBDintersecting each other atO.Also, diagonals of a rhombus bisect each other at right angles.IfAC=12 cm,AO= 6 cmApplying Pythagoras theorem in right-angled triangleAOB.we get: $A B^{2}=A O^{2}+B O^{2}$ $\Rightarrow B O^{2}=A B^{2}-A O^{2}$ $\Rightarrow B O^{2}=10^{2}-6^{2}=100-36=64$ $\Right...
Read More →In an equilateral triangle ABC, if AD ⊥ BC, then which of the following is true?
Question: In an equilateral triangleABC, ifADBC, then which of the following is true? (a) 2AB2= 3AD2(b) 4AB2= 3AD2(c) 3AB2= 4AD2(d) 3AB2= 2AD2 Solution: (c) 3AB2= 4AD2Applying Pythagoras theorem in right-angled trianglesABDandADC, we get: $A B^{2}=A D^{2}+B D^{2}$ $\Rightarrow A B^{2}=\left(\frac{1}{2} A B\right)^{2}+A D^{2} \quad\left(\because \triangle \mathrm{ABC}\right.$ is equilateral and $\left.A D=\frac{1}{2} \mathrm{AB}\right)$ $\Rightarrow A B^{2}=\frac{1}{4} A B^{2}+A D^{2}$ $\Rightarr...
Read More →A solid is in the shape of a cone surmounted on a hemisphere,
Question: A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them is being 3.5 cm and the total height of solid is 9.5 cm. Find the volume of the solid. (Use = 22/7). Solution: Height of cone = 9.5 3.5 = 6 cmVolume of the solid = Volume of cone + Volume of hemisphere $=\frac{1}{3} \pi r^{2} h+\frac{2}{3} \pi r^{3}$ $=\frac{1}{3} \times \frac{22}{7} \times(3.5)^{2} \times 6+\frac{2}{3} \times \frac{22}{7} \times(3.5)^{3}$ $=77+89.83$ $=166.83 \mathrm{~cm}^{3}$...
Read More →In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is
Question: In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is Figure (a) right-angle(b) isosceles(c) scalene(d) obtuse-angled Solution: (b) isoscelesIn an isosceles triangle, the perpendicular from the vertex to the base bisects the base....
Read More →A solid is composed of a cylinder with hemispherical ends.
Question: A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and the radius of each of the hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs 10 per dm2. Solution: We have a solid composed of cylinder with hemispherical ends. Radius of the two curved surfaces $(r)=7 \mathrm{~cm}$ Height of cylinder is $h$. Total height of the body $(h+2 r)=104 \mathrm{~cm}$ So, total surface area is given by, Total surface area ...
Read More →In a △ABC, it is given that AD is the internal bisector of ∠A.
Question: In a △ABC, it is given that AD is the internal bisector of A. If AB = 10 cm, AC = 14 cm and BC = 6 cm, the CD = ?(a) 4.8 cm(b) 3.5 cm(c) 7 cm(d) 10.5 cm Solution: By using angle bisector theore in △ABC, we have $\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\mathrm{BD}}{\mathrm{DC}}$ $\Rightarrow \frac{10}{14}=\frac{6-x}{x}$ $\Rightarrow 10 x=84-14 x$ $\Rightarrow 24 x=84$ $\Rightarrow x=3.5$ Hence, the correct answer is option (b)....
Read More →A vessel is a hollow cylinder fitted with a hemispherical
Question: A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylinder is $\frac{14}{3} \mathrm{~m}$ and the diameter of hemisphere is $3.5 \mathrm{~m}$. Calculate the volume and the internal surface area of the solid. Solution: Given that: Radius of the same base $r=\frac{3.5}{2}=1.75 \mathrm{~m}$ Height of the cylinder $h=\frac{14}{3} \mathrm{~m}$ The volume of the vessel is given by $V=\pi r^{2} h+\frac{2}{3} \pi r^{3}$ $=3.14 \times 1.75^{2} \t...
Read More →A boiler is in the form of a cylinder 2 m long
Question: A boiler is in the form of a cylinder 2 m long with hemispherical ends each of 2 metre diameter. Find the volume of the boiler. Solution: Given that: Height of the cylinder Radius of the cylinder and hemisphere are same and is given by $r=\frac{d}{2}=\frac{2}{2}=1 \mathrm{~m}$ The volume of the cylinder is cylinder is $V_{1}=\pi r^{2} h$ $=\frac{22}{7} \times 1^{2} \times 2$ $=\frac{22}{7} \times 2 \mathrm{~m}^{3}$ There are two hemispheres at each ends of the cylinder, therefore the v...
Read More →In a ∆ABC, it is given that AD is the internal bisector of ∠A.
Question: In a ∆ABC, it is given thatADis the internal bisector of A. IfBD= 4 cm,DC= 5 cm andAB= 6 cm, thenAC= ? (a) 4.5 cm(b) 8 cm(c) 9 cm(d) 7.5 cm Solution: (d) 7.5 cmIt is given thatADbisects angleA.Therefore, applying angle bisector theorem, we get: $\frac{B D}{D C}=\frac{A B}{A C}$ $\Rightarrow \frac{4}{5}=\frac{6}{x}$ $\Rightarrow x=\frac{5 \times 6}{4}=7.5$ Hence,AC= 7.5 cm...
Read More →Draw the structures of major monohalo products in each of the following reactions:
Question: Draw the structures of major monohalo products in each of the following reactions: (i) (ii) (iii) (iv) (v) (vi) Solution: (i) (ii) (iii) (iv) (v) (vi)...
Read More →In a ∆ABC, it is given that AB = 6 cm, AC = 8 cm and AD is the bisector of ∠A.
Question: In a ∆ABC, it is given thatAB= 6 cm,AC= 8 cm andADis the bisector of A. Then,BD:DC=? (a) 3 : 4(b) 9 : 16(c) 4 : 3 (d) $\sqrt{3}: 2$ Solution: (a) 3 : 4 In $\triangle A B D$ and $\triangle A C D$, we have: $\angle B A D=\angle C A D$ Now, $\frac{B D}{D C}=\frac{A B}{A C}=\frac{6}{8}=\frac{3}{4}$ $B D: D C=3: 4$...
Read More →A tent is in the form of a cylinder of diameter 20 m and height 2.5 m,
Question: A tent is in the form of a cylinder of diameter 20 m and height 2.5 m, surmounted by a cone of equal base and height 7.5 m. Find the capacity of the tent and the cost of the canvas at Rs 100 per square metre. Solution: Given that: Radius of the base $r=\frac{d}{2}=\frac{20}{2}=10 \mathrm{~m}$ Height of the cylinder $h_{1}=2.5 \mathrm{~m}$ Height of the cone $h_{2}=7.5 \mathrm{~m}$ Slant height of the cone $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{10^{2}+7.5^{2}}$ $=12.5 \mathrm{~m}$ The total cap...
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