In the given figure, the length of BC is
Question: In the given figure, the length of BC is (a) 7 cm(b) 10 cm(c) 14 cm(d) 15 cm Solution: We know that tangent segments to a circle from the same external point are congruent.Therefore, we haveAF = AE = 4 cmBF = BD = 3 cmEC = AC AE = 11 4 = 7 cmCD = CE = 7 cm BC = BD + DC = 3 + 7 = 10 cmHence, the correct answer is option (b)....
Read More →Solve this
Question: The quantities $x=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}, y=\frac{E}{B}$ and $z=\frac{1}{C R}$ are defined where $C$-capacitance, $R$-Resistance, $l$-length, $E$-Electric field, $B$-magnetic field and $\varepsilon_{0}, \mu_{0}$, - free space permittivity and permeability respectively. Then :$x, y$ and $z$ have the same dimension.Only $x$ and $z$ have the same dimension.Only $x$ and $y$ have the same dimension.Only $y$ and $z$ have the same dimension.Correct Option: 1 Solution: (1) We...
Read More →The slant height of the frustum of a cone is 5 cm.
Question: The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, write the height of the frustum. Solution: Slant height of the Frustum = 5 cm i.e. $l=5 \mathrm{~cm}$. $r_{1}-r_{2}=4 \mathrm{~cm}$ $l=\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}$ $5=\sqrt{h^{2}+(4)^{2}}$ Squaring both sides we get $25=h^{2}+4^{2}$ $25=h^{2}+16$ $25-16=4^{2}$' or $h^{2}=9 \mathrm{~cm}$ $4=3 \mathrm{~cm}$ Height of the Frustum = 3 cm $r_{1}-r_{2}=4 \mat...
Read More →The slant height of the frustum of a cone is 5 cm.
Question: The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, write the height of the frustum. Solution: Slant height of the Frustum = 5 cm i.e. $l=5 \mathrm{~cm}$. $r_{1}-r_{2}=4 \mathrm{~cm}$ $l=\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}$ $5=\sqrt{h^{2}+(4)^{2}}$ Squaring both sides we get $25=h^{2}+4^{2}$ $25=h^{2}+16$ $25-16=4^{2}$' or $h^{2}=9 \mathrm{~cm}$ $4=3 \mathrm{~cm}$ Height of the Frustum = 3 cm $r_{1}-r_{2}=4 \mat...
Read More →A quantity $x$ is given by
Question: A quantity $x$ is given by $\left(I F v^{2} / W L^{4}\right)$ in terms of moment of inertia $I$, force $F$, velocity $v$, work $W$ and Length $L$. The dimensional formula for $x$ is same as that of:planck's constantforce constantenergy densitycoefficient of viscosityCorrect Option: , 3 Solution: (3) Dimension of Force $F=\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}$ Dimension of velocity $V=\mathrm{L}^{1} \mathrm{~T}^{-1}$ Dimension of work $=\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~...
Read More →If PA and PB are two tangents to a circle with centre O, such that ∠AOB = 110°, find ∠APB.
Question: IfPAandPBare two tangents to a circle with centreO, such that AOB= 110, find APB. (a) 55(b) 60(c) 70(d) 90 Solution: (c) 70 Given, $\mathrm{PA}$ and $\mathrm{PB}$ are tangents to a circle with centre $\mathrm{O}$, with $\angle \mathrm{AOB}=110^{\circ}$. Now, we know that tangents drawn from an external point are perpendicular to the radius at the point of contact. So, $\angle \mathrm{OAP}=90^{\circ}$ and $\angle \mathrm{OBP}=90^{\circ}$. $\Rightarrow \angle \mathrm{OAP}+\angle \mathrm{...
Read More →Dimensional formula for thermal conductivity is (here $K$ denotes the temperature :
Question: Dimensional formula for thermal conductivity is (here $K$ denotes the temperature :$\mathrm{MLT}^{-2} \mathrm{~K}$$\mathrm{MLT}^{-2} \mathrm{~K}^{-2}$$\mathrm{MLT}^{-3} \mathrm{~K}$$\mathrm{MLT}^{-3} \mathrm{~K}^{-1}$Correct Option: , 4 Solution: (4) From formula, $\frac{d Q}{d t}=k A \frac{d T}{d x}$ $\Rightarrow k=\frac{\left(\frac{d Q}{d t}\right)}{A\left(\frac{d T}{d x}\right)}$ $[k]=\frac{\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]}{\left[\mathrm{L}^{2}\right]\left[\mathrm{KL}^{...
Read More →A cylinder and a cone are of the same base
Question: A cylinder and a cone are of the same base radius and of same height. Find the ratio of the value of the cylinder to that of the cone Solution: Since, cylinder and a cone both are have same radius and height. Therefore, $\frac{V_{1}}{V_{2}}=\frac{\pi r^{2} h}{\frac{1}{3} \pi r^{2} h}$ $\frac{V_{1}}{V_{2}}=\frac{1}{\frac{1}{3}}$ $\frac{V_{1}}{V_{2}}=\frac{3}{1}$ $V_{1}: V_{2}=3: 1$...
Read More →In the given figure, AT is a tangent to the circle with centre O,
Question: In the given figure,ATis a tangent to the circle with centreO, such thatOT= 4 cm and OTA= 30. Then, AT = ? (a) 4 cm(b) 2 cm (c) $2 \sqrt{3} \mathrm{~cm}$ (d) $4 \sqrt{3} \mathrm{~cm}$ Solution: (c) $2 \sqrt{3} \mathrm{~cm}$ $O A \perp A T$ So, $\frac{A T}{O T}=\cos 30^{\circ}$ $\Rightarrow \frac{A T}{4}=\frac{\sqrt{3}}{2}$ $\Rightarrow A T=\left(\frac{\sqrt{3}}{2} \times 4\right)$ $\Rightarrow A T=2 \sqrt{3}$...
Read More →The surface area of a sphere is 616 cm2 .
Question: The surface area of a sphere is 616 cm2. Find its radius. Solution: The surface area of sphere = 616k cm2 We know that $4 \pi r^{2}=616$ $r^{2}=\frac{616}{4 \pi}$ Taking squire root both the side $\sqrt{r^{2}}=\sqrt{\frac{616}{4 \pi}}$ $r=7 \mathrm{~cm}$...
Read More →Amount of solar energy received on the earth's surface per unit area per unit time
Question: Amount of solar energy received on the earth's surface per unit area per unit time is defined a solar constant. Dimension of solar constant is :$\mathrm{ML}^{2} \mathrm{~T}^{-2}$$\mathrm{ML}^{0} \mathrm{~T}^{-3}$$\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-1}$$\mathrm{MLT}^{-2}$Correct Option: , 2 Solution: (2) Solar constant $=\frac{\text { Energy }}{\text { Time Area }}$ Dimension of Energy, $E=\mathrm{ML}^{2} \mathrm{~T}^{-2}$ Dimension of Time $=\mathrm{T}$ Dimension of Area $=\ma...
Read More →If the areas of circular bases of a frustum
Question: If the areas of circular bases of a frustum of a cone are 4 cm2and 9 cm2respectively and the height of the frustum is 12 cm. What is the volume of the frustum? Solution: Area of circular bases of frustum is $A_{1}=4 \mathrm{~cm}^{2}$ $A_{2}=9 \mathrm{~cm}^{2}$ The height of frustumh= 12 cm Now, the volume of frustum $V=\frac{h}{3}\left\{A_{1}+A_{2}+\sqrt{A_{1} A_{2}}\right\}$ $=\frac{12}{3}\{4+9+\sqrt{4 \times 9}\}$ $=4\{13+6\}$ $V=76 \mathrm{~cm}^{3}$...
Read More →In the given figure, O is the centre of the circle, PQ is a chord and PT is the tangent at P.
Question: In the given figure, O is the centre of the circle, PQ is a chord and PT is the tangent at P. If POQ = 70∘, then TPQ is equal to (a) 35∘(b) 45∘(c) 55∘(d) 70∘ Solution: We know that the radius and tangent are perperpendular at their point of contactSince, OP = OQ∵POQ is a isosceles right triangleNow, In isoceles right triangle POQPOQ + OPQ + OQP = 180∘ [Angle sum property of a triangle]⇒ 70∘+ 2OPQ = 180∘⇒ OPQ = 55∘Now, TPQ + OPQ = 90∘⇒ TPQ = 35∘Hence, the correct answer is option (a)....
Read More →If the slant height of the frustum of a cone is 6 cm
Question: If the slant height of the frustum of a cone is 6 cm and the perimeters of its circular bases are 24 cm and 12 cm respectively. What is the curved surface area of the frustum? Solution: The parameter of upper base $=2 \pi r_{1}$ $2 \pi r_{1}=12$ $r_{1}=\frac{6}{\pi} \mathrm{cm}$ The parameter of lower base $=2 \pi r_{2}$ $2 \pi r_{2}=24$ $r_{2}=\frac{12}{\pi} \mathrm{cm}$ The surface area of frustum $=\pi\left(\frac{6}{\pi}+\frac{12}{\pi}\right) \times 6$ $=\pi \times \frac{18}{\pi} \t...
Read More →If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is :
Question: If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is :$\left[\mathrm{P}^{2} \mathrm{AT}^{-2}\right]$$\left[\mathrm{PA}^{-1} \mathrm{~T}^{-2}\right]$$\left[\mathrm{PA}^{1 / 2} \mathrm{~T}^{-1}\right]$$\left[\mathrm{P}^{1 / 2} \mathrm{AT}^{-1}\right]$Correct Option: , 3 Solution: (3) Energy, $E \propto A^{a} T^{b} P^{c}$ or, $\quad E=k A^{a} T^{b} P^{c}$ $\ldots$ (i) where $k$ is a dimensionless constant and $a, b$ a...
Read More →In the given figure, O is the centre of a circle. AOC is its diameter, such that ∠ACB = 50°.
Question: In the given figure,Ois the centre of a circle.AOCis its diameter, such that ACB= 50. IfATis the tangent to the circle at the pointA,then BAT= ? (a) 40(b) 50(c) 60(d) 65 Solution: (b)50 $\angle A B C=90^{\circ}$ (A ngle in a semicircle) In $\triangle A B C$, we have : $\angle A C B+\angle C A B+\angle A B C=180^{\circ}$ $\Rightarrow 50^{0}+\angle C A B+90^{0}=180^{\circ}$ $\Rightarrow \angle C A B=\left(180^{\circ}-140^{0}\right)$ $\Rightarrow \angle C A B=40^{\circ}$ Now, $\angle C A ...
Read More →If r1 and r2 denote the radii of the circular bases
Question: If r1and r2denote the radii of the circular bases of the frustum of a cone such that r1 r2, then write the ratio of the height of the cone of which the frustum is a part to the height fo the frustum. Solution: Since, $\Delta V O^{\prime} B \sim \Delta V O A$ Therefore, In $\Delta V O^{\prime} B-\Delta V O A$ $\frac{O^{\prime} B}{O A}=\frac{O^{\prime} V}{O V}$ $\frac{r_{2}}{r_{1}}=\frac{h-h_{1}}{h}$ $\frac{r_{2}}{r_{1}}=1-\frac{h_{1}}{h}$ $\frac{h_{1}}{h}=1-\frac{r_{2}}{r_{1}}$ $=\frac{...
Read More →In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm.
Question: In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA = 17 cm, then the length of AC (in cm) is(a) 9 cm(b) 15 cm (c) $\sqrt{353} \mathrm{~cm}$ (d) 25 cm Solution: We know that the radius and tangent are perperpendular at their point of contactIn right triangle AOBBy using Pythagoras theorem, we haveOA2= AB2+ OB2⇒ 172= AB2+ 82⇒289 = AB2+ 64⇒AB2= 225⇒ AB = 15 cmThe tangents drawn from the external point are equal.Therefore, the length of AC is 15 cm...
Read More →If speed V, area A and force F are chosen as fundamental units,
Question: If speed $\mathrm{V}$, area $\mathrm{A}$ and force $\mathrm{F}$ are chosen as fundamental units, then the dimension of Young's modulus will be :$\mathrm{FA}^{2} \mathrm{~V}^{-1}$$\mathrm{FA}^{2} \mathrm{~V}^{-3}$$\mathrm{FA}^{2} \mathrm{~V}^{-2}$$\mathrm{FA}^{-1} \mathrm{~V}^{0}$Correct Option: , 4 Solution: (4) Young's modulus, $Y=\frac{\text { stress }}{\text { strain }}$ $\Rightarrow Y=\frac{\mathrm{F}}{\mathrm{A}} / \frac{\Delta \ell}{\ell_{0}}=\mathrm{FA}^{-1} \mathrm{~V}^{0}$...
Read More →A hemisphere and a cone have equal bases.
Question: A hemisphere and a cone have equal bases. If their heights are also equal, then what is the ratio of their curved surfaces? Solution: The base of the cone and hemisphere are equal. So radius of the two is also equal. and Height of the hemisphere = height of the cone Then the slant height of the cone $I=\sqrt{r^{2}+h^{2}}$ $=\sqrt{r^{2}+r^{2}}$ $=\sqrt{2 r^{2}}$ $=r \sqrt{2}$ ............(i) Now, the curved surface area of Hemisphere $=2 \pi r^{2}$ and The curved surface area of cone Pu...
Read More →In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm.
Question: In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of AB is(a) 8 cm(b) 14 cm(c) 16 cm (d) $\sqrt{136} \mathrm{~cm}$ Solution: We know that the radius and tangent are perperpendular at their point of contactIn right triangle AOPAO2= OP2+ PA2⇒ 102= 62+ PA2⇒ PA2= 64⇒ PA = 8 cmSince, the perpendicular drawn from the centre bisect the chord. PA = PB = 8 cmNow, AB = AP + PB = 8 + 8 =...
Read More →If a chord AB subtends an angle of 60∘ at the centre of a circle,
Question: If a chord AB subtends an angle of 60∘at the centre of a circle, then the angle between the tangents to the circle drawn from A and B isl to (a) 30∘(b) 60∘(c) 90∘(d) 120∘ Solution: We know that the radius and tangent are perperpendular at their point of contact∵OBC = OAC = 90∘Now, In quadrilateral ABOCACB + OAC + OBC + AOB = 360∘ [Angle sum property of a quadrilateral]⇒ ACB + 90∘+ 90∘+ 60∘= 360∘⇒ ACB + 240∘= 360∘⇒ ACB = 120∘Hence, the correct answer is option (d)....
Read More →Two cones have their heights in the ratio 1 : 3 and radii 3 : 1.
Question: Two cones have their heights in the ratio 1 : 3 and radii 3 : 1. What is the ratio of their volumes? Solution: Let the radius of the cone is 3xandx, And the height of the cone isyand 3y. Then, Volume of the first cone $v_{1}=\frac{1}{3} \pi r^{2} h$ $=\frac{1}{3} \pi(3 x)^{2} y$ $=\frac{1}{3} \pi 9 x^{2} y$ $=3 \pi x^{2} y$.............$(i)$ Volume of the second cone $v_{2}=\frac{1}{3} \pi(x)^{2} \times 3 y$ $=\pi x^{2} y$............(ii) Then the radius of their volume Or $\frac{v_{1}...
Read More →In the given figure, AB and AC are tangents to a circle with centre O such that ∠BAC = 40∘
Question: In the given figure, AB and AC are tangents to a circle with centre O such that BAC = 40∘.Then BOC is equal to (a) 80∘(b) 100∘(c) 120∘(d) 140∘ Solution: We know that the radius and tangent are perperpendular at their point of contact∵OBA = OCA = 90∘Now, In quadrilateral ABOCBAC + OCA + OBA + BOC = 360∘ [Angle sum property of a quadrilateral]⇒ 40∘+ 90∘+ 90∘+ BOC = 360∘⇒ 220∘+ BOC = 360∘⇒ BOC = 140∘Hence, the correct answer is option (d)....
Read More →If ' C and ' V ' represent capacity and voltage respectively then
Question: If ' $C^{\prime}$ and ' $\mathrm{V}$ ' represent capacity and voltage respectively then what are the dimensions of $\lambda$ where $C / V=\lambda$ ?$\left[M^{-2} L^{-4} I^{3} T^{7}\right]$$\left[M^{-2} L^{-3} I^{2} T^{6}\right]$$\left[M^{-1} L^{-3} I^{-2} T^{-7}\right]$$\left[M^{-3} L^{-4} I^{3} T^{7}\right]$Correct Option: 1, Solution: (1) $\because v=\frac{w}{q}$ and $c=\frac{q}{v}$ dimension of $\frac{c}{v}$ $\Longrightarrow \frac{q}{v^{2}}$ $\Rightarrow \frac{q}{w^{2}} \times q^{2}...
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