When the distance is kept fixed,
Question: When the distance is kept fixed, speed and time vary directly with each other. Solution: False When the distance is kept fixed, speed and time vary indirectly/inversely with each other. Since, if we increase speed, then taken time will less and vice-versa....
Read More →When the speed is kept fixed,
Question: When the speed is kept fixed, time and distance vary inversely with each other. Solution: False When the speed is kept fixed, time and distance vary directly with each other....
Read More →Two quantities x and y are said
Question: Two quantities x and y are said to vary directly with each other, if for some rational number k, xy =k. Solution: FalseTwo quantities x and y are said to vary directly with each other, if $x_{y}=\mathrm{k}$ (constant)...
Read More →The common ratio of a finite GP is 3, and its last term is 486.
Question: The common ratio of a finite GP is 3, and its last term is 486. If the sum of these terms is 728, find the first term. Solution: 'Tn' represents the $\mathrm{n}^{\text {th }}$ term of a G.P. series. $T_{n}=a r^{n-1}$ $\Rightarrow 486=a(3)^{n-1}$ $\left.\Rightarrow 486=a\left(3^{n} \div 3\right)\right)$ $\Rightarrow 486 \times 3=a\left(3^{n}\right)$ $\Rightarrow 1458=a\left(3^{n}\right) \ldots \ldots \ldots .(i)$ Sum of a G.P. series is represented by the formula, $\mathrm{Sn}=\mathrm{a...
Read More →Devangi travels 50 m distance in 75 steps,
Question: Devangi travels 50 m distance in 75 steps, then the distance travelled in 375 steps is _______ km. Solution: Distance travelled in 75 steps = 50m Distance covered in 1 step = 50/75m Distance covered in 375 steps = (50/75)375 = 18750/75 = 250m =250/1000 km = 0.25km In questions from 43 to 59, state whether the statements are true (T) or false (F)....
Read More →If 45 persons can complete a work in 20 days,
Question: If 45 persons can complete a work in 20 days, then the time taken by 75 persons will be ______ hours. Solution: 45 persons complete a work in 20 days 1 person can complete work in = 45 20 = 900 days Time taken by 75 persons = 900/75 = 12 days = 1224 = 288 hr...
Read More →If the solve the problem
Question: If $y=\left(\sin ^{-1} x\right)^{2}$, prove that: $\left(1-x^{2}\right) y_{2}-x y_{1}-2=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $y=\left(\sin ^{-1} x\right)^{2} \ldots \ldots$ equation 1 to prove : $\left(1-x^{2}\right) y_{2}-x y_{1}-2=0$ We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ $\mat...
Read More →If the area occupied by 15 postal
Question: If the area occupied by 15 postal stamps is 60 cm2, then the area occupied by 120 such postal stamps will be _______. Solution: Area occupied by 15 postal stamps = 60cm2 Area occupied by 1 postal stamps = 60/15 = 4 cm2 Area occupied by 120 such postal stamps = 4120 =480 cm2...
Read More →If x varies inversely as y and x = 4
Question: If x varies inversely as y and x = 4 when y = 6, then when x = 3 the value of y is _______. Solution: If x varies inversely as y, then xy = k 46 = k k=24 Now, if x = 3, then y is; y = k/x = 24/3 = 8 38. In direct proportion, a1/b1=a2/b2 a1/b1= a2/b2= k 39. In case of inverse proportion,a2/a1= b2/b1...
Read More →If the thickness of a pile of 12 cardboard
Question: If the thickness of a pile of 12 cardboard sheets is 45 mm, then the thickness of a pile of 240 sheets is _______ cm. Solution: Thickness of pile of 12 cardboard sheets = 45mm Thickness of 1 cardboard sheet = 45/12 mm Hence, thickness of a pile of 240 sheets = (45/12)240 = 45 20 = 900mm = 900/10 cm = 90cm...
Read More →How many terms of the series 2 + 6 + 18 + …. + must be taken to make the
Question: How many terms of the series 2 + 6 + 18 + . + must be taken to make the sum equal to 728? Solution: Sum of a G.P. series is represented by the formula $S_{n}=a \frac{r^{n}-1}{r-1}$ hen r1. Sn represents the sum of the G.P. series upto nth terms, a represents the first term, r represents the common ratio and n represents the number of terms. Here, a = 2 r = (ratio between the n term and n-1 term) 6 2 = 3 $S_{n}=728$ $\therefore 728=2 \times \frac{3^{n}-1}{3-1}$ $\Rightarrow 728=2 \times...
Read More →An auto rickshaw takes 3 hours to cover a distance of 36 km.
Question: An auto rickshaw takes 3 hours to cover a distance of 36 km. If its speed is increased by 4 km/h, the time taken by it to cover the same distance is __________. Solution: Distance covered by auto rickshaw in 3 hours = 36km Speed = 36/3 = 12 km/hr If we increase the speed by 4km/hr, then the total speed becomes = 12+4 = 16km/hr Now, the time taken by auto rickshaw will be = 36/16 = (3660)/16 = 135 min = 120 + 15 = 2 hour 15min...
Read More →A car is travelling 48 km in one hour.
Question: A car is travelling 48 km in one hour. The distance travelled by the car in 12 minutes is _________. Solution: Distance travelled by car in one hour = 48km Distance travelled in one minute = 48/60 km Distance travelled in 12 minute = (48/60)12 = 48/5 = 9.6km...
Read More →If two quantities p and q vary inversely
Question: If two quantities p and q vary inversely with each other, then- of their corresponding values remains constant. Solution: If two quantities p and q vary inversely with each other, then product of their corresponding values remains constant....
Read More →If two quantities x and y vary directly with each other,
Question: If two quantities x and y vary directly with each other, then of their corresponding values remains constant. Solution: If two quantities x and y vary directly with each other, then ratio of their corresponding values remains constant. [see definition of direct proportion]...
Read More →If the solve the problem
Question: If $y=\left(\sin ^{-1} x\right)^{2}$, prove that: $\left(1-x^{2}\right) y_{2}-x y_{1}-2=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $y=\left(\sin ^{-1} x\right)^{2} \ldots \ldots$ equation 1 to prove : $\left(1-x^{2}\right) y_{2}-x y_{1}-2=0$ We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ $\mat...
Read More →If on increasing a, b decreases
Question: If on increasing a, b decreases in such a manner that remainsand positive, then a and b are said to vary inversely with each other. Solution: If on increasing a, b decreases in such a manner that ab remains constant and positive, then a and b are said to vary inversely with each other. [see definition of inverse proportion]...
Read More →Prove the following
Question: On increasing $\mathrm{a}$, b increases in such a manner that $\frac{a}{b}$ remains--and positive, then $\mathrm{a}$ and $\mathrm{b}$ are said to vary directly with each other. Solution: On increasing $a$, $b$ increases in such a manner that $\frac{a}{b}$ remains constant and positive, then $a$ and $b$ are said to vary directly with each other....
Read More →Find the sum of the geometric series 3 + 6 + 12 + … + 1536.
Question: Find the sum of the geometric series 3 + 6 + 12 + + 1536. Solution: Tn represents the $n^{\text {th }}$ term of a G.P. series. $r=6 \div 3=2$ $\mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ $\Rightarrow 1536=3 \times 2^{n-1}$ $\Rightarrow 1536 \div 3=2^{n} \div 2$ $\Rightarrow 1536 \div 3 \times 2=2^{n}$ $\Rightarrow 1024=2^{n}$ $\Rightarrow 2^{10}=2^{n}$ n = 10 Sum of a G.P. series is represented by the formula, $S_{n}=a \frac{r^{n}-1}{r-1}$ hen r1. Sn represents the sum of the G...
Read More →When the speed remains constant,
Question: When the speed remains constant, the distance travelled isproportional to the time. Solution: When the speed remains constant, the distance travelled is directly proportional to the time. e.g. If 10 km cover in 10 min with uniform speed, then 20 km cover in 20 min with same speed....
Read More →If x varies directly as y,
Question: If x varies directly as y, then Solution: If x and y varies directly, then; x/y = k If x=12 and y = 48, then k = 12/48 = Now, if x=6, and k = 1/4, then y will be; 6/y = y = 64=24...
Read More →If x varies inversely as y,
Question: If x varies inversely as y, then Solution: If x varies inversely as y, then; xy = k If x = 60 and y = 10 Then, xy = 6010 = 600 = k Hence, x2 = 600 x = 600/2 = 300...
Read More →If the solve the problem
Question: If $y=\sin (\sin x)$, prove that $: \frac{d^{2} y}{d x^{2}}+\tan x \cdot \frac{d y}{d x}+y \cos ^{2} x=0$ Solution: Given, $y=\sin (\sin x) \ldots \ldots .$ equation 1 To prove: $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+\tan \mathrm{x} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cos ^{2} \mathrm{x}=0$ We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ As $\fra...
Read More →If 12 pumps can empty a reservoir in 20 hours,
Question: If 12 pumps can empty a reservoir in 20 hours, then time required by 45 such pumps to empty the same reservoir is ______ hours. Solution: Time taken by 12 pumps to empty a reservoir = 20 hr Time taken by 1 pump to empty the reservoir = 20 12 = 240 hr Hence, time taken by 45 pumps to empty the reservoir = 240/45 = (24060)/45 = 14400/45 = 320 min = 569 + 20 min = 5 hour 20 min...
Read More →In a GP, the ratio of the sum of the first three terms is to first six terms is 125
Question: In a GP, the ratio of the sum of the first three terms is to first six terms is 125: 152. Find the common ratio. Solution: Sum of a G.P. series is represented by the formula, $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{\mathrm{r}^{\mathrm{n}}-1}{\mathrm{r}-1}$ hen r1. Sn represents the sum of the G.P. series upto nth terms, a represents the first term, r represents the common ratio and n represents the number of terms. Sum of first 3 terms $={ }^{\mathrm{a}} \times \frac{\mathrm{r}^{3}-1...
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