The sum of n terms of a progression is
Question: The sum of $n$ terms of a progression is $\left(2^{n}-1\right)$. Show that it is a GP and find its common ratio Solution: In this question, we will try to rewrite the given sum of the progression like the formula for the sum a G.P. series. It is given that $S_{n}=\left(2^{n}-1\right)$ The formula for the sum of a G.P. series is, $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{\mathrm{r}^{\mathrm{n}}-1}{\mathrm{r}-1}$ By solving the 2 equations together, we can say that $\left(2^{n}-1\right)=...
Read More →Find the sum of the series :
Question: Find the sum of the series : NOTE: The following terms are not G.P. series, but we can convert them to form one. (i) $8+88+888+\ldots$ To $n$ terms (ii) $3+33+333+\ldots$. To n terms (iii) $0.7+0.77+0.777+\ldots$. To $\mathrm{n}$ terms Solution: The expression can be rewritten as [Taking 8 as a common factor] 8(1+ 11 + 111+ to n terms) [Multiplying and dividing the expression by 9] $=\frac{8}{9}(9+99+999+\ldots$ to $n$ terms $)$ $=\frac{8}{9}((10-1)+(100-1)+(1000-1)+\ldots$ to $n$ term...
Read More →Two quantities are said to vary——–with each other,
Question: Two quantities are said to varywith each other, if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant. Solution: Two quantities are said to vary inversely with each other, if increase in one cause a decrease in the other in such a manner that the product of their corresponding values remains constant....
Read More →Two quantities are said to vary——— with each other,
Question: Two quantities are said to vary with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. Solution: Two quantities are said to vary directly with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant....
Read More →If the solve the problem
Question: If $x=\cos \theta, y=\sin ^{3} \theta$. Prove that $y \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=3 \sin ^{2} \theta\left(5 \cos ^{2} \theta-1\right)$ Solution: The idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$, i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$ We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm...
Read More →x and y are said to vary directly
Question: x and y are said to vary directly with each other, if for, some positive number k,-= k. Solution: $x$ and $y$ are said to vary directly wifh'ether, if for some positive number $k, \frac{x}{y}=k$....
Read More →Both x and y are said to vary
Question: Both x and y are said to varywith each other, if for some positive number k, xy =k. Solution: Both x and y are said to vary inversely with each other, if for some positive number k,xy = k. [see condition of inverse proportion]...
Read More →When two quantities
Question: When two quantities $\mathrm{x}$ and $\mathrm{y}$ are in---proportion or vary----they are written as $x \propto \frac{1}{y}$ Solution: When two quantities $x$ and $y$ are in inverse proportion or vary inversely, they are written a $x \propto \frac{1}{y}$ [see definition of inverse proportion]...
Read More →When two quantities x and y
Question: When two quantities $x$ and $y$ are in---proportion or vary---they are written as $x \propto y$ Solution: When two quantities $\mathrm{x}$ and $\mathrm{y}$ are in direct proportion or vary directly, they are written as $x \propto y$ [see definition of direct proportion]...
Read More →Both x and y vary directly with each other
Question: Both x and y vary directly with each other and when x is 10, y is 14, which of the following is not a possible pair of corresponding values of x and y? (a) 25 and 35 (b) 35 and 25 (c) 35 and 49 (d) 15 and 21 Solution: (b) 35 and 25 Explanation: x and y are directly proportional. x y If x = 10 and y = 14, then; 10 14 or 5 7 Now, if we compare, (a) 25 35 = 57 (b) 35 25 = 7 5 (c) 35 49 = 5 7 (d) 15 21 = 5 7 Therefore, option (b) is not a possible pair of corresponding values of x and y. I...
Read More →If the solve the problem
Question: If $x=a(1+\cos \theta), y=a(\theta+\sin \theta)$ Prove that $\frac{d^{2} y}{d x^{2}}=\frac{-1}{a}$ at $\theta=\frac{\pi}{2}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $\mathrm{dy} / \mathrm{d} \theta=\mathrm{f}^{\prime}(\theta)$ and $\mathrm{d} \mathrm{x} / \mathrm{d} \theta=\mathrm{g}^{\prime}(\theta)$ We can Write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\...
Read More →If the distance travelled by a rickshaw in one hour is 10 km,
Question: If the distance travelled by a rickshaw in one hour is 10 km, then the distance travelled by the same rickshaw with the same speed in one minute is: (a) 250/9 m (b) 500/9 m (c) 1000 m (d) 500/3 m Solution: (d) 500/3 m Explanation: Distance travelled = 10km Time taken = 1 hr In one minute, distance covered = 10/60 km = (101000)/60 m = 500/3 m...
Read More →If two quantities p and q vary inversely
Question: If two quantities p and q vary inversely with each other, then: (a) p/q remains constant. (b) p + q remains constant. (c) p q remains constant. (d) p q remains constant. Solution: (c) If two quantities p and q vary inversely with each other, then p x q remains constant. Since, in inverse proportion, an increase in p cause a proportional decrease in q and vice-versa. Hence, option (c) is correct....
Read More →Evaluate :
Question: Evaluate NOTE: In an expression like this ⇒ $\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{X}$ n represents the upper limit, 1 represents the lower limit , x is the variable expression which we are finding out the sum of and i represents the index of summarization. (i) $\sum_{n=1}^{10}\left(2+3^{n}\right)$ (ii) $\sum_{k=1}^{n}\left[2^{k}+3^{(k-1)}\right]$ (iii) $\sum_{n=1}^{8} 5^{n}$ Solution: We can write this as $\left(2+3^{1}\right)+\left(2+3^{2}\right)+\left(2+3^{3}\right)+\ldots$ to 10...
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 (a) x/y remains constant. (b) x y remains constant. (c) x + y remains constant. (d) x y remains constant. Solution: (a) x/y remains constant. If x and y vary directly, then x/y = k...
Read More →100 persons had food provision for 24 days.
Question: 100 persons had food provision for 24 days. If 20 persons left the place, the provision will last for (a) 30 days (b) 96/5 days (c) 120 days (d) 40 days Solution: (a) 30 days Explanation: 100 persons have food provision for = 24 days 1 person will have food provision for = 24100 = 2500 days If 20 persons left the place, then total left = 100-20 = 80 persons Hence, 80 persons have food provision for = 2400/80 = 30 days...
Read More →Meenakshee cycles to her school at an average speed
Question: Meenakshee cycles to her school at an average speed of 12 km/h and takes 20 minutes to reach her school. If she wants to reach her school in 12 minutes, her average speed should be (a) 20/3 km/h (b) 16 km/h (c) 20 km/h (d) 15 km/h Solution: (c) 20 km/h Explanation: Speed = 12km/h Time taken = 20 minutes = 20/60 hr = 1/3 hr Distance covered = S T = 12 1/3 = 4km Speed required to cover 4km in 12 minutes = (4/12)60 = 20 km/hr...
Read More →Both x and y are in direct proportion,
Question: Both x and y are in direct proportion, then 1/x and 1/y are: (a) in indirect proportion. (b) in inverse proportion. (c) neither in direct nor in inverse proportion. (d) sometimes in direct and sometimes in inverse proportion. Solution: (b) in inverse proportion....
Read More →If the solve the problem
Question: If $x=a(1-\cos \theta), y=a(\theta+\sin \theta)$, prove that $\frac{d^{2} y}{d x^{2}}=-\frac{1}{a}$ at $\theta=\frac{\pi}{2}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$ We can Write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}...
Read More →Which of the following vary inversely
Question: Which of the following vary inversely with each other? (a) speed and distance covered. (b) distance covered and taxi fare. (c) distance travelled and time taken. (d) speed and time taken. Solution: (d) speed and time taken. Explanation: By the formula of speed we know; Speed = distance/time As we can see here, when the time decreases, the speed increases. Hence, they inversely proportional....
Read More →Which quantities in the previous
Question: Which quantities in the previous question vary inversely with each other? (a) x and y (b) p and q (c) r and s (d) u and v Solution: (d) u and v Explanation: In option (d), we can see when the value of u is increasing, the value of v is decreasing. Hence, u and v are inversely proportional....
Read More →In which of the following case,
Question: In which of the following case, do the quantities vary directly with each other? Solution: Option (a) Explanation: In option (a), the values of x is directly proportional to values of y, such as; y = 4x If we put the values of x = 0.5, 2, 8 and 32, we get the values of y as 2, 8, 32 and 128 respectively....
Read More →By travelling at a speed of 48 kilometres per hour,
Question: By travelling at a speed of 48 kilometres per hour, a car can finish a certain journey in 10 hours. To cover the same distance in 8 hours, the speed of the car should be (a) 60 km/h (b) 80 km/h (c) 30 km/h (d) 40 km/h Solution: (a) 60 km/h Explanation: Speed of car = 48 km/hr Time taken = 10 hr As we know, Distance = speed time = 48 10 = 480 km Speed required by car to cover 480 km in 8 hours = 480/8 = 60 km/hr....
Read More →A truck needs 54 litres of diesel for covering
Question: A truck needs 54 litres of diesel for covering a distance of 297 km. The diesel required by the truck to cover a distance of 550 km is (a) 100 litres (b) 50 litres (c) 25.16 litres (d) 25 litres Solution: (a) 100 litres Explanation: Distance covered by truck using 54 litres diesel = 297 km Distance covered by truck using 1 litre diesel = 297/54 = 5.5 km Hence, for 550 km, diesel required = 550/5.5 = 100 litres...
Read More →The number of teeth and the age of a person vary
Question: The number of teeth and the age of a person vary (a) directly with each other (b) inversely with each other (c) neither directly nor inversely with each other (d) sometimes directly,and sometimes inversely with each other Solution: (d) The number of teeth and the age of a person vary sometimes directly and sometimes inversely with each other, we cannot predict about the number of teeth with exactly the age of a person. It change with person-to-person. Hence, option (d) is correct....
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