Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a \cos \theta$ and $y=b \sin \theta$ Solution: as $x=a \cos \theta$ and $y=b \sin \theta$ Then, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\frac{\mathrm{d}(\operatorname{acos} \theta)}{\mathrm{d} \theta}=-\mathrm{a} \sin \theta$ $\frac{\mathrm{dy}}{\mathrm{d} \theta}=\frac{\mathrm{d}(\mathrm{b} \sin \theta)}{\mathrm{d} \theta}=\mathrm{b} \cos \theta$ $\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\fra...
Read More →The reciprocal of a positive
Question: The reciprocal of a positive rational number is_______. Solution: The reciprocal of a positive rational number is positive rational number. Let us take positive rational number $2 / 3$ The reciprocal of this positive rational number is $3 / 2$ (positive rational number)...
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(1+i)^{3}-(1-i)^{3}$ Solution: Given: $(1+i)^{3}-(1-i)^{3} \ldots(i)$ We know that $(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$ $(a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}$ By applying the formulas in eq. (i), we get $(1)^{3}+3(1)^{2}(i)+3(1)(i)^{2}+(i)^{3}-\left[(1)^{3}-3(1)^{2}(i)+3(1)(i)^{2}-(i)^{3}\right]$ $=1+3 i+3 i^{2}+i^{3}-\left[1-3 i+3 i^{2}-i^{3}\right]$ $=1+3 i+3 i^{2}+i^{3}-1+3 i-3 i^{2}+i^{3}$ $=6 i+2 i^{...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ Solution: $x=a(\theta+\sin \theta)$ Differentiating it with respect to $\theta$, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}(1+\cos \theta) \ldots \ldots(1)$ And, $y=a(1-\cos \theta)$ Differentiating it with respect to $\theta$, $\frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{a}(0+\sin \theta)$ $\frac{d y}{d \theta}=a \sin \theta$ .....(2) Using equation (1) and (2), $\frac{\mathrm{dy}}...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ Solution: $x=a(\theta+\sin \theta)$ Differentiating it with respect to $\theta$, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}(1+\cos \theta) \ldots \ldots(1)$ And, $y=a(1-\cos \theta)$ Differentiating it with respect to $\theta$, $\frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{a}(0+\sin \theta)$ $\frac{d y}{d \theta}=a \sin \theta$ .....(2) Using equation (1) and (2), $\frac{\mathrm{dy}}...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A can do a piece of work in 15 days. B is 50% more efficient than A. A can finish it in (a) 10 days (b) $7 \frac{1}{2}$ days (c) 12 days (d) $10 \frac{1}{2}$ days Solution: (a) 10 days Work done by $\mathrm{A}$ in 1 day $=\frac{1}{15}$ $\mathrm{B}$ is $50 \%$ more efficient than $\mathrm{A}$. $\therefore$ Work done by B in 1 day $=\frac{150}{100} \times \frac{1}{15}=\frac{1}{10}$ Thus, B can complete the work in 10 days....
Read More →Between the numbers
Question: Between the numbers $\frac{15}{20}$ and $\frac{35}{40}$, the greater number is__________. Solution: The LCM of the denominators 20 and 40 is 40 $\therefore(15 / 20)=[(15 \times 2) /(20 \times 2)]=(30 / 40)$ and $(35 / 40)=[(35 \times 1) /(40 \times 1)]=(35 / 40)$ Now, $3035$ $\Rightarrow(30 / 40)(35 / 40)$ Hence, $(15 / 20)(35 / 40)$ $\therefore 35 / 40$ is greater. So. between the numbers $(15 / 20)$ and $(35 / 40)$, the greater number is $(35 / 40)$....
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(1+2 i)^{-3}$ Solution: Given: $(1+2 i)^{-3}$ Above equation can be re - written as $=\frac{1}{(1+2 i)^{3}}$ Now, rationalizing $=\frac{1}{(1+2 i)^{3}} \times \frac{(1-2 i)^{3}}{(1-2 i)^{3}}$ $=\frac{(1-2 i)^{3}}{(1+2 i)^{3}(1-2 i)^{3}}$ We know that, $(a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}$ $(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$ $=\frac{(1)^{3}-3(1)^{2}(2 i)+3(1)(2 i)^{2}-(2 i)^{3}}{\left[(1)^{3}+3(1)^{2}(2 ...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: 3 men or 5 women can do a work in 12 days. How long will 6 men and 5 women take to do it? (a) 6 days (b) 5 days (c) 4 days (d) 3 days Solution: (c) 4 days Three men can complete the work in 12 days. Thus, one man can complete the work in 36 days. Rate of work done by one man in 1 day $=\frac{1}{36}$ Similarly, rate of work done by one woman in 1 day $=\frac{1}{5 \times 12}=\frac{1}{60}$ Now, six men will do $\frac{6}{36}$, i.e., $\frac{1}{6}$ unit of work i...
Read More →The equivalent rational number
Question: The equivalent rational number of $\frac{7}{9}$, whose denominator is 45 is_________. Solution: Form the question it is given that equivalent of $7 / 9=$ Numerator $/ 45$ To get 45 in the denominator multiply both numerator and denominator by 5 Then, $=(7 \times 5) /(9 \times 5)$ $=35 / 45$ So, the equivalent rational number of $7 / 9$, whose denominator is 45 is $(35 / 45)$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A and B together can do a piece of work in 12 days; B and C can do it in 20 days while C and A can do it in 15 days. A, B and C all working together can do it in (a) 6 days (b) 9 days (c) 10 days (d) $10 \frac{1}{2}$ days Solution: (c) 10 days $(\mathrm{A}+\mathrm{B})$ can do a work in 12 days. $(\mathrm{B}+\mathrm{C})$ can do a work in 20 days. $(\mathrm{C}+\mathrm{A})$ can do a work in 15 days. Now, we have : Work done by $(\mathrm{A}+\mathrm{B})$ in 1 da...
Read More →The equivalent of
Question: The equivalent of $\frac{5}{7}$ whose numerator is 45 , is -. Solution: Form the question it is given that equivalent of $5 / 7=45 /$ denominator To get 45 in the numerator multiply both numerator and denominator by 9 Then, $=(5 \times 9) /(7 \times 9)$ $=45 / 63$ So, the equivalent of $5 / 7$, whose numerator is 45 is $(45 / 63)$...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a t^{2}$ and $y=2 a t$ Solution: Given that $x=a t^{2}, y=2 a t$ So, $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\mathrm{d}\left(\mathrm{at}^{2}\right)}{\mathrm{dt}}=2 \mathrm{at}$ $\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{d}(2 \mathrm{at})}{\mathrm{dt}}=2 \mathrm{a}$ Therefore, $\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{2 a}{2 a t}=\frac{1}{t}$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: The rates of working of A and B are in the ratio 3 : 4. The number of days taken by them to finish the work are in the ratio (a) 3 : 4 (b) 9 : 16 (c) 4 : 3 (d) 16 : 9 Solution: (c) 4 : 3 The number of days taken for working is the reciprocal of the rate of work. i.e., number of days taken $=\frac{1}{\text { rate of work }}=\frac{1}{3 / 4}=\frac{4}{3}$...
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(2+i)^{-2}$ Solution: Given: $(2+i)^{-2}$ Above equation can be re written as $=\frac{1}{(2+i)^{2}}$ Now, rationalizing $=\frac{1}{(2+i)^{2}} \times \frac{(2-i)^{2}}{(2-i)^{2}}$ $=\frac{(2-i)^{2}}{(2+i)^{2}(2-i)^{2}}$ $=\frac{4+i^{2}-4 i}{\left(4+i^{2}+4 i\right)\left(4+i^{2}-4 i\right)}\left[\because(\mathrm{a}-\mathrm{b})^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}-2 \mathrm{ab}\right]$ $=\frac{4-1-4 i}{(4-1+4 i)(4-1-4 i)}\l...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A alone can finish a piece of work in 10 days which B alone can do in 15 days. If they work together and finish it, then out of total wages of Rs 3000, A will get (a) Rs 1200 (b) Rs 1500 (c) Rs 1800 (d) Rs 2000 Solution: (c) Rs 1800 Since the wage distribution will follow the work distribution ratio, we have: Work done by $\mathrm{A}$ in 1 day $=\frac{1}{10}$ Work done by $B$ in 1 day $=\frac{1}{15}$ Net work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\fr...
Read More →Which of the following statements is always true?
Question: Which of the following statements is always true? (a) $(x-y) / 2$ is a rational number between $x$ and $y$. (b) $(x+y) / 2$ is a rational number between $x$ and $y$. (c) $(x \times y) / 2$ is a rational number between $x$ and $y$. (d) $(x \div y) / 2$ is a rational number between $x$ and $y$. Solution: (b) $(x+y) / 2$ is a rational number between $x$ and $y$ Let us assume the value of $x$ and $y$ is 6 and 9 respectively Then, $=(6+9) / 2$ $=14 / 2$ $=7$ Hence, the value 7 is lies betwe...
Read More →Solve this
Question: If $y(\cos x)^{(\cos x)^{(\cos x) \ldots \infty}}$, prove that $\frac{d y}{d x}=\frac{y^{2} \tan x}{(1-y \log \cos x)}$. Solution: Here, $y=(\cos x)^{(\cos x)^{(\cos x)^{-1}}}$ $y=(\cos x)^{y}$ By taking log on both sides, $\log y=\log (\cos x)^{y}$ $\log y=y(\log \cos x)$ Differentiating both sides with respect to $x$ by using the product rule, $\frac{1}{y} \frac{d y}{d x}=y \frac{d(\log \cos x)}{d x}+\log \cos x \frac{d y}{d x}$ $\frac{1}{y} \frac{d y}{d x}=\frac{y}{\cos x} \frac{d(\...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A works twice as fast as B. If both of them can together finish a piece of work in 12 days, then B alone can do it in (a) 24 days (b) 27 days (c) 36 days (d) 48 days Solution: (c) 36 days Let A take $x$ days to complete the work. Then B takes $2 x$ days to complete the work. A's 1 day's work $=\frac{1}{x}$ B's 1 day's work $=\frac{1}{2 x}$ A and B take 12 days to complete the work. Net work done by $(\mathrm{A}+\mathrm{B})$ in 1 day $=\frac{1}{12}=\frac{1}{...
Read More →(X+Y)/2 is a rational number
Question: $(x+y) / 2$ is a rational number (a) Between x and y (b) Less than x and y both (c) Greater than x and y both (d) Less than x but greater than y Solution: (a) Between $x$ and $y$ Let us assume the value of $x$ and $y$ is 4 and 8 respectively Then, $=(4+8) / 2$ $=12 / 2$ $=6$ Hence, the value 6 is lies between 4 and 8 ....
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(-2+\sqrt{-3})^{-1}$ Solution: Given: $(-2+\sqrt{-3})^{-1}$ We can re- write the above equation as $=\frac{1}{-2+\sqrt{-3}}$ $=\frac{1}{-2+\sqrt{3 i^{2}}}\left[\because i^{2}=-1\right]$ $=\frac{1}{-2+i \sqrt{3}}$ Now, rationalizing $=\frac{1}{-2+i \sqrt{3}} \times \frac{-2-i \sqrt{3}}{-2-i \sqrt{3}}$ $=\frac{-2-i \sqrt{3}}{(-2+i \sqrt{3})(-2-i \sqrt{3})} \ldots(\mathrm{i})$ Now, we know that, $(a+b)(a-b)=\left(a^{2}-...
Read More →Solve this
Question: If $y=e^{x^{x^{x}}}+x^{e^{x}}+e^{x^{x^{e}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}^{\mathrm{e}^{\mathrm{x}}}} \cdot \mathrm{x}^{\mathrm{e}^{\mathrm{x}}}\left\{\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}+\mathrm{e}^{\mathrm{x}} \cdot \log \mathrm{x}\right\}+\mathrm{x}^{\mathrm{e}^{\mathrm{x}}} \cdot \mathrm{e}^{\mathrm{e}^{\mathrm{x}}}$ $\left\{\frac{1}{x}+e^{x} \cdot \log x\right\}+e^{x^{x^{e}}} x^{x^{e}} \cdot x^{e-1}\{1+e \log x\}$ Solution: Here, $y=e^{e...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: To complete a work, A takes 50% more time than B. If together they take 18 days to complete the work, how much time shall B take to do it? (a) 30 days (b) 35 days (c) 40 days (d) 45 days Solution: (a) 30 days Let $B$ take $x$ days to complete the work. Then A takes $\left(x+\frac{50}{100} x\right)=1.5 x$ A's 1 day's work $=\frac{1}{1.5 x}=\frac{2}{3 x}$ $B^{\prime} s 1$ day's work $=\frac{1}{x}$ $(A+B)$ takes 18 days to complete the work. $(A+B)^{\prime} s ...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A can do a job in 16 days and B can do the same job in 12 days. With the help of C, they can finish the job in 6 days only. Then, C alone can finish it in (a) 34 days (b) 22 days (c) 36 dyas (d) 48 days Solution: (d) 48 days A can do a job in 16 days. B can do the job in 12 days. Suppose $\mathrm{C}$ can do the job in $\mathrm{x}$ days. A's 1 day work $=\frac{1}{16}$ B's 1 day work $=\frac{1}{12}$ C's 1 day work $=\frac{1}{x}$ $\mathrm{A}, \mathrm{B}$ and $...
Read More →Between two given rational numbers,
Question: Between two given rational numbers, we can find (a) one and only one rational number (b) only two rational numbers (c) only ten rational numbers (d) infinitely many rational numbers Solution: (d) We can find infinite many rational numbers between two given rational numbers....
Read More →