A person standing at the junction (crossing) of two straight paths represented

Question: A person standing at the junction (crossing) of two straight paths represented by the equations 2x 3y+ 4 = 0 and 3x+ 4y 5 = 0 wants to reach the path whose equation is 6x 7y+ 8 = 0 in the least time. Find equation of the path that he should follow. Solution: The equations of the given lines are 2x 3y+ 4 = 0 (1) 3x+ 4y 5 = 0 (2) 6x 7y+ 8 = 0 (3) The person is standing at the junction of the paths represented by lines (1) and (2). On solving equations $(1)$ and $(2)$, we obtain $x=-\frac...

What is the key difference between primary and secondary sewage treatment?

Question: What is the key difference between primary and secondary sewage treatment? Solution:...

Prove that for any prime positive integer p,

Question: Prove that for any prime positive integer $p, \sqrt{p}$ is an irrational number. Solution: Let us assume that $\sqrt{p}$ is rational . Then, there exist positive co primes a and $\mathrm{b}$ such that $\sqrt{p}=\frac{a}{b}$ $p=\left(\frac{a}{b}\right)^{2}$ $\Rightarrow p b^{2}=a^{2}$ $\Rightarrow p b^{2}=a^{2}$ $\Rightarrow p \mid a^{2}$ $\Rightarrow p \mid a$ $\Rightarrow a=p c$ for some positive integer $c$ $\Rightarrow b^{2} p=a^{2}$ $\Rightarrow b^{2} p=p^{2} c^{2}(\because a=p c)$...

Differentiate the function with respect to x.

Question: Differentiate the function with respect tox. $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$ Solution: Let $y=\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$ Taking logarithm on both the sides, we obtain $\log y=\log \sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$ $\Rightarrow \log y=\frac{1}{2} \log \left[\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}\right]$ $\Rightarrow \log y=\frac{1}{2}[\log \{(x-1)(x-2)\}-\log \{(x-3)(x-4)(x-5)\}]$ $\Rightarrow \log y=\frac{1}{2}[\log (x-1)+\log (x-2)-\log (x-3)-\log (x-4...

Prove that the product of the lengths of the perpendiculars drawn from the points

Question: Prove that the product of the lengths of the perpendiculars drawn from the points$\left(\sqrt{a^{2}-b^{2}}, 0\right)$ and $\left(-\sqrt{a^{2}-b^{2}}, 0\right)$ to the line $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ is $b^{2}$. Solution: The equation of the given line is $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ Or, $b x \cos \theta+a y \sin \theta-a b=0$ $\ldots(1)$ Length of the perpendicular from point $\left(\sqrt{a^{2}-b^{2}}, 0\right)$ to line (1) is $p_{1}=\fra...

What is sewage? In which way can sewage be harmful to us?

Question: What is sewage? In which way can sewage be harmful to us? Solution: Sewage is the municipal waste matter that is carried away in sewers and drains. It includes both liquid and solid wastes, rich in organic matter and microbes. Many of these microbes are pathogenic and can cause several water- borne diseases. Sewage water is a major cause of polluting drinking water. Hence, it is essential that sewage water is properly collected, treated, and disposed....

Differentiate the function with respect to x.

Question: Differentiate the function with respect tox. $\cos x \cdot \cos 2 x \cdot \cos 3 x$ Solution: Let $y=\cos x \cdot \cos 2 x \cdot \cos 3 x$ Taking logarithm on both the sides, we obtain $\log y=\log (\cos x \cdot \cos 2 x \cdot \cos 3 x)$ $\Rightarrow \log y=\log (\cos x)+\log (\cos 2 x)+\log (\cos 3 x)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \frac{d y}{d x}=\frac{1}{\cos x} \cdot \frac{d}{d x}(\cos x)+\frac{1}{\cos 2 x} \cdot \frac{d}{d x}(\cos 2 x)+\frac{1...

Find the smallest number which when increased by 17

Question: Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468. Solution: TO FIND: Smallest number which when increased by 17 is exactly divisible by both 520 and 468. L.C.M OF 520 and 468 $520=2^{3} \times 5 \times 13$ $468=2^{2} \times 3^{2} \times 13$ LCM of 520 and $468=2^{3} \times 3^{2} \times 5 \times 13$ = 4680 Hence 4680 is the least number which exactly divides 520 and 468 i.e. we will get a remainder of 0 in this case. But we need the Smallest n...

Find the smallest number which when increased by 17

Question: Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468. Solution: TO FIND: Smallest number which when increased by 17 is exactly divisible by both 520 and 468. L.C.M OF 520 and 468 $520=2^{3} \times 5 \times 13$ $468=2^{2} \times 3^{2} \times 13$ LCM of 520 and $468=2^{3} \times 3^{2} \times 5 \times 13$ = 4680 Hence 4680 is the least number which exactly divides 520 and 468 i.e. we will get a remainder of 0 in this case. But we need the Smallest n...

A ray of light passing through the point (1, 2) reflects on the x-axis

Question: A ray of light passing through the point (1, 2) reflects on thex-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A. Solution: Let the coordinates of point A be (a, 0). Draw a line (AL) perpendicular to thex-axis. We know that angle of incidence is equal to angle of reflection. Hence, let $\angle B A L=\angle C A L=\Phi$ Let $\angle \mathrm{CAX}=\theta$ $\therefore \angle \mathrm{OAB}=180^{\circ}-(\theta+2 \Phi)=180^{\circ}-\left[\theta+2\l...

Differentiate the following w.r.t. x:

Question: Differentiate the following w.r.t.x: $\cos \left(\log x+e^{x}\right), x0$ Solution: Let $y=\cos \left(\log x+e^{x}\right)$ By using the chain rule, we obtain $\frac{d y}{d x}=-\sin \left(\log x+e^{x}\right) \cdot \frac{d}{d x}\left(\log x+e^{x}\right)$ $=-\sin \left(\log x+e^{x}\right) \cdot\left[\frac{d}{d x}(\log x)+\frac{d}{d x}\left(e^{x}\right)\right]$ $=-\sin \left(\log x+e^{x}\right) \cdot\left(\frac{1}{x}+e^{x}\right)$ $=-\left(\frac{1}{x}+e^{x}\right) \sin \left(\log x+e^{x}\r...

Question: ​What can you say about the prime factorisations of the denominators of the following rationals:(i) 43.123456789 (ii) $43 . \overline{123456789}$ (iii) $27 . \overline{142857}$ (iv) 0.120120012000120000 ... Solution: (i) Since 43.123456789 has terminating decimal expansion. So, its denominator is of the form 2m 5n, wherem,nare non-negative integers. (ii) Since $43.123456789$ has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5 . (iii) Since $27 . \ov...

Question: ​What can you say about the prime factorisations of the denominators of the following rationals:(i) 43.123456789 (ii) $43 . \overline{123456789}$ (iii) $27 . \overline{142857}$ (iv) 0.120120012000120000 ... Solution: (i) Since 43.123456789 has terminating decimal expansion. So, its denominator is of the form 2m 5n, wherem,nare non-negative integers. (ii) Since $43.123456789$ has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5 . (iii) Since $27 . \ov...

Differentiate the following w.r.t. x:

Question: Differentiate the following w.r.t.x: $\frac{\cos x}{\log x}, x0$ Solution: Let $y=\frac{\cos x}{\log x}$ By using the quotient rule, we obtainc $\frac{d y}{d x}=\frac{\frac{d}{d x}(\cos x) \times \log x-\cos x \times \frac{d}{d x}(\log x)}{(\log x)^{2}}$ $=\frac{-\sin x \log x-\cos x \times \frac{1}{x}}{(\log x)^{2}}$ $=\frac{-[x \log x \cdot \sin x+\cos x]}{x(\log x)^{2}}, x0$...

Given that HCF (306. 657) = 9,

Question: Given that HCF (306. 657) = 9, find LCM (306, 657). Solution: GIVEN: HCF of two numbers 306 and 657 is 9. TO FIND: L.C.M of number We know that, L.C.M $\times$ H.C.F $=$ First Number $\times$ Second Number L.C.M $\times 9=306 \times 657$ L.C.M $=\frac{306 \times 657}{9}$ L.C.M = 22338...

Find equation of the line which is equidistant from parallel lines

Question: Find equation of the line which is equidistant from parallel lines 9x+ 6y 7 = 0 and 3x+ 2y+ 6 = 0. Solution: The equations of the given lines are 9x+ 6y 7 = 0 (1) 3x+ 2y+ 6 = 0 (2) Let P (h,k) be the arbitrary point that is equidistant from lines (1) and (2). The perpendicular distance of P (h,k) from line (1) is given by $d_{1}=\frac{|9 h+6 k-7|}{(9)^{2}+(6)^{2}}=\frac{|9 h+6 k-7|}{\sqrt{117}}=\frac{|9 h+6 k-7|}{3 \sqrt{13}}$ The perpendicular distance of P (h,k) from line (2) is give...

Differentiate the following w.r.t. x:

Question: Differentiate the following w.r.t.x: $\log (\log x), x1$ Solution: Let $y=\log (\log x)$ By using the chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x}[\log (\log x)]$ $=\frac{1}{\log x} \cdot \frac{d}{d x}(\log x)$ $=\frac{1}{\log x} \cdot \frac{1}{x}$ $=\frac{1}{x \log x}, x1$...

Can two numbers have 16 as their HCF and 380 as their LCM?

Question: Can two numbers have 16 as their HCF and 380 as their LCM? Give reason. Solution: TO FIND: can two numbers have 16 as their H.C.F and 380 as their L.C.M On dividing 380 by 16 we get 23 as the quotient and 12 as the remainder, Since L.C.M is not exactly divisible by the H.C.F, two numbers cannot have 16 as their H.C.F and 380 as their L.C.M...

The HCF of two numbers is 145 and their LCM is 2175.

Question: The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other. Solution: GIVEN: LCM and HCF of two numbers are 2175 and 145 respectively. If one number is 725 TO FIND: Other number We know that, L.C.M $\times$ H.C.F $=$ First Number $\times$ Second Number $2175 \times 145=725 \times$ Second Number Second Number $=\frac{2175 \times 145}{725}$ Second Number $=435$...

Differentiate the following w.r.t. x:

Question: Differentiate the following w.r.t.x: $\sqrt{e^{\sqrt{x}}}, x0$ Solution: Let $y=\sqrt{e^{\sqrt{x}}}$ Then, $y^{2}=e^{\sqrt{x}}$ By differentiating this relationship with respect tox, we obtain $y^{2}=e^{\sqrt{x}}$ $\Rightarrow 2 y \frac{d y}{d x}=e^{\sqrt{x}} \frac{d}{d x}(\sqrt{x})$ [By applying the chain rule] $\Rightarrow 2 y \frac{d y}{d x}=e^{\sqrt{x}} \frac{1}{2} \cdot \frac{1}{\sqrt{x}}$ $\Rightarrow \frac{d y}{d x}=\frac{e^{\sqrt{x}}}{4 y \sqrt{x}}$ $\Rightarrow \frac{d y}{d x}...

If sum of the perpendicular distances of a variable point

Question: If sum of the perpendicular distances of a variable point P (x,y) from the linesx+y 5 = 0 and 3x 2y+ 7 = 0 is always 10. Show that P must move on a line. Solution: The equations of the given lines are x+y 5 = 0 (1) 3x 2y+ 7 = 0 (2) The perpendicular distances of P (x,y) from lines (1) and (2) are respectively given by $d_{1}=\frac{|x+y-5|}{\sqrt{(1)^{2}+(1)^{2}}}$ and $d_{2}=\frac{|3 x-2 y+7|}{\sqrt{(3)^{2}+(-2)^{2}}}$ i.e., $d_{1}=\frac{|x+y-5|}{\sqrt{2}}$ and $d_{2}=\frac{|3 x-2 y+7|...

The HCF to two numbers is 16 and their product is 3072.

Question: The HCF to two numbers is 16 and their product is 3072. Find their LCM. Solution: GIVEN: HCF of two numbers is 16. If product of numbers is 3072 TO FIND: L.C.M of numbers We know that, L.C.M $\times$ H.C.F $=$ First Number $\times$ Second Number L.C.M $\times 16=3072$ L.C.M $=\frac{3072}{16}$ L.C.M = 192...

The LCM and HCF of two numbers are 180 and 6 respectively.

Question: The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number. Solution: GIVEN: LCM and HCF of two numbers are 180 and 6 respectively. If one number is 30 TO FIND: Other number We know that, L.C.M $\times$ H.C.F $=$ First Number $\times$ Second Number $180 \times 6=30 \times$ Second Number Second Number $=\frac{180 \times 6}{30}$ Second Number $=36$...

A circular field has a circumference of 360 km.

Question: A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again? Solution: GIVEN: A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60, and 72 km a day, round the field. TO FIND: When they meet again.In order to calculate the time when they meet, we first find out the time taken by each cyclist in covering the distance. Number of days $1^{\text {s...

Name any two species of fungus, which are used in the production of the antibiotics.

Question: Name any two species of fungus, which are used in the production of the antibiotics. Solution: Antibiotics are medicines that are produced by certain micro-organisms to kill other disease-causing micro-organisms. These medicines are commonly obtained from bacteria and fungi The species of fungus used in the production of antibiotics are:...