Find two irrational numbers between
Question: Find two irrational numbers between 0.5 and 0.55. Solution: Let $a=0.5$ $b=0.55$ Hereaandbare rational number. So we observe that in first decimal placeaandbhave same digit .Soab. Hence two irrational numbers are $0.510100100010000 \ldots$ and $0.5202002000200002 \ldots$ lying between $0.5$ and $0.55$....
Read More →Find, if
Question: Find $\frac{d y}{d x}$, if $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}},-1 \leq x \leq 1$ Solution: It is given that, $y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}}$ $\therefore \frac{d y}{d x}=\frac{d}{d x}\left[\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^{2}}\right]$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}\left(\sin ^{-1} x\right)+\frac{d}{d x}\left(\sin ^{-1} \sqrt{1-x^{2}}\right)$ $\Rightarrow \frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{\sqrt{1-\left(\sqrt{1-x^{2}}\right)^{2}}} \cdot \frac{...
Read More →Give an example of each, of two irrational numbers whose:
Question: Give an example of each, of two irrational numbers whose:(i) difference is a rational number.(ii) difference is an irrational number.(iii) sum is a rational number.(iv) sum is an irrational number.(v) product is an rational number.(vi) product is an irrational number.(vii) quotient is a rational number.(viii) quotient is an irrational number. Solution: (i) Let $\sqrt{2}, 1+\sqrt{2}$ And, so $1+\sqrt{2}-\sqrt{2}=1$ Therefore, $\sqrt{2}$ and $1+\sqrt{2}$ are two irrational numbers and th...
Read More →A point R with x-coordinate 4 lies on the line segment joining the pointsP
Question: A point R withx-coordinate 4 lies on the line segment joining the pointsP (2, 3, 4) and Q (8, 0, 10). Find the coordinates of the point R. [Hint suppose $\mathrm{R}$ divides $\mathrm{PQ}$ in the ratio $k$. 1. The coordinates of the point $\mathrm{R}$ are given by $\left(\frac{8 k+2}{k+1}, \frac{-3}{k+1}, \frac{10 k+4}{k+1}\right)$ ] Solution: The coordinates of points P and Q are given as P (2, 3, 4) and Q (8, 0, 10). Let R divide line segment PQ in the ratiok:1. Hence, by section form...
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Question: Find $\frac{d y}{d x}$, if $y=12(1-\cos t), x=10(t-\sin t),-\frac{\pi}{2}t\frac{\pi}{2}$ Solution: It is given that, $y=12(1-\cos t), x=10(t-\sin t)$ $\therefore \frac{d x}{d t}=\frac{d}{d t}[10(t-\sin t)]=10 \cdot \frac{d}{d t}(t-\sin t)=10(1-\cos t)$ $\frac{d y}{d t}=\frac{d}{d t}[12(1-\cos t)]=12 \cdot \frac{d}{d t}(1-\cos t)=12 \cdot[0-(-\sin t)]=12 \sin t$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\right)}=\frac{12 \sin t}{10(1-\cos t)}=\...
Read More →Find the coordinates of a point on y-axis which are at a distance of
Question: Find the coordinates of a point ony-axis which are at a distance of$5 \sqrt{2}$ from the point $P(3,-2,5)$. Solution: If a point is on they-axis, thenx-coordinate and thez-coordinate of the point are zero. Let $\mathrm{A}(0, b, 0)$ be the point on the $y$-axis at a distance of $5 \sqrt{2}$ from point $\mathrm{P}(3,-2,5)$. Accordingly, $\mathrm{AP}=5 \sqrt{2}$ $\therefore \mathrm{AP}^{2}=50$ $\Rightarrow(3-0)^{2}+(-2-b)^{2}+(5-0)^{2}=50$ $\Rightarrow 9+4+b^{2}+4 b+25=50$ $\Rightarrow b^...
Read More →if the
Question: $x^{x^{2}-3}+(x-3)^{x^{2}}$, for $x3$ Solution: Let $y=x^{x^{2}-3}+(x-3)^{x^{2}}$ Also, let $u=x^{x^{2}-3}$ and $v=(x-3)^{x^{2}}$ $\therefore y=u+v$ Differentiating both sides with respect tox,we obtain $\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ ...(1) $u=x^{x^{2}-3}$ $\therefore \log u=\log \left(x^{x^{2}-3}\right)$ $\log u=\left(x^{2}-3\right) \log x$ Differentiating with respect tox, we obtain $\frac{1}{u} \cdot \frac{d u}{d x}=\log x \cdot \frac{d}{d x}\left(x^{2}-3\right)+\...
Read More →If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10)
Question: If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (4, 3b, 10) and R (8, 14, 2c), then find the values ofa,bandc. Solution: It is known that the coordinates of the centroid of the triangle, whose vertices are $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)$ and $\left(x_{3}, y_{3}, z_{3}\right)$, are $\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}, \frac{z_{1}+z_{2}+z_{3}}{3}\right)$. Therefore, coordinates of the centroid...
Read More →Find three different irrational numbers between the rational numbers
Question: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$. Solution: Let $x=\frac{5}{7}=0 . \overline{714285}$ and $y=\frac{9}{11}=0 . \overline{81}$ Here we observe that in the first decimalxhas digit 7 andyhas 8. Soxy. In the second decimal placex has digit 1. So, if we considering irrational numbers $a=0.72072007200072 \ldots$ $b=0.73073007300073 \ldots$ $c=0.74074007400074 \ldots$ We find that $xabcy$ Hence are required irrational numbers...
Read More →Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).
Question: Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0). Solution: Let AD, BE, and CF be the medians of the given triangle ABC. Since AD is the median, D is the mid-point of BC. $\therefore$ Coordinates of point $D=\left(\frac{0+6}{2}, \frac{4+0}{2}, \frac{0+0}{2}\right)=(3,2,0)$ $\mathrm{AD}=\sqrt{(0-3)^{2}+(0-2)^{2}+(6-0)^{2}}=\sqrt{9+4+36}=\sqrt{49}=7$ Since $B E$ is the median, $E$ is the mid-point of $A C$. $\therefore$ Coordinates of p...
Read More →Find a rational number and also an irrational number lying between the numbers
Question: Find a rational number and also an irrational number lying between the numbers 0.3030030003 ... and 0.3010010001 ... Solution: Let $a=0.3030030003 \ldots$ $b=0.3010010001 \ldots$ Here decimal representation ofaandbare non-terminating and non-repeating. Soaandbare irrational numbers. We observe that in first two decimal place ofaandbhave the same digit but digit in the third place of their decimal representation is distinct. Therefore,ab. Hence one rational number is $0.3011$ lying betw...
Read More →Find one irrational number between
Question: Find one irrational number between $0.2101$ and $0.222 \ldots=0 . \overline{2}$. Solution: Let $a=0.2101$ $b=0.2222 \ldots=0 . \overline{2}$ Hereaandbare rational numbers .Sinceahas terminating andbhas repeating decimal. We observe that in second decimal placeahas 1 andbhas 2. Soab. Hence one irrational number is $0.220100100010000 \ldots$ lying between $0.2101$ and $0.2222 \ldots$...
Read More →Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.
Question: Three vertices of a parallelogram ABCD are A (3, 1, 2), B (1, 2, 4) andC (1, 1, 2). Find the coordinates of the fourth vertex. Solution: The three vertices of a parallelogram ABCD are given as A (3, 1, 2), B (1, 2, 4), and C (1, 1, 2). Let the coordinates of the fourth vertex be D (x,y,z). We know that the diagonals of a parallelogram bisect each other. Therefore, in parallelogram ABCD, AC and BD bisect each other. $\therefore$ Mid-point of $A C=$ Mid-point of $B D$ $\Rightarrow\left(\...
Read More →if the
Question: $x^{x}+x^{a}+a^{x}+a^{a}$, for some fixed $a0$ and $x0$ Solution: Let $y=x^{x}+x^{a}+a^{x}+a^{a}$ Also, let $x^{x}=u, x^{a}=v, a^{x}=w$, and $a^{a}=s$ $\therefore y=u+v+w+s$ $\Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}+\frac{d w}{d x}+\frac{d s}{d x}$ ...(1) $u=x^{x}$ $\Rightarrow \log u=\log x^{x}$ $\Rightarrow \log u=x \log x$ Differentiating both sides with respect tox, we obtain' $\frac{1}{u} \frac{d u}{d x}=\log x \cdot \frac{d}{d x}(x)+x \cdot \frac{d}{d x}(\log x...
Read More →Give two rational numbers lying between
Question: Give two rational numbers lying between 0.515115111511115...0.5353353335... Solution: Let $a=0.515115111511115 \ldots$ and $b=0.535335333533335 \ldots$ Here the decimal representation ofaandbare non-terminating and non-repeating. So we observe that in first decimal placeaandbhave the same digitbut digit in the second place of their decimal representation are distinct. And the numberahas 1 andbhas 3. Soab. Hence two rational numbers are $0.5152,0.532$ lying between $0.515115111511115 \l...
Read More →Find the coordinates of the points which trisect the line segment joining the points
Question: Find the coordinates of the points which trisect the line segment joining the points P (4, 2, 6) and Q (10, 16, 6). Solution: Let A and B be the points that trisect the line segment joining points P (4, 2, 6) and Q (10, 16, 6) Point A divides PQ in the ratio 1:2. Therefore, by section formula, the coordinates of point A are given by $\left(\frac{1(10)+2(4)}{1+2}, \frac{1(-16)+2(2)}{1+2}, \frac{1(6)+2(-6)}{1+2}\right)=(6,-4,-2)$ Point B divides PQ in the ratio 2:1. Therefore, by section...
Read More →Give two rational numbers lying between
Question: Give two rational numbers lying between 0.232332333233332... and 0.212112111211112. Solution: Let $a=0.232332333233332 \ldots$ $b=0.212112111211112 \ldots$ Here the decimal representation ofaandbare non-terminating and non-repeating. So we observe that in first decimal place ofaandbhave the same digitbut digit in the second place of their decimal representation are distinct. And the numberahas 3 andbhas 1. Soab. Hence two rational numbers are $0.222,0.221$ lying between $0.232332333233...
Read More →Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1)
Question: Using section formula, show that the points $A(2,-3,4), B(-1,2,1)$ and $C\left(0, \frac{1}{3}, 2\right)$ are collinear. Solution: The given points are $\mathrm{A}(2,-3,4), \mathrm{B}(-1,2,1)$, and $\mathrm{C}\left(0, \frac{1}{3}, 2\right)$ Let P be a point that divides AB in the ratiok:1. Hence, by section formula, the coordinates of P are given by $\left(\frac{k(-1)+2}{k+1}, \frac{k(2)-3}{k+1}, \frac{k(1)+4}{k+1}\right)$ Now, we find the value ofkat which point P coincides with point ...
Read More →Find the ratio in which the YZ-plane divides the line segment formed by joining the points
Question: Find the ratio in which the YZ-plane divides the line segment formed by joining the points(2, 4, 7) and (3, 5, 8). Solution: Let the YZ planedivide the line segment joining points (2, 4, 7) and (3, 5, 8) in the ratiok:1. Hence, by section formula, the coordinates of point of intersection are given by $\left(\frac{k(3)-2}{k+1}, \frac{k(-5)+4}{k+1}, \frac{k(8)+7}{k+1}\right)$ On the YZ plane, thex-coordinate of any point is zero. $\frac{3 k-2}{k+1}=0$ $\Rightarrow 3 k-2=0$ $\Rightarrow k...
Read More →In the following equations, find which variables x, y, z etc.
Question: In the following equations, find which variablesx, y, zetc. represent rational or irrational numbers: (i) $x^{2}=5$ (ii) $y^{2}=9$ (iii) $z^{2}=0.04$ (iv) $u^{2}=\frac{17}{4}$ (v) $v^{2}=3$ (vi) $w^{2}=27$ (vii) $t^{2}=0.4$ Solution: (i) Given that $x^{2}=5$ Now we have to find the value ofx Since $x^{2}=5$ $\Rightarrow x=\sqrt{5}$ So itxis an irrational number (ii) Given that $y^{2}=9$ Now we have to find the value ofy $y^{2}=9$ $\Rightarrow y=\sqrt{9}$ $\Rightarrow y=3$ Soyis a ratio...
Read More →Given that P (3, 2, –4), Q (5, 4, –6) and R (9, 8, –10) are collinear.
Question: Given that P (3, 2, 4), Q (5, 4, 6) and R (9, 8, 10) are collinear.Find the ratio in which Q divides PR. Solution: Let point Q (5, 4, 6) divide the line segment joining points P (3, 2, 4) and R (9, 8, 10) in the ratiok:1. Therefore, by section formula, $(5,4,-6)=\left(\frac{k(9)+3}{k+1}, \frac{k(8)+2}{k+1}, \frac{k(-10)-4}{k+1}\right)$ $\Rightarrow \frac{9 k+3}{k+1}=5$ $\Rightarrow 9 k+3=5 k+5$ $\Rightarrow 4 k=2$ $\Rightarrow k=\frac{2}{4}=\frac{1}{2}$Thus, point Q divides PR in the r...
Read More →Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6)
Question: Find the coordinates of the point which divides the line segment joining the points (2, 3, 5) and (1, 4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally. Solution: (i) The coordinates of point $R$ that divides the line segment joining points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ internally in the ratio $m: n$ are $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+m y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$ Let R (x,y,z) be the point t...
Read More →if the
Question: $(\sin x-\cos x)^{(\sin x-\cos x)}, \frac{\pi}{4}x\frac{3 \pi}{4}$ Solution: Let $y=(\sin x-\cos x)^{(\sin x-\cos x)}$ Taking logarithmon both the sides, we obtain $\log y=\log \left[(\sin x-\cos x)^{(\sin x-\cos x)}\right]$ $\Rightarrow \log y=(\sin x-\cos x) \cdot \log (\sin x-\cos x)$ Differentiating both sides with respect tox, we obtain $\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}[(\sin x-\cos x) \log (\sin x-\cos x)]$ $\Rightarrow \frac{1}{y} \frac{d y}{d x}=\log (\sin x-\cos x) \c...
Read More →Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0)
Question: Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (4, 0, 0) is equal to 10. Solution: Let the coordinates of P be (x,y,z). The coordinates of points A and B are (4, 0, 0) and (4, 0, 0) respectively. It is given that PA + PB = 10. $\Rightarrow \sqrt{(x-4)^{2}+y^{2}+z^{2}}+\sqrt{(x+4)^{2}+y^{2}+z^{2}}=10$ $\Rightarrow \sqrt{(x-4)^{2}+y^{2}+z^{2}}=10-\sqrt{(x+4)^{2}+y^{2}+z^{2}}$ On squaring both sides, we obtain $\Rightarrow(x-4)^{2}+y^{2}+z^{2}=...
Read More →, for some constant a and b.
Question: $\cos (a \cos x+b \sin x)$, for some constant $a$ and $b$. Solution: Let $y=\cos (a \cos x+b \sin x)$ By using chain rule, we obtain $\frac{d y}{d x}=\frac{d}{d x} \cos (a \cos x+b \sin x)$ $\Rightarrow \frac{d y}{d x}=-\sin (a \cos x+b \sin x) \cdot \frac{d}{d x}(a \cos x+b \sin x)$ $=-\sin (a \cos x+b \sin x) \cdot[a(-\sin x)+b \cos x]$ $=(a \sin x-b \cos x) \cdot \sin (a \cos x+b \sin x)$...
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