Evaluate: √625/729
Question: Evaluate: $\sqrt{\frac{625}{729}}$ Solution: $\sqrt{\frac{625}{729}}=\frac{\sqrt{625}}{\sqrt{729}}$ Using long division method: $\sqrt{625}=25$ $\sqrt{729}=27$ $\therefore \sqrt{\frac{625}{729}}=\frac{\sqrt{625}}{\sqrt{729}}=\frac{25}{27}$...
Read More →To construct a triangle similar to a given
Question: To construct a triangle similar to a given $\triangle \mathrm{ABC}$ with its sides $\frac{7}{3}$ of the corresponding sides of $\triangle \mathrm{ABC}$, draw a ray $B X$ making acute angle with $B C$ and $X$ lies on the opposite side of $A$ with respect of $B C$. The points $B_{1}$, $B_{2}, \ldots, B_{7}$ are located at equal distances on $B X, B_{3}$ is joined to $C$ and then a line segment $B_{6} C^{\prime}$ is drawn parallel to $B_{3} C$, where $C^{\prime}$ lines on $B C$ produced. ...
Read More →If f (x) = |x − 2| write whether f' (2) exists or not.
Question: If $f(x)=|x-2|$ write whether $f^{\prime}(2)$ exists or not. Solution: Given: $f(x)=|x-2|= \begin{cases}x-2, x2 \\ -x+2, x \leq 2\end{cases}$ Now, $(\mathrm{LHD}$ at $x=2)$ $\lim _{x \rightarrow 2^{-}} \frac{f(x)-f(2)}{x-2}$ $=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{2-h-2}$ $=\lim _{h \rightarrow 0} \frac{(-2+h+2)-0}{-h}$ $=-1$ (RHD atx= 2) $\lim _{x \rightarrow 2^{+}} \frac{f(x)-f(2)}{x-2}$ $=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{2+h-2}$ $=\lim _{h \rightarrow 0} \frac{2+h-2-0...
Read More →If f (x) = |x − 2| write whether f' (2) exists or not.
Question: If $f(x)=|x-2|$ write whether $f^{\prime}(2)$ exists or not. Solution: Given: $f(x)=|x-2|= \begin{cases}x-2, x2 \\ -x+2, x \leq 2\end{cases}$ Now, $(\mathrm{LHD}$ at $x=2)$ $\lim _{x \rightarrow 2^{-}} \frac{f(x)-f(2)}{x-2}$ $=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{2-h-2}$ $=\lim _{h \rightarrow 0} \frac{(-2+h+2)-0}{-h}$ $=-1$ (RHD atx= 2) $\lim _{x \rightarrow 2^{+}} \frac{f(x)-f(2)}{x-2}$ $=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{2+h-2}$ $=\lim _{h \rightarrow 0} \frac{2+h-2-0...
Read More →Evaluate: √121/256
Question: Evaluate: $\sqrt{\frac{121}{256}}$ Solution: $\sqrt{\frac{121}{256}}=\frac{\sqrt{121}}{\sqrt{256}}$ Using division method: $\sqrt{121}=11$ $\therefore \sqrt{\frac{121}{256}}=\frac{\sqrt{121}}{\sqrt{256}}=\frac{11}{16}$...
Read More →By geometrical construction,
Question: By geometrical construction, it is possible to divide a line segment in the ratio $\sqrt{3}: \frac{1}{\sqrt{3}}$ Solution: True Given, ratio $=\sqrt{3}: \frac{1}{\sqrt{3}}$ $\therefore \quad$ Required ratio $=3: 1 \quad$ [multiply $\sqrt{3}$ in each term] So, $\sqrt{3}: \frac{1}{\sqrt{3}}$ can be simplified as $3: 1$ and 3 as well as 1 both are positive integer. Hence, the geometrical constrution is possible to divide a line segment in the ratio 3 : 1...
Read More →To draw a pair of tangents to a circle
Question: To draw a pair of tangents to a circle which are inclined to each other at an angle of 60, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be (a) 135 (b) 90 (c) 60 (d) 120 Solution: (d)The angle between them should be 120 because in that case the figure formed by the intersection point of pair of tangent, the two end points of those-two radii tangents are drawn) and the centre of the circle is a quadrilateral. From figure it...
Read More →Evaluate:√64/225
Question: Evaluate: $\sqrt{\frac{64}{225}}$ Solution: $\sqrt{\frac{64}{225}}=\frac{\sqrt{64}}{\sqrt{225}}$ Using long division method: $\sqrt{64}=8$ $\sqrt{225}=15$ $\therefore \sqrt{\frac{64}{225}}=\frac{\sqrt{64}}{\sqrt{225}}=\frac{8}{15}$...
Read More →If f (x) is differentiable at x = c,
Question: If $f(x)$ is differentiable at $x=c$, then write the value of $\lim _{x \rightarrow c} f(x)$ Solution: Given: $f(x)$ is differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. or, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ Consider, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$ $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x...
Read More →If f (x) is differentiable at x = c,
Question: If $f(x)$ is differentiable at $x=c$, then write the value of $\lim _{x \rightarrow c} f(x)$ Solution: Given: $f(x)$ is differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. or, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ Consider, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$ $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x...
Read More →To construct a triangle similar to a given
Question: To construct a triangle similar to a given $\triangle A B C$ with its sides $\frac{8}{5}$ of the corresponding sides of $\triangle A B C$ draw a ray $B X$ such that $\angle C B X$ is an acute angle and $X$ is on the opposite side of $A$ with respect to $B C$. The minimum number of points to be located at equal distances on ray BX is (a) 5 (b) 8 (c) 13 (d) 3 Solution: (b) To construct a triangle similar to a given triangle, with its sides $\frac{m}{n}$ of the corresponding sides of give...
Read More →If f (x) is differentiable at x = c,
Question: If $f(x)$ is differentiable at $x=c$, then write the value of $\lim _{x \rightarrow c} f(x)$. Solution: Given: $f(x)$ is differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. or, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ Consider, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$ $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{...
Read More →If f (x) is differentiable at x = c,
Question: If $f(x)$ is differentiable at $x=c$, then write the value of $\lim _{x \rightarrow c} f(x)$. Solution: Given: $f(x)$ is differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. or, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ Consider, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$ $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{...
Read More →Evaluate: √16/81
Question: Evaluate: $\sqrt{\frac{16}{81}}$ Solution: $\sqrt{\frac{16}{81}}=\frac{\sqrt{16}}{\sqrt{81}}$ $\sqrt{16}=4$ and $\sqrt{81}=9$ $\therefore \sqrt{\frac{16}{81}}=\frac{\sqrt{16}}{\sqrt{81}}=\frac{4}{9}$...
Read More →To construct a triangle similar to a given
Question: To construct a triangle similar to a given $\triangle \mathrm{ABC}$ with its sides $\frac{3}{7}$ of the corresponding sides of $\triangle \mathrm{ABC}$, first draw a ray $B X$ such that $\angle C B X$ is an acute angle and $X$ lies on the opposite side of $A$ with respect to $B C$. Then, locate points $B_{1}, B_{2}, B_{3} \ldots$ on $B X$ at equal distances and next step is to join (a) B 10 to C (b) $B_{13}$ to $C$ (c) $B_{7}$ to $C$ (d) $B_{4}$ to $C$ Solution: (c)Here, we locate poin...
Read More →To divide a line segment AB in the ratio 5 : 6,
Question: To divide a line segment AB in the ratio 5 : 6, draw a ray AY such that BAX is an acute angle, then draw a ray BY parallel to AY and the points A1, A2, A3, and B1, B2, B3, are located to equal distances on ray AY and BY, respectively. Then, the points joined are (a) A5and A6 (b) A6and B5 (c) A4andB5 (d) A5and B4 Solution: (a)Given a line segment AB and we have to divide it in the ratio 5:6. Steps of construction Draw a ray AX making an acute BAX. Draw a ray BY parallel to AX by making ...
Read More →Find the length of each side of a square whose area is equal to the area of a rectangle of
Question: Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres. Solution: Area of the rectangle $=(13.6 \times 3.4)=46.24$ sq $\mathrm{m}$ Thus, area of the square is $46.24 \mathrm{sq} \mathrm{m}$. Length of each side of the square $=\sqrt{46.24} \mathrm{~m}$ Using long division method: $\sqrt{46.24}=6.8$ Thus, the length of a side of the square is 6.8 metres....
Read More →Find the length of each side of a square whose area is equal to the area of a rectangle of
Question: Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres. Solution: Area of the rectangle $=(13.6 \times 3.4)=46.24$ sq $\mathrm{m}$ Thus, area of the square is $46.24 \mathrm{sq} \mathrm{m}$. Length of each side of the square $=\sqrt{46.24} \mathrm{~m}$ Using long division method:...
Read More →Find the length of each side of a square whose area is equal to the area of a rectangle of
Question: Find the length of each side of a square whose area is equal to the area of a rectangle of length 13.6 metres and breadth 3.4 metres. Solution: Area of the rectangle $=(13.6 \times 3.4)=46.24$ sq $\mathrm{m}$ Thus, area of the square is $46.24 \mathrm{sq} \mathrm{m}$. Length of each side of the square $=\sqrt{46.24} \mathrm{~m}$ Using long division method:...
Read More →To divide a line segment AB in the ratio 4 : 7,
Question: To divide a line segment AB in the ratio 4 : 7, a ray AX is drawn first such that BAX is an acute angle and then pointsA1A2, A3, are located at equal distances on the ray AY and the point B is joined to (a) A12 (b) A11 (c)A12 (d) A9 Solution: (b)Here, minimum 4+7 = 11 points are located at equal distances on the ray AX, and then B is joined to last point is A11...
Read More →Evaluate √0.9correct up to two places of decimal.
Question: Evaluate $\sqrt{0.9}$ correct up to two places of decimal. Solution: Usinglong division method: $\therefore \sqrt{0.9}=0.948$ $\Rightarrow \sqrt{0.9}=0.95 \quad$ ( correct up to two decimal places)...
Read More →Give an example of a function which is continuos but not differentiable at at a point.
Question: Give an example of a function which is continuos but not differentiable at at a point. Solution: Consider a function, $f(x)= \begin{cases}x, x0 \\ -x, x \leq 0\end{cases}$ This mod function is continuous at $x=0$ but not differentiable at $x=0$. Continuity at $x=0$, we have: $(\mathrm{LHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0}-(0-h)$ $=0$ (RHL atx= 0) $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} f(0+h)$...
Read More →To divide a line segment AB in the ratio 5 : 7,
Question: To divide a line segment AB in the ratio 5 : 7, first a ray AX is drawn, so that BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is (a) 8 (b) 10 (c) 11 (d) 12 Solution: (d)We know that, to divide a line segment AB in the ratio m: n, first draw a ray AX which makes an acute angle BAX, then marked m + n points at equal distance. Here, m = 5, n = 7 So, minimum number of these points = m+n = 5 + 7 = 12....
Read More →Give an example of a function which is continuos but not differentiable at at a point.
Question: Give an example of a function which is continuos but not differentiable at at a point. Solution: Consider a function, $f(x)= \begin{cases}x, x0 \\ -x, x \leq 0\end{cases}$ This mod function is continuous at $x=0$ but not differentiable at $x=0$. Continuity at $x=0$, we have: $(\mathrm{LHL}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} f(x)$ $=\lim _{h \rightarrow 0} f(0-h)$ $=\lim _{h \rightarrow 0}-(0-h)$ $=0$ (RHL atx= 0) $\lim _{x \rightarrow 0^{+}} f(x)$ $=\lim _{h \rightarrow 0} f(0+h)$...
Read More →Evaluate √2.8 correct up to two places of decimal.
Question: Evaluate $\sqrt{2.8}$ correct up to two places of decimal Solution: Using long division method: $\therefore \sqrt{2.8}=1.673$ $\Rightarrow \sqrt{2.8}=1.67 \quad$ (correct up to two decimal places)...
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