A is a point at a distance 13 cm
Question: A is a point at a distance 13 cm from the centre 0 of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the ΔABC. Solution: Given Two tangents are drawn from an external point A to the circle with centre 0, $O A=13 \mathrm{~cm}$ Tangent $B C$ is drawn at a point $R$. radius of circle equals $5 \mathrm{~cm}$. To find perimeter of $\triangle ...
Read More →Evaluate √3 up to two places of decimal.
Question: Evaluate $\sqrt{3}$ up to two places of decimal. Solution: Using long division method: $\sqrt{3}=1.732$ $\Rightarrow \sqrt{3}=1.73 \quad$ (correct up to two decimal places)...
Read More →Is every continuous function differentiable?
Question: Is every continuous function differentiable? Solution: No, function may be continuous at a point but may not be differentiable at that point. For example: function $f(x)=|x|$ is continuous at $x=0$ but it is not differentiable at $x=0$....
Read More →Evaluate:
Question: Evaluate: $\sqrt{0.2916}$ Solution: Using long division method: $\therefore \sqrt{0.2916}=0.54$...
Read More →Is every differentiable function continuous?
Question: Is every differentiable function continuous? Solution: Yes, if a function is differentiable at a point then it is necessary continuous at that point. Proof : Let a function $f(x)$ be differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. Let $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ In order to prove that $f(x)$ is continous at $x=c$, it is sufficient to show that $\lim f(x)=f(c)$ $\lim _{x \rightarrow c} f(x)=\lim _{x \right...
Read More →If an isosceles ΔABC in which AB = AC = 6 cm,
Question: If an isosceles ΔABC in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle. Solution: In a circle, ΔABC is inscribed. Join OB, OC and OA. Conside $\triangle A B O$ and $\triangle A C O$ $A B=A C$[given] $B O=C O$ [radii of same circle] $A O$ is common. $\therefore$ $\triangle A B O \cong \triangle A C O$ [by SSS congruence rule] $\Rightarrow$ $\angle 1=\angle 2$ [CPOT] Now, in $\triangle A B M$ and $\triangle A C M$, $A B=A C$ [given] $\angle 1...
Read More →Is every differentiable function continuous?
Question: Is every differentiable function continuous? Solution: Yes, if a function is differentiable at a point then it is necessary continuous at that point. Proof : Let a function $f(x)$ be differentiable at $x=c$. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. Let $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ In order to prove that $f(x)$ is continous at $x=c$, it is sufficient to show that $\lim f(x)=f(c)$ $\lim _{x \rightarrow c} f(x)=\lim _{x \right...
Read More →Evaluate:
Question: Evaluate: $\sqrt{1.0816}$ Solution: Using long division method: $\therefore \sqrt{1.0816}=1.04$...
Read More →Evaluate:
Question: Evaluate: $\sqrt{10.0489}$ Solution: Using long division method: $\therefore \sqrt{10.0489}=3.17$...
Read More →Define differentiability of a function at a point.
Question: Define differentiability of a function at a point. Solution: Let $f(x)$ be a real valued function defined on an open interval $(a, b)$ and let $c \in(a, b)$. Then $f(x)$ is said to be differentiable or derivable at $x=c$ iff $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. or, $f^{\prime}(c)=\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$...
Read More →Evaluate:
Question: Evaluate: $\sqrt{9.8596}$ Solution: Using long division method: $\therefore \sqrt{9.8596}=3.14$...
Read More →Evaluate:
Question: Evaluate: $\sqrt{75.69}$ Solution: Using long division method: $\therefore \sqrt{75.69}=8.7$...
Read More →The set of points at which the function
Question: The set of points at which the function $f(x)=\frac{1}{\log |x|}$ is not differentiable, is_____________ Solution: The given function is $f(x)=\frac{1}{\log |x|}$. For $f(x)$ to be defined, $x \neq 0$ and $\log |x| \neq 0$ $\Rightarrow x \neq 0$ and $|x| \neq 1$ $\Rightarrow x \neq 0$ and $x \neq \pm 1$ Thus, the functionf(x) is not defined whenx= 1,x= 0 andx= 1 We know that, the logarithmic function is differentiable at each point in its domain. Every constant function is differentiab...
Read More →Evaluate:
Question: Evaluate: $\sqrt{156.25}$ Solution: Using long division method: $\therefore \sqrt{156.25}=12.5$...
Read More →Evaluate:
Question: Evaluate: $\sqrt{33.64}$ Solution: Using long division method: $\therefore \sqrt{33.64}=5.8$...
Read More →The set of points where
Question: The set of points wheref(x) = |sinx| is not differentiable, is ____________. Solution: Let $g(x)=|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ Now, $L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{g(0-h)-g(0)}{-h}$ $\Rightarrow L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{-(-h)-0}{-h}$ $\Rightarrow L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{h}{-h}$ $\Rightarrow L g^{\prime}(0)=-1$ And $R g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h}$ $\Rightarrow R g^{\prime}(0)=\...
Read More →Evaluate:
Question: Evaluate: $\sqrt{1.69}$ Solution: Using long division method: $\therefore \sqrt{1.69}=1.3$...
Read More →The set of points where
Question: The set of points wheref(x) = |sinx| is not differentiable, is ____________. Solution: Let $g(x)=|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ Now, $L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{g(0-h)-g(0)}{-h}$ $\Rightarrow L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{-(-h)-0}{-h}$ $\Rightarrow L g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{h}{-h}$ $\Rightarrow L g^{\prime}(0)=-1$ And $R g^{\prime}(0)=\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h}$ $\Rightarrow R g^{\prime}(0)=\...
Read More →The tangent at a point C of a circle
Question: The tangent at a point C of a circle and a diameter AB when extended intersect at P. If PCA = 110, find CBA. Solution: Here, AB is a diameter of the circle from point C and a tangent is drawn which meets at a point P. Join OC. Here, OC is radius. Since, tangent at any point of a circle is perpendicular to the radius through point of contact circle. $\therefore \quad O C \perp P C$ Now, $\angle P C A=110^{\circ}$ [given] \$\Rightarrow \quad \angle P C O+\angle O C A=110^{\circ}$ $\Right...
Read More →The area of a square field is 60025 m
Question: The area of a square field is 60025 m2. A man cycles along its boundary at 18 km/h. In how much time will he return to the starting point? Solution: Area of the square field $=60025 \mathrm{~m}^{2}$ Length of each side of the square field $=\sqrt{60025}=245 \mathrm{~m}$ Perimeter of the field $=4 \times 245=980 \mathrm{~m}$ $=\frac{980}{1000} \mathrm{~km}$ The man is cycling at a speed of 18 km/h. Time $=\frac{\text { Distance travelled }}{\text { Speed }}$ $=\frac{980 / 1000}{18}$ $=\...
Read More →Find the greatest number of five digits which is a perfect square.
Question: Find the greatest number of five digits which is a perfect square. Also find the square root of the number so obtained. Solution: Greatest number of five digits =99999Using the long division method:99999 is not a perfect square. According to the long division method, the obtained square root is between 316 and 317. Squaring the smaller number, i.e. 316, will give us the perfect square that would be less than 99999. $316^{2}=99856$ 99856 is the required number. Its square root is 316....
Read More →The set of points where
Question: The set of points wheref(x) = cos |x| is differentiable, is ____________. Solution: We know $|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ $\therefore f(x)=\cos |x|= \begin{cases}\cos x, x \geq 0 \\ \cos (-x), x0\end{cases}$ $\Rightarrow f(x)=\cos |x|= \begin{cases}\cos x, x \geq 0 \\ \cos x, x0\end{cases}$ $[\cos (-\theta)=\cos \theta]$ We know that, cosine function is differentiable in its domain. So,f(x) is differentiable forallx 0 andx 0. Let us check the differentiability of...
Read More →The set of points where
Question: The set of points wheref(x) = cos |x| is differentiable, is ____________. Solution: We know $|x|= \begin{cases}x, x \geq 0 \\ -x, x0\end{cases}$ $\therefore f(x)=\cos |x|= \begin{cases}\cos x, x \geq 0 \\ \cos (-x), x0\end{cases}$ $\Rightarrow f(x)=\cos |x|= \begin{cases}\cos x, x \geq 0 \\ \cos x, x0\end{cases}$ $[\cos (-\theta)=\cos \theta]$ We know that, cosine function is differentiable in its domain. So,f(x) is differentiable forallx 0 andx 0. Let us check the differentiability of...
Read More →Find the least number of four digits which is a perfect square.
Question: Find the least number of four digits which is a perfect square. Also find the square root of the number so obtained. Solution: Smallest number of four digits =1000Using the long division method:1000 is not a perfect square. By the long division method, the obtained square root is between 31 and 32. Squaring the next integer (32) will give us the next perfect square. $32^{2}=1024$ Thus, 1024 is the smallest four digit perfect square. Also, $\sqrt{1024}=32$...
Read More →Find the least number which must be added to 8400 to obtain a perfect square.
Question: Find the least number which must be added to 8400 to obtain a perfect square. Find this perfect square and its square root. Solution: Using the long division method:The next natural number that is a perfect square can be obtained by squaring the next natural number of the obtained quotient, i.e. 91. Therefore square of $(91+1)=92^{2}=8464$ Number that should be added to the given number to make it a perfect square $=8464-8400=64$ The perfect square thus obtained is 8464 and its square ...
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