Tick (✓) the correct answer
Question: Tick (✓) the correct answer Which of the following numbers is not a perfect square? (a) 1156 (b) 4787 (c) 2704 (d) 3969 Solution: (b) 4787By the property of squares, a number ending in 2, 3,7 or 8 is not a perfect square....
Read More →Draw a parallelogram ABCD in which BC = 5 cm,
Question: Draw a parallelogram $A B C D$ in which $B C=5 \mathrm{~cm}, A B=3 \mathrm{~cm}$ and $\angle A B C=60^{\circ}$, divide it into triangles $B C D$ and ABD by the diagonal BD. Construct the triangles BD'C' similar to $\triangle B D C$ with Scale factor $\frac{1}{4}$. Draw the line segment D'A' parallel to DA, where A' lies on extended side BA. IS A'BC'D' a parallelogram? Solution: Steps of construction Draw a line segment AB = 3 cm. Now, draw a ray BY making an acute ABY = 60. With B as c...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Which of the following numbers is not a perfect square? (a) 1843 (b) 3721 (c) 1024 (d) 1296 Solution: (a) 1843 According to the property of squares, a number ending in 2, 3, 7 and 8 is not a perfect square....
Read More →Two line segments AB and AC
Question: Two line segments $\mathrm{AB}$ and $\mathrm{AC}$ include an angle of $60^{\circ}$, where $\mathrm{AB}=5 \mathrm{~cm}$ and $\mathrm{AC}=7 \mathrm{~cm}$. Locate points $\mathrm{P}$ and $Q$ on $A B$ and $A C$, respectively such that $A P=\frac{3}{4} A B$ and $A Q=\frac{1}{4} A C$. Join $P$ and $Q$ and measure the length PQ. Solution: Given that, AB = 5 cm and AC = 7 cm Also, $A P=\frac{3}{4} A B$ and $A Q=\frac{1}{4} A C$...(i) From Eq. (i), $A P=\frac{3}{4} \cdot A B=\frac{3}{4} \times ...
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Which of the following numbers is not a perfect square? (a) 1444 (b) 3136 (c) 961 (d) 2222 Solution: (d) 2222 According to the property of squares, a number ending in 2, 3, 7 or 8 is not a perfect square....
Read More →Tick (✓) the correct answer
Question: Tick (✓) the correct answer Which of the following numbers is not a perfect square? (a) 7056 (b) 3969 (c) 5478 (d) 4624 Solution: (c) 5478According to the properties of squares, a number ending in 2, 3, 7 or 8 is not a perfect square....
Read More →Write the derivative
Question: Write the derivative of $f(x)=|x|^{3}$ at $x=0$. Solution: Given: $f(x)=|x|^{3}= \begin{cases}x^{3}, x \geq 0 \\ -x^{3}, x0\end{cases}$ $(\mathrm{LHD}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}}{-h}$ $=0$ (RHD atx= 0) $\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{x \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}-0}{h}$ $=0$ and $f(0)=...
Read More →Write the derivative
Question: Write the derivative of $f(x)=|x|^{3}$ at $x=0$. Solution: Given: $f(x)=|x|^{3}= \begin{cases}x^{3}, x \geq 0 \\ -x^{3}, x0\end{cases}$ $(\mathrm{LHD}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}}{-h}$ $=0$ (RHD atx= 0) $\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{x \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}-0}{h}$ $=0$ and $f(0)=...
Read More →Write the derivative
Question: Write the derivative of $f(x)=|x|^{3}$ at $x=0$. Solution: Given: $f(x)=|x|^{3}= \begin{cases}x^{3}, x \geq 0 \\ -x^{3}, x0\end{cases}$ $(\mathrm{LHD}$ at $x=0)$ $\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}}{-h}$ $=0$ (RHD atx= 0) $\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$ $=\lim _{x \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{x}$ $=\lim _{h \rightarrow 0} \frac{h^{3}-0}{h}$ $=0$ and $f(0)=...
Read More →Evaluate:
Question: Evaluate: $\sqrt{98} \times \sqrt{162}$ Solution: We have: $\sqrt{98} \times \sqrt{162}$ $=\sqrt{98 \times 162}$ $=\sqrt{2 \times 7 \times 7 \times 2 \times 9 \times 9}$ $=2 \times 7 \times 9$ $=126$...
Read More →Construct a tangent to a circle of radius 4 cm
Question: Construct a tangent to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre. Solution: Given, a point M is at a distance of 6 cm from the centre of a circle of radius 4 cm. Steps of construction 1. Draw a circle of radius $4 \mathrm{~cm}$. Let centre of this circle is $\mathrm{O}$. 2. Join OM' and bisect it. Let $M$ be mid-point of OM'. 3. Taking $M$ as centre and $M O$ as radius draw a circle to intersect circle $(0,4)$ at two points, $P$ and $Q$. 4. Joi...
Read More →Evaluate:
Question: Evaluate: $\frac{\sqrt{1183}}{\sqrt{2023}}$ Solution: We have: $\frac{\sqrt{1183}}{\sqrt{2023}}$ $=\sqrt{\frac{1183}{2023}}$ $=\sqrt{\frac{169}{289}}$ $=\frac{\sqrt{169}}{\sqrt{289}}$ $=\frac{\sqrt{13 \times 13}}{\sqrt{17 \times 17}}$ $=\frac{13}{17}$...
Read More →Write the points of non-differentiability
Question: Write the points of non-differentiability of $f(x)=|\log | x||$. Solution: We have,f(x) = |log |x|| $|x|=\left\{\begin{array}{cl}-x -\inftyx-1 \\ -x -1x0 \\ x 0x1 \\ x 1x\infty\end{array}\right.$ $\log |x|=\left\{\begin{array}{cc}\log (-x) -\inftyx-1 \\ \log (-x) -1x0 \\ \log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $|\log | x||=\left\{\begin{array}{lc}\log (-x) -\inftyx-1 \\ -\log (-x) -1x0 \\ -\log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $(\mathrm{LHD}$ at $x=-1)=\lim ...
Read More →Write the points of non-differentiability
Question: Write the points of non-differentiability of $f(x)=|\log | x||$. Solution: We have,f(x) = |log |x|| $|x|=\left\{\begin{array}{cl}-x -\inftyx-1 \\ -x -1x0 \\ x 0x1 \\ x 1x\infty\end{array}\right.$ $\log |x|=\left\{\begin{array}{cc}\log (-x) -\inftyx-1 \\ \log (-x) -1x0 \\ \log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $|\log | x||=\left\{\begin{array}{lc}\log (-x) -\inftyx-1 \\ -\log (-x) -1x0 \\ -\log (x) 0x1 \\ \log (x) 1x\infty\end{array}\right.$ $(\mathrm{LHD}$ at $x=-1)=\lim ...
Read More →Evaluate:
Question: Evaluate: $\frac{\sqrt{80}}{\sqrt{405}}$ Solution: We have: $\frac{\sqrt{80}}{\sqrt{405}}$ $=\sqrt{\frac{80}{405}}$ $=\sqrt{\frac{16}{81}}$ $=\frac{\sqrt{16}}{\sqrt{81}}$ $=\frac{4}{9}$...
Read More →Draw a ΔABC in which BC = 6 cm,
Question: Draw a $\triangle A B C$ in which $B C=6 \mathrm{~cm}, C A=5 \mathrm{~cm}$ and $A B=4 \mathrm{~cm}$. Construct a triangle similar to it and of scale factor $\frac{5}{3}$ Solution: Steps of construction Draw a line segment BC = 6 cm. Taking Sand C as centres, draw two arcs of radii 4 cm and 5 cm intersecting each other at A. Join BA and CA. ΔABC is the required triangle. From B, draw any ray BX downwards making at acute angle. Mark five points B1, B2,B3, B4and B5on BX, such thatBB, = B,...
Read More →Evaluate:
Question: Evaluate: $\sqrt{3 \frac{33}{289}}$ Solution: $\sqrt{3 \frac{33}{289}}=\sqrt{\frac{900}{289}}=\frac{\sqrt{900}}{\sqrt{289}}$ Using long division method: $\sqrt{289}=17$ And $\sqrt{900}=\sqrt{2 \times 2 \times 5 \times 5 \times 3 \times 3}=2 \times 5 \times 3=30$ $\therefore \sqrt{3 \frac{33}{289}}=\frac{30}{17}=1 \frac{13}{17}$...
Read More →Draw a right ΔABC in which BC = 12 cm,
Question: Draw a right ΔABC in which BC = 12 cm, AB = 5 cm and B = 90.Construct a triangle similar to it and of scale factor Is the new triangle also a right triangle? Solution: Steps of construction Draw a line segment BC = 12 cm, From 6 draw a line AB = 5 cm which makes right angle at B. 3. Join $A C, \triangle A B C$ is the given right triangle. 4. From $B$ draw an acute $\angle C B Y$ downwards. 5. On ray BY, mark three points $B_{1}, B_{2}$ and $B_{3}$, such that $B B_{1}=B_{1} B_{2}=B_{2} ...
Read More →Write the points where
Question: Write the points where $f(x)=\left|\log _{e} x\right|$ is not differentiable. Solution: Given: $f(x)=\left|\log _{e} x\right|= \begin{cases}-\log _{e} x, 0x1 \\ \log _{e} x, x \geq 1\end{cases}$ Clearly $f(x)$ is differentiable for all $x1$ and for all $x1$. So, we have to check the differentiability at $x=1$. We have, (LHD at $x=1$ ) $\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ $=\lim _{x \rightarrow 1^{-}} \frac{-\log x-\log 1}{x-1}$ $=-\lim _{x \rightarrow 1^{-}} \frac{\log x...
Read More →Draw a line segment of length 7 cm.
Question: Draw a line segment of length 7 cm. Find a point P on it which1divides it in the ratio 3:5. Solution: Steps of construction Draw a line segment AB = 7 cm. Draw a ray AX, making an acute BAX Along AX, mark 3+ 5= 8 pointsA1, A2, A3, A4, A5, A6,A7, A8such thatAA1= A1A2= A2A3= A3A4= A4A5= A5A6= A6A7= A7A8 Join A8B From A3, draw A3C || A8B meeting AB at C.[by making an angle equal to BA8A at A3] Then, C is the point on AB which divides it in the ratio 3 : 5, $\therefore$$\frac{A C}{C B}=\fr...
Read More →Evaluate:
Question: Evaluate: $\sqrt{4 \frac{73}{324}}$ Solution: $\sqrt{4 \frac{73}{324}}=\sqrt{\frac{1369}{324}}=\frac{\sqrt{1369}}{\sqrt{324}}$ Using long division method: $\sqrt{1369}=37$ $\sqrt{324}=\sqrt{2 \times 2 \times 9 \times 9}=2 \times 9=18$ $\therefore \sqrt{4 \frac{73}{324}}=\frac{37}{18}=2 \frac{1}{18}$...
Read More →Write the points where
Question: Write the points where $f(x)=\left|\log _{e} x\right|$ is not differentiable. Solution: Given: $f(x)=\left|\log _{e} x\right|= \begin{cases}-\log _{e} x, 0x1 \\ \log _{e} x, x \geq 1\end{cases}$ Clearly $f(x)$ is differentiable for all $x1$ and for all $x1$. So, we have to check the differentiability at $x=1$. We have, (LHD at $x=1$ ) $\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}$ $=\lim _{x \rightarrow 1^{-}} \frac{-\log x-\log 1}{x-1}$ $=-\lim _{x \rightarrow 1^{-}} \frac{\log x...
Read More →Evaluate:
Question: Evaluate: $\sqrt{3 \frac{13}{36}}$ Solution: $\sqrt{3 \frac{13}{36}}$ $=\sqrt{\frac{121}{36}}$ $=\frac{\sqrt{121}}{\sqrt{36}}$ $=\frac{\sqrt{11 \times 11}}{\sqrt{6 \times 6}}$ $=\frac{11}{6}$ $=1 \frac{5}{11}$...
Read More →A pair of tangents can be constructed
Question: A pair of tangents can be constructed to a circle inclined at an angle of 170. Solution: True If the angle between the pair of tangents is always greater than 0 or less than $180^{\circ}$, then we can construct a pair of tangents to a circle. Hence, we can drawn a pair of tangents to a circle inclined at an angle of $170^{\circ}$....
Read More →A pair of tangents can be constructed
Question: A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of 3 cm from the centre. Solution: False Since, the radius of the circle is 3.5 cm i.e., r = 3.5 cm and a point P situated at a distance of 3 cm from the centre i.e.,d= 3 cm We see that, r d i.e., a point P lies inside the circle. So, no tangent can be drawn to a circle from a point lying inside it....
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