Draw a circle of radius 4 cm.
Question: Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents. Solution: In order to draw the pair of tangents, we follow the following stepsSteps of construction Take a point 0 on the plane of the paper and draw a circle of radius OA = 4 cm. Produce OA to B such that OA = AB = 4 cm. Taking A as the centre draw a circle of rad...
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Question: Tick (✓) the correct answer Which of the following is the square of an even number? (a) 196 (b) 441 (c) 625 (d) 529 Solution: (a) 196 Square of an even number is always an even number....
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Question: Tick (✓) the correct answer $\sqrt{2 \frac{1}{4}}=?$ (a) $2 \frac{1}{2}$ (b) $1 \frac{1}{2}$ (c) $1 \frac{1}{4}$ (d) none of these Solution: (b) $1 \frac{1}{2}$ $\sqrt{2 \frac{1}{4}}=\sqrt{\frac{9}{4}}=\frac{\sqrt{9}}{\sqrt{4}}=\frac{3}{2}=1 \frac{1}{2}$...
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Question: Tick (✓) the correct answer $\frac{\sqrt{288}}{\sqrt{128}}=?$ (a) $\frac{\sqrt{3}}{2}$ (b) $\frac{3}{\sqrt{2}}$ (C) $\frac{3}{2}$ (d) $1.49$ Solution: (C) $\frac{3}{2}$ $\frac{\sqrt{288}}{\sqrt{128}}=\sqrt{\frac{288}{128}}=\sqrt{\frac{2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3}{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}}=\sqrt{\frac{3 \times 3}{2 \times 2}}=\frac{\sqrt{3 \times 3}}{\sqrt{2 \times 2}}=\frac{3}{2}$...
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Question: Tick (✓) the correct answer $\sqrt{0.9} \times \sqrt{1.6}=?$ (a) $0.12$ (b) $1.2$ (c) $0.75$ (d) 12 Solution: (b) $1.2$ $\sqrt{0.9} \times \sqrt{1.6}=\sqrt{1.44}=1.2$...
Read More →Draw a ΔABC in which AB = 5 cm,
Question: Draw a $\triangle \mathrm{ABC}$ in which $\mathrm{AB}=5 \mathrm{~cm}, \mathrm{BC}=6 \mathrm{~cm}$ and $\angle \mathrm{ABC}=60^{\circ}$. Draw a ΔABC in which AB = 5 cm,Construct a triangle similar to ABC with scale factor $\frac{5}{7}$ Justify the construction. Solution: Steps of construction 1. Draw a line segment $A B=5 \mathrm{~cm}$. 2. From point $B$, draw $\angle A B Y=60^{\circ}$ on which take $B C=6 \mathrm{~cm}$. 3. Join $A C, \triangle A B C$ is the required triangle. 4. From $...
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Question: Tick (✓) the correct answer $\sqrt{0.1}=?$ (a) $0.1$ (b) $0.01$ (c) $0.316$ (d) none of these Solution: (c) $0.316$ Using long division method: $\therefore \sqrt{0.1}=0.316$...
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Question: If $f(x)=\sqrt{x^{2}+9}$, write the value of $\lim _{x \rightarrow 4} \frac{f(x)-f(4)}{x-4}$. Solution: Given: $f(x)=\sqrt{x^{2}+9}$ Now, $f(4)=\sqrt{16+9}$ $=\sqrt{25}$ $=5$ So, $\frac{f(x)-f(4)}{x-4}=\frac{\sqrt{x^{2}+9}-5}{x-4}$ On rationalising the numerator, we get $\frac{f(x)-f(4)}{x-4}=\frac{\sqrt{x^{2}+9}-5}{x-4} \times \frac{\sqrt{x^{2}+9}+5}{\sqrt{x^{2}+9}+5}$ $=\frac{x^{2}+9-25}{(x-4)\left(\sqrt{x^{2}+9}+5\right)}$ $=\frac{x^{2}-16}{(x-4)\left(\sqrt{x^{2}+9}+5\right)}$ $=\fr...
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Question: Tick (✓) the correct answer $\sqrt{0.9}=?$ (a) $0.3$ (b) $0.03$ (c) $0.33$ (d) $0.94$ Solution: (d) $0.94$ $\sqrt{0.9}=0.94$...
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Question: If $f(x)=\sqrt{x^{2}+9}$, write the value of $\lim _{x \rightarrow 4} \frac{f(x)-f(4)}{x-4}$. Solution: Given: $f(x)=\sqrt{x^{2}+9}$ Now, $f(4)=\sqrt{16+9}$ $=\sqrt{25}$ $=5$ So, $\frac{f(x)-f(4)}{x-4}=\frac{\sqrt{x^{2}+9}-5}{x-4}$ On rationalising the numerator, we get $\frac{f(x)-f(4)}{x-4}=\frac{\sqrt{x^{2}+9}-5}{x-4} \times \frac{\sqrt{x^{2}+9}+5}{\sqrt{x^{2}+9}+5}$ $=\frac{x^{2}+9-25}{(x-4)\left(\sqrt{x^{2}+9}+5\right)}$ $=\frac{x^{2}-16}{(x-4)\left(\sqrt{x^{2}+9}+5\right)}$ $=\fr...
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Question: Tick (✓) the correct answer What least number must be added to 15370 to make it a perfect square? (a) 4 (b) 6 (c) 8 (d) 9 Solution: (b) 6 $15370+6=15376$ $\sqrt{15376}=124$...
Read More →Draw an isosceles triangle ABC in
Question: Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5 cm. Construct a triangle PQR similar to AABC in which PQ = 8 cm. Also justify the construction. Solution: Let $\triangle \mathrm{PQR}$ and $\triangle \mathrm{ABC}$ are similar triangles, then its scale factor between the corresponding sides is $\frac{P Q}{A B}=\frac{8}{6}=\frac{4}{3}$ Steps of construction Draw a line segment BC = 5 cm. Construct OQ the perpendicular bisector of line segment BC meeting BC at P. Taking B ...
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Question: Tick (✓) the correct answer What least number must be added to 526 to make it a perfect square? (a) 3 (b) 2 (c) 1 (d) 6 Solution: (a) 3 $526+3=529$ $529=23^{2}$...
Read More →Write the value of the derivative
Question: Write the value of the derivative of $f(x)=|x-1|+|x-3|$ at $x=2$. Solution: Given: $f(x)=|x-1|+|x-3|$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-(x-1)-(x-3), x1 \\ x-1-(x-3), 1 \leq x3 \\ (x-1)+(x-3), x \geq 3\end{array}\right.$ $\Rightarrow f(x)= \begin{cases}-2 x+4, x1 \\ 2, 1 \leq x3 \\ 2 x-4, x \geq 3\end{cases}$ We check differentiability atx= 2 (LHD 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 \rightar...
Read More →Write the value of the derivative
Question: Write the value of the derivative of $f(x)=|x-1|+|x-3|$ at $x=2$. Solution: Given: $f(x)=|x-1|+|x-3|$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-(x-1)-(x-3), x1 \\ x-1-(x-3), 1 \leq x3 \\ (x-1)+(x-3), x \geq 3\end{array}\right.$ $\Rightarrow f(x)= \begin{cases}-2 x+4, x1 \\ 2, 1 \leq x3 \\ 2 x-4, x \geq 3\end{cases}$ We check differentiability atx= 2 (LHD 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 \rightar...
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Question: Tick (✓) the correct answer What least number must be subtracted from 176 to make it a perfect square? (a) 16 (b) 10 (c) 7 (d) 4 Solution: (c) 7 $(176-7)=169$ $\sqrt{169}=13$...
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Question: Tick (✓) the correct answer Which of the following is a pythagorean triplet? (a) (2, 3, 5) (b) (5, 7, 9) (c) (6, 9, 11) (d) (8, 15, 17) Solution: (d) (8,15,17) This can be understood from the property of Pythagorean triplets. According to this property, for a natural number $m,\left(2 m, m^{2}-1, m^{2}+1\right)$ is a Pythagorean triplet. Here, m = 4 2m = 8 m2- 1=15and m2+ 1 = 17...
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Question: If $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely, write the value of $\lim _{x \rightarrow c} f(x)$. Solution: Given: $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely. Then, $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$ Now, $\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)+f(c)\right]$ $=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)\right]+f(c)$ $=\lim _{x \righ...
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Question: Tick (✓) the correct answer Ifnis odd, then (1 + 3 + 5 + 7 +... tonterms) is equal to (a) (n2+ 1) (b) (n2 1) (c)n2 (d) (2n2+ 1) Solution: (c)n2...
Read More →Draw two concentric circles of radii 3 cm and 5 cm.
Question: Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation. Solution: Given, two concentric circles of radii 3 cm and 5 cm with centre 0. We have to draw pair of tangents from point P on outer circle to the other. Steps of construction Draw two concentric circles with centre 0 and radii 3 cm and 5 cm. Taking any point P on outer circle. Join OP. Bisec...
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Question: Tick (✓) the correct answer The square of a proper fraction is (a) larger than the fraction (b) smaller than the fraction (c) equal to the fraction (d) none of these Solution: (b) smaller than the fraction...
Read More →Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
Question: Write the number of points wheref(x) = |x| + |x 1| is continuous but not differentiable. Solution: Given: $f(x)=|x|+|x-1|$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-x-(x-1), x0 \\ x-(x-1), 0 \leq x1 \\ x+(x-1), x \geq 1\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-2 x+1, x0 \\ 1, 0 \leq x1 \\ 2 x-1, x \geq 1\end{array}\right.$ When $x0$, we have: $f(x)=-2 x+1$ which, being a polynomial function is continuous and differentiable. When $0 \leq x1$, we have: $f(x)=1$ whic...
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Question: Tick (✓) the correct answer Which of the following cannot be the unit digit of a perfect square number? (a) 6 (b) 1 (c) 9 (d) 8 Solution: (d) 8 According to the property of squares, a perfect square cannot have 2, 3, 7 or 8 as the unit digit....
Read More →Write the number of points where f (x) = |x| + |x − 1| is continuous but not differentiable.
Question: Write the number of points wheref(x) = |x| + |x 1| is continuous but not differentiable. Solution: Given: $f(x)=|x|+|x-1|$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-x-(x-1), x0 \\ x-(x-1), 0 \leq x1 \\ x+(x-1), x \geq 1\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{lc}-2 x+1, x0 \\ 1, 0 \leq x1 \\ 2 x-1, x \geq 1\end{array}\right.$ When $x0$, we have: $f(x)=-2 x+1$ which, being a polynomial function is continuous and differentiable. When $0 \leq x1$, we have: $f(x)=1$ whic...
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Question: Tick (✓) the correct answer Which of the following numbers is not a perfect square? (a) 3600 (b) 6400 (c) 81000 (d) 2500 Solution: (c) 81000 According to the property of squares, a number ending in odd number of zeroes is not a perfect square....
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