Find the square root of number by using the method of prime factorisation:
Question: Find the square root of number by using the method of prime factorisation:225 Solution: By prime factorisation method: $225=3 \times 3 \times 5 \times 5$ $\sqrt{225}=3 \times 5=15$...
Read More →Find the value of using the diagonal method:
Question: Find the value of using the diagonal method:(256)2 Solution: $256^{2}=65536$...
Read More →Find the value of using the diagonal method:
Question: Find the value of using the diagonal method:(137)2 Solution: $137^{2}=18769$...
Read More →Find the value of using the diagonal method:
Question: Find the value of using the diagonal method:(86)2 Solution: $86^{2}=7396$...
Read More →If AB is a chord of a circle with centre 0,
Question: If AB is a chord of a circle with centre 0, AOC is a diameter and AT is the tangent at A as shown in figure. Prove that BAT = ACB. Solution: Since, AC is a diameter line, so angle in semi-circle makes an angle 90. $\begin{array}{lll}\therefore \angle A B C=90^{\circ} \text { [by property] }\end{array}$ In $\triangle A B C, \quad \angle C A B+\angle A B C+\angle A C B=180^{\circ}$ $\left[\because\right.$ sum of all interior angles of any triangle is $\left.180^{\circ}\right]$ $\Rightarr...
Read More →The number of points
Question: The number of points in $[-\pi, \pi]$ where $f(x)=\sin ^{-1}(\sin x)$ is not differentiable is._____________ Solution: $f(x)=\sin ^{-1}(\sin x)= \begin{cases}-x-\pi, -\pi \leq x \leq-\frac{\pi}{2} \\ x, -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \\ \pi-x, \frac{\pi}{2} \leq x \leq \pi\end{cases}$ Let us check the differentiability of the function at $x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$. At $x=-\frac{\pi}{2}$ $L f^{\prime}\left(-\frac{\pi}{2}\right)=\lim _{x \rightarrow-\frac{\pi}{2}^...
Read More →Find the value of using the diagonal method:
Question: Find the value of using the diagonal method:(67)2 Solution: $67^{2}=4489$...
Read More →Find the value of using the column method:
Question: Find the value of using the column method:(96)2 Solution: Using column method: Here, $a=9$ b = 6 $\therefore 96^{2}=9216$...
Read More →Find the value of using the column method:
Question: Find the value of using the column method:(52)2 Solution: Using the column method:Here, a = 5 b = 2 $\therefore 52^{2}=2704$...
Read More →Find the value of using the column method:
Question: Find the value of using the column method:(35)2 Solution: Using the column method: Here, a = 3 and b = 5 $\therefore 35^{2}=1225$...
Read More →The set of points where
Question: The set of points wheref(x) =x [x] not differentiable is ____________. Solution: Let $g(x)=x$ and $h(x)=[x]$. Every polynomial function is differentiable for all $x \in \mathrm{R}$. So, $g(x)=x$ is differentiable for all $x \in \mathrm{R}$. Also, the function $h(x)=[x]$ is discontinuous at all integral values of $x$ i.e. $x \in$ Z. So, $h(x)=[x]$ is not differentiable at all integral values of $x$ i.e. $x \in Z$. Now, $f(x)=g(x)-h(x)=x-[x]$ So, the function $f(x)=x-[x]$ is differentiab...
Read More →The set of points where
Question: The set of points wheref(x) =x [x] not differentiable is ____________. Solution: Let $g(x)=x$ and $h(x)=[x]$. Every polynomial function is differentiable for all $x \in \mathrm{R}$. So, $g(x)=x$ is differentiable for all $x \in \mathrm{R}$. Also, the function $h(x)=[x]$ is discontinuous at all integral values of $x$ i.e. $x \in$ Z. So, $h(x)=[x]$ is not differentiable at all integral values of $x$ i.e. $x \in Z$. Now, $f(x)=g(x)-h(x)=x-[x]$ So, the function $f(x)=x-[x]$ is differentiab...
Read More →Find the value of using the column method:
Question: Find the value of using the column method: (23)2 Solution: Using the column method: a = 2 b = 3 $\therefore 23^{2}=529$...
Read More →Write (T) for true and (F) for false for each of the statements given below:
Question: Write (T) for true and (F) for false for each of the statements given below: (i) The number of digits in a perfect square is even. (ii) The square of a prime number is prime. (iii) The sum of two perfect squares is a perfect square. (iv) The difference of two perfect squares is a perfect square. (v) The product of two perfect squares is a perfect square. Solution: (i) F The number of digits in a square can also be odd. For example: 121 (ii) F A prime number is one that is not divisible...
Read More →The function g(x) = |x – 1| + |x + 1| is not differentiable
Question: The functiong(x) = |x 1| + |x+ 1| is not differentiable atx= ____________. Solution: $|x-1|= \begin{cases}x-1, x \geq 1 \\ -(x-1), x1\end{cases}$ $|x+1|= \begin{cases}x+1, x \geq-1 \\ -(x+1), x-1\end{cases}$ $\therefore g(x)=|x-1|+|x+1|= \begin{cases}-(x-1)-(x+1), x-1 \\ -(x-1)+x+1, -1 \leq x1 \\ x-1+x+1, x \geq 1\end{cases}$ $\Rightarrow g(x)=|x-1|+|x+1|= \begin{cases}-2 x, x-1 \\ 2, -1 \leq x1 \\ 2 x, x \geq 1\end{cases}$ When $x-1, g(x)=-2 x$ which being a polynomial function is con...
Read More →The function g(x) = |x – 1| + |x + 1| is not differentiable
Question: The functiong(x) = |x 1| + |x+ 1| is not differentiable atx= ____________. Solution: $|x-1|= \begin{cases}x-1, x \geq 1 \\ -(x-1), x1\end{cases}$ $|x+1|= \begin{cases}x+1, x \geq-1 \\ -(x+1), x-1\end{cases}$ $\therefore g(x)=|x-1|+|x+1|= \begin{cases}-(x-1)-(x+1), x-1 \\ -(x-1)+x+1, -1 \leq x1 \\ x-1+x+1, x \geq 1\end{cases}$ $\Rightarrow g(x)=|x-1|+|x+1|= \begin{cases}-2 x, x-1 \\ 2, -1 \leq x1 \\ 2 x, x \geq 1\end{cases}$ When $x-1, g(x)=-2 x$ which being a polynomial function is con...
Read More →Fill in the blanks:
Question: Fill in the blanks: (i) The square of an even number is ......... (ii) The square of an odd number is ......... (iii) The square of a proper fraction is ......... than the given fraction. (iv)n2= the sum of firstn......... natural numbers. Solution: (i) The square of an even number iseven.(ii) The square of an odd number isodd.(iii) The square of a proper fraction issmallerthan the given fraction. (iv) $n^{2}=$ the sum of first $n \underline{\text { odd }}$ natural numbers....
Read More →The function g(x) = |x – 1| + |x + 1| is not differentiable
Question: The functiong(x) = |x 1| + |x+ 1| is not differentiable atx= ____________. Solution: $|x-1|= \begin{cases}x-1, x \geq 1 \\ -(x-1), x1\end{cases}$ $|x+1|= \begin{cases}x+1, x \geq-1 \\ -(x+1), x-1\end{cases}$ $\therefore g(x)=|x-1|+|x+1|= \begin{cases}-(x-1)-(x+1), x-1 \\ -(x-1)+x+1, -1 \leq x1 \\ x-1+x+1, x \geq 1\end{cases}$ $\Rightarrow g(x)=|x-1|+|x+1|= \begin{cases}-2 x, x-1 \\ 2, -1 \leq x1 \\ 2 x, x \geq 1\end{cases}$ When $x-1, g(x)=-2 x$ which being a polynomial function is con...
Read More →From an external point P, two tangents,
Question: From an external point P, two tangents, PA and PB are drawn to a circle with centre 0. At one point E on the circle tangent is drawn which intersects PA and PB at C and D, respectively. If PA = 10 cm, find the perimeter of the trianlge PCD. Solution: Two tangents PA and PB are drawn to a circle with centre 0 from an external point P Perimeter of $\triangle P C D=P C+C D+P D$ $=P C+C E+E D+P D$ $=P C+C A+D B+P D$ $=P A+P B$ $=2 P A=2(10)$ $=20 \mathrm{~cm}$ $[\because C E=C A, D E=D B, ...
Read More →Evaluate: (i) 88 × 92
Question: Evaluate: (i) 88 92 (ii) 78 82 Solution: (i) $88 \times 92=(90-2) \times(90+2)=\left(90^{2}-2^{2}\right)=8100-4=8096$ (ii) $78 \times 82=(80-2) \times(80+2)=\left(80^{2}-2^{2}\right)=6400-4=6396$...
Read More →Evaluate: (i) 69 × 71
Question: Evaluate: (i) 69 71 (ii) 94 106 Solution: (i) $69 \times 71=(70-1) \times(70+1)=\left(70^{2}-1^{2}\right)=4900-1=4899$ (ii) $94 \times 106=(100-6) \times(100+6)=\left(100^{2}-6^{2}\right)=10000-36=9964$...
Read More →Let s denotes the semi-perimeter
Question: Let s denotes the semi-perimeter of a Δ ABC in which BC = a, CA = b and AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively. Prove that BD = s b. Solution: A circle is inscribed in the A ABC, which touches the BC, CA and AB. Given, $\quad B C=a, C A=b$ and $A B=c$ By using the property, tangents are drawn from an external point to the circle are equal in length. $\therefore \quad B D=B F=x \quad$ [say] $D C=C E=y$[say] and $A E=A F=Z$ [say] NOW, $B C+C A+A B=a+b+c...
Read More →Using the formula
Question: Using the formula (ab)2= (a2 2ab+b2), evaluate: (i) (196)2 (ii) (689)2 (iii) (891)2 Solution: (i) $(196)^{2}=(200-4)^{2}=200^{2}-2(200 \times 4)+4^{2}=40000-1600+16=38416$ (ii) $(689)^{2}=(700-11)^{2}=700^{2}-2(700 \times 11)+11^{2}=490000-15400+121=474721$ (iii) $(891)^{2}=(900-9)^{2}=900^{2}-2(900 \times 9)+9^{2}=810000-16200+81=793881$...
Read More →Using the formula
Question: Using the formula (a+b)2= (a2+ 2ab+b2), evaluate: (i) (310)2 (ii) (508)2 (iii) (630)2 Solution: (i) $310^{2}=(300+10)^{2}=\left(300^{2}+2(300 \times 10)+10^{2}\right)=90000+6000+100=96100$ (ii) $508^{2}=(500+8)^{2}=\left(500^{2}+2(500 \times 8)+8^{2}\right)=250000+8000+64=258064$ (iii) $630^{2}=(600+30)^{2}=\left(600^{2}+2(600 \times 30)+30^{2}\right)=360000+36000+900=396900$...
Read More →Evaluate:
Question: Evaluate: (i) (38)2 (37)2 (ii) (75)2 (74)2 (iii) (92)2 (91)2 (iv) (105)2 (104)2 (v) (141)2 (140)2 (vi)(218)2 (217)2 Solution: Given: $\left[(n+1)^{2}-n^{2}\right]=(n+1)+n$ (i) $(38)^{2}-(37)^{2}=38+37=75$ (ii) $(75)^{2}-(74)^{2}=75+74=149$ (iii) $(92)^{2}-(91)^{2}=92+91=183$ (iv) $(105)^{2}-(104)^{2}=105+104=209$ (v) $(141)^{2}-(140)^{2}=141+140=281$ (vi) $(218)^{2}-(217)^{2}=218+217=435$...
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