Write a pythagorean triplet whose smallest member is
Question: Write a pythagorean triplet whose smallest member is (i) 6 (ii) 14 (iii) 16 (iv) 20 Solution: For every number $m1$, the Pythagorean triplet is $\left(2 m, m^{2}-1, m^{2}+1\right)$. Using the above result: (i)$2 m=6$ $m=3, m^{2}=9$ $m^{2}-1=9-1=8$ $m^{2}+1=9+1=10$ Thus, the Pythagorean triplet is $[6,8,10]$ (ii)$2 m=14$ $m=7, m^{2}=49$ $m^{2}-1=49-1=48$ $m^{2}+1=49+1=50$ Thus, the Pythagorean triplet is $[14,48,50]$. (iii)$2 m=16$ $m=8, m^{2}=64$ $m^{2}-1=64-1=63$ $m^{2}+1=64+1=65$ Thu...
Read More →If a hexagon ABCDEF circumscribe a circle,
Question: If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF =BC + DE + FA Solution: Given A hexagon ABCDEF circumscribe a circle. To prove $A B+C D+E F=B C+D E+F A$ Proof $\quad A B+C D+E F=(A Q+Q B)+(C S+S D)+(E U+U F)$ $=A P+B R+C R+D T+E T+F P$ $=(A P+F P)+(B R+C R)+(D T+E T)$ $A B+C D+E F=A F+B C+D E$ $A Q=A P$ $Q B=B R$ $C S=C R$ $D S=D T$ $E U=E T$ [tangents drawn from an external point to a circle are equal] Hence proved....
Read More →The function f(x) = |x + 1| is not differentiable
Question: The function $f(x)=|x+1|$ is not differentiable at $x=$___________ Solution: The given function is $f(x)=|x+1|$. $f(x)=|x+1|= \begin{cases}x+1, x \geq-1 \\ -(x+1), x-1\end{cases}$ Now, $(x+1)$ and $-(x+1)$ are polynomial functions which are differentiable at each $x \in \mathrm{R} .$ So, $f(x)$ is differentiable for all $x-1$ and for all $x-1$. So, we need to check the differentiability of $f(x)$ at $x=-1$. We have, $L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1-h)-f(-1)}{-h}$ $\...
Read More →Express 81 as the sum of 9 odd numbers.
Question: (i) Express 81 as the sum of 9 odd numbers. (ii) Express 100 as the sum of 10 odd numbers. Solution: Sum of first $\mathrm{n}$ odd natural numbers $=n^{2}$ (i) Expressing 81 as a sum of 9 odd numbers: $81=(9)^{2}$ $n=9$ $81=1+3+5+7+9+11+13+15+17$ (ii) Expressing 100 as a sum of 10 odd numbers: $100=(10)^{2}$ $n=10$ $100=1+3+5+7+9+11+13+15+17+19$...
Read More →The function f(x) = |x + 1| is not differentiable
Question: The function $f(x)=|x+1|$ is not differentiable at $x=$___________ Solution: The given function is $f(x)=|x+1|$. $f(x)=|x+1|= \begin{cases}x+1, x \geq-1 \\ -(x+1), x-1\end{cases}$ Now, $(x+1)$ and $-(x+1)$ are polynomial functions which are differentiable at each $x \in \mathrm{R} .$ So, $f(x)$ is differentiable for all $x-1$ and for all $x-1$. So, we need to check the differentiability of $f(x)$ at $x=-1$. We have, $L f^{\prime}(-1)=\lim _{h \rightarrow 0} \frac{f(-1-h)-f(-1)}{-h}$ $\...
Read More →Prove that a diameter AB of a circle
Question: Prove that a diameter AB of a circle bisects all those chords which are parallel to the tangent at the point A. Solution: Given, AB is a diameter of the circle. A tangent is drawn from point A. Draw a chord CD parallel to the tangent MAN. So. $C D$ is a chord of the circle and $O A$ is a radius of the circle. $\angle M A O=90^{\circ}$ [tanment at anv noint of a circle is perpendicular to the radius through the point of contact] $\angle C E O=\angle M A O \quad$ [corresponding angles] $...
Read More →Without adding, find the sum:
Question: Without adding, find the sum: (i) (1 + 3 + 5 + 7 + 9 + 11 + 13) (ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) (iii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23) Solution: Sum of first $\mathrm{n}$ odd numbers $=n^{2}$ (i) $(1+3+5+7+9+11+13)=7^{2}=49$ (ii) $(1+3+5+7+9+11+13+15+17+19)=10^{2}=100$ (iii) $(1+3+5+7+9+11+13+15+17+19+21+23)=12^{2}=144$...
Read More →Prove that the tangents drawn at
Question: Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord. Solution: To prove 1 = 2, let PQ be a chord of the circle. Tangents are drawn at the points R and Q. Let P be another point on the circle, then, join PQ and PR. Since, at point Q, there is a tangent. $\therefore$$\angle 2=\angle P \quad$ [angles in alternate segments are equal] Since, at point $R$, there is a tangent. $\therefore$ $\angle 1=\angle P$ [angles in alternate segments are equa...
Read More →Which of the following are squares of odd numbers?
Question: Which of the following are squares of odd numbers? (i) 484 (ii) 961 (iii) 7396 (iv) 8649 (v) 4225 Solution: According to the property of squares, the square of an odd number is also an odd number. Using this property, we will determine which of the numbers in the given list of squares is a square of an odd number.(i) 484.This is an even number. Thus, it is not a square of an odd number.(ii) 961This is an odd number. Thus, it is a square of an odd number.(iii) 7396This is an even number...
Read More →Which of the following are squares of odd numbers?
Question: Which of the following are squares of odd numbers? (i) 484 (ii) 961 (iii) 7396 (iv) 8649 (v) 4225 Solution: According to the property of squares, the square of an odd number is also an odd number. Using this property, we will determine which of the numbers in the given list of squares is a square of an odd number.(i) 484.This is an even number. Thus, it is not a square of an odd number.(ii) 961This is an odd number. Thus, it is a square of an odd number.(iii) 7396This is an even number...
Read More →The set of points where the function
Question: The set of points where the function $f(x)=|2 x-1| \sin x$ is differentiable, is (a) $R$ (b) $R-\left\{\frac{1}{2}\right\}$ (c) $(0, \infty)$ (d) none of these Solution: Let $g(x)=|2 x-1|$ and $h(x)=\sin x$ We know that, the trigonometric functions are differentiable in their respective domain. So, $h(x)=\sin x$ is differentiable for all $x \in \mathrm{R}$. Now, $g(x)=|2 x-1|= \begin{cases}2 x-1, x \geq \frac{1}{2} \\ -(2 x-1), x\frac{1}{2}\end{cases}$ $(2 x-1)$ and $-(2 x-1)$ are poly...
Read More →Which of the following are squares of even numbers?
Question: Which of the following are squares of even numbers? (i) 196 (ii) 441 (iii) 900 (iv) 625 (v) 324 Solution: The square of an even number is always even. Thus, even numbers in the given list of squares will be squares of even numbers.(i) 196This is an even number. Thus, it must be a square of an even number.(ii) 441This is an odd number. Thus, it is not a square of an even number.(iii) 900This is an even number. Thus, it must be a square of an even number.(iv) 625This is an odd number. Th...
Read More →The set of points where the function
Question: The set of points where the function $f(x)=|2 x-1| \sin x$ is differentiable, is (a) $R$ (b) $R-\left\{\frac{1}{2}\right\}$ (c) $(0, \infty)$ (d) none of these Solution: Let $g(x)=|2 x-1|$ and $h(x)=\sin x$ We know that, the trigonometric functions are differentiable in their respective domain. So, $h(x)=\sin x$ is differentiable for all $x \in \mathrm{R}$. Now, $g(x)=|2 x-1|= \begin{cases}2 x-1, x \geq \frac{1}{2} \\ -(2 x-1), x\frac{1}{2}\end{cases}$ $(2 x-1)$ and $-(2 x-1)$ are poly...
Read More →A chord PQ of a circle is parallel to
Question: A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ. Solution: Given Chord PQ is parallel to tangent at R. To prove R bisects the arc PRQ Proof$\angle 1=\angle 2$ [alternate interior angles] $\angle 1=\angle 3$ [angle between tangent and chord is equal to angle made by chord in alternate segment] $\therefore$ $\angle 2=\angle 3$ $P R=Q R \quad$ sides opoosite to equal angles are equal $\Rightarrow \quad P R=Q R$ So, $R$ b...
Read More →Give reason to show that none of the numbers given below is a perfect square:
Question: Give reason to show that none of the numbers given below is a perfect square: (i) 5372 (ii) 5963 (iii) 8457 (iv) 9468 (v) 360 (vi) 64000 (viii) 2500000 Solution: By observing the properties of square numbers, we can determine whether a given number is a square or not.(i) 5372 A number that ends with 2 is not a perfect square. Thus, the given number is not a perfect square.(ii) 5963 Anumberthat ends with 3 is not a perfect square. Thus, the given number is not a perfect square.(iii) 845...
Read More →Find the largest number of 3 digits which
Question: Find the largest number of 3 digits which is a perfect square. Solution: The largest 3 digit number is 999. The number whose square is 999 is 31.61. Thus, the square of any number greater than 31.61 will be a 4 digit number. Therefore, the square of 31 will be the greatest 3 digit perfect square. $31^{2}=31 \times 31=961$...
Read More →In figure, common tangents AB and CD
Question: In figure, common tangents AB and CD to two circles intersect at E. Prove that AB = CD. Solution: Given Common tangents AB and CD to two circles intersecting at E. To proveAB = CD Proof $E A=E C$$\ldots(1)$ [the lengths of tanoents drawn from an internal point to a circle are equal] $E B=E D$ ....(ii) On adding Eqs. (i) and (ii), we get $E A+E B=E C+E D$ $\Rightarrow$ $A B=C D$ Hence proved....
Read More →Find the largest number of 2 digits which is a perfect square.
Question: Find the largest number of 2 digits which is a perfect square. Solution: The first three digit number (100) is a perfect square. Its square root is 10. The number before 10 is 9. Square of $9=(9)^{2}=81$ Thus, the largest 2 digit number that is a perfect square is 81 ....
Read More →By what least number should the given number be divided to get a perfect square number?
Question: By what least number should the given number be divided to get a perfect square number? In each case, find the number whose square is the new number. (i) 1575 (ii) 9075 (iii) 4851 (iv) 3380 (v) 4500 (vi) 7776 (vii) 8820 (viii) 4056 Solution: (i) Resolving 1575 into prime factors: $1575=3 \times 3 \times 5 \times 5 \times 7=3^{2} \times 5^{2} \times 7$ Thus, to get a perfect square, the given number should be divided by 7 New number obtained $=\left(3^{2} \times 5^{2}\right)=(3 \times 5...
Read More →The function
Question: The function $f(x)=e^{|x|}$ is (a) continuous every where but not differentiable atx= 0(b) continuous and differentiable everywhere(c) not continuous atx= 0(d) none of these Solution: The given function is $f(x)=e^{|x|}$. We know If $f$ is continuous on its domain $D$, then $|f|$ is also continuous on $D$. Now, the identity function $p(x)=x$ is continuous everywhere. So, $g(x)=|p(x)|=|x|$ is also continuous everywhere. Also, the exponential function $a^{x}, a0$ is continuous everywhere...
Read More →The function
Question: The function $f(x)=e^{|x|}$ is (a) continuous every where but not differentiable atx= 0(b) continuous and differentiable everywhere(c) not continuous atx= 0(d) none of these Solution: The given function is $f(x)=e^{|x|}$. We know If $f$ is continuous on its domain $D$, then $|f|$ is also continuous on $D$. Now, the identity function $p(x)=x$ is continuous everywhere. So, $g(x)=|p(x)|=|x|$ is also continuous everywhere. Also, the exponential function $a^{x}, a0$ is continuous everywhere...
Read More →in figure, AB and CD are common tangents
Question: in figure, AB and CD are common tangents to two circles of equal radii. Prove that AB = CD. Solution: Given AB and CD are tangents to two circles of equal radii. To prove AB = CD Construction Join $O A, O C, O^{\prime} B$ and $O^{\prime} D$ Proof Now, $\angle O A B=90^{\circ}$ [tangent at any point of a circle is perpendicular to radius through the point of contact] Thus, $A C$ is a straight line. Also, $\angle O A B+\angle O C D=180^{\circ}$ $\therefore \quad A B \| C D$ Similarly, $B...
Read More →In figure, AB and CD are common tangents
Question: In figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD Solution: Given AS and CD are common tangent to two circles of unequal radius To prove AB = CD Construction Produce $A B$ and $C D$, to intersect at $P$. Proof $P A=P C$ [the length of tangents drawn from an internal point to a circle are equal] Also, $P B=P D$ [the lengths of tangents drawn from an internal point to a circle are equal] $\therefore$$P A-P B=P C-P D$ $A B=C D$ Hence proved....
Read More →By what least number should the given number be multiplied to get a perfect square number?
Question: By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number. (i) 3975 (ii) 2156 (iii) 3332 (iv) 2925 (v) 9075 (vi) 7623 (vii) 3380 (viii) 2475 Solution: 1. Resolving 3675 into prime factors: $3675=3 \times 5 \times 5 \times 7 \times 7$ Thus, to get a perfect square, the given number should be multiplied by 3. New number $=\left(3^{2} \times 5^{2} \times 7^{2}\right)=(3 \times 5 \times 7)^{2}=(10...
Read More →Solve this
Question: Let $f(x)=\left\{\begin{array}{ll}1, x \leq-1 \\ |x|, -1x1 \\ 0, x \geq 1\end{array}\right.$ Then, $f$ is (a) continuous atx= 1(b) differentiable atx= 1(c) everywhere continuous(d) everywhere differentiable Solution: (b) differentiable at $x=-1$ $f(x)= \begin{cases}1, x \leq-1 \\ |x|, -1x1 \\ 0, x \geq 1\end{cases}$ Differentiabilty at $x=-1$ $(\mathrm{LHD} x=-1)$ $\lim _{x \rightarrow-1^{-}} \frac{f(x)-f(-1)}{x+1}$ $=\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1}$ $=\lim _{x \rightarr...
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