If z and w are two complex numbers
Question: If z and w are two complex numbers such that |zw| = 1 and arg (z) arg (w) = /2, then show that z̅w = i. Solution: Let z = |z| (cos 1+ I sin 1) and w = |w| (cos 2+ I sin 2) Given |zw| = |z| |w| = 1 Also arg (z) arg (w) = /2 ⇒1 2= /2 Now, z̅w = |z| (cos 1 I sin 1) |w| (cos 2+ I sin 2)g | = 1 = |z| |w| (cos (-1) + I sin (-1)) (cos 2+ I sin 2) = 1 [cos (2 1) + I sin (2 1)] = [cos (-/2) + I sin (-/2)] = 1 [0 i] = i Hence proved...
Read More →Write the complex number
Question: Write the complex number $z=\frac{1-i}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ in polar from. Solution: According to the question, We have, $z=\frac{1-\mathrm{i}}{\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}}$ $=\frac{\sqrt{2}\left[\frac{1}{\sqrt{2}}-i \frac{1}{\sqrt{2}}\right]}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ $=\frac{\sqrt{2}\left[\cos \frac{\pi}{4}-i \sin \frac{\pi}{4}\right]}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ $=\sqrt{2}\left[\cos \left(-\frac{\pi}{4}-\frac{\p...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Function $f(x)=a^{x}$ is increasing or $R$, if A. $a0$ B. $a0$ C. $a1$ D. $a0$ Solution: Let $\mathrm{x}_{1}\mathrm{x}_{2}$ and both are real number $a^{x_{1}}a^{x_{2}}$ $\Rightarrow f\left(x_{1}\right)f\left(x_{2}\right)$ $\Rightarrow x_{1}x_{2} \in$ only possible on $a1$...
Read More →Find the complex number satisfying
Question: Find the complex number satisfying the equation z + 2 |(z + 1)| + i = 0. Solution: According to the question, We have, z + 2 |(z + 1)| + i = 0 (1) Substituting z = x + iy, we get ⇒ x + iy + 2 |x + iy + 1| + i = 0 $\Rightarrow \mathrm{x}+\mathrm{i}(1+\mathrm{y})+\sqrt{2}\left[\sqrt{(\mathrm{x}+1)^{2}+\mathrm{y}^{2}}\right]=0$ $\Rightarrow \mathrm{x}+\mathrm{i}(1+\mathrm{y})+\sqrt{2} \sqrt{\left(\mathrm{x}^{2}+2 \mathrm{x}+1+\mathrm{y}^{2}\right)}=0$ Comparing real and imaginary parts to...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: The function $f(x)=\frac{\lambda \sin x+2 \cos x}{\sin x+\cos x}$ is increasing, if A. $\lambda1$ B. $\lambda1$ C. $\lambda2$ D. $\lambda2$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=\frac{\lambda \sin x+2 \cos x}{\sin x+\cos x}$ For increasing function $f^{\prime}(x)0$ $\frac{\...
Read More →Solve the system of equations Re (z2) = 0,
Question: Solve the system of equations Re (z2) = 0, |z| = 2. Solution: According to the question, We have, Re (z2) = 0, |z| = 2 Let z = x + iy. Then, |z| = (x2+ y2) Given in the question, (x2+ y2) = 2 ⇒x2+ y2= 4 (i) z2= x2+ 2ixy y2 = (x2 y2) + 2ixy Now, Re (z2) = 0 ⇒x2 y2= 0 (ii) Equating (i) and (ii), we get ⇒x2= y2= 2 ⇒x = y = 2 Hence, z = x + iy = 2 i2 = 2 + i2, 2 i2, 2 + i2 and 2 i2 Hence, we have four complex numbers....
Read More →Find the equation of the line through the intersection of the lines
Question: Find the equation of the line through the intersection of the lines $x-7 y+5=$ 0 and $3 x+y-7=0$ and which is parallel to $x$-axis. Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right)$. Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $x-7 y+5=0 \ldots$ (i) $3 x+y-7=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (i) by 3, we get $3 x-21 y+15=0$ ..........(iii) On subtracting eq. (iii)...
Read More →If for complex numbers z1 and z2,
Question: If for complex numbers z1and z2, arg (z1) arg (z2) = 0, then show that |z1 z2| = |z1| |z2|. Solution: According to the question, Let z1= |z1| (cos 1+ I sin 1) and z2= |z2| (cos 2+ I sin 2) We have, arg (z1) arg (z2) = 0 ⇒1 2= 0 ⇒1= 2 We also have, z2= |z2| (cos 1+ I sin 1) ⇒z1 z2= ((|z1|cos 1 |z2| cos 1) + i (|z1| sin 1 |z2| sin 1)) $\Rightarrow\left|z_{1}-z_{2}\right|=\sqrt{\left(\left|z_{1}\right| \cos \theta_{1}-\left|z_{2}\right| \cos \theta_{1}\right)^{2}+\left(\left|z_{1}\right| ...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: The function $f(x)=\frac{x}{1+|x|}$ is A. strictly increasing B. strictly decreasing C. neither increasing nor decreasing D. none of these Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ $f(x)=\frac{x}{1+|x|}$ For $x0$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{...
Read More →If |z1| = |z2| = ….. = |zn| = 1,
Question: If |z1| = |z2| = .. = |zn| = 1, then show that |z1+ z2+ z3+ . + zn|= | 1/z1+ 1/z2+ 1/z3+ + 1/zn| Solution: According to the question, We have, |z1| = |z2| = = |zn| = 1 ⇒|z1|2= |z2|2= = |zn|2= 1 ⇒z1z̅1= z2z̅2= z3z̅3= = znz̅n= 1 $\Rightarrow \mathrm{z}_{1}=\frac{1}{\overline{\mathrm{z}_{1}}}, \mathrm{z}_{2}=\frac{1}{\overline{\mathrm{z}_{2}}}, \ldots, \mathrm{z}_{\mathrm{n}}=\frac{1}{\overline{\mathrm{z}_{\mathrm{n}}}}$ Now, $\Rightarrow\left|\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}+...
Read More →If z1, z2 and z3, z4 are two pairs of conjugate
Question: If z1, z2and z3, z4are two pairs of conjugate complex numbers, then find arg(z1/z4) + arg(z2/z3). Solution: According to the question, We have, z1and z2are conjugate complex numbers. The negative side of the real axis = r1(cos 1 i sin 1) = r1[cos (-1) + I sin (-1)] Similarly, z3= r2(cos 2 i sin 2) ⇒z4= r2[cos (-2) + I sin (-2)] $\Rightarrow \arg \left(\frac{z_{1}}{z_{4}}\right)+\arg \left(\frac{z_{2}}{z_{3}}\right)=\arg \left(z_{1}\right)-\arg \left(z_{4}\right)+\arg \left(z_{2}\right)...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: If the function $f(x)=\cos |x|-2 a x+b$ increases along the entire number scale, then A. $a=b$ B. $a=\frac{1}{2} b$ C. $a \leq-\frac{1}{2}$ D. $a-\frac{3}{2}$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=\cos |x|-2 a x+b$ $\frac{d(f(x))}{d x}=-\sin x-2 a=f^{\prime}(x)$ For incr...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: In the interval $(1,2)$, function $f(x)=2|x-1|+3|x-2|$ is A. increasing B. decreasing C. constant D. none of these Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly decreasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=2(x-1)+3(2-x)$ $\Rightarrow f(x)=-x+4$ $\frac{d(f(x))}{d x}=f^{\prime}(x)=-1$ Therefore $f^{\prime}(x)0$ Hence decreasing...
Read More →Find the equation of the line through the intersection of the lines
Question: Find the equation of the line through the intersection of the lines 2x 3y = 0 and $4 x-5 y=2$ and which is perpendicular to the line $x+2 y+1=0$ Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right) .$ Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $2 x-3 y=0 \ldots$ (i) $4 x-5 y=2 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (i) by 2, we get $4 x-6 y=0 \ldots$ (iii) On subtracting eq...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Every invertible function is A. monotonic function B. constant function C. identity function D. not necessarily monotonic function Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ If $f(x)$ is strictly increasing function on interval $[a, b]$, then $f^{-1}$ exist and it is also a strictly increasin...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Function $f(x)=|x|-|x-1|$ is monotonically increasing when A. $x0$ B. $x1$ C. $x1$ D. $0x1$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for $a l l ~ x \in(a, b)$ Given:- For $x0$ $f(x)=-1$ for $0x1$ $f(x)=2 x-1$ for $x1$ $f(x)=1$ Hence $f(x)$ will increasing in $0x1$...
Read More →Find the equation of the line through the intersection of the lines
Question: Find the equation of the line through the intersection of the lines $5 x-3 y=1$ and $2 x+3 y=23$ and which is perpendicular to the line $5 x-3 y=1$ Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right) .$ Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $5 x-3 y=1 \ldots(i)$ $2 x+3 y=23 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Adding eq. (i) and (ii) we get $5 x-3 y+2 x+3 y=1+23$ $\Rightarrow 7 x=24...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: $f(x)=2 x-\tan ^{-1} x-\log \left\{x+\sqrt{x^{2}+1}\right\}$ is monotonically increasing when A. $x0$ B. $x0$ C. $x \in R$ D. $X \in R-\{0\}$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=2 x-\tan ^{-1} x-\log \left\{x+\sqrt{x^{2}+1}\right\}$ $\frac{d f(x)}{d x}=2-\frac{1}{1+x^{2}}...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: If the function $f(x)=k x^{3}-9 x^{2}+9 x+3$ is monotonically increasing in every interval, then A. $k3$ B. $k \leq 3$ C. $k3$ D. $k3$ Solution: Formula:- (i) $a x^{2}+b x+c0$ for all $x \Rightarrow a0$ and $b^{2}-4 a c0$ (ii) $a x^{2}+b x+c0$ for $a l l x \Rightarrow a0$ and $b^{2}-4 a c0$ (iii) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(...
Read More →Find the equation of the line drawn through the point of intersection of the
Question: Find the equation of the line drawn through the point of intersection of the lines $x-y=1$ and $2 x-3 y+1=0$ and which is parallel to the line $3 x+4 y=12$ Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right)$. Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $x-y=1 \ldots(1)$ $2 x-3 y+1=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (i) by 2, we get $2 x-2 y=2$ or $2 x-2 y-2=0 \ldots$...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Function $f(x)=2 x^{3}-9 x^{2}+12 x+29$ is monotonically decreasing when A. $x2$ B. $x2$ C. $x3$ D. $1x2$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on ( $a, b$ ) to be strictly decreasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=2 x^{3}-9 x^{2}+12 x+29$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\mathrm{f}^{\prime}(\mathrm{x})=6(\mathrm{x}...
Read More →If for complex numbers z1 and z2,
Question: If for complex numbers z1and z2, arg (z1) arg (z2) = 0, then show that |z1 z2| = |z1| |z2|. Solution: According to the question, Let z1= |z1| (cos 1+ I sin 1) and z2= |z2| (cos 2+ I sin 2) We have, arg (z1) arg (z2) = 0 ⇒1 2= 0 ⇒1= 2 We also have, z2= |z2| (cos 1+ I sin 1) ⇒z1 z2= ((|z1|cos 1 |z2| cos 1) + i (|z1| sin 1 |z2| sin 1)) $\Rightarrow\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|=\sqrt{\left(\left|\mathrm{z}_{1}\right| \cos \theta_{1}-\left|\mathrm{z}_{2}\right| \cos \theta_{1}\...
Read More →If |z1| = |z2| = ….. = |zn| = 1,
Question: If |z1| = |z2| = .. = |zn| = 1, then show that |z1+ z2+ z3+ . + zn|= | 1/z1+ 1/z2+ 1/z3+ + 1/zn| Solution: According to the question, We have, |z1| = |z2| = = |zn| = 1 ⇒|z1|2= |z2|2= = |zn|2= 1 ⇒z1z̅1= z2z̅2= z3z̅3= = znz̅n= 1 $\Rightarrow \mathrm{z}_{1}=\frac{1}{\overline{\mathrm{z}_{1}}}, \mathrm{z}_{2}=\frac{1}{\overline{\mathrm{z}_{2}}}, \ldots, \mathrm{z}_{\mathrm{n}}=\frac{1}{\overline{\mathrm{z}_{\mathrm{n}}}}$ Now, $\Rightarrow\left|\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}+...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Function $f(x)=x^{3}-27 x+5$ is monotonically increasing when A. $x-3$ B. $|x|3$ C. $x \leq-3$ D. $|x| \geq 3$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=x^{3}-27 x+5$ $\frac{d(f(x))}{d x}=3 x^{2}-27=f^{\prime}(x)$ for increasing function $f^{\prime}(x)0$ $3 x^{2}-270$ $\Rightar...
Read More →Find the equation of the line drawn through the point of intersection of the
Question: Find the equation of the line drawn through the point of intersection of the lines $x+y=9$ and $2 x-3 y+7=0$ and whose slope is $\frac{-2}{3}$ Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right)$. Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $x+y=9 \ldots$ (i) $2 x-3 y+7=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (i) by 2, we get $2 x+2 y=18$ or $2 x+2 y-18=0 \ldots$ (iii) On ...
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