The sum of two opposite angles of a parallelogram is 130°.
Question: The sum of two opposite angles of a parallelogram is 130. Find the measure of each of its angles. Solution: Let $A B C D$ be a parallelogram and let the sum of its opposite angles be $130^{\circ}$. $\angle A+\angle C=130^{\circ}$ $T$ he opposite angles are equal in a parallelogram. $\therefore \angle A=\angle C=x^{\circ}$ $\Rightarrow x+x=130$ $\Rightarrow 2 x=130$ $\Rightarrow x=\frac{130}{2}$ $\Rightarrow x=65$ $\therefore \angle A=65^{\circ}$ and $\angle C=65^{\circ}$ $\angle A+\ang...
Read More →Find the modulus of each of the following:
Question: Find the modulus of each of the following: $(3+\sqrt{-5})$ Solution: Given: $z=(3+\sqrt{-5})$ The above can be re - written as $z=3+\sqrt{(-1) \times 5}$ $z=3+i \sqrt{5}\left[\because i^{2}=-1\right]$ Now, we have to find the modulus of (3 + i5) So, $|z|=|3+i \sqrt{5}|=\sqrt{(3)^{2}+(\sqrt{5})^{2}}=\sqrt{9+5}=\sqrt{14}$ Hence, the modulus of $(3+\sqrt{-5})$ is $\sqrt{14}$...
Read More →Two adjacent angles of a parallelogram are (3x − 4)° and (3x + 16)°.
Question: Two adjacent angles of a parallelogram are (3x 4) and (3x+ 16). Find the value ofxand hence find the measure of each of its angles. Solution: Let $A B C D$ be a parallelogram. $L$ et $\angle A=(3 x-4)^{\circ}$ $\angle B=(3 x+16)^{\circ}$ $\therefore \angle A+\angle B=180^{\circ} \quad\left[\right.$ since the sum of adjacent angles of a parallelogram is $\left.180^{\circ}\right]$ $\Rightarrow 3 x-4+3 x+16=180$ $\Rightarrow 3 x-4+3 x+16=180$ $\Rightarrow 6 x+12=180$ $\Rightarrow 6 x=168$...
Read More →The population of India in
Question: The population of India in $2004-05$ is a rational number. Solution: True. Population of India can always be a whole number. Hence, it is also a rational number....
Read More →Two adjacent angles of a parallelogram are in the ratio 4 : 5.
Question: Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles. Solution: Let $A B C D$ be the parallelogram. Then, $\angle A$ and $\angle B$ are its adjacent angles. Let $\angle A=(4 x)^{\circ}$ $\angle B=(5 x)^{\circ}$ $\therefore \angle A+\angle B=180^{\circ} \quad\left[s\right.$ ince sum of the adjacent angles of a parallelogram is $\left.180^{\circ}\right]$ $\Rightarrow 9 x=180$ $\Rightarrow x=\frac{180}{9}$ $\Rightarrow x=20$ $\therefore \an...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when If $x=10(t-\sin t), y=12(1-\cos t)$, find $\frac{d y}{d x}$ Solution: Here, $x=10(t-\sin t) y=12(1-\cos t)$ $\frac{\mathrm{dx}}{\mathrm{dt}}=10(1-\cos \mathrm{t})$ (1) $\frac{d y}{d t}=12(\sin t) \ldots \ldots(2)$ $\frac{d y}{d x}=\frac{\frac{d y}{d x}}{\frac{d x}{d t}}=\frac{12(\sin t)}{10(1-c o s t)} \mid$ from equation 1 and 2 $\frac{d y}{d x}=\frac{12 \sin \frac{t}{2} \cdot \cos t / 2}{10 \sin ^{2} t / 2}$ $\frac{\mathrm{dy}}{\mathrm{dx}...
Read More →All positive rational numbers
Question: All positive rational numbers lie between 0 and 1000 . Solution: False. There are infinite positive rational number on the right side of 0 on the number line....
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $(2-5 i)^{2}$ Solution: Given: $z=(2-5 i)^{2}$ First we calculate $(2-5 \mathrm{i})^{2}$ and then we find the conjugate $(2-5 i)^{2}=(2)^{2}+(5 i)^{2}-2(2)(5 i)$ $=4+25 i^{2}-20 i$ $=4+25(-1)-20 i\left[\because i^{2}=-1\right]$ $=4-25-20 i$ $=-21-20 i$ Now, we have to find the conjugate of $(-21-20 \mathrm{i})$ So, the conjugate of $(-21-20 \mathrm{i})$ is $(-21+20 \mathrm{i})$...
Read More →0 is a rational number.
Question: 0 is a rational number. Solution: True. Because, $0 / 1$ is a rational number....
Read More →Two adjacent angles of a parallelogram are equal.
Question: Two adjacent angles of a parallelogram are equal. What is the measure of each of these angles? Solution: Let the required angle be $x^{\circ}$. As the adjacent angles are equal, we have: $x+x=180 \quad\left(s\right.$ ince the sum of adjacent angles of a parallelogram is $\left.180^{\circ}\right)$ $\Rightarrow 2 x=180$ $\Rightarrow x=\frac{180}{2}$ $\Rightarrow x=90^{\circ}$ Hence, the measure of each of the angles is $90^{\circ}$....
Read More →-3/4 is smaller than
Question: $-3 / 4$ is smaller than $-2$. Solution: False. Express each of the given rational numbers with 4 as the common denominator. Now, $-3 / 4=[(-3 \times 1) /(4 \times 1)]=(-3 / 4)$ $-2 / 1=[(-2 \times 4) /(1 \times 4)]=(-8 / 4)$ Then, $-3 / 4-8 / 4$ Hence, $-3 / 4-2$ So, $-3 / 4$ is greater than $-2$....
Read More →ABCD is a parallelogram in which ∠A = 110°.
Question: ABCDis a parallelogram in which A= 110. Find the measure of each of the angles B,Cand D. Solution: It is given that $A B C D$ is a parallelogram in which $\angle A$ is equal to $110^{\circ}$. Sum of the adjacent angles of a parallelogram is $180^{\circ}$. $\therefore \angle A+\angle B=180^{\circ}$ $\Rightarrow 110^{\circ}+\angle B=180^{\circ}$ $\Rightarrow \angle B=\left(180^{\circ}-110^{\circ}\right)$ $\Rightarrow \angle B=70^{\circ}$ $\therefore \angle B=70^{\circ}$ Also, $\angle B+\...
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $-\sqrt{-1}$ Solution: Given: $z=-\sqrt{-1}$ The above can be re - written as $Z=-\sqrt{i^{2}}\left[\because \mathrm{i}^{2}=-1\right]$ $z=0-i$ So, the conjugate of $z=(0-i)$ is $\bar{z}=0+i$ Or $\bar{z}=i$...
Read More →Subtraction of rational
Question: Subtraction of rational number is commutative. Solution: False. Subtraction of rational number is not commutative. Let $x$ and $y$ are any two rational number, Then, $x-y \neq y-x$...
Read More →Rational numbers are closed
Question: Rational numbers are closed under addition and multiplication but not under subtraction. Solution: False. Rational numbers are closed under addition, subtraction and multiplication....
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following: $\sqrt{2}$ Solution: Given: $z=\sqrt{2}$ The above can be re - written as $z=\sqrt{2}+0 i$ Here, the imaginary part is zero So, the conjugate of $z=\sqrt{2}+0 i$ is $\bar{z}=\sqrt{2}-0 i$ Or $\bar{z}=\sqrt{2}$...
Read More →Between any two rational numbers
Question: Between any two rational numbers there are exactly ten rational numbers. Solution: False. Between any two rational numbers there are infinite rational numbers....
Read More →In the adjacent figure, the bisectors of ∠A and ∠B meet in a point P.
Question: In the adjacent figure, the bisectors of Aand Bmeet in a pointP. If C= 100 and D= 60, find the measure of APB. Solution: Sum of the angles of a quadrilateral is 360. $\therefore \angle A+\angle B+60^{\circ}+100^{\circ}=360^{\circ}$ $\angle A+\angle B=360-100-60=200^{\circ}$ or $\frac{1}{2}(\angle A+\angle B)=100^{\circ} \quad \ldots(1)$ Sum of the angles of a triangle is $180^{\circ}$. In $\triangle A P B$ : $\frac{1}{2}(\angle A+\angle B)+\angle P=180^{\circ} \quad$ (because $A P$ and...
Read More →If x and y are negative rational
Question: If $x$ and $y$ are negative rational numbers, then so is $x+y$. Solution: True. For example, Let $x=-1 / 3$ and $y=-2 / 3$ Then, $=x+y$ $=(-1 / 3)+(-2 / 3)$ $=-1 / 3-2 / 3$ $=-3 / 3$...
Read More →Find the conjugate of each of the following:
Question: Find the conjugate of each of the following $\sqrt{-3}$ Solution: Given: $z=\sqrt{-3}$ The above can be re written as $z=\sqrt{(-1) \times 3}$ $z=\sqrt{3 i^{2}}\left[\because \dot{\mathrm{i}}^{2}=-1\right]$ $z=0+\mathrm{i} \sqrt{3}$ So, the conjugate of $z=0+i \sqrt{3}$ is $\bar{z}=0-i \sqrt{3}$ Or $\bar{z}=-i \sqrt{3}=-\sqrt{-3}$...
Read More →For rational numbers x and y,
Question: For rational numbers $x$ and $y$, if $xy$ then $x-y$ is a positive rational number. Solution: False. Because, for rational numbers $x$ and $y$, if $xy$ then $x-y$ is a negative rational number. Example, let $x=2$ and $y=3$ Then, $=X-Y$ $=2-3$ $=-1$...
Read More →Two angles of a quadrilateral measure 85° and 75° respectively.
Question: Two angles of a quadrilateral measure 85 and 75 respectively. The other two angles are equal. Find the measure of each of these equal angles. Solution: Let the two unknown angles measure $x^{\circ}$ each. Sum of the angles of a quadrilateral is $360^{\circ}$. $\therefore 85+75+x+x=360$ $160+2 x=360$ $2 x=360-160=200$ $x=100$ Each of the equal angle measures $100^{\circ}$....
Read More →For any rational number
Question: For any rational number $x, x+(-1)=-x$. Solution: False. The correct form is for any rational number $x,(x) \times(-1)=-x$....
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when If $x=a\left(\frac{1+t^{2}}{1-t^{2}}\right)$ and $y=\frac{2 t}{1-t^{2}}$, find $\frac{d y}{d x}$ Solution: Here, $x=a\left(\frac{1+t^{2}}{1-t^{2}}\right)$ differentiating bove function with respect to $t$, we have, $\frac{d x}{d t}=a\left[\frac{\left(1-t^{2}\right) \frac{d\left(1+t^{2}\right)}{d t}-\left(1+t^{2}\right) \frac{d\left(1-t^{2}\right)}{d t}}{\left(1-t^{2}\right)^{2}}\right]$ $\frac{d x}{d t}=a\left[\frac{\left(1-t^{2}\right)(2 t)-\left(1+t^{2}\r...
Read More →Three angles of a quadrilateral are equal and the measure of the fourth angle is 120°.
Question: Three angles of a quadrilateral are equal and the measure of the fourth angle is 120. Find the measure of each of the equal angles. Solution: Three angles of a quadrilateral are equal and the measure of the fourth angle is 120. Find the measure of each of the equal angles. $\therefore x+x+x+120=360$ $3 x+120=360$ $3 x=240$ $x=\frac{240}{3}=80$ Each of the equal angles measure 80....
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