A horse is placed for grazing inside a rectangular field 70 m by 52 m.
Question: A horse is placed for grazing inside a rectangular field 70 m by 52 m. It is tethered to one corner by a rope 21 m long. On how much area can it graze? How much area is left ungrazed? Solution: Radius of the quadrant of the circle = 21 mThe shaded portion shows the part of the field the horse can graze. Area of the grazed field = Area of the quadrant OPQ $=\frac{1}{4}$ of the circle having radius OP $=\frac{1}{4} \pi \mathrm{r}^{2}$ $=\frac{1}{4} \times \frac{22}{7} \times 21 \times 21...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:y2 7y+ 12 Solution: $y^{2}-7 y+12$ $=y^{2}-7 y+\left(\frac{7}{2}\right)^{2}-\left(\frac{7}{2}\right)^{2}+12 \quad\left[\right.$ Adding and subtracting $\left.\left(\frac{7}{2}\right)^{2}\right]$ $=\left(y-\frac{7}{2}\right)^{2}-\frac{49}{4}+\frac{48}{4} \quad[$ Completing the square $]$ $=\left(y-\frac{7}{2}\right)^{2}-\frac{1}{4}$ $=\left(y-\frac{7}{2}\right)^{2}-\left(\frac{1}{2}\right)...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:4x2 12x+ 5 Solution: $4 x^{2}-12 x+5$ $=4\left(x^{2}-3 x+\frac{5}{4}\right) \quad\left[\right.$ Making the coefficient of $\left.x^{2}=1\right]$ $=4\left[x^{2}-3 x+\left(\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}+\frac{5}{4}\right] \quad\left[\right.$ Adding and subtracting $\left.\left(\frac{3}{2}\right)^{2}\right]$ $=4\left[\left(x-\frac{3}{2}\right)^{2}-\frac{9}{4}+\frac{5}{4}...
Read More →ABCD is a rhombus in which altitude
Question: ABCD is a rhombus in which altitude from D to side AB bisects AB. Find the angles of the rhombus. Solution: Let sides of a rhombus be $\quad A B=B C=C D=D A=x$ Now, join DB. In $\triangle A L D$ and $\triangle B L D, \quad \angle D L A=\angle D L B=90^{\circ}$ [since, $D L$ is a perpendicular bisector of $A B$ ] $A L=B L=\frac{x}{2}$ and $\quad D L=D L \quad$ [common side] $\therefore \quad \Delta A L D \cong \triangle B L D \quad$ [by SAS congruence rule] $A D=B D$ [by CPCT] Now, in $...
Read More →A rope by which a cow is tethered is increased from 16 m to 23 m.
Question: A rope by which a cow is tethered is increased from 16 m to 23 m. How much additional ground does it have now graze? Solution: r1= 16 mr2= 23 m Amount of additional ground available $=$ Area of the bigger circle $-$ Area of the smaller circle $=\pi\left(r_{1}^{2}-r_{2}^{2}\right)$ $=\pi\left(23^{2}-16^{2}\right)$ $=\pi(23+16)(23-16)$ $=858 \mathrm{~m}^{2}$...
Read More →A rope by which a cow is tethered is increased from 16 m to 23 m.
Question: A rope by which a cow is tethered is increased from 16 m to 23 m. How much additional ground does it have now graze? Solution: r1= 16 mr2= 23 m Amount of additional ground available $=$ Area of the bigger circle $-$ Area of the smaller circle $=\pi\left(r_{1}^{2}-r_{2}^{2}\right)$ $=\pi\left(23^{2}-16^{2}\right)$ $=\pi(23+16)(23-16)$ $=858 \mathrm{~m}^{2}$...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:a2+ 2a 3 Solution: $a^{2}+2 a-3$ $=a^{2}+2 a+\left(\frac{2}{2}\right)^{2}-\left(\frac{2}{2}\right)^{2}-3 \quad\left[\right.$ Adding and subtracting $\left(\frac{2}{2}\right)^{2}$, that is, $\left.1^{2}\right]$ $=a^{2}+2 a+1^{2}-1^{2}-3$ $=(a+1)^{2}-4 \quad[$ Completing the square $]$ $=(a+1)^{2}-2^{2}$ $=[(a+1)-2][(a+1)+2]$ $=(a+1-2)(a+1+2)$ $=(a-1)(a+3)$...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:a2 14a 51 Solution: $a^{2}-14 a-51$ $=a^{2}-14 a+\left(\frac{14}{2}\right)^{2}-\left(\frac{14}{2}\right)^{2}-51 \quad\left[\right.$ Adding and subtracting $\left(\frac{14}{2}\right)^{2}$, that is, $\left.7^{2}\right]$ $=a^{2}-14 a+7^{2}-7^{2}-51$ $=(a-7)^{2}-100 \quad[$ Completing the square $]$ $=(a-7)^{2}-10^{2}$ $=[(a-7)-10][(a-7)+10]$ $=(a-7-10)(a-7+10)$ $=(a-17)(a+3)$...
Read More →If the determinant
Question: If the determinant $\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ splits into exactly $k$ determinants of order 3 , each element of which contains only one term, then $k=$____________ Solution: Let $\Delta=\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ $=\left|\begin{array}{ccc}x+a p+u l \\ y+b q+v m \\ z+c r+w n\end{a...
Read More →The angle between two altitudes
Question: The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60. Find the angles of the parallelogram. Solution: Let the parallelogram be ABCD, in which ADC and ABC are obtuse angles. Now, DE and DF are two altitudes of parallelogram and angle between them is 60. Now, $B E D F$ is a quadrilateral, in which $\angle B E D=\angle B F D=90^{\circ}$ $\therefore$$\angle F B E=360^{\circ}-(\angle F D E+\angle B E D+\angle B F D)$ $=360^{\cir...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:x2+ 12x+ 20 Solution: $x^{2}+12 x+20$ $=x^{2}+12 x+\left(\frac{12}{2}\right)^{2}-\left(\frac{12}{2}\right)^{2}+20 \quad\left[\right.$ Adding and subtracting $\left(\frac{12}{2}\right)^{2}$, that is, $\left.6^{2}\right]$ $=x^{2}+12 x+6^{2}-6^{2}+20$ $=(x+6)^{2}-16 \quad[$ Completing the square $]$ $=(x+6)^{2}-4^{2}$ $=[(x+6)-4][(x+6)+4]$ $=(x+6-4)(x+6+4)$ $=(x+2)(x+10)$...
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:p2+ 6p 16 Solution: $\mathrm{p}^{2}+6 \mathrm{p}-16$ $=\mathrm{p}^{2}+6 \mathrm{p}+\left(\frac{6}{2}\right)^{2}-\left(\frac{6}{2}\right)^{2}-16 \quad\left[\right.$ Adding and subtracting $\left(\frac{6}{2}\right)^{2}$, that is, $\left.3^{2}\right]$ $=\mathrm{p}^{2}+6 \mathrm{p}+3^{2}-9-16$ $=(\mathrm{p}+3)^{2}-25 \quad[$ Completing the square $]$ $=(\mathrm{p}+3)^{2}-5^{2}$ $=[(\mathrm{p}...
Read More →If the determinant
Question: If the determinant $\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ splits into exactly $k$ determinants of order 3 , each element of which contains only one term, then $k=$____________ Solution: Let $\Delta=\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ $\Delta=\left|\begin{array}{ccc}x+a p+u l+f \\ y+b q+v m+g \\ z+c r+w n+h\end{array}\right|$ $=\left|\begin{array}{ccc}x+a p+u l \\ y+b q+v m \\ z+c r+w n\end{a...
Read More →The value of the determinant
Question: The value of the determinant $\left|\begin{array}{lll}\sin A \cos A \sin A+\cos B \\ \sin B \cos A \sin B+\cos B \\ \sin C \cos A \sin C+\cos B\end{array}\right|$ is__________ Solution: Let $\Delta=\left|\begin{array}{ccc}\sin A \cos A \sin A+\cos B \\ \sin B \cos A \sin B+\cos B \\ \sin C \cos A \sin C+\cos B\end{array}\right|$ $\Delta=\left|\begin{array}{lll}\sin A \cos A \sin A+\cos B \\ \sin B \cos A \sin B+\cos B \\ \sin C \cos A \sin C+\cos B\end{array}\right|$ $=\left|\begin{arr...
Read More →Find the area of a quadrant of a circle whose circumference is 88 cm.
Question: Find the area of a quadrant of a circle whose circumference is 88 cm. Solution: Let the radius of the circle ber.Now, Circumference $=88$ $\Rightarrow 2 \pi r=88$ $\Rightarrow r=14 \mathrm{~cm}$ Now, Area of quadrant $=\frac{1}{4} \pi r^{2}=\frac{1}{4} \times \frac{22}{7} \times(14)^{2}=154 \mathrm{~cm}^{2}$ Hence, the area of the quadrant of the circle is 154 cm2....
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:4y2+ 12y+ 5 Solution: $4 y^{2}+12 y+5$ $=4\left(y^{2}+3 y+\frac{5}{4}\right) \quad\left[\right.$ Making the coefficient of $\left.y^{2}=1\right]$ $=4\left[y^{2}+3 y+\left(\frac{3}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}+\frac{5}{4}\right] \quad\left[\right.$ Adding and subtracting $\left.\left(\frac{3}{2}\right)^{2}\right]$ $=4\left[\left(y+\frac{3}{2}\right)^{2}-\frac{9}{4}+\frac{5}{4...
Read More →Solve this
Question: If $\left|\begin{array}{lll}x+1 x+2 x+a \\ x+2 x+3 x+b \\ x+3 x+4 x+c\end{array}\right|=0$, then $a, b, c$ are in___________ Solution: Given: $\left|\begin{array}{lll}x+1 x+2 x+a \\ x+2 x+3 x+b \\ x+3 x+4 x+c\end{array}\right|=0$ $\left|\begin{array}{lll}x+1 x+2 x+a \\ x+2 x+3 x+b \\ x+3 x+4 x+c\end{array}\right|=0$ Applying $C_{1} \rightarrow C_{1}-C_{2}$ $\Rightarrow\left|\begin{array}{lll}x+1-x-2 x+2 x+a \\ x+2-x-3 x+3 x+b \\ x+3-x-4 x+4 x+c\end{array}\right|=0$ $\Rightarrow\left|\b...
Read More →ABCD is a trapezium in which AB || DC
Question: ABCD is a trapezium in which AB || DC and A = B = 45. Find angles C and D of the trapezium. Solution: Given, ABCD is a trapezium and whose parallel sides in the figure are AB and DC. Since, AB || CD and BC is transversal, then sum of two cointerior angles is 180....
Read More →If A and B are square matrices
Question: If $A$ and $B$ are square matrices of order 3 and $|A|=5,|B|=5$, then $|3 A B|=$________ Solution: Given:AandBare square matrices of order 3|A| = 5|B| = 5 Now, $|3 A B|=|3 A||B| \quad(\because|A B|=|A||B|$, if they are square matrices of same order $)$ $=|3 A| \times 5 \quad(\because|B|=5)$ $=3^{3}|A| \times 5 \quad(\because$ Order of $A$ is $3 \times 3)$ $=135|A|$ $=135 \times(5) \quad(\because|A|=5)$ $=675$ Hence, $|3 A B|=\underline{675}$....
Read More →Factorize each of the following quadratic polynomials by using the method of completing the square:
Question: Factorize each of the following quadratic polynomials by using the method of completing the square:p2 10q+ 21 Solution: $p^{2}-10 q+21$ $=\mathrm{q}^{2}-10 \mathrm{q}+\left(\frac{10}{2}\right)^{2}-\left(\frac{10}{2}\right)^{2}+21 \quad$ Adding and subtracting $\left(\frac{10}{2}\right)^{2}$, that is, $\left.5^{2}\right]$ $=\mathrm{q}^{2}-2 \times \mathrm{q} \times 5+5^{2}-5^{2}+21$ $=(\mathrm{q}-5)^{2}-4 \quad[$ Completing the square $]$ $=(q-5)^{2}-2^{2}$ $=[(q-5)-2][(q-5)+2]$ $=(q-5-...
Read More →The short and long hands of a clock are 4 cm and 6 cm long respectively.
Question: The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 2 days. Solution: In 2 days, the short hand will complete 4 rounds.Length of the short hand = 4 cm Distance covered by the short hand $=4 \times 2 \pi \times 4=32 \pi \mathrm{cm}$ In the same 2 days, the long hand will complete 48 rounds.Length of the long hand = 6 cm Distance covered by the long hand $=48 \times 2 \pi \times 6=576 \pi \mathrm{cm}$ Total distanc...
Read More →One angle of a quadrilateral is of 108°
Question: One angle of a quadrilateral is of 108 and the remaining three angles are equal. Find each of the three equal angles. Thinking Process The sum of all the angles In a quadrilateral is 360, use this result and simplify it. Solution: Let each of the three equal angles be x. Now, sum of angles of a quadrilateral = 360 = 108 + x + x + x = 360 = 3x = 360 108 x = 252/3 = x = 84 x = 84 Hence, each of the three equal angles is 84....
Read More →The short and long hands of a clock are 4 cm and 6 cm long respectively.
Question: The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 2 days. Solution: In 2 days, the short hand will complete 4 rounds.Length of the short hand = 4 cm Distance covered by the short hand $=4 \times 2 \pi \times 4=32 \pi \mathrm{cm}$ In the same 2 days, the long hand will complete 48 rounds.Length of the long hand = 6 cm Distance covered by the long hand $=48 \times 2 \pi \times 6=576 \pi \mathrm{cm}$ Total distanc...
Read More →Opposite angles of a quadrilateral
Question: Opposite angles of a quadrilateral ABCD are equal. If AB = 4 cm, determine CD. Solution: Given, opposite angles of a quadrilateral are equal. So, ABCD is a parallelogram and we know that, in a parallelogram opposite sides are also equal. CD = AB = 4cm...
Read More →Solve this
Question: $\left|\begin{array}{ccc}0 x y z x-z \\ y-x 0 y-z \\ z-x z-y 0\end{array}\right|=$_________ Solution: $\Delta=\left|\begin{array}{ccc}0 x y z x-z \\ y-x 0 y-z \\ z-x z-y 0\end{array}\right|$ Expanding along $R_{1}$, we get $=0(0-(y-x)(z-y))-(y-z)(0-(x-z)(z-y))+(z-x)(x y z(y-z)-0)$ $=0-(y-x)(-(x-z)(z-y))+(z-x)(x y z(y-z))$ $=(y-x)(x-z)(z-y)+(z-x) x y z(y-z)$ $=(y-z)(z-x)(y-x+x y z)$ Hence, $\left|\begin{array}{ccc}0 x y z x-z \\ y-x 0 y-z \\ z-x z-y 0\end{array}\right|=\underline{(y-z)(...
Read More →