If angles A, B, C and D of the quadrilateral
Question: If angles $A, B, C$ and $D$ of the quadrilateral $A B C D$, taken in order are in the ratio $3: 7: 6: 4$, then $A B C D$ is a (a) rhombus (b) parallelogram (c) trapezium (d) kite Solution: (c)Given, ratio of angles of quadrilateral ABCD is 3 : 7 : 6 : 4. Let angles of quadrilateral ABCD be 3x, 7x, 6x and 4x, respectively. We know that, sum of all angles of a quadrilateral is 360. 3x + 7x + 6x + 4x = 360 = 20x = 360 = x=360/20 = 18 $\therefore \quad$ Angles of the quadrilateral are $\an...
Read More →Resolve each of the following quadratic trinomial into factor:
Question: Resolve each of the following quadratic trinomial into factor:2x2+ 5x+ 3 Solution: The given expression is $2 \mathrm{x}^{2}+5 \mathrm{x}+3$. (Coefficient of $x^{2}=2$, coefficient of $x=5$ and constant term =3) We will split the coefficient of $\mathrm{x}$ into two parts such that their sum is 5 and their product equals th product of the coefficient of $\mathrm{x}^{2}$ and the constant term, i.e., $2 \times 3=6$. Now, $2+3=5$ and $2 \times 3=6$ Replacing the middle term $5 \mathrm{x}$...
Read More →Let A = [aij] be a square matrix
Question: Let $A=\left[a_{i j}\right]$ be a square matrix of order 3 with $|A|=2$ and let $C=\left[c_{i j}\right]$, where $c_{i j}=$ cofactor of $a_{i j}$ in $A$. Then, $|C|=$__________ Solution: Given: $|A|=2$ Order of $A=3$ As we know, $|\operatorname{adj}(A)|=|A|^{n-1} \quad$ where $n$ is the order of $A$ Also, $\operatorname{adj}(A)=C^{T}$ $\Rightarrow|\operatorname{adj}(A)|=\left|C^{T}\right|$ $\Rightarrow|\operatorname{adj}(A)|=|C| \quad\left(\because\left|C^{T}\right|=|C|\right)$ $\Righta...
Read More →The sum of the radii of two circles is 7 cm, and the difference of their circumferences is 8 cm.
Question: The sum of the radii of two circles is 7 cm, and the difference of their circumferences is 8 cm. Find the circumference of the circles. Solution: Let the radii of the two circles ber1cm andr2cm.Now,Sum of the radii of the two circles = 7 cm $r_{1+} r_{2}=7 \quad \ldots(i)$ Difference of the circumferences of the two circles = 88 cm $\Rightarrow 2 \pi \mathrm{r}_{1}-2 \pi \mathrm{r}_{2}=8$ $\Rightarrow 2 \pi\left(\mathrm{r}_{1}-\mathrm{r}_{2}\right)=8$ $\Rightarrow\left(\mathrm{r}_{1}-\...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:(a+ 7)(a 10) + 16 Solution: (a + 7)(a 10) + 16 $=a^{2}-10 a+7 a-70+16$ $=a^{2}-3 a-54$ To factorise $a^{2}-3 a-54$, we will find two numbers $p$ and $q$ such that $p+q=-3$ and $p q=-54$. Now, $6+(-9)=-3$ and $6 \times(-9)=-54$ Splitting the middle term $-3 a$ in the given quadratic as $-9 a+6 a$, we get : $a^{2}-3 a-54=a^{2}-9 a+6 a-54$ $=\left(a^{2}-9 a\right)+(6 a-54)$ $=a(a-9)+6(a-9)$ $=(a+6)(a-9)$...
Read More →The quadrilateral formed by joining the
Question: The quadrilateral formed by joining the mid-points of the side of quadrilateral PQRS, taken in order, is a rhombus, if (a)PQRS is a rhombus (b)PQRS is a parallelogram (c)diagonals of PQRS are perpendicular (d)diagonals of PQRS are equal Solution: (d) Given, the quadrilateral $A B C D$ is a rhombus. So, sides $A B, B C, C D$ and $A D$ are equal. Now, in $\triangle P Q S$, we have $D$ and $C$ are the mid-points of $P Q$ and $P S$. So, $\quad D C=\frac{1}{2} Q S$ [by mid-point theorem] .....
Read More →If A = diag (2, 3, 4)
Question: If $A=\operatorname{diag}(2,3,4)$, then $\left|A^{2}\right|=$ Solution: Given: $A=\operatorname{diag}(2,3,4)$ $\left|A^{2}\right|=|A|^{2} \quad\left(\because\left|A^{n}\right|=|A|^{n}\right)$ $=(2 \times 3 \times 4)^{2}$ $=(24)^{2}$ $=576$ Hence, $\left|A^{2}\right|=\underline{576}$....
Read More →The quadrilateral formed by joining
Question: The quadrilateral formed by joining the mid-points of the sides of a quadrilateral $\mathrm{PQRS}$, taken in order, is a rectangle, if (a) PQRS is a rectangle (b) PQRS is a parallelogram (c) diagonals of PQRS are perpendicular (d) diagonals of $P Q R S$ are equal Solution: (c) Since, the quadrilateral $A B C D$ formed by joining the mid-points of quadrilateral $P Q R S$ is a rectangle. $A C=B D[$ since, diagonals of a rectangle are equal] $\Rightarrow P Q=Q R$ Thus, quadrilateral $P Q ...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:(a2 5a)2 36 Solution: $\left(a^{2}-5 a\right)^{2}-36$ $=\left(a^{2}-5 a\right)^{2}-6^{2}$ $=\left[\left(a^{2}-5 a\right)-6\right]\left[\left(a^{2}-5 a\right)+6\right]$ $=\left(a^{2}-5 a-6\right)\left(a^{2}-5 a+6\right)$ In order to factorise $a^{2}-5 a-6$, we will find two numbers $p$ and $q$ such that $p+q=-5$ and $p q=-6$ Now, $(-6)+1=-5$ and $(-6) \times 1=-6$ Splitting the middle term $-5$ in the given quadratic as $-6 a+a$, we g...
Read More →A B C D is a rhombus such that
Question: $A B C D$ is a rhombus such that $\angle A C B=40^{\circ}$, then $\angle A D B$ is (a) $40^{\circ}$ (b) $45^{\circ}$ (c) $50^{\circ}$ (d) $60^{\circ}$ Solution: (c) Given, $A B C D$ is a rhombus such that $\angle A C B=40^{\circ} \Rightarrow \angle O C B=40^{\circ}$ Since, $A D \| B C$ $\angle D A C=\angle B C A=40^{\circ}$ [alternate interior angles] Also, $\angle A O D=90^{\circ}$ [diagonals of a rhombus are perpendicular to each other] We know that, sum of all angles of a triangle $...
Read More →In the above question,
Question: In the above question, $a_{11} C_{21}+a_{12} C_{22}+a_{13} C_{23}=$_______ Solution: Given:|A| = 5 As we know,Sum of products of elements of row (or column) with their corresponding cofactors = Value of the determinantandSum of products of elements of row (or column) with the cofactors of any other row (or column) = 0 Thus, $a_{11} C_{21}+a_{12} C_{22}+a_{13} C_{23}=0$ Hence, $a_{11} C_{21}+a_{12} C_{22}+a_{13} C_{23}=\underline{0}$....
Read More →The length of a chain used as the boundary of a semicircular park is 108 m.
Question: The length of a chain used as the boundary of a semicircular park is 108 m. Find the area of the park. Solution: Let the radius of the park ber.Length of chain = Perimeter of the semicircular park⇒ 108 = Length of the arc + Diameter $\Rightarrow 108=\frac{1}{2} \times 2 \pi r+2 r$ $\Rightarrow 108=r\left(\frac{22}{7}+2\right)$ $\Rightarrow 108=\frac{36}{7} r$ $\Rightarrow r=21 \mathrm{~m}$ Now, Area of park $=\frac{1}{2} \pi r^{2}=\frac{1}{2} \times \frac{22}{7} \times(21)^{2}=693 \mat...
Read More →Let A = [aij] be a 3 × 3 matrix such that
Question: Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix such that $|A|=5 .$ If $C_{i j}=$ Cofactor of $a_{i j}$ in $A .$ Then $a_{11} C_{11}+a_{12} C_{12}+a_{13} C_{13}=$__________ Solution: Given:|A| = 5 As we know,Sum of products of elements of row (or column) with their corresponding cofactors = Value of the determinantandSum of products of elements of row (or column) with the cofactors of any other row (or column) = 0 Thus, $a_{11} C_{11}+a_{12} C_{12}+a_{13} C_{13}=|A|=5$ Hence, $a_...
Read More →A diagonal of a rectangle is inclined to
Question: A diagonal of a rectangle is inclined to one side of the rectangle at $25^{\circ}$. The acute angle between the diagonals is (a) $55^{\circ}$ (b) $50^{\circ}$ (c) $40^{\circ}$ (d) $25^{\circ}$ Solution: (b) We know that, diagonals of a rectangle are equal in length. Alternate Method Given, in a rectangle $A B C D$,...
Read More →A wire when bent in the form of an equilateral triangle encloses an area of
Question: A wire when bent in the form of an equilateral triangle encloses an area of $121 \sqrt{3} \mathrm{~cm}^{2} .$ The same wire is bent to form a circle. Find the area enclosed by the circle. Solution: Area of an equilateral triangle $=\frac{\sqrt{3}}{4} \times(\text { Side })^{2}$ $\Rightarrow 121 \sqrt{3}=\frac{\sqrt{3}}{4} \times(\text { Side })^{2}$ $\Rightarrow 121 \times 4=(\text { Side })^{2}$ $\Rightarrow$ Side $=22 \mathrm{~cm}$ Perimeter of an equilateral triangle $=3 \times$ Sid...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:y2+ 5y 36 Solution: To factorise $y^{2}+5 y-36$, we will find two numbers $p$ and $q$ such that $p+q=5$ and $p q=-36$. Now, $9+(-4)=5$ and $9 \times(-4)=-36$ Splitting the middle term 5y in the given quadratic as $-4 y+9 y$, we get: $\mathrm{y}^{2}+5 \mathrm{y}-36=\mathrm{y}^{2}-4 \mathrm{y}+9 \mathrm{y}-36$ $=\left(y^{2}-4 y\right)+(9 y-36)$ $=y(y-4)+9(y-4)$ $=(y+9)(y-4)$...
Read More →A wire when bent in the form of an equilateral triangle encloses an area of
Question: A wire when bent in the form of an equilateral triangle encloses an area of $121 \sqrt{3} \mathrm{~cm}^{2} .$ The same wire is bent to form a circle. Find the area enclosed by the circle. Solution: Area of an equilateral triangle $=\frac{\sqrt{3}}{4} \times(\text { Side })^{2}$ $\Rightarrow 121 \sqrt{3}=\frac{\sqrt{3}}{4} \times(\text { Side })^{2}$ $\Rightarrow 121 \times 4=(\text { Side })^{2}$ $\Rightarrow$ Side $=22 \mathrm{~cm}$ Perimeter of an equilateral triangle $=3 \times$ Sid...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2 4x 21 Solution: To factorise $\mathrm{x}^{2}-4 \mathrm{x}-21$, we will find two numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=-4$ and $\mathrm{pq}=-21$. Now, $3+(-7)=-4$ and $3 \times(-7)=-21$ Splitting the middle term $-4 \mathrm{x}$ in the given quadratic as $-7 \mathrm{x}+3 \mathrm{x}$, we get: $\mathrm{x}^{2}-4 \mathrm{x}-21=\mathrm{x}^{2}-7 \mathrm{x}+3 \mathrm{x}-21$ $=\left(\mathrm{x}^{2}-7 \mathrm{...
Read More →Three angles of a quadrilateral
Question: Three angles of a quadrilateral are $75^{\circ}, 90^{\circ}$ and $75^{\circ}$, then the fourth angle is (a) $90^{\circ}$ (b) $95^{\circ}$ (c) $105^{\circ}$ (d) $120^{\circ}$ Solution: (d)Given, A = 75, B = 90 and C = 75 We know that, sum of all the1 angles of a quadrilateral is 360. A+ B + C + D = 360 = 75 + 90+ 75+ D = 360 D = 360 (75 + 90 + 75) = 360 -240 = 120 Hence, the fourth angle of a quadrilateral is 120....
Read More →If |A| denotes the value of the determinant
Question: If $|A|$ denotes the value of the determinant of a square matrix of order 3, then $|-2 \mathrm{~A}|=$ ____________ Solution: Given:A is a 3 3 matrix Now, $|-2 A|=(-2)^{3}|A| \quad(\because$ Order of $A$ is $3 \times 3)$ $=-8|A|$ Hence, $|-2 A|=-8|\underline{A}|$....
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2+ 14a+ 48 Solution: To factorise $\mathrm{a}^{2}+14 \mathrm{a}+48$, we will find two numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=14$ and pq $=48$. Now, $8+6=14$ and $8 \times 6=48$ Splitting the middle term 14a in the given quadratic as $8 a+6 a$, we get: $\mathrm{a}^{2}+14 \mathrm{a}+48=\mathrm{a}^{2}+8 \mathrm{a}+6 \mathrm{a}+48$ $=\left(\mathrm{a}^{2}+8 \mathrm{a}\right)+(6 \mathrm{a}+48)$ $=\mathrm{a}...
Read More →A copper wire when bent in the form of a square encloses an area of 484 cm2.
Question: A copper wire when bent in the form of a square encloses an area of 484 cm2. The same wire is not bent in the form of a circle. Find the area enclosed by the circle. Solution: Area of the circle = 484 cm2 Area of the square $=\mathrm{Side}^{2}$ $\Rightarrow 484=$ Side $^{2}$ $\Rightarrow 22^{2}=$ Side $^{2}$ $\Rightarrow$ Side $=22 \mathrm{~cm}$ Perimeter of the square $=4 \times$ Side Perimeter of the square $=4 \times 22$ = 88 cm Length of the wire = 88 cmCircumference of the circle ...
Read More →If I is the identity matrix of order 10,
Question: If $I$ is the identity matrix of order 10, then determinant of $I$ is________ Solution: Given:Order ofIis 10Det(I) = 1, whereIis the identity matrix of ordern.Hence,determinant ofIis1....
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:a2+ 2a 3 Solution: To factorise $\mathrm{a}^{2}+2 \mathrm{a}-3$, we will find two numbers $\mathrm{p}$ and $\mathrm{q}$ such that $\mathrm{p}+\mathrm{q}=2$ and $\mathrm{pq}=-3$. Now, $3+(-1)=2$ and $3 \times(-1)=-3$ Splitting the middle term $2 \mathrm{a}$ in the given quadratic as $-\mathrm{a}+3 \mathrm{a}$, we get: $a^{2}+2 a-3=a^{2}-a+3 a-3$ $=\left(a^{2}-a\right)+(3 a-3)$ $=a(a-1)+3(a-1)$ $=(a+3)(a-1)$...
Read More →Factorize each of the following algebraic expression:
Question: Factorize each of the following algebraic expression:x2 11x 42 Solution: To factorise $x^{2}-11 x-42$, we will find two numbers $p$ and $q$ such that $p+q=-11$ and $p q=-42$. Now, $3+(-14)=-22$ and $3 \times(-14)=42$ Splitting the middle term $-11 \mathrm{x}$ in the given quadratic as $-14 \mathrm{x}+3 \mathrm{x}$, we get: $\mathrm{x}^{2}-11 \mathrm{x}-42=\mathrm{x}^{2}-14 \mathrm{x}+3 \mathrm{x}-42$ $=\left(x^{2}-14 x\right)+(3 x-42)$ $=x(x-14)+3(x-14)$ $=(x+3)(x-14)$...
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