Solve the following system of equations by matrix method:
Question: Solve the following system of equations by matrix method: (i) $5 x+7 y+2=0$ $4 x+6 y+3=0$ (ii) $5 x+2 y=3$ $3 x+2 y=5$ (iii) $3 x+4 y-5=0$ $x-y+3=0$ (iv) $3 x+y=19$ $3 x-y=23$ (v) $3 x+7 y=4$ $x+2 y=-1$ (vi) $3 x+y=23$ $5 x+3 y=12$ Solution: (i) The given system of equations can be written in matrix form as folllows: $\left[\begin{array}{ll}5 7 \\ 4 6\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}-2 \\ -3\end{array}\right]$ $\mathrm{AX}=\mathrm{B...
Read More →A cube of side 6 cm is cut into a number of cubes, each of side 2 cm.
Question: A cube of side 6 cm is cut into a number of cubes, each of side 2 cm. The number of cubes formed is(a) 6(b) 9(c) 12(d) 27 Solution: (d) 27 Volume of the given cube $=(6 \times 6 \times 6) \mathrm{cm}^{3}$ Volume of each small cube $=(2 \times 2 \times 2) \mathrm{cm}^{3}$ Number of cubes formed $=\frac{\text { Volume of the given cube }}{\text { Volume of each small cube }}$ $=\left(\frac{6 \times 6 \times 6}{2 \times 2 \times 2}\right)$ $=27$...
Read More →Do the following pair of linear equations
Question: Do the following pair of linear equations have no solution? Justify your answer. (i) 2x + 4y = 3 and 12y + 6x = 6 (ii) x = 2y and y = 2x (iii) 3x + y 3 = 0 and 2x + y = 2 Solution: Condition for no solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ (i) Yes, given pair of equations, $2 x+4 y=3$ and $12 y+6 x=6$ Here,$a_{1}=2, b_{1}=4, c_{1}=-3$ $a_{2}=6, b_{2}=12, c_{2}=-6$ $\therefore$ $\frac{a_{1}}{a_{2}}=\frac{2}{6}=\frac{1}{3}, \frac{b_{1}}{b_{2}}=\frac{4}{1...
Read More →The four angles of a quadrilateral are as 3 : 5 : 7 : 9.
Question: The four angles of a quadrilateral are as 3 : 5 : 7 : 9. Find the angles. Solution: Let the angles be in the ratio $3 x: 5 x: 7 x: 9 x$. Since, the sum of all the angles of a quadrilateral is $360^{\circ}$, we have: $3 x+5 x+7 x+9 x=360^{\circ}$ $\Rightarrow 24 x=360^{\circ}$ $\Rightarrow x=15^{\circ}$ Thus, the angles are: $3 x=45^{\circ}$ $5 x=75^{\circ}$ $7 x=105^{\circ}$ $9 x=135^{\circ}$...
Read More →How many bags of grain can be stored in a cuboidal granary (8 m × 6 m × 3 m),
Question: How many bags of grain can be stored in a cuboidal granary (8 m 6 m 3 m), if each bag occupies a space of 0.64 m3?(a) 8256(b) 90(c) 212(d) 225 Solution: (d) 225 Volume of the cuboidal granary $=(8 \mathrm{~m} \times 6 \mathrm{~m} \times 3 \mathrm{~m})$ Volume of each bag $=0.64 \mathrm{~m}^{3}$ Number of bags that can be stored in the cuboidal granary $=\frac{\text { Volume of the cuboidal granary }}{\text { Volume of each bag }}$ $=\left(\frac{8 \times 6 \times 3}{0.64}\right)$ $=225$...
Read More →Three angles of a quadrilateral are equal.
Question: Three angles of a quadrilateral are equal. Fourth angle is of measure 150. What is the measure of equal angles. Solution: Let $x$ be the measure of the equal angle $s$ of the quadrilateral. Since, the sum of all the angles of a quadrilateral is $360^{\circ}$, we have: $x^{\circ}+x^{\circ}+x^{\circ}+150^{\circ}=360^{\circ}$ $\Rightarrow 3 x^{\circ}=360^{\circ}-150^{\circ}$ $\Rightarrow x^{\circ}=210^{\circ}$ $\therefore$ The measure of each angle is $70^{\circ}$...
Read More →Two angles of a quadrilateral are of measure 65° and the other two angles are equal.
Question: Two angles of a quadrilateral are of measure 65 and the other two angles are equal. What is the measure of each of these two angles? Solution: Let $x$ be the measure of each angle. Since, the sum of all the angles of a quadrilateral is $360^{\circ}$, we have: $65^{\circ}+65^{\circ}+x^{\circ}+x^{\circ}=360^{\circ}$ $\Rightarrow 2 x^{\circ}+130^{\circ}=360^{\circ}$ $\Rightarrow 2 x^{\circ}=230^{\circ}$ $\Rightarrow x^{\circ}=115^{\circ}$ $\therefore$ The measure of each angle is $115^{\c...
Read More →If each edge of a cube is increased by 50%, the percentage increase in the surface area is
Question: If each edge of a cube is increased by 50%, the percentage increase in the surface area is(a) 50%(b) 75%(c) 100%(d) 125% Solution: (d) 125%Let the original edge of the cube beaunits.Then, the original surface area of the cube = 6a2units New edge of the cube = 150% ofa $=\frac{150 a}{100}$ $=\frac{3 a}{2}$ Hence, new surface area $=6 \times\left(\frac{3 a}{2}\right)^{2}$ $=\frac{27 a^{2}}{2}$ Increase in area $=\left(\frac{27 a^{2}}{2}-6 a^{2}\right)$ $=\frac{15 a^{2}}{2}$ $\%$ increase...
Read More →A quadrilateral has all its four angles of the same measure.
Question: A quadrilateral has all its four angles of the same measure. What is the measure of each? Solution: Let $x$ be the measure of each angle. Since, the sum of all the angles of a quadrilateral is $360^{\circ}$, we have: $x^{\circ}+x^{\circ}+x^{\circ}+x^{\circ}=360^{\circ}$ $\Rightarrow 4 x^{\circ}=360^{\circ}$ $\Rightarrow x^{\circ}=90^{\circ}$ $\therefore$ The measure of each angle is $90^{\circ}$....
Read More →The father’s age is six times his son’s age.
Question: The fathers age is six times his sons age. Four years hence, the age of the father will be four times his sons age. The present ages (in year) of the son and the father are, respectively (a) 4 and 24 (b) 5 and 30 (c) 6 and 36 (d) 3 and 24 Solution: (c) Let x yr be the present age of father and y yr be the present age of son. Four years hence, it has relation by given condition, (x + 4) = 4(y + 4) ⇒ x-4y = 12 (i) and x = 6y (ii) On putting the value of x from Eq. (ii) in Eq. (i), we get...
Read More →A quadrilateral has three acute angles each measures 80°.
Question: A quadrilateral has three acute angles each measures 80. What is the measure of the fourth angle? Solution: Let $x$ be the fourth angle. Since, the sum of all angles of a quadrilateral is $360^{\circ}$, we have: $80^{\circ}+80^{\circ}+80^{\circ}+x=360^{\circ}$ $\Rightarrow 240^{\circ}+x=360^{\circ}$ $\Rightarrow x=120^{\circ}$ $\therefore$ The fourth angle is $120^{\circ}$....
Read More →The three angles of a quadrilateral are respectively equal to 110°, 50° and 40°.
Question: The three angles of a quadrilateral are respectively equal to 110, 50 and 40. Find its fourth angle. Solution: Let $x$ be the the fourth angle. Since, the sum of all the angles of a quadrilateral is $360^{\circ}$, we have: $110^{\circ}+50^{\circ}+40^{\circ}+x=360^{\circ}$ $\Rightarrow 200^{\circ}+x=360^{\circ}$ $\Rightarrow x=160^{\circ}$ $\therefore$ The fourth angle is $160^{\circ}$....
Read More →The angles of a quadrilateral are 110°, 72°, 55° and x°.
Question: The angles of a quadrilateral are 110, 72, 55 andx. Find the value ofx. Solution: The sum of anlges of a quadilateral is $360^{\circ}$. So, we get $110^{\circ}+72^{\circ}+55^{\circ}+x=360^{\circ}$ $\Rightarrow 237^{\circ}+x=360^{\circ}$ $\Rightarrow x=360^{\circ}-237^{\circ}$ $\therefore x=123^{\circ}$...
Read More →Aruna has only ₹ 1 and ₹ 2 coins with her.
Question: Aruna has only ₹ 1 and ₹ 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹ 75, then the number of ₹ 1 and ₹ 2 coins are, respectively (a) 35 and 15 (b) 35 and 20 (c) 15 and 35 (d) 25 and 25 Solution: (d)Let number of ₹ 1 coins = x and number of ₹ 2 coins = y Now, by given conditions x+y=50 (i) Also, x1+y2=75 ⇒ x + 2y = 75 (ii) On subtracting Eq. (i) from Eq. (ii), we get (x + 2y) (x + y) = 75 50 ⇒ y = 25 When y = 25, then x = 25...
Read More →If x = a and y = b is the solution
Question: If x = a and y = b is the solution of the equations x- y = 2 and x + y = 4, then the values of a and b are, respectively (a) 3 and 5 (b) 5 and 3 (c) 3 and 1 (d) 1 and 3 Solution: (c)Since, x = a and y = b is the solution of the equations x y = 2 and x+ y = 4, then these values will satisfy that equations a-b= 2 ,..(i) and a + b = 4 (ii) On adding Eqs. (i) and (ii), we get 2a = 6 a = 3 and b = 1...
Read More →In Fig. 16.19, ABCD is a quadrilateral.
Question: In Fig. 16.19,ABCDis a quadrilateral. (i) Name a pair of adjacent sides. (ii) Name a pair of opposite sides. (iii) How many pairs of adjacent sides are there? (iv) How many pairs of opposite sides are there? (v) Name a pair of adjacent angles. (vi) Name a pair of opposite angles. (vii) How many pairs of adjacent angles are there? (viii) How many pairs of opposite angles are there? Solution: (i) $(\mathrm{AB}, \mathrm{BC})$ or $(\mathrm{BC}, \mathrm{CD})$ or $(\mathrm{CD}, \mathrm{DA})$...
Read More →Complete each of the following, so as to make a true statement:
Question: Complete each of the following, so as to make a true statement: (i) A quadrilateral has ....... sides. (ii) A quadrilateral has ...... angles. (iii) A quadrilateral has ..... vertices, no three of which are ..... (iv) A quadrilateral has .... diagonals. (v) The number of pairs of adjacent angles of a quadrilateral is ....... (vi) The number of pairs of opposite angles of a quadrilateral is ....... (vii) The sum of the angles of a quadrilateral is ...... (viii) A diagonal of a quadrilat...
Read More →A pair of linear equations
Question: A pair of linear equations which has a unique solution x 2 and y = 3 is (a) x + y = 1 and 2x 3y = 5 (b) 2x+ 5y= -11 and 4x + 10y = -22 (c) 2x y = 1 and 3x + 2y = 0 (d) x 4y -14 = 0 and 5x y -13 = 0 Solution: (b)If x = 2, y = 3 is a unique solution of any pair of equation, then these values must satisfy that pair of equations. From option (b), LHS = 2x + 5y = 2(2) + 5(- 3) = 4 15 = 11 = RHS and LHS = 4x + 10y = 4(2) + 10(- 3)= 8 30 = 22 = RHS...
Read More →In a quadrilateral, define each of the following:
Question: In a quadrilateral, define each of the following: (i) Sides (ii) Vertices (iii) Angles (iv) Diagonals (v) Adjacent angles (vi) Adjacent sides (vii) Opposite sides (viii) Opposite angles (ix) Interior (x) Exterior Solution: (i) In a quadrilateral $\mathrm{ABCD}$, the four line segments $\mathrm{AB}, \mathrm{BC}, \mathrm{CD}$ and DA are called its sides. (ii) The vertices of a quadrilateral are the corner $s$ of the quadrilateral. A quadrilateral has four vertices. (iii) The meeting poin...
Read More →One equation of a pair of dependent
Question: One equation of a pair of dependent linear equations is 5x+ 7y 2 = 0. The second equation can be (a) 10x + 14y + 4=0 (b)-10x-14y + 4 =0 (c) -10x + 14y + 4 = 0 (d) 10x-14y + 4=0 Solution: (d)Condition for dependent linear equations $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}=\frac{1}{k}$...(i) Given equation of line is, $-5 x+7 y-2=0$ Here, $a_{1}=-5, b_{1}=7, c_{1}=-2$ From Eq. (i), $-\frac{5}{a_{2}}=\frac{7}{b_{2}}=-\frac{2}{c_{2}}=\frac{1}{k}$ [say] $\Rightarrow$ $a_...
Read More →Define the following terms:
Question: Define the following terms: (i) Quadrilateral (ii) Convex Quadrilateral Solution: (i) A quadrilateral is a polygon that has four sides (or edges) and four vertices (or corners). It can be any four-sided closed shape.(ii) A convex quadrilateral is a mathematical figure whose every internal angle is less than or equal to 180 degrees....
Read More →The sum of length, breadth and height of a cuboid is 19 cm and its diagonal is
Question: The sum of length, breadth and height of a cuboid is $19 \mathrm{~cm}$ and its diagonal is $5 \sqrt{5} \mathrm{~cm}$. Its surface area is (a) 361 cm2(b) 125 cm2(c) 236 cm2(d) 486 cm2 Solution: (c) 236 cm2Letl, bandhbe the length, breadth and height of the cuboid.Then, $l+b+h=19$ $\Rightarrow(l+b+h)^{2}=(19)^{2}$ Therefore, $\left(l^{2}+b^{2}+h^{2}\right)+2(l b+b h+l h)=361$ $\Rightarrow(5 \sqrt{5})^{2}+2(l b+b h+l h)=361$ $\Rightarrow 2(l b+b h+l h)=(361-125)$ $\Rightarrow 2(l b+b h+l ...
Read More →The value of c for which the
Question: The value of c for which the pair of equations cx- y = 2 and 6x 2y = 3will have infinitely many solutions is (a) 3 (b) 3 (c)-12 (d) no value Solution: (d) Condition for infinitely many solutions $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ $\ldots$ (i) The given lines are $c x-y=2$ and $6 x-2 y=3$ Here, $a_{1}=c_{1} b_{1}=-1, c_{1}=-2$ and $a_{2}=6, b_{2}=-2, c_{2}=-3$ From Eq. (i), $\frac{c}{6}=\frac{-1}{-2}=\frac{-2}{-3}$ Here, $\frac{c}{6}=\frac{1}{2}$ and $\frac{c}...
Read More →If the areas of three adjacent faces of a cuboid are x, y and z, respectively, the volume of the cuboid is
Question: If the areas of three adjacent faces of a cuboid arex,yandz, respectively, the volume of the cuboid is(a)xyz(b) 2xyz (c) $\sqrt{x y z}$ (d) $3 \sqrt{x y z}$ Solution: (c) $\sqrt{x y z}$ Let the length of the cuboid =lbreadth of the cuboid =band height of the cuboid =hSince, the areas of the three adjacent faces arex, yandz, we have: $l b=x$ $b h=y$ $l h=z$ Therefore, $l b \times b h \times l h=x y z$ $\Rightarrow l^{2} b^{2} h^{2}=x y z$ $\Rightarrow l b h=\sqrt{x y z}$ Hence, the volu...
Read More →What is a regular polygon?
Question: What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides Solution: A polygon that has equal sides and equal angles is called a regular polygon.(i) Equilateral triangle:(ii) Square:(iii) Regular hexagon:...
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