The radii of the base of a cylinder and a cone are in the ratio 3 : 4.
Question: The radii of the base of a cylinder and a cone are in the ratio 3 : 4. If their heights are in the ratio 2 : 3, the ratio between their volumes is(a) 9 : 8(b) 3 : 4(c) 8 : 9(d) 4 : 3 Solution: (a) 9 : 8Let the radii of the base of the cylinder and cone be 3rand 4rand their heights be 2hand 3h,respectively. Then, ratio of their volumes $=\frac{\pi(3 r)^{2} \times(2 h)}{\frac{1}{3} \pi(4 r)^{2} \times(3 h)}$ $=\frac{9 r^{2} \times 2 \times 3}{16 r^{2} \times 3}$ $=\frac{9}{8}$ $=9: 8$...
Read More →Two adjacent angles of a parallelogram are as 1 : 2.
Question: Two adjacent angles of a parallelogram are as 1 : 2. Find the measures of all the angles of the parallelogram. Solution: Let the angle be $\mathrm{A}$ and $\mathrm{B}$. The $a$ ngles are in the ratio of $1: 2$. $M$ easures of $\angle \mathrm{A}$ and $\angle \mathrm{B}$ are $\mathrm{x}^{\circ}$ and $2 \mathrm{x}^{\circ}$. Then, $\angle \mathrm{C}=\angle \mathrm{A}$ and $\angle \mathrm{D}=\angle \mathrm{B}(o$ pposite angles of $a$ parallelogram are congruent $)$ As we know that the sum o...
Read More →On increasing the radii of the base and the height of a cone by 20%, its volume will increase by
Question: On increasing the radii of the base and the height of a cone by 20%, its volume will increase by(a) 20%(b) 40%(c) 60%(d) 72.8% Solution: (d) 72.8%Let the original radius of the cone berand height beh. Then, original volume $=\frac{1}{3} \pi r^{2} h$ Let $\frac{1}{3} \pi r^{2} h=V$ New radius $=120 \%$ of $r$ $=\frac{120 r}{100}$ $=\frac{6 r}{5}$ New height = 120% ofh $=\frac{120 h}{100}$ $=\frac{6 h}{5}$ Hence, the new volume $=\frac{1}{3} \pi \times\left(\frac{6 \mathrm{r}}{5}\right)^...
Read More →The measure of one angle of a parallelogram is 70°.
Question: The measure of one angle of a parallelogram is 70. What are the measures of the remaining angles? Solution: Given that one angle of the parallelogram is $70^{\circ}$. Since opposite angles have same value, if one is $70^{\circ}$, then the one directly opposite will also be $70^{\circ}$. So, let one angle be $x^{\circ}$. $\mathrm{x}^{\circ}+70^{\circ}=180^{\circ}$ (the sum of adjacent angles of a parallelogram is $180^{\circ}$ ) $\mathrm{x}^{\circ}=180^{\circ}-70^{\circ}$ $\mathrm{x}^{\...
Read More →If an angle of a parallelogram is two-third of its adjacent angle,
Question: If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram. Solution: Two adjacent angles of a parallelogram add up to $180^{\circ}$. Let $x$ be the angle. $\therefore x+\frac{2 x}{3}=180^{\circ}$ $\frac{5 x}{3}==180^{\circ}$ $x=72^{\circ}$ $\frac{2 x}{3}=\frac{2 \times 72^{\circ}}{3}=108^{\circ}$ Thus, two of the angles in the parallelogram are $108^{\circ}$ and the other two are $72^{\circ}$....
Read More →Two opposite angles of a parallelogram are (3x − 2)° and (50 − x)°.
Question: Two opposite angles of a parallelogram are (3x 2) and (50 x). Find the measure of each angle of the parallelogram. Solution: $O$ ppostie angles of a parallelogram are congurent. $\therefore(3 x-2)^{\circ}=(50-x)^{\circ}$ $3 x^{\circ}-2^{\circ}=50^{\circ}-x^{\circ}$ $3 x^{\circ}+x^{\circ}=50^{\circ}+2^{\circ}$ $4 x^{\circ}=52^{\circ}$ $x^{\circ}=13^{\circ}$ Putting the value of $x$ in one angle : $3 x^{\circ}-2^{\circ}=39^{\circ}-2^{\circ}$ $=37^{\circ}$ $O$ pposite angles are congurent...
Read More →The area of the base of a right circular cone is 154 cm2 and its height is 14 cm.
Question: The area of the base of a right circular cone is 154 cm2and its height is 14 cm. Its curved surface area is (a) $154 \sqrt{5} \mathrm{~cm}^{2}$ (b) $154 \sqrt{7} \mathrm{~cm}^{2}$ (c) $77 \sqrt{7} \mathrm{~cm}^{2}$ (d) $77 \sqrt{5} \mathrm{~cm}^{2}$ Solution: (a) $154 \sqrt{5} \mathrm{~cm}^{2}$ Area of the base of the of a right circular cone $=\pi r^{2}$ Therefore, $\pi r^{2}=154$ $\Rightarrow \frac{22}{7} \times r^{2}=154$ $\Rightarrow r^{2}=\left(154 \times \frac{7}{22}\right)$ $\Ri...
Read More →For which values of a and b will the following
Question: For which values of a and b will the following pair of linear equations has infinitely many solutions? x + 2y = 1 (a -b)x+(a + b)y = a + b 2 Solution: Given pair of linear equations are x+2y=1 (a-b)x+(a+b)y=a+b-2 on comparing with ax+by+c=0,we get $a_{1}=1, b_{4}=2$ and $c_{4}=-1 \quad$ [from Eq. (i)] $a_{2}=(a-b), b_{2}=(a+b) \quad$ [from Eq. (ii)] and $\quad c_{2}=-(a+b-2)$ For infinitely many solutions of the the pairs of linear equations, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\f...
Read More →The area of the base of a right circular cone is 154 cm2 and its height is 14 cm.
Question: The area of the base of a right circular cone is 154 cm2and its height is 14 cm. Its curved surface area is (a) $154 \sqrt{5} \mathrm{~cm}^{2}$ (b) $154 \sqrt{7} \mathrm{~cm}^{2}$ (c) $77 \sqrt{7} \mathrm{~cm}^{2}$ (d) $77 \sqrt{5} \mathrm{~cm}^{2}$ Solution: (a) $154 \sqrt{5} \mathrm{~cm}^{2}$ Area of the base of the of a right circular cone $=\pi r^{2}$ Therefore, $\pi r^{2}=154$ $\Rightarrow \frac{22}{7} \times r^{2}=154$ $\Rightarrow r^{2}=\left(154 \times \frac{7}{22}\right)$ $\Ri...
Read More →In the following figure RISK and CLUE are parallelograms.
Question: In the following figureRISKandCLUEare parallelograms. Find the measure ofx. Solution: In the parallelogram RISK : $\angle \mathrm{ISK}+\angle \mathrm{RKS}=180^{\circ}$ (sum of adjacent angles of a parallelogram is $180^{\circ}$ ) $\angle \mathrm{ISK}=180^{\circ}-120^{\circ}=60^{\circ}$ Similarly, in parallelogram CLUE : $\angle \mathrm{CEU}=\angle \mathrm{CLU}=70^{\circ}$ (opposite angles of a parallelogram are equal) In the triangle : $x+\angle \mathrm{ISK}+\angle \mathrm{CEU}=180^{\c...
Read More →In the following figures GUNS and RUNS are parallelograms.
Question: In the following figuresGUNSandRUNSare parallelograms. Findxandy. Solution: (i) Opposite sides are equal in a parallelogram. $\therefore 3 \mathrm{y}-1=26$ $3 \mathrm{y}=27$ $\mathrm{y}=9$ Similarly, $3 \mathrm{x}=18$ x = 6 (ii) Diagonals bisect each other in a parallelogram. $\therefore \mathrm{y}-7=20$ $\mathrm{y}=27$ $\mathrm{x}-\mathrm{y}=16$ $\mathrm{x}-27=16$ $\mathrm{x}=43$...
Read More →The diameter of the base of a cone is 42 cm and its volume is 12936 cm3.
Question: The diameter of the base of a cone is 42 cm and its volume is 12936 cm3. Its height is(a) 28 cm(b) 21 cm(c) 35 cm(d) 14 cm Solution: (a) 28 cmLethbe the height of the cone.Diameter of the cone = 42 cmRadius of the cone = 21 cm Then, volume of the cone $=\frac{1}{3} \pi r^{2} h$ $=\frac{1}{3} \times \frac{22}{7} \times 21 \times 21 \times h$ $=22 \times 21 \times h$ Therefore, $22 \times 21 \times h=12936$ $\Rightarrow h=\left(\frac{12936}{22 \times 21}\right)$ $\Rightarrow h=28 \mathrm...
Read More →In the adjacent figure HOPE is a parallelogram.
Question: In the adjacent figureHOPEis a parallelogram. Find the angle measuresx,yandz. State the geometrical truths you use to find them. Solution: $\angle \mathrm{HOP}+70^{\circ}=180^{\circ} \quad(l$ inear pair $)$ $\angle \mathrm{HOP}=180^{\circ}-70^{\circ}=110^{\circ}$ $\mathrm{x}=\angle \mathrm{HOP}=110^{\circ}(o$ pposite angles of a parallelogram are equal $)$ $\angle \mathrm{EHP}+\angle \mathrm{HEP}=180^{\circ} \quad\left(s\right.$ um of adjacent angles of a parallelogram is $\left.180^{\...
Read More →For which value (s) of k will the pair of equations
Question: For which value (s) of k will the pair of equations kx+3y = k 3, 12 x + ky =k has no solution? Solution: Given pair of linear equations is kx + 3y = k 3 and 12x + ky = k On comparing with ax + by + c = 0, we get a1= k, b1= 3 and c1= -(k-3) a2= 12,b2= k and c2= -k For no solution of the pair of linear equations, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ $\Rightarrow \quad \frac{k}{12}=\frac{3}{k} \neq \frac{-(k-3)}{-k}$ Taking first two parts, we get $\Rightarro...
Read More →Can the following figures be parallelograms.
Question: Can the following figures be parallelograms. Justify your answer. Solution: (i) No. This is because the opposite angles are not equal. (ii) Yes. This is because the opposite sides are equal. (iii) No, This is because the diagonals do not bisect each other....
Read More →The radius of the base of a cone is 5 cm and its height is 12 cm. Its curved surface area is
Question: The radius of the base of a cone is 5 cm and its height is 12 cm. Its curved surface area is(a) 60 cm2(b) 65 cm2(c) 30 cm2(d) None of these Solution: (b) 65 cm2Given:r= 5 cm,h= 12 cm Slant height of the cone, $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{(5)^{2}+(12)^{2}}$ $=\sqrt{25+144}$ $=\sqrt{169}$ $=13 \mathrm{~cm}$ Hence, the curved surface area of the cone $=\pi r l$ $=(\pi \times 5 \times 13) \mathrm{cm}^{2}$ $=65 \pi \mathrm{cm}^{2}$...
Read More →The radius of the base of a cone is 5 cm and its height is 12 cm. Its curved surface area is
Question: The radius of the base of a cone is 5 cm and its height is 12 cm. Its curved surface area is(a) 60 cm2(b) 65 cm2(c) 30 cm2(d) None of these Solution: (b) 65 cm2Given:r= 5 cm,h= 12 cm Slant height of the cone, $l=\sqrt{r^{2}+h^{2}}$ $=\sqrt{(5)^{2}+(12)^{2}}$ $=\sqrt{25+144}$ $=\sqrt{169}$ $=13 \mathrm{~cm}$ Hence, the curved surface area of the cone $=\pi r l$ $=(\pi \times 5 \times 13) \mathrm{cm}^{2}$ $=65 \pi \mathrm{cm}^{2}$...
Read More →For which value(s) of λ,
Question: For which value(s) of , do the pair of linear equations x + y =2and x + y = 1 have (i) no solution? (ii) infinitely many solutions? (iii) a unique solution? Solution: The given pair of linear equations is x + y = 2and x + y = 1 a1= , b1= 1, c1= 2 a2=1, b2=c2=-1 (i) For no solution, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ $\Rightarrow$$\frac{\lambda}{1}=\frac{1}{\lambda} \neq \frac{-\lambda^{2}}{-1}$ $\Rightarrow \quad \lambda^{2}-1=0$ $\Rightarrow \quad(\lamb...
Read More →The heights of two circular cylinders of equal volume are in the ratio 1 : 2.
Question: The heights of two circular cylinders of equal volume are in the ratio 1 : 2. The ratio of their radii is (a) $1: \sqrt{2}$ (b) $\sqrt{2}: 1$ (c) 1 : 2(d) 1 : 4 Solution: (b) $\sqrt{2}: 1$ Let the radii of the two cylinders berandRand their heights behand 2h, respectively.Since the volumes of the cylinders are equal, therefore: $\pi \times r^{2} \times h=\pi \times R^{2} \times 2 h$ $\Rightarrow \frac{r^{2}}{R^{2}}=\frac{2}{1}$ $\Rightarrow\left(\frac{r}{R}\right)^{2}=\frac{2}{1}$ $\Ri...
Read More →PQRSTU is a regular hexagon.
Question: PQRSTUis a regular hexagon. Determine each angle of ΔPQT. Solution: A regular hexagon is made up of 6 equilateral triangles. So, $\angle \mathrm{PQT}=60^{\circ}$ and $\angle \mathrm{QTP}=30^{\circ}$ Since the sum of the angles of $\triangle \mathrm{PQT}$ is $180^{\circ}$, we have: $\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{T}=180^{\circ}$ $\Rightarrow \angle \mathrm{P}+60^{\circ}+30^{\circ}=180^{\circ}$ $\Rightarrow \angle \mathrm{P}=180^{\circ}-90^{\circ}$ $\angle \mathrm{P}+...
Read More →The line represented by x = 7
Question: The line represented by x = 7 is parallel to the X-axis, justify whether the statement is true or not. Solution: Not true, by graphically, we observe that x = 7 line is parallel to y-axis and perpendicular to X-axis....
Read More →The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3.
Question: The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. The ratio of their volumes is(a) 27 : 20(b) 20 : 27(c) 4 : 9(d) 9 : 4 Solution: (b) 20 : 27Let the radii of the two cylinders be 2rand 3rand their heights be 5hand 3h, respectively. Then, ratio of their volumes $=\frac{\pi \times(2 \mathrm{r})^{2} \times 5 h}{\pi \times(3 r)^{2} \times 3 h}$ $=\frac{4 r^{2} \times 5}{9 r^{2} \times 3}$ $=\frac{20}{27}$ $=20: 27$...
Read More →For all real values of c,
Question: For all real values of c, the pair of equations x 2y = 8 and 5x lOy = c have a unique solution. Justify whether it is true or false. Solution: False, the given pair of linear equations x-2y-8=0 5x-10y=c Here, $a_{1}=1, b_{1}=-2, c_{1}=-8$ $a_{2}=5, b_{2}=-10, c_{2}=-c$ Now, $\frac{a_{1}}{a_{2}}=\frac{1}{5}, \frac{b_{1}}{b_{2}}=\frac{-2}{-10}=\frac{1}{5}$ $\frac{c_{1}}{c_{2}}=\frac{-8}{-c}=\frac{8}{c}$ But if $c=40$ (real value), then the ratio $\frac{c_{1}}{c_{2}}$ becomes $\frac{1}{5}...
Read More →Determine the number of sides of a polygon whose exterior
Question: Determine the number of sides of a polygon whose exterior and interior angles are in the ratio 1 : 5. Solution: Let $n$ be the number of sides of a polygon. Let $x$ and $5 x$ be the exterior and interior angles. Since the sum of an interior and the corresponding exterior angle is $180^{\circ}$, we have: $x+5 x=180^{\circ}$ $\Rightarrow 6 x=180^{\circ}$ $\Rightarrow x=30^{\circ}$ The polygon has $n$ sides. So, sum of all the exterior angles $=(30 \mathrm{n})^{\circ}$ We know that the su...
Read More →The ratio between the radius of the base and the height of a cylinder is 2 : 3.
Question: The ratio between the radius of the base and the height of a cylinder is 2 : 3. If its volume is 1617 cm3, the total surface area of the cylinder is(a) 308 cm2(b) 462 cm2(c) 540 cm2(d) 770 cm2 Solution: (d) 770 cm2Let the common multiple be x.Let the radius of the cylinder be 2xcm and its height be 3xcm. Then, volume of the cylinder $=\pi r^{2} h$ $=\frac{22}{7} \times(2 x)^{2} \times 3 x$ Therefore, $\frac{22}{7} \times(2 x)^{2} \times 3 x=1617$ $\Rightarrow \frac{22}{7} \times 4 x^{2...
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