eSaral - JEE | NEET | Class 8-10 Exam Prep App

The angle between the altitudes of a parallelogram,

Question: The angle between the altitudes of a parallelogram, through the same vertex of an obtuse angle of the parallelogram is 60. Find the angles of the parallelogram. Solution: Draw a parallelogram $\mathrm{ABCD}$. Drop a perpendicular from $\mathrm{B}$ to the side $\mathrm{AD}$, at the point $\mathrm{E} .$ Drop perpendicular from $\mathrm{B}$ to the side $\mathrm{CD}$, at the point $\mathrm{F}$. In the quadrilateral $\mathrm{BEDF}:$ $\angle \mathrm{EBF}=60^{\circ}, \angle \mathrm{BED}=90^{\...

The volume of a hemisphere is 19404 cm3.

Question: The volume of a hemisphere is 19404 cm3. The total surface area of the hemisphere is(a) 4158 cm2(b) 16632 cm2(c) 8316 cm2(d) 3696 cm2 Solution: (a) 4158 cm2 Volume of hemisphere $=\frac{2}{3} \pi r^{3}$ Therefore, $\frac{2}{3} \pi r^{3}=19404$ $\Rightarrow \frac{2}{3} \times \frac{22}{7} \times r^{3}=19404$ $\Rightarrow r^{3}=\left(19404 \times \frac{21}{44}\right)$ $\Rightarrow r^{3}=(21)^{3}$ $\Rightarrow r=21 \mathrm{~cm}$ Hence, the total surface area of the hemisphere $=3 \pi r^{2...

The volume of a hemisphere is 19404 cm3.

Question: The volume of a hemisphere is 19404 cm3. The total surface area of the hemisphere is(a) 4158 cm2(b) 16632 cm2(c) 8316 cm2(d) 3696 cm2 Solution: (a) 4158 cm2 Volume of hemisphere $=\frac{2}{3} \pi r^{3}$ Therefore, $\frac{2}{3} \pi r^{3}=19404$ $\Rightarrow \frac{2}{3} \times \frac{22}{7} \times r^{3}=19404$ $\Rightarrow r^{3}=\left(19404 \times \frac{21}{44}\right)$ $\Rightarrow r^{3}=(21)^{3}$ $\Rightarrow r=21 \mathrm{~cm}$ Hence, the total surface area of the hemisphere $=3 \pi r^{2...

Find the angles marked with a question mark shown in Fig. 17.27

Question: Find the angles marked with a question mark shown in Fig. 17.27 Solution: In $\triangle \mathrm{CEB}:$ $\angle \mathrm{ECB}+\angle \mathrm{CBE}+\angle \mathrm{BEC}=180^{\circ} \quad$ (angle sum property of a triangle) $40^{\circ}+90^{\circ}+\angle \mathrm{EBC}=180^{\circ}$ $\therefore \angle \mathrm{EBC}=50^{\circ}$ Also, $\angle \mathrm{EBC}=\angle \mathrm{ADC}=50^{\circ}$ (opposite angle of a parallelogram) In $\triangle \mathrm{FDC}:$ $\angle \mathrm{FDC}+\angle \mathrm{DCF}+\angle ...

A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere.

Question: A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere. The radius of the sphere is(a) 2.1 cm(b) 1.05 cm(c) 1.5 cm(d) 2 cm Solution: (a) 2.1 cmRadius of cone = 2.1 cmHeight of cone = 8.4 cm Volume of cone $=\frac{1}{3} \pi \mathrm{r}^{2} \mathrm{~h}=\frac{1}{3} \pi(2.1)^{2} \times 8.4$ Volume of the sphere $=\frac{4}{3} \pi r^{3}$ Therefore,Volume of cone = Volume of sphere $\Rightarrow \frac{1}{3} \pi \times(2.1)^{2} \times 8.4=\frac{4}{3} \pi r^{3...

Two straight paths are represented

Question: Two straight paths are represented by the equations x-3y = 2 and 2x + 6y =5. Check whether the paths cross each other or not. Solution: Given linear equations are $x-3 y-2=0$ $\ldots$ (i) and $\quad-2 x+6 y-5=0 \quad \ldots$ (ii) On comparing both the equations with $a x+b y+c=0$, we get $a_{1}=1, b_{1}=-3$ and $\quad c_{1}=-2$[from Eq. (i)] $a_{2}=-2, b_{2}=6$ and $\quad c_{2}=-5 \quad$ [from Eq. (ii)] Here, $\frac{a_{1}}{a_{2}}=\frac{1}{-2}$ $\frac{b_{1}}{b_{2}}=\frac{-3}{6}=-\frac{1...

A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere.

Question: A metallic cone of base radius 2.1 cm and height 8.4 cm is melted and moulded into a sphere. The radius of the sphere is(a) 2.1 cm(b) 1.05 cm(c) 1.5 cm(d) 2 cm Solution: (a) 2.1 cmRadius of cone = 2.1 cmHeight of cone = 8.4 cm Volume of cone $=\frac{1}{3} \pi \mathrm{r}^{2} \mathrm{~h}=\frac{1}{3} \pi(2.1)^{2} \times 8.4$ Volume of the sphere $=\frac{4}{3} \pi r^{3}$ Therefore,Volume of cone = Volume of sphere $\Rightarrow \frac{1}{3} \pi \times(2.1)^{2} \times 8.4=\frac{4}{3} \pi r^{3...

In a parallelogram ABCD, the diagonals bisect each other at O.

Question: In a parallelogramABCD, the diagonals bisect each other atO. If ABC= 30, BDC= 10 and CAB= 70. Find:DAB, ADC, BCD, AOD, DOC, BOC, AOB, ACD, CAB, ADB, ACB, DBCand DBA. Solution: $\angle A B C=30^{\circ}$ $\therefore \angle A D C=30^{\circ}$ (opposite angle of the parallelogram) and $\angle B D A=\angle \mathrm{ADC}-\angle \mathrm{BDC}=30^{\circ}-10^{\circ}=20^{\circ}$ $\angle B A C=\angle A C D=70^{\circ}$ (alternate angle) In $\triangle \mathrm{ABC}:$ $\angle \mathrm{CAB}+\angle \mathrm...

A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone of base radius 8 cm.

Question: A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone of base radius 8 cm. The height of the cone is(a) 12 cm(b) 14 cm(c) 15 cm(d) 18 cm Solution: DISCLAIMER : The answer to the question does not match the options given.External diameter = 8 cmInternal diameter = 4 cmLet the external and internalradii of the hollow metallic sphere be R and r, respectively.Then, External radius $=\frac{8}{2}=4 \mathrm{~cm}$ Internal Radius $=\f...

A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone of base radius 8 cm.

Question: A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone of base radius 8 cm. The height of the cone is(a) 12 cm(b) 14 cm(c) 15 cm(d) 18 cm Solution: DISCLAIMER : The answer to the question does not match the options given.External diameter = 8 cmInternal diameter = 4 cmLet the external and internalradii of the hollow metallic sphere be R and r, respectively.Then, External radius $=\frac{8}{2}=4 \mathrm{~cm}$ Internal Radius $=\f...

Find the values of p in (i) to (iv) and

Question: Find the values of p in (i) to (iv) and p and q in (v) for the following pair of equations (i) 3x y 5 = 0 and 6x 2y p = 0, if the lines represented by these equations are parallel. (ii) x + py = 1 and px y = 1 if the pair of equations has no solution. (iii) 3x + 5y = 7 and 2px 3y= 1, if the lines represented by these equations are intersecting at a unique point. (iv) 2x + 3y 5 = 0 and px 6y 8 = 0, if the pair of equations has a unique solution. (v) 2x + 3y = 7 and 2px + py = 28 qy, if ...

The ratio between the volume of two spheres is 8 : 27.

Question: The ratio between the volume of two spheres is 8 : 27. What is the ratio between their surface areas?(a) 2 : 3(b) 4 : 5(c) 5 : 6(d) 4 : 9 Solution: (d) 4 : 9Let the radii of the spheres be R and r, respectively. Then, ratio of their volumes $=\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi \mathrm{r}^{3}}$ Therefore, $\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{8}{27}$ $\Rightarrow \frac{R^{3}}{r^{3}}=\frac{8}{27}$ $\Rightarrow\left(\frac{R}{r}\right)^{3}=\left(\frac{2}{3}\r...

Two adjacent angles of a parallelogram are (3x − 4)° and (3x + 10)°.

Question: Two adjacent angles of a parallelogram are (3x 4) and (3x+ 10). Find the angles of the parallelogram. Solution: We know that the adjacent angle $s$ of a parallelogram are supplementry. Hence, $(3 x+10)^{\circ}$ and $(3 x-4)^{\circ}$ are supplementry. $(3 x+10)^{\circ}+(3 x-4)^{\circ}=180^{\circ}$ $6 x^{\circ}+6^{\circ}=180^{\circ}$ $6 x^{\circ}=174^{\circ}$ $x=29^{\circ}$ First angle $=(3 \mathrm{x}+10)^{\circ}=\left(3 \times 29^{\circ}+10^{\circ}\right)=97^{\circ}$ Second angle $=(3 \...

The shorter side of a parallelogram is 4.8 cm and the longer side is half as much again as the shorter side.

Question: The shorter side of a parallelogram is 4.8 cm and the longer side is half as much again as the shorter side. Find the perimeter of the parallelogram. Solution: Given: Shorter side $=4.8 \mathrm{~cm}$ Longer side $=\frac{4.8}{2}+4.8=7.2 \mathrm{~cm}$ Perimeter $=$ Sum of all the sides $=4.8+4.8+7.2+7.2$ $=24 \mathrm{~cm}$...

The diameter of a sphere is 14 cm. Its volume is

Question: The diameter of a sphere is 14 cm. Its volume is(a) 1428 cm3(b) 1439 cm3 (c) $1437 \frac{1}{3} \mathrm{~cm}^{3}$ (d) 1440 cm3 Solution: (c) $1437 \frac{1}{3} \mathrm{~cm}^{3}$ Volume of the sphere $=\frac{4}{3} \pi r^{3}$ $=\left(\frac{4}{3} \times \frac{22}{7} \times 7 \times 7 \times 7\right) \mathrm{cm}^{3}\left[d=14 \mathrm{~cm} \Rightarrow r=\frac{14}{2} \mathrm{~cm}=7 \mathrm{~cm}\right]$ $=\frac{4312}{3} \mathrm{~cm}^{3}$ $=1437 \frac{1}{3} \mathrm{~cm}^{3}$...

The perimeter of a parallelogram is 150 cm.

Question: The perimeter of a parallelogram is 150 cm. One of its sides is greater than the other by 25 cm. Find the length of the sides of the parallelogram. Solution: Opposite sides of a parallelogram are same. Le $t$ two sides of the parallelogram be $x$ and $y$. Given : $x=y+25$ Also, $x+y+x+y=150 \quad($ Perimeter $=$ Sum of all the sides of a paralle $\log$ ram $)$ $y+25+y+y+25+y=150$ $4 y=150-50$ $4 y=100$ $y=\frac{100}{4}=25$ $\therefore x=y+25=25+25=50$ Thus, the length $s$ of the sides ...

Two adjacent sides of a parallelogram are 4 cm and 3 cm respectively.

Question: Two adjacent sides of a parallelogram are 4 cm and 3 cm respectively. Find its perimeter. Solution: We know that the opposite sides of a parallelogram are equal. Two sides are given, i.e. $4 \mathrm{~cm}$ and $3 \mathrm{~cm}$. Therefore, the rest of the sies will also be $4 \mathrm{~cm}$ and $3 \mathrm{~cm}$. $\therefore P$ erimeter $=S$ um of all the sides of $a$ parallelogram $=4+3+4+3$ $=14 \mathrm{~cm}$...

The height of a conical tent is 14 m and its floor area is 346.5 m2.

Question: The height of a conical tent is 14 m and its floor area is 346.5 m2. How much canvas, 1.1 wide, will be required for it?(a) 490 m(b) 525 m(c) 665 m(d) 860 m Solution: (b) 525 m Area of the floor of a conical tent $=\pi r^{2}$ Therefore, $\pi r^{2}=346.5$ $\Rightarrow \frac{22}{7} \times r^{2}=346.5$ $\Rightarrow r^{2}=\left(\frac{3465}{10} \times \frac{7}{22}\right)$ $\Rightarrow r^{2}=\frac{441}{4}$ $\Rightarrow r^{2}=\left(\frac{21}{2}\right)^{2}$ $\Rightarrow r=\frac{21}{2} \mathrm{...

The height of a conical tent is 14 m and its floor area is 346.5 m2.

Question: The height of a conical tent is 14 m and its floor area is 346.5 m2. How much canvas, 1.1 wide, will be required for it?(a) 490 m(b) 525 m(c) 665 m(d) 860 m Solution: (b) 525 m Area of the floor of a conical tent $=\pi r^{2}$ Therefore, $\pi r^{2}=346.5$ $\Rightarrow \frac{22}{7} \times r^{2}=346.5$ $\Rightarrow r^{2}=\left(\frac{3465}{10} \times \frac{7}{22}\right)$ $\Rightarrow r^{2}=\frac{441}{4}$ $\Rightarrow r^{2}=\left(\frac{21}{2}\right)^{2}$ $\Rightarrow r=\frac{21}{2} \mathrm{...

All the angles of a quadrilateral are equal to each other.

Question: All the angles of a quadrilateral are equal to each other. Find the measure of each. Is the quadrilateral a parallelogram? What special type of parallelogram is it? Solution: Let the angle be $\mathrm{x}$. $A$ ll the angles are equal. $\therefore x+x+x+x=360^{\circ}$ $4 x=360^{\circ}$ $x=90^{\circ}$ So, each angle is $90^{\circ}$ and quadrilateral is a parallelogram. It is a rectangle....

The sum of two opposite angles of a parallelogram is 130°.

Question: The sum of two opposite angles of a parallelogram is 130. Find all the angles of the parallelogram. Solution: Let the angles be A, B, C and D. It is given that the sum of two opposite angles is $130^{\circ}$. $\therefore \angle \mathrm{A}+\angle \mathrm{C}=130^{\circ}$ $\angle \mathrm{A}+\angle \mathrm{A}=130^{\circ}$ (opp o site angle $s$ of a parallelogram are same) $\angle \mathrm{A}=65^{\circ}$ and $\angle \mathrm{C}=65^{\circ}$ The s um of adjacent angles of a paralle $\log$ ram i...

A metallic cylinder of radius 8 cm and height 2 cm is melted and converted into a right circular cone of height 6 cm.

Question: A metallic cylinder of radius 8 cm and height 2 cm is melted and converted into a right circular cone of height 6 cm. The radius of the base of this cone is(a) 4 cm(b) 5 cm(c) 6 cm(d) 8 cm Solution: (d) 8 cmRadius of the cylinder = 8 cmHeight of the cylinder = 2 cmHeight of the cone = 6 cmVolume of the cylinder = Volume of the coneTherefore, $\pi \times 8 \times 8 \times 2=\frac{1}{3} \pi \times r^{2} \times 6$ $\Rightarrow 128=r^{2} \times 2$ $\Rightarrow r^{2}=\frac{128}{2}$ $\Righta...

A metallic cylinder of radius 8 cm and height 2 cm is melted and converted into a right circular cone of height 6 cm.

Question: A metallic cylinder of radius 8 cm and height 2 cm is melted and converted into a right circular cone of height 6 cm. The radius of the base of this cone is(a) 4 cm(b) 5 cm(c) 6 cm(d) 8 cm Solution: (d) 8 cmRadius of the cylinder = 8 cmHeight of the cylinder = 2 cmHeight of the cone = 6 cmVolume of the cylinder = Volume of the coneTherefore, $\pi \times 8 \times 8 \times 2=\frac{1}{3} \pi \times r^{2} \times 6$ $\Rightarrow 128=r^{2} \times 2$ $\Rightarrow r^{2}=\frac{128}{2}$ $\Righta...

ABCD is a parallelogram in which ∠A = 70°.

Question: ABCDis a parallelogram in which A=70. Compute B,CandD. Solution: Opposite angles of a parallelogram are equal. $\therefore \angle \mathrm{C}=70^{\circ}=\angle \mathrm{A} .$ $\angle \mathrm{B}=\angle \mathrm{D}$ Also, the sum of the adjacent angles of a parallelogram is $180^{\circ}$. $\therefore \angle \mathrm{A}+\angle \mathrm{B}=180^{\circ}$ $70^{\circ}+\angle \mathrm{B}=180^{\circ}$ $\angle \mathrm{B}=110^{\circ}$ $\therefore \angle \mathrm{B}=110^{\circ}, \angle \mathrm{C}=70^{\cir...

In a parallelogram ABCD, ∠D = 135°,

Question: In a parallelogramABCD, D= 135, determine the measure ofAandB. Solution: In $a$ parallelogram, opposite angles have the same value. $\therefore \angle \mathrm{D}=\angle \mathrm{B}$ $=135^{\circ}$ Also, $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{D}=360^{\circ}$ $\angle \mathrm{A}+\angle \mathrm{D}=180^{\circ}$ (opposite angles have the same value) $\angle \mathrm{A}=180^{\circ}-135^{\circ}=45^{\circ}$ $\angle \mathrm{A}=45^{\circ}$...

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