Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Statement-1 (A): If volumes of two spheres are in the ratio $125: 64$, then their surface areas are in the ratio $25: 16$
Statement-2 $(R)$ : If volumes of two spheres are $V_1, V_2$ and their surface areas are $S_1, S_2$ respectively, then $\frac{S_1}{S_2}=\left(\frac{V_1}{V_2}\right)^{2 / 3}$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(a)
Let $V_1, V_2$ be volumes and $S_1, S_2$ be surface areas of two spheres of radii $r_1, r_2$ respectively. Then,$
\begin{array}{ll}
& V_1=\frac{4}{3} \pi r_1^3, V_2=\frac{4}{3} \pi r_2^3, S_1=4 \pi r_1^2 \text { and } S_2=4 \pi r_2{ }^2 \\
\therefore & \frac{V_1}{V_2}=\frac{r_1^3}{r_2{ }^3} \text { and } \frac{S_1}{S_2}=\frac{r_1^2}{r_2^2} \\
\Rightarrow & \left(\frac{V_1}{V_2}\right)^{1 / 3}=\left(\frac{r_1}{r_2}\right) \text { and }\left(\frac{S_1}{S_2}\right)^{1 / 2}=\frac{r_1}{r_2} \Rightarrow\left(\frac{S_1}{S_2}\right)^{1 / 2}=\left(\frac{V_1}{V_2}\right)^{1 / 3} \Rightarrow \frac{S_1}{S_2}\left(\frac{V_1}{V_2}\right)^{2 / 3}\end{array}$
Thus, statement-2 is true. Replacing $V_1: V_2$ by $125: 64$, we obtain$
\frac{S_1}{S_2}=\left(\frac{125}{64}\right)^{2 / 3}=\left(\frac{5^3}{4^3}\right)^{2 / 3}=\left(\frac{5}{4}\right)^2=\frac{25}{16}$
Thus, statement- 1 is also true and it is a direct consequence of statement-2. Hence, option (a) is correct.

View full question & answer→

MCQ 21 Mark

Statement-1 (A): $\Lambda$ conical bessel of base radius 5 cm and height 24 cm is full of water. If the turter is ampited intert eylimitrical nessel of internal radius 10 cm . then the water level rises by 2 cm .
Statement-2 (R): Volumes of a cylinder and a cone of hase radius $r$ and height $h$ are given bu $V_1=\pi r^2 h$ and $V_2=\frac{1}{3} \pi r^2 h$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(a)
We have,
$r_1=$ radius of the conical vessel $=5 cm, h=$ height of the conical vessel $=24 cm$ and, $r_2=$ radius of the cylindrical vessel $=10 cm$.
Suppose water level rises upto the height $h$ in the cylindrical vessel. Clearly, Volume of water in conical vessel = Volume of water in cylindrical vessel.
$\frac{1}{3} \pi \times 5^2 \times 24=\pi \times 10^2 \times h \Rightarrow h=2 cm$
Thus, statement -1 is true. Clearly, statement- 2 is also true and it is a correct explanation for statement-1.

View full question & answer→

MCQ 31 Mark

Statement-1 (A): If a cylinder, a cone and a hemisphere are of equal bases and have the same height, then the volume of the cylinder is equal to the sum of the volumes of cone and the hemisphere.
Statement-2 (R): If a cylinder, a cone and a hemisphere are of equal base and have the same height, then their volumes are in the ratio 3 : 1 : 2.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(a)
Let the equal radii of cylinder, cone and hemisphere be $r$. The cone, cylinder and hemisphere are of the same height. Therefore, cylinder and cone are of the same height equal to $r$. Let $V_1, V_2$ and $V_3$ denote the volume of cylinder, cone and hemisphere. Then,
$V_1=\pi r^3, V_2=\frac{1}{3} \pi r^3 \text { and } V_3=\frac{2}{3} \pi r^3 \Rightarrow V_1: V_2: V_3=1: \frac{1}{3}: \frac{2}{3}=3: 1: 2$
So, statement-2 is true. It is evident from (i) that $V_1=V_2+V_3$. So, statement-1 is also true. Also, statement-2 is a correct explanation for statement-1.

View full question & answer→

MCQ 41 Mark

Statement-1 (A): If the volumes of two cubes are in the ratio $V_1: V_2$, then their surface areas are in the ratio $V_1^{2 / 3}: V_2^{2 / 3}$.
Slatement-2 (R): If surface areas of two cubes are in the ratio $S_1: S_2$, then their volumes are in the ratio $S_1^{3 / 2}: S_2^{3 / 2}$.

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
✓
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: B.

Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.

(b)
Let the lengths of edges of two cubes be $a$ and $b$ units respectively. Then,$
\begin{aligned}
& S_1=6 a^2, S_2=6 b^2, V_1=a^3 \text { and } V_2=b^3 \\
\Rightarrow & \frac{S_1}{S_2}=\frac{a^2}{b^2} \text { and } \frac{V_1}{V_2}=\frac{a^3}{b^3} \\
\Rightarrow \quad & \left(\frac{S_1}{S_2}\right)^{1 / 2}=\frac{a}{b} \text { and }\left(\frac{V_1}{V_2}\right)^{1 / 3}=\frac{a}{b} \Rightarrow\left(\frac{S_1}{S_2}\right)^{1 / 2}=\left(\frac{V_1}{V_2}\right)^{1 / 3} \Rightarrow \frac{V_1}{V_2}=\frac{S_1^{3 / 2}}{S_2^{3 / 2}} \text { and } \frac{S_1}{S_2}=\frac{V_1^{2 / 3}}{V_2^{2 / 3}}
\end{aligned}$
Thus, if $S_1: S_2$ is known, then $V_1: V_2=S_1^{3 / 2}: S_2^{3 / 2}$. So, statement-2 is true.
If $V_1: V_2$ is known, then $S_1: S_2=V_1^{2 / 3}: V_2^{2 / 3}$. So, statement-1 is true.

View full question & answer→

MCQ 51 Mark

Statement-1 (A). If two metallic right circular cones of equal height and base radii 3 cm and 4 cm are melted recast into a solid sphere radius 5 cm, then the cones are of height 20 cm each.
Statement-2 (R): If two right circular cones of same height h and base radii $r_1, r_2$ are melted and recast into a solid sphere of radius r, then $h=\frac{4 r^2}{r_1+r_2}$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
✓
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: C.

Statement-1 is True, Statement-2 is False.

(c)
Clearly,
Sum of the volumes of two cones $=$ Volume of the sphere
$\Rightarrow \quad \frac{1}{3} \pi r_1{ }^2 h+\frac{1}{3} \pi r_2{ }^2 h=\frac{4}{3} \pi r^3 \Rightarrow h\left(r_1^2+r_2^2\right)=4 r^3 \Rightarrow h=\frac{4 r^3}{r_1^2+r_2^2}$
So, statement-2 is no true. Putting $r=5, r_1=3, r_2=4$ in (i), we obtain $h=20 cm$.
So, statement-1 is true.

View full question & answer→

MCQ 61 Mark

Statement-1 (A): Two cubes each of volume $125 cm^3$ are rejoined end to end to form a cuboid, the surface area of the resulting cuboid is $250 cm^2$
Statement-2 (R): If n cubes each of volume $a^3$ cubic units are joined end to end to form a cuboid. Then, the surface area of the resulting cuboid is $2(2 n+1) a^2$ square units.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(a)
Volume of each cube is $a^3$ cubic units. Therefore, length of an edge of each cube is a units. If $n$ cubes are joined end to end, then they form a cuboid of length $=n a$, breadth $=a$ and height $=a$. Let $S$ be the surface area of the cuboid formed. Then,
$S=2(n a \times a+a \times a+a \times n a)=2(2 n+1) a^2 \text { sq. units }$
So, statement- 2 is true. Replacing $n$ by 2 and a by 5 cm , we obtain $S=250 cm^2$.
So, statement-1 is also true and statement-2 is correct explanation for statement-1. Hence, option (a) is correct.

View full question & answer→

Maths Assertion (A) & Reason (B) MCQ

Test yourself on this topic