2 Marks Questions

Question 12 Marks

The radius and slant height of a cone are in the ratio $4: 7$. If its curved surface area is $792 \ cm^2$, find its radius. $($Use $\left.\pi=\frac{22}{7}\right)$.

Answer

Let the radius of cone $(r)=4 x \ cm$ and the slant height of the cone $(l)=7 x \ cm$
Curved surface area of cone $=\pi r l$
$\therefore \pi r l=792 \ cm^2$
$\Rightarrow \frac{22}{7} \times 4 \times \times 7 x =792$
$\Rightarrow x ^2=\frac{792}{22 \times 4}=9$
$\Rightarrow x=3 \ cm$
$\therefore$ Radius of the cone $=4 \times 3=12 \ cm$

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Question 22 Marks

The largest sphere is carved out of a solid cube of side $21 \ cm$. Find the volume of the sphere.

Answer

Given: Side of cube $21 \ cm$
Formulas used:
Volume of sphere = $\frac{4}{3} \pi r^3$
Calculation:

The largest sphere that can be carved out of a cube of side $21 \ cm$ will have the diameter equal to $21 \ cm$.
$21$ Radius of sphere = $\frac{21}{2} \ cm$
Volume of sphere $=\frac{4}{3} \times \frac{22}{7} \times \frac{21}{2} \times \frac{21}{2}$
$\Rightarrow 11 \times 21 \times 21$
$\Rightarrow 4851 \ cm^3$
$\therefore$ Radius of the cone $=4 \times 3=12 \ cm$

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Question 32 Marks

Simplify: $\left[5\left(8^{\frac{1}{3}}+27^{\frac{1}{3}}\right)^3\right]^{\frac{1}{4}}$

Answer

$={\left[5\left(8^{\frac{1}{3}}+27^{\frac{1}{3}}\right)^3\right]^{\frac{1}{4}}}$
$=\left[5\left(\left(2^3\right)^{\frac{1}{3}}+\left(3^3\right)^{\frac{1}{3}}\right)^3\right]^{\frac{1}{4}}$
$=\left[5\left((2)^{\frac{1}{3} \times 3}+(3)^{\frac{1}{3} \times 3}\right)^3\right]^{\frac{1}{4}}$
$=\left[5(2+3)^3\right]^{\frac{1}{4}}$
$=\left[5(5)^3\right]^{\frac{1}{4}}$
$=\left[5^4\right]^{\frac{1}{4}}$
$=5$

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Question 42 Marks

Simplify: $\left(\frac{5^{-1} \times 7^2}{5^2 \times 7^{-4}}\right)^{7 / 2} \times\left(\frac{5^{-2} \times 7^3}{5^3 \times 7^{-6}}\right)^{-5 / 2}$

Answer

We have,
$\left(\frac{5^{-1} \times 7^2}{5^2 \times 7^{-4}}\right)^{\frac{7}{2}} \times\left(\frac{5^{-2} \times 7^3}{5^3 \times 7^{-5}}\right)^{-\frac{5}{2}}$
=$\left(\frac{7^{2+4}}{5^{2+1}}\right)^{\frac{7}{2}} \times\left(\frac{7^{3+5}}{5^{3+2}}\right)^{-\frac{5}{2}}$
=$\left(\frac{7^6}{5^3}\right)^{\frac{7}{2}} \times\left(\frac{7^8}{5^5}\right)^{-\frac{5}{2}}$
=$\frac{7^{6 \times \frac{7}{2}}}{5^{3 \times \frac{7}{2}}} \times \frac{7^{8 \times-\frac{5}{2}}}{5^{5 \times-\frac{5}{2}}}$
=$\frac{7^{21-20}}{5^{\frac{21}{2}-\frac{25}{2}}}=\frac{7}{5^{-\frac{4}{2}}}$
$=7 \times 5^{\frac{4}{2}}=7 \times 5^2$
$=7 \times 25=175$

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Question 52 Marks

In which quadrant will the point lie, if:
(i) The y-coordinate is 3 and the x-coordinate is -4?
(ii) The x-coordinate is -5 and the y-coordinate is-3?
(iii) The y-coordinate is 4 and the x-coordinate is 5?
(iv) The y-coordinate is 4 and the x-coordinate is -4?

Answer

(i) II
(ii) III
(iii) I
(iv) II

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Question 62 Marks

If a point $C$ lies between two points $A$ and $B$ such that $A C=B C$, then prove that $A C=\frac{1}{2} A B$. Explain by drawing the figure.

Answer

Given, $AC = BC$
AC + AC = BC + AC ........[ AC are added to both the side ]
2 AC = AB .........[ BC + AC coincides with AB ]
$\therefore AC =\frac{1}{2} AB$

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Question 72 Marks

Read the following two statements which are taken as axioms:
i. If two lines intersect each other, then the vertically opposite angles are not equal.
ii. If a ray stands on a line, then the sum of two adjacent angles so formed is equal to $180^{\circ}$.
Is this system of axioms consistent? Justify your answer.

Answer

It is known that, if two lines intersect each other, then the vertically opposite angles are equal. It is a theorem, therefore, given Statement I is false and not an axiom.
Also, we know that, if a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. It is an axiom.
Therefore, given statement parallel is true and an axiom.
Thus, in given statements, first is false and second is an axiom.Therefore, given system of axioms is not consistent.

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