Question
solve :$2 x-3 y=14 ; 5 x+2 y=16$
✓
Answer
$x=4, y=-2$
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Examine each of the following statements and comment:
If two coins are tossed at the same time, there are 3 possible outcomes-two heads, two tails, or one of each. Therefore, for each outcome, the probability of occurrence is $\frac{1}{3}$.
→If two coins are tossed at the same time, there are 3 possible outcomes-two heads, two tails, or one of each. Therefore, for each outcome, the probability of occurrence is $\frac{1}{3}$.
Prove that angles inscribed in the same arc are congruent.
Given: In a circle with center $C, \angle P Q R$ and $\angle P S R$ are inscribed in same arc PQR. Arc PTR is intercepted by the angles.
To prove: $\angle P Q R \cong \angle P S R$.
Proof:
$ m \angle PQR =\frac{1}{2} \times[ m (\operatorname{arc} PTR )]$
$m \angle \square=\frac{1}{2} \times[ m (\operatorname{arc} PTR )] \ldots . . . \text { (ii) } \square$
$m \angle \square= m \angle PSR \quad \ldots . . .[ By \text { (i) and (ii) }]$
$\therefore \angle PQR \cong \angle PSR $
→Given: In a circle with center $C, \angle P Q R$ and $\angle P S R$ are inscribed in same arc PQR. Arc PTR is intercepted by the angles.
To prove: $\angle P Q R \cong \angle P S R$.
Proof:
$ m \angle PQR =\frac{1}{2} \times[ m (\operatorname{arc} PTR )]$
$m \angle \square=\frac{1}{2} \times[ m (\operatorname{arc} PTR )] \ldots . . . \text { (ii) } \square$
$m \angle \square= m \angle PSR \quad \ldots . . .[ By \text { (i) and (ii) }]$
$\therefore \angle PQR \cong \angle PSR $
Write the first five terms of the following sequences whose $n^{th}$ terms are:
$\text{a}_\text{n}=\frac{2\text{n}-3}{6}$
→$\text{a}_\text{n}=\frac{2\text{n}-3}{6}$
If two dice are rolled simultaneously, find the probability of the following events.
The sum of the digits on the upper faces is 33.
The sum of the digits on the upper faces is 33.
The first term a and common difference d are given. Find first four terms of A.P :
→$a=-1, d=-\frac{1}{2}$
Ratio of areas of two triangles with equal heights is 2 : 3. If base of the smaller triangle is 6 cm then what is the corresponding base of the bigger triangle?
Show that $\sec x+\tan x=\sqrt{\frac{1+\sin x}{1-\sin x}}$
→In the figure, if the chord $P Q$ and chord $R S$ intersect at point $T$, prove that: $m \angle S T Q=\frac{1}{2}[m(\operatorname{arc} P R)+$ $m (\operatorname{arc} S Q)]$ for any measure of $\angle S T Q$ by filling out the boxes
Proof: $m \angle STQ = m \angle SPQ +\square \ldots$....[Theorem of the external angle of a triangle $]$ $=\frac{1}{2} m (\operatorname{arc~SQ})+\square \quad \ldots . . .[$ Inscribed angle theorem $]$
$=\frac{1}{2}[\square+\square]$
→Proof: $m \angle STQ = m \angle SPQ +\square \ldots$....[Theorem of the external angle of a triangle $]$ $=\frac{1}{2} m (\operatorname{arc~SQ})+\square \quad \ldots . . .[$ Inscribed angle theorem $]$
$=\frac{1}{2}[\square+\square]$
Using prime factorization, find the HCF and LCM of:
$24, 36, 40$
→$24, 36, 40$
On comparing the ratios $\frac{\text{a}_1}{\text{a}_2},\frac{\text{b}_1}{\text{b}_2}$ and $\frac{\text{c}_1}{\text{c}_2},$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.
→$5x - 4y + 8 = 0$
$7x + 6y - 9 = 0$