TS Polycet (Polytechnic) 2024 Previous Question Paper with Answers And Model Papers With Complete Analysis

Polycet Mock Test

46) If $\frac{a_1}{a_2}\neq\frac{b_1}{b_2},$ where a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0 are two linear equations, $\frac{a_1}{a_2}\neq\frac{b_1}{b_2},$ అయ్యేటట్లుగాa₁x+b₁y+c₁=0 మరియు a₂x+b₂y+c₂=0 అనేవి రెండు రేఖీయ సమీ కరణాలైతే, ఆ సమీ కరణాలు

A) have a unique solution ఏకైక సాధన కలిగి ఉంటాయి
B) have infinitely many solutions అనంతమైన సాధనలు కలిగి ఉంటాయి
C) have finite solutions పరిమితమైన సాధనలు కలిగి ఉంటాయి
D) have no solution ఏ సాధనలను కలిగి ఉండవు

View Answer

A) have a unique solution ఏకైక సాధన కలిగి ఉంటాయి

Explanation:Given the two linear equations:
$a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0$
And the condition that $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ .
Solution:
The two equations represent two straight lines in the coordinate plane. The condition $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ means that the lines are not parallel and will intersect at exactly one point.
– When two lines intersect at exactly one point, the system of linear equations has a unique solution.
– This condition rules out the possibility of the lines being parallel (which would mean no solution) or coincident (which would mean infinitely many solutions).
Therefore, the correct answer is:
– 1. Have a unique solution.

47) The value of p, for which the pair of equations 3x+4y+2=0 and 9x+ py+8=0 represents parallel lines, is 3x +4y+ 2 = 0 మరియు 9x + py + 8 = 0 అను సమీకరణాల జత సమాంతర రేఖలను సూచించిన, P విలువ

A) 2
B) 4
C) 6
D) 12

View Answer

D) 12

Explanation:To determine the value of $p$ for which the pair of equations represents parallel lines, we first examine the general condition for parallel lines.
Given equations:
– 1. $3x + 4y + 2 = 0$
– 2. $9x + py + 8 = 0$
Step 1: Write the equations in slope-intercept form $y = mx + c$ , where $m$ is the slope.
For the first equation 3x + 4y + 2 = 0, solve for $y$ : $4y = -3x - 2$
$y = -\frac{3}{4}x - \frac{1}{2}$ So, the slope $m_1$ of the first line is $-\frac{3}{4}$ .
For the second equation $9x + py + 8 = 0$ , solve for $y$ : $py = -9x - 8$
$y = -\frac{9}{p}x - \frac{8}{p}$ So, the slope $m_2$ of the second line is $-\frac{9}{p}$ .
Step 2: Condition for parallel lines
For the lines to be parallel, their slopes must be equal, i.e., $m_1 = m_2$
$-\frac{3}{4} = -\frac{9}{p}$
Step 3: Solve for $p$
$\frac{3}{4} = \frac{9}{p}$ Cross multiply: $3p = 36$
$p = 12$
Therefore, the value of $p$ is 12.
The correct answer is:4. 12.

48) Which of the following equation represent the situation where Kiran bought 5 oranges, 7 apples and Harish bought 2 oranges, 12 apples for same amount of total money? కిరణ్ 5 నారింజలు, 7 యాపిల్స్ మరియు హరీశ్ 2 నారింజలు, 12 యాపిల్స్ విడివిడిగా ఒకే మొత్తానికి కొన్నారు. కింది సమీకరణాలలో ఈ విషయాన్ని సూచించే సమీకరణము ఏది?

A) 5x+12y=2x+7y
B) 5x+7y=2x+12y
C) 5x-7y=2x-12y
D) 5x+2y=7x+12y

View Answer

B) 5x+7y=2x+12y

Explanation:Let’s denote:
– $x$ as the price of one orange.
– $y$ as the price of one apple.
Given:
– Kiran bought 5 oranges and 7 apples.
– Harish bought 2 oranges and 12 apples.
– They spent the same amount of money.
Thus, the total money spent by Kiran is: $5x + 7y$ And the total money spent by Harish is: $2x + 12y$
Since they spent the same amount, we can set these two expressions equal to each other: $5x + 7y = 2x + 12y$
Simplifying the equation: $5x - 2x = 12y - 7y$
$3x = 5y$
This is the correct equation representing the given situation.
Therefore, the correct answer is:
2. $5x + 7y = 2x + 12y$ .

49) If $\frac2{\sqrt x}+\frac3{\sqrt y}\;=\;2$ and $\frac4{\sqrt x}-\frac9{\sqrt y}\;=\;-1$ , then $\frac2{\sqrt x}+\frac3{\sqrt y}\;=\;2$ మరియు $\frac4{\sqrt x}-\frac9{\sqrt y}\;=\;-1$ అయిన,

A) x=4, y=3
B) x=2, y=9
C) x=4, y=9
D) x=2, y=3

View Answer

C) x=4, y=9

Explanation:To solve the given system of equations quickly, follow this shortcut method:
Given equations:
– 1. $\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2$
– 2. $\frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1$
Step 1: Let’s assume new variables for simplification:
Let:
– $p = \frac{1}{\sqrt{x}}$
– $q = \frac{1}{\sqrt{y}}$
Now the equations become:
– $2p + 3q = 2$
– $4p - 9q = -1$
Step 2: Use substitution or elimination to solve for $p$ and $q$ .
Multiply the first equation by 2 to align coefficients of $p$ for elimination:
$(2p + 3q = 2) \times 2 \quad \Rightarrow \quad 4p + 6q = 4 \quad \text{(Equation 3)}$
Now subtract the second equation from Equation 3:
(4p + 6q) – (4p – 9q) = 4 – (-1)
4p + 6q – 4p + 9q = 5
15q = 5
$q = \frac{1}{3}$
Step 3: Substitute $q = \frac{1}{3}$ into the first equation:
$2p + 3 \times \frac{1}{3} = 2$
2p + 1 = 2
$2p = 1$
$p = \frac{1}{2}$
Step 4: Solve for $x$ and $y$ :
Recall:
– $p = \frac{1}{\sqrt{x}}$ and $q = \frac{1}{\sqrt{y}}$
– So $\sqrt{x} = \frac{1}{p} = 2$ and $\sqrt{y} = \frac{1}{q} = 3$
Thus:
– $x = 4$
– $y = 9$
Final Answer:
– $x = 4$ , $y = 9$

50) The pair of equations x+y=5 and 2x+2y=k has infinitely many solutions if k = x + y = 5 మరియు 2x + 2y = k అను సమీకరణాల జత అనంతమైన సాధనలను కలిగి ఉండాలి అనిన, k =

A) 4
B) 6
C) 8
D) 10

View Answer

D) 10

Explanation:To determine the value of $k$ for which the system of equations has infinitely many solutions, we need to examine the two equations:
– 1. $x + y = 5$
– 2. $2x + 2y = k$
Step 1: Express both equations in a similar form
The second equation can be simplified by dividing both sides by 2:
$\frac{2x + 2y}{2} = \frac{k}{2} \quad \Rightarrow \quad x + y = \frac{k}{2}$
Now, the system of equations becomes:
– 1. $x + y = 5$
– 2. $x + y = \frac{k}{2}$
Step 2: Condition for infinitely many solutions
For the system to have infinitely many solutions, the two equations must represent the same line. This happens when their right-hand sides are equal. Therefore, we set:
$5 = \frac{k}{2}$
Step 3: Solve for $k$
Multiply both sides by 2 to solve for $k$ :
$10 = k$
Final Answer:
The value of $k$ for which the system of equations has infinitely many solutions is 10.