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

11)The distance between the points $\left(x_1,\;y_1\right)$ and $\left(x_2,\;y_2\right)$ is
$\left(x_1,\;y_1\right)$ మరియు $\left(x_2,\;y_2\right)$ బిందువుల మధ్య దూరము కనుగొనుటకు సూత్రము

A) $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

B) $\sqrt{\left(x_2+x_1\right)^2+\left(y_2+y_1\right)^2}$

C) $\sqrt{\left(x_2-x_1\right)^2-\left(y_2-y_1\right)^2}$

D) $\sqrt{\left(x_2+x_1\right)^2-\left(y_2+y_1\right)^2}$

View Answer

A) $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

Explanation:🎯 Shortcut Method to Find the Distance Between Two Points
The distance formula between two points $(x_1, y_1)$

and $(x_2, y_2)$

in a plane is:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Explanation:
– The difference between the x-coordinates is $(x_2 - x_1)$

.
– The difference between the y-coordinates is $(y_2 - y_1)$

.
– Then, apply the Pythagorean theorem to find the distance.
Final Answer:
$\boxed{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

12)The coordinates of the point which divides the line segment joining the points (4,-3) and (8, 5) in the ratio 3:1 internally is
బిందువులు (4, −3) మరియు (8, 5) లచే ఏర్పడు రేఖాఖండమును 3:1 నిష్పత్తిలో అంతరంగా విభజించు బిందువు నిరూపకాలు

A) (3,7)

B) (7,3)

C) (-7,-3)

D) (-3,-7)

View Answer

B) (7,3)
Explanation:🎯 Shortcut Method: Finding Coordinates of a Point Dividing a Line Segment in a Given Ratio
To find the coordinates of a point that divides a line segment joining two points $(x_1, y_1)$

and $(x_2, y_2)$

in the ratio m : n, use the section formula:
$\left( x, y \right) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)$

Given:
– Point 1: (4, -3)
– Point 2: (8, 5)
– Ratio: 3 : 1
Step 1: Apply the Section Formula
$x = \frac{3(8) + 1(4)}{3 + 1} = \frac{24 + 4}{4} = \frac{28}{4} = 7$

$y = \frac{3(5) + 1(-3)}{3 + 1} = \frac{15 - 3}{4} = \frac{12}{4} = 3$

Final Answer:
The coordinates of the point are (7, 3).
$\boxed{(7, 3)}$

13)Which of the following is not a formula for arithmetic mean?
ఈ క్రింది వానిలో అంకగణిత సగటునకు సూత్రము కానిది ఏది?

A) $\frac{\Sigma f_ix_i}{\Sigma f_i}$

B) $a+\frac{\Sigma f_id_i}{\Sigma f_i}$

C) $a+\left[\frac{\Sigma f_i\mu_i}{\Sigma f_i}\right]\times h$

D) $l+\left[\frac{f_1-f_0}{2f_1\;-\;f_0\;-\;f_2}\right]\times h$

View Answer

D) $l+\left[\frac{f_1-f_0}{2f_1\;-\;f_0\;-\;f_2}\right]\times h$

Explanation:🎯 Shortcut Method for Identifying the Formula for Arithmetic Mean
We are asked to identify which of the following is NOT a formula for the arithmetic mean.
Let’s quickly review each option:
– 1. $\frac{\Sigma f_i x_i}{\Sigma f_i}$

This is the weighted mean formula. It is a standard formula for the arithmetic mean when you have data with frequencies. Correct formula.
– 2. $a + \frac{\Sigma f_i d_i}{\Sigma f_i}$

This formula is used for grouped data where $a$

is the assumed mean, and $d_i$

is the deviation of values from the assumed mean. Correct formula.
– 3. $a + \left[ \frac{\Sigma f_i \mu_i}{\Sigma f_i} \right] \times h$

This is another formula for grouped data. Here, $\mu_i$

is the deviation from the actual mean, and $h$

is the class width. Correct formula.
– 4. $l + \left[ \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h$

This formula is used to calculate the mode in grouped data, not the mean. Incorrect formula for arithmetic mean.
Conclusion:
The formula for the mode is in Option 4. Therefore, the correct answer is:
$\boxed{4. \, l + \left[\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right] \times h}$

This is NOT a formula for the arithmetic mean; it is for the mode.

14)Rain fall of a town in a week is 4 cm, 5 cm, 12 cm, 3 cm, 6 cm, 8 cm and 4 cm, then the average rainfall per day is
ఒక వారములో ఒక పట్టణపు వర్షపాతం 4 సెం.మీ., 5 సెం.మీ., 12 సెం.మీ., 3 సెం.మీ., 6 సెం.మీ., 8 సెం.మీ. మరియు 4 సెం.మీ. అయిన, ఒక రోజులో సరాసరి వర్షపాతము

A) 4 cm

B) 5 cm

C) 6 cm

D) 7 cm

View Answer

C) 6 cm
Explanation:🎯 Shortcut Method for Finding the Average Rainfall per Day
To find the average rainfall per day, we use the formula for the arithmetic mean:
$\text{Average Rainfall} = \frac{\text{Total Rainfall}}{\text{Number of Days}}$

Given rainfall for the week:
– 4 cm, 5 cm, 12 cm, 3 cm, 6 cm, 8 cm, and 4 cm.
Step 1: Calculate the total rainfall
$\text{Total Rainfall} = 4 + 5 + 12 + 3 + 6 + 8 + 4 = 42 \, \text{cm}$

Step 2: Divide the total rainfall by the number of days (7 days in a week)
$\text{Average Rainfall} = \frac{42}{7} = 6 \, \text{cm}$

Final Answer: $\boxed{6 \, \text{cm}}$

15)Mode of the data 9, 10, 19, 7, 11, 5, 6, 7, 8, 14, 10, 7, 6 is
9, 10, 19, 7, 11, 5, 6, 7, 8, 14, 10, 7, 6 అనే దత్తాంశం యొక్క బాహుళకము

A) 6

B) 7

C) 10

D) 19

View Answer

B) 7
Explanation:🎯 Shortcut Method for Finding the Mode
The mode of a dataset is the value that appears most frequently. Let’s go through the steps to find the mode.
Given data:
9, 10, 19, 7, 11, 5, 6, 7, 8, 14, 10, 7, 6
Step 1: Count the frequency of each number
– 9 appears 1 time
– 10 appears 2 times
– 19 appears 1 time
– 7 appears 3 times
– 11 appears 1 time
– 5 appears 1 time
– 6 appears 2 times
– 8 appears 1 time
– 14 appears 1 time
Step 2: Identify the number with the highest frequency
– The number 7 appears 3 times, which is the highest frequency.
Final Answer: $\boxed{7}$

16)If A, B and C are interior angles of triangle ABC, then the value of $\cos\left(\frac{A+B}2\right)$ is
A, B మరియు C లు త్రిభుజం ABC లోని అంతర కోణాలైన, $\cos\left(\frac{A+B}2\right)$ యొక్క విలువ

A) $\cos\left(\frac{A-B}2\right)$

B) $\sin\left(\frac{A+B}2\right)$

C) $\sin\frac C2$

D) $\cos\frac B2$

View Answer

C) $\sin\frac C2$

Explanation:🎯 Shortcut Method for the Given Expression:
We are given that $A$

, $B$

, and $C$

are the interior angles of a triangle, i.e., $A + B + C = 180^\circ$

.
We need to find the value of:
$\cos\left(\frac{A + B}{2}\right)$

Step 1: Use the fact that $A + B + C = 180^\circ$

Since the sum of the angles in a triangle is $180^\circ$

, we can substitute $A + B = 180^\circ - C$

. Thus,
$\frac{A + B}{2} = \frac{180^\circ - C}{2} = 90^\circ - \frac{C}{2}$

Step 2: Use the cosine of a difference
Now we have:
$\cos\left(90^\circ - \frac{C}{2}\right)$

Using the co-function identity: $\cos(90^\circ - x) = \sin(x)$

We get:
$\cos\left(90^\circ - \frac{C}{2}\right) = \sin\frac{C}{2}$

Final Answer: $\boxed{\sin\frac{C}{2}}$

This is the quickest method using the fact that the sum of the angles in a triangle is $180^\circ$

and applying trigonometric identities.

17)The value of cos 54° cos 36° – sin 54° sin 36° is
cos 54° cos 36° – sin 54° sin 36° యొక్క విలువ

A) 0

B) 1

C) $\frac{\sqrt3}2$

D) $\frac1{\sqrt2}$

View Answer

A) 0
Explanation:🎯 Shortcut Method for Solving $\cos 54^\circ \cos 36^\circ - \sin 54^\circ \sin 36^\circ$

The expression $\cos A \cos B - \sin A \sin B$

is a well-known cosine identity:
$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Step 1: Apply the cosine addition formula
Using the identity $\cos(A + B) = \cos A \cos B - \sin A \sin B$

, we can rewrite the given expression as:
$\cos(54^\circ + 36^\circ) = \cos 90^\circ$

Step 2: Simplify
We know that:
$\cos 90^\circ = 0$

Final Answer: $\boxed{0}$

18)Which of the following rational number have terminating decimal?
కింది వాటిలో ఏ అకరణీయ సంఖ్య అంతమయ్యే దశాంశాన్ని కలిగి ఉంటుంది?

A) $\frac7{250}$

B) $\frac{16}{225}$

C) $\frac5{18}$

D) $\frac2{21}$

View Answer

A) $\frac7{250}$

Explanation:🎯 Shortcut Method to Determine if a Rational Number Has a Terminating Decimal
A rational number has a terminating decimal if and only if, when simplified, its denominator has no prime factors other than 2 and 5. In other words, the denominator of the fraction must be of the form $2^m \times 5^n$

, where $m$

and $n$

are non-negative integers.
Let’s check each option:
– 1. $\frac{7}{250}$

– The denominator is 250, which factors as:
$250 = 2 \times 5^3$

– Since the denominator contains only the primes 2 and 5, this fraction will have a terminating decimal.
– 2. $\frac{16}{225}$

– The denominator is 225, which factors as:
$225 = 3^2 \times 5^2$

– Since 225 has a factor of 3, it does not meet the condition for a terminating decimal.
– 3. $\frac{5}{18}$

– The denominator is 18, which factors as:
$18 = 2 \times 3^2$

– Since 18 has a factor of 3, it does not meet the condition for a terminating decimal.
– 4. $\frac{2}{21}$

– The denominator is 21, which factors as:
$21 = 3 \times 7$

– Since 21 has a factor of 3, it does not meet the condition for a terminating decimal.
Final Answer: $\boxed{\frac{7}{250}}$

19)H.C.F. of 2023, 2024, 2025 is –

A) 2024

B) 2023

C) 0

D) 1

View Answer

D) 1
Explanation:🎯 Shortcut Method to Find the HCF of 2023, 2024, and 2025
The HCF (Highest Common Factor) of three numbers is the largest number that divides all of them exactly.
Step 1: Check the divisibility of 2023, 2024, and 2025
– The numbers 2023, 2024, and 2025 are consecutive numbers, which means:
– 2024 is one more than 2023, and one less than 2025.
– Consecutive numbers always have an HCF of 1, because there is no number greater than 1 that can divide all three.
Step 2: Conclude the HCF
Since these are consecutive numbers, their HCF is 1.
Final Answer: $\boxed{1}$

20)A boy observed the top of an electric pole at an angle of elevation of 60° when the observation point is 6 meters away from the.foot of the pole, then the height of the pole is
ఒక బాలుడు ఒక విద్యుత్ స్థంభం అడుగు భాగం నుండి 6 మీ. దూరంలో ఉన్న బిందువు నుండి విద్యుత్ స్థంభం పై భాగాన్ని 60° ఊర్ధ్వ కోణంతో పరిశీలించిన, ఆ స్థంభం ఎత్తు

A) 6 m

B) $6\sqrt2\;m$

C) $6\sqrt3\;m$

D) $\frac6{\sqrt3}m$

View Answer

C) $6\sqrt3\;m$

Explanation:🎯 Shortcut Method Using Trigonometry (Tangent)
We are given the following information:
– The angle of elevation to the top of the electric pole is $60^\circ$

.
– The distance from the observation point to the foot of the pole is 6 meters.
To find the height of the pole, we can use the tangent of the angle of elevation. The tangent function is given by:
$\tan(\theta) = \frac{\text{height of the pole}}{\text{distance from the pole}}$