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

Polycet Mock Test

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 –
2023, 2024, 2025 ల యొక్క గ.సా.భా.

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:
$\tan(\theta) = \frac{\text{height of the pole}}{\text{distance from the pole}}$
In this case:
– $\theta = 60^\circ$
– The distance from the pole = 6 meters
– We need to find the height of the pole.
So,
$\tan(60^\circ) = \frac{\text{height of the pole}}{6}$
Since $\tan(60^\circ) = \sqrt{3}$ , we have:
$\sqrt{3} = \frac{\text{height of the pole}}{6}$
Step 2: Solve for the height of the pole
Multiplying both sides by 6:
$\text{height of the pole} = 6 \times \sqrt{3}$
Thus, the height of the pole is:
$\text{height of the pole} = 6\sqrt{3} \, \text{meters}$
Final Answer: $\boxed{6\sqrt{3} \, \text{m}}$