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

26) Which of the following two sets are equal sets? క్రింది రెండు సమితులలో ఏవి సమాన సమితులు?

A) A = {5, 6}, B = {5}
B) A = {5, 6}, B = {5, 6, 7}
C) A = {5, 6, 7}, B = {7, 5, 6}
D) A = {5, 6, 8}, B = {5, 6, 7}

View Answer

C) A = {5, 6, 7}, B = {7, 5, 6}

Explanation:🎯 Understanding Equal Sets
Two sets are equal sets if and only if they have exactly the same elements. The order of the elements in a set does not matter.
Step-by-step analysis of each option:
– 1. A = {5, 6}, B = {5}
– Set A has the elements 5 and 6.
– Set B has only the element 5.
– These are not equal sets, because Set A has an additional element (6) that Set B does not have.
– 2. A = {5, 6}, B = {5, 6, 7}
– Set A has the elements 5 and 6.
– Set B has the elements 5, 6, and 7.
– These are not equal sets, because Set B has an extra element (7) that Set A does not have.
– 3. A = {5, 6, 7}, B = {7, 5, 6}
– Set A has the elements 5, 6, and 7.
– Set B has the same elements, 5, 6, and 7, but in a different order.
– These are equal sets, because they contain exactly the same elements, just in a different order. The order does not matter in sets.
4. A = {5, 6, 8}, B = {5, 6, 7}
– Set A has the elements 5, 6, and 8.
– Set B has the elements 5, 6, and 7.
– These are not equal sets, because Set A contains 8, while Set B contains 7. They have different elements.
Final Answer: $\boxed{3}$

27) Exponential form of $\log_b\;a\;=\;c$ $\log_b\;a\;=\;c$ యొక్క ఘాత రూపము

A) b^a=c
B) a^c=b
C) a^b=c
D) b^c=a

View Answer

D) b^c=a

Explanation:🎯 Understanding the Exponential Form of Logarithms
The logarithmic equation $\log_b a = c$ means that the exponent $c$ to which the base $b$ must be raised to produce $a$ is $c$ .
This can be rewritten in exponential form as:
$b^c = a$
Explanation:
– $\log_b a = c$ implies that $b$ raised to the power $c$ gives $a$ .
– So, in exponential form, we have:
$b^c = a$
Final Answer: $\boxed{b^c = a}$

28) If a right circular cylinder has base radius 14 cm and height 21 cm, then its curved surface area is ______. $\left(use\;\pi=\frac{22}7\right)$ ఒక క్రమ వృత్తాకార స్థూపము యొక్క భూవ్యాసార్ధము 14 సెం. మీ. మరియు ఎత్తు 21 సెం.మీ. అయిన, ఆ స్థూపము వక్రతల వైశాల్యము _____. $\left(\pi=\frac{22}7\;అని\;తీసుకొనుము\right)$

A) 924 cm²
B) 2772 cm²
C) 3080 cm²
D) 1848 cm²

View Answer

D) 1848 cm²

Explanation:🎯 Shortcut Method to Find the Curved Surface Area of a Cylinder
The formula for the curved surface area (CSA) of a right circular cylinder is:
$\text{CSA} = 2 \pi r h$
Where:
– $r$ = radius of the base
– $h$ = height of the cylinder
– $\pi = \frac{22}{7}$
Given:
– Radius $r = 14 \, \text{cm}$
– Height $h = 21 \, \text{cm}$
– $\pi = \frac{22}{7}$
Step 1: Substitute the values into the formula
$\text{CSA} = 2 \times \frac{22}{7} \times 14 \times 21$
Step 2: Simplify the calculation
$\text{CSA} = 2 \times \frac{22}{7} \times 14 \times 21 = 2 \times \frac{22 \times 14 \times 21}{7}$
$= 2 \times \frac{22 \times 294}{7}$
$= 2 \times 924 = 1848 \, \text{cm}^2$
Final Answer: $\boxed{1848 \, \text{cm}^2}$

29) The volume of a right circular cone with radius 6 cm and height 7 cm is ________. $\left(use\;\pi=\frac{22}7\right)$ 6 సెం.మీ. భూవ్యాసార్ధము మరియు 7 సెం.మీ. ఎత్తు కలిగిన ఒక క్రమ వృత్తాకార శంఖువు యొక్క
ఘనపరిమాణము _______ $\left(\pi=\frac{22}7\;అని\;తీసుకొనుము\right)$

A) 264 cm³
B) 792 cm³
C) 301 cm³
D) 616 cm³

View Answer

A) 264 cm³

Explanation:🎯 Shortcut Method to Find the Volume of a Right Circular Cone
The formula for the volume of a right circular cone is:
$\text{Volume} = \frac{1}{3} \pi r^2 h$
Where:
– $r$ = radius of the base
– $h$ = height of the cone
– $\pi = \frac{22}{7}$
Given:
– Radius $r = 6 \, \text{cm}$
– Height $h = 7 \, \text{cm}$
– $\pi = \frac{22}{7}$
Step 1: Substitute the values into the formula
$\text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 6^2 \times 7$
Step 2: Simplify the calculation
$\text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 36 \times 7$
Cancel the 7’s:
$= \frac{1}{3} \times 22 \times 36$
Now, multiply:
$= \frac{1}{3} \times 792 = 264 \, \text{cm}^3$
Final Answer: $\boxed{264 \, \text{cm}^3}$

30) The value of $\sin^215^\circ\;+\;\cos^215^\circ$ is $\sin^215^\circ\;+\;\cos^215^\circ$ యొక్క విలువ

A) 0
B) 1
C) $\frac12$
D) $\frac1{\sqrt2}$

View Answer

B) 1

Explanation:🎯 Simplifying the Expression $\sin^2 15^\circ + \cos^2 15^\circ$
We are asked to find the value of:
$\sin^2 15^\circ + \cos^2 15^\circ$
Step 1: Use the Pythagorean identity
The Pythagorean identity states that:
$\sin^2 \theta + \cos^2 \theta = 1$
This identity holds for any angle $\theta$ , including $15^\circ$ .
Step 2: Apply the identity to the given expression
$\sin^2 15^\circ + \cos^2 15^\circ = 1$
Final Answer: $\boxed{1}$
This is the simplest and quickest method, using the basic trigonometric identity.