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

Polycet Mock Test

31)Two boys on the either side of a temple of 45 meters height observe its top at the angles of elevation 30° and 60° respectively. Find the distance between the two boys?
45 మీ. ఎత్తుగల ఒక గుడి పై భాగాన్ని, దాని ఇరువైపులానున్న ఇద్దరు బాలురు 30° మరియు 60° ఊర్ధ్వకోణాలతో పరిశీలించారు. ఆ ఇద్దరు బాలుర మధ్య దూరం ఎంత?

A) 60√3 m

B) 40√3 m

C) $\frac{60}{\sqrt3}m$

D) $\frac{40}{\sqrt3}m$

View Answer

A) 60√3 m
Explanation:🎯 Shortcut Method to Find the Distance Between Two Boys
We are given:
– Height of the temple $h = 45 \, \text{m}$

– Angle of elevation from the first boy $\theta_1 = 30^\circ$

– Angle of elevation from the second boy $\theta_2 = 60^\circ$

The two boys are on opposite sides of the temple, and we need to find the distance between them.
Step 1: Use tan to relate the distances from each boy to the temple.
From the tan function:
– For the first boy:
$\tan 30^\circ = \frac{h}{d_1}$

where $d_1$

is the distance from the first boy to the temple.
$\tan 30^\circ = \frac{45}{d_1}$

Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$

:
$\frac{1}{\sqrt{3}} = \frac{45}{d_1}$

$d_1 = 45 \sqrt{3}$

– For the second boy:
$\tan 60^\circ = \frac{h}{d_2}$

where $d_2$

is the distance from the second boy to the temple.
$\tan 60^\circ = \frac{45}{d_2}$

Since $\tan 60^\circ = \sqrt{3}$

:
$\sqrt{3} = \frac{45}{d_2}$

$d_2 = \frac{45}{\sqrt{3}} = 15\sqrt{3}$

Step 2: Find the total distance between the two boys.
The total distance between the two boys is the sum of $d_1$

and $d_2$

:
$\text{Distance between the two boys} = d_1 + d_2 = 45 \sqrt{3} + 15 \sqrt{3} = 60 \sqrt{3}$

Final Answer: $\boxed{60\sqrt{3} \, \text{m}}$

32)If a cylinder and a cone have bases of equal radii and are of equal heights, then that their volumes are in the ratio of _____
ఒక స్థూపము మరియు శంఖువు సమాన భూవ్యాసార్ధమును మరియు ఎత్తును కల్గి ఉన్నాయి. అయినచో, వాటి ఘనపరిమాణముల నిష్పత్తి_____.

A) 1:2

B) 2:3

C) 3:1

D) 1:4

View Answer

C) 3:1
Explanation:🎯 Shortcut Method to Find the Volume Ratio of a Cylinder and a Cone
Given:
– The radius of the base ( $r$

) and the height ( $h$

) of both the cylinder and the cone are equal.
Volume Formula for Cylinder:
$V_{\text{cylinder}} = \pi r^2 h$

Volume Formula for Cone:
$V_{\text{cone}} = \frac{1}{3} \pi r^2 h$

Finding the Ratio of Volumes:
The ratio of the volume of the cylinder to the volume of the cone is:
$\frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h}$

Simplifying the expression:
$\frac{V_{\text{cylinder}}}{V_{\text{cone}}} = 3$

Thus, the volume of the cylinder is 3 times the volume of the cone.
Final Answer: $\boxed{3:1}$

33)If two cubes each of volume 64 cm3 are joined end to end together, then the surface area of the resulting cuboid is ____.
64 ఘనపు సెం.మీ. ఘనపరిమాణము గల రెండు సమఘనములు అంచులు తాకునట్లు అమర్చబడినవి. అయిన, ఏర్పడిన క్రొత్త ఘనాకృతి యొక్క సంపూర్ణతల వైశాల్యము ____.

A) 128 cm²

B) 160 cm²

C) 192 cm²

D) 384 cm²

View Answer

B) 160 cm²
Explanation:🎯 Shortcut Method to Find the Surface Area of a Cuboid
Given:
– Volume of each cube = $64 \, \text{cm}^3$

– The cubes are joined end to end to form a cuboid.
Step 1: Find the side length of each cube
The volume of a cube is given by the formula:
$V_{\text{cube}} = a^3$

Where $a$

is the side length of the cube.
For each cube: $64 = a^3$

Taking the cube root of both sides: $a = \sqrt[3]{64} = 4 \, \text{cm}$

So, the side length of each cube is $4 \, \text{cm}$

.
Step 2: Dimensions of the resulting cuboid
When the two cubes are joined end to end:
– The length of the cuboid = 4 + 4 = 8, cm
– The width of the cuboid = 4 cm (the same as the side of the cube)
– The height of the cuboid = 4cm (the same as the side of the cube)
Step 3: Surface Area of the Cuboid
The surface area A of a cuboid is given by the formula:
A = 2(lw + lh + wh)
Substitute the dimensions of the cuboid:
– l = 8cm
– w = 4cm
– h = 4cm
$A = 2 \left( (8 \times 4) + (8 \times 4) + (4 \times 4) \right)$

$A = 2 \left( 32 + 32 + 16 \right)$

$A = 2 \times 80 = 160 \, \text{cm}^2$

Final Answer: $\boxed{160 \, \text{cm}^2}$

34)If ABC is a right triangle right angled at ‘C’ and let BC = a, CA =b, AB = c and let p be the length of perpendicular from C on AB, then ________.
లంబకోణ త్రిభుజము ABC లో లంబకోణము శీర్షము ‘C” వద్ద కలదు. BC= a, CA=b, AB= c మరియు ‘C’ నుండి AB కి గీసిన లంబము పొడవు p అయిన _____

A) cp = ab

B) $\frac1{p^2}=\frac1{a^2}-\frac1{b^2}$

C) $a^2+b^{2\;}=\;p^2$

D) None
ఏదీ కాదు

View Answer

A) cp = ab
Explanation:🎯 Given Information:
– Triangle ABC is a right triangle right-angled at C.
– The sides of the triangle are:
– BC = a (perpendicular side),
– CA = b (perpendicular side),
– AB = c (hypotenuse).
– p is the length of the perpendicular from C on AB.
We need to identify the correct relationship involving $p$

, $a$

, $b$

, and $c$

.
Step 1: Using the Area of the Triangle
The area of the triangle can be calculated in two ways:
– 1. Using the base and height (since it’s a right triangle):
$\text{Area} = \frac{1}{2} \times a \times b$

– 2. Using the hypotenuse and the perpendicular p from C to the hypotenuse AB:
$\text{Area} = \frac{1}{2} \times c \times p$

Since both expressions represent the area of the same triangle, equating them gives us:
$\frac{1}{2} \times a \times b = \frac{1}{2} \times c \times p$

Simplifying:
$a \times b = c \times p$

Thus, the relationship is:
$\boxed{c \times p = a \times b}$

35)Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on the top of the dice is 13?
రెండు పాచికలు ఒకేసారి దొర్లించడం జరిగింది. రెండు పాచికలపై కనిపించే చుక్కల మొత్తం 13 అవ్వడానికి సంభావ్యత ఎంత?

A) 1

B) 1/2

C) 2/3

D) 0

View Answer

D) 0
Explanation:🎯 Understanding the Problem
We are rolling two dice and we need to find the probability that the sum of the numbers on the top of the dice is 13.
Step 1: Possible Outcomes when Rolling Two Dice
Each die has 6 faces, numbered from 1 to 6. Therefore, the total number of possible outcomes when two dice are rolled is:
$6 \times 6 = 36$

Step 2: Sum of 13
To get a sum of 13, let’s check the possible combinations:
– The highest sum you can get when rolling two dice is $6 + 6 = 12$

, which means it’s impossible to get a sum of 13.
Step 3: Conclusion
Since there is no way to get a sum of 13 with two dice, the number of favorable outcomes is 0.
Step 4: Calculating the Probability
The probability is calculated as:
$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{0}{36} = 0$

Final Answer:
$\boxed{0}$

36)In ΔABC, DE || BC. If AD=x, DB = x-2, AE = x+2 and EC=x-1, then the value of x = ______.
ΔABC లో DE || BC, AD=x, DB=x−2, AE =x+2 మరియు EC = x – 1 అయిన, ‘x’ విలువ ______.

A) 3

B) 2

C) 1

D) 4

View Answer

D) 4
Explanation:We are given that $DE \parallel BC$

in triangle ABC, which implies that by the Basic Proportionality Theorem (also known as Thales’ Theorem), the ratios of the lengths of the corresponding sides of the two triangles are equal.
In this case, we are given:
– AD = x,
– DB = x – 2,
– AE = x + 2,
– EC = x – 1.

From the Basic Proportionality Theorem, we know that:
$\frac{AD}{DB} = \frac{AE}{EC}$
Substituting the values of $AD$ , $DB$ , $AE$ , and $EC$ :
$\frac{x}{x-2} = \frac{x+2}{x-1}$
Step 1: Cross Multiply
$x(x-1) = (x-2)(x+2)$
Step 2: Expand both sides
$x^2 - x = x^2 - 4$
Step 3: Simplify the equation
$x^2 - x = x^2 - 4$
Cancel $x^2$ from both sides:
$-x = -4$
Step 4: Solve for $x$
$x = 4$
Final Answer:
$\boxed{4}$

37)If two concentric circles of radii 5 cm and 3 cm are drawn, then the length of the chord of the larger circle which touches the smaller circle is ______.
5 సెం.మీ. మరియు 3 సెం.మీ. వ్యాసార్థాలతో రెండు ఏక కేంద్ర వృత్తాలు గీయబడ్డాయి. అయిన, చిన్న వృత్తాన్ని స్పృశించే పెద్ద వృత్తము యొక్క జ్యా పొడవు ________.

A) 4 cm

B) 6 cm

C) 8 cm

D) 10 cm

View Answer

C) 8 cm
Explanation:To solve this problem quickly, we can use the following shortcut method:
Step-by-Step Process:
– We are given two concentric circles, one with radius 5 cm (larger circle) and the other with radius 3 cm (smaller circle).
– The chord of the larger circle touches the smaller circle, meaning the perpendicular distance from the center to the chord is equal to the radius of the smaller circle (3 cm).
Now, let’s calculate the length of the chord of the larger circle that touches the smaller circle.
Using the Right Triangle Approach:
– Consider the center of the circles as the origin $O$

.
– The radius of the larger circle $R = 5$

cm.
– The radius of the smaller circle $r = 3$

cm.
– The distance from the center to the chord (which is perpendicular to the chord) is equal to the radius of the smaller circle $r = 3$

cm.
– Let the half-length of the chord be $x$

.
Using the Pythagorean theorem in the right triangle formed by the radius of the larger circle, the perpendicular from the center to the chord, and half of the chord length, we get:
$R^2 = r^2 + x^2$

Substituting the values:

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\[x = \sqrt{16} = 4 \, \text{cm}$
Full Length of the Chord:
The full length of the chord is twice the half-length:
$\text{Length of the chord} = 2 \times x = 2 \times 4 = 8 \, \text{cm}$
Final Answer:
$\boxed{8 \, \text{cm}}$
Thus, the length of the chord is 8 cm.
However, since 8 cm is not listed in the given options, I recommend double-checking the options provided in the problem or revisiting the context. Based on the formula, the correct answer is 8 cm.
[/su_spoiler]
</div>
<div class="mcq-question" data-answer="B"><b>38)The area of a sector whose radius is 7 cm with the angle 72° is _____.
$\left(use\;\pi=\frac{22}7\right) $
వృత్త వ్యాసార్ధము 7 సెం.మీ మరియు సెక్టరు కోణము 72° అయిన, సెక్టరు వైశాల్యము ___________.$\left(\pi=\frac{22}7\;అని\;తీసుకొనుము\right) $</b>
<div class="mcq-options">
A) 38 cm<sup>2</sup>
B) 30.8 cm<sup>2</sup>
C) 28.8 cm<sup>2</sup>
D) 57 cm<sup>2</sup>
</div>
[su_spoiler title="View Answer" style="fancy" icon="arrow"]
B) 30.8 cm<sup>2</sup>
Explanation:To find the area of a sector with a given radius and angle, we can use the following formula:
Formula for the Area of a Sector:
$\text{Area of Sector} = \frac{\theta}{360} \times \pi r^2$
Where:
- $ \theta $ is the angle of the sector (in degrees),
- $ r $ is the radius of the circle,
- $ \pi $ is the constant (use $ \pi = \frac{22}{7} $ as given in the question).
Given:
- Radius, $ r = 7 \, \text{cm} $,
- Angle, $ \theta = 72^\circ $,
- $ \pi = \frac{22}{7} $.
Step-by-Step Calculation:
- 1. Substitute the values into the formula:
$\text{Area of Sector} = \frac{72}{360} \times \frac{22}{7} \times 7^2$
- 2. Simplify:
- $\text{Area of Sector} = \frac{72}{360} \times \frac{22}{7} \times 49$
- 3. Simplify the fraction $ \frac{72}{360} $:
$\frac{72}{360} = \frac{1}{5}$
So, the formula becomes:
$\text{Area of Sector} = \frac{1}{5} \times \frac{22}{7} \times 49$
- 4. Simplify further:
$\text{Area of Sector} = \frac{1}{5} \times \frac{22 \times 49}{7}\]
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Final Answer:
$\boxed{30.8 \, \text{cm}^2}$
Thus, the area of the sector is 30.8 cm².

39)A girl of height 90 cm is walking away from the base of a lamp post at a speed of 120 cm / sec. If the lamp post is 360 cm above the ground, then the length of her shadow after 4 seconds is ____.
90 సెం.మీ. ఎత్తు గల ఒక బాలిక దీపస్తంభము నుండి దూరముగా 120 సెం.మీ./సె. వేగముతో నడుచుచున్నది. దీపస్తంభము ఎత్తు 360 సెం.మీ. అయిన, 4 సెకండ్ల తరువాత ఏర్పడే ఆ బాలిక నీడ పొడవు _____.

A) 90 cm

B) 120 cm

C) 160 cm

D) 180 cm

View Answer

C) 160 cm
Explanation:To solve this problem, we can use the concept of similar triangles. Here’s how we approach it step-by-step:
Given:
– Height of the lamp post = 360 cm.
– Height of the girl = 90 cm.
– The girl is walking away from the base of the lamp post at a speed of 120 cm/sec.
– Time = 4 seconds.
Step-by-Step Calculation:
– 1. Calculate the distance the girl has walked:
The girl walks at 120 cm/sec, and she walks for 4 seconds. So, the distance she has walked is:
$\text{Distance walked} = \text{Speed} \times \text{Time} = 120 \, \text{cm/sec} \times 4 \, \text{sec} = 480 \, \text{cm}$

– 2. Understand the concept of similar triangles:
The girl, the lamp post, and the tip of her shadow form two similar triangles:
– One triangle is formed by the lamp post and the tip of the shadow.
– The second triangle is formed by the girl and the tip of her shadow.
– 3. Use the property of similar triangles:
Since the triangles are similar, the ratio of the corresponding sides will be equal. The height of the lamp post is 360 cm, the height of the girl is 90 cm, and the distance of the girl from the lamp post is 480 cm.
Let the length of the shadow be $x$

. We can set up the proportion:
$\frac{\text{Height of lamp post}}{\text{Height of girl}} = \frac{\text{Distance from lamp post to tip of shadow}}{\text{Distance from girl to tip of shadow}}$

Substituting the known values:
$\frac{360}{90} = \frac{480 + x}{x}$

– 4. Solve for $x$

:
$\frac{360}{90} = \frac{480 + x}{x}$

Simplifying:
$4 = \frac{480 + x}{x}$

Multiply both sides by x :
4x = 480 + x
Subtract x from both sides:
3x = 480
Divide by 3:
$x = \frac{480}{3} = 160 \, \text{cm}$

Final Answer:
The length of her shadow after 4 seconds is 160 cm.

40)If the ratio of corresponding sides of two similar triangles is 2:3, then the ratio of areas of these triangles is _____.
రెండు సరూప త్రిభుజాల అనురూప భుజాల నిష్పత్తి 2:3 అయితే, ఈ త్రిభుజాల వైశాల్యాల నిష్పత్తి ____.

A) 2 : 3

B) √2 : √3

C) 4 : 9

D) 16 : 81

View Answer

C) 4 : 9
Explanation:For two similar triangles, the ratio of their areas is the square of the ratio of the corresponding sides.
Given:
– The ratio of the corresponding sides of two similar triangles is $\frac{2}{3}$

.
Formula for the ratio of areas:
The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. So:
$\text{Ratio of areas} = \left(\frac{\text{Corresponding sides}}{\text{Corresponding sides}}\right)^2$

$\text{Ratio of areas} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$

Final Answer:
The ratio of the areas of the two triangles is $\frac{4}{9}$

, which corresponds to the option 3. 4 : 9.

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