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

Polycet Mock Test

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:
$5^2 = 3^2 + x^2$
$25 = 9 + x^2$
$x^2 = 25 - 9 = 16$
$x = \sqrt{16} = 4 \, \text{cm}$
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.

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 ___________. $\left(\pi=\frac{22}7\;అని\;తీసుకొనుము\right)$

A) 38 cm²
B) 30.8 cm²
C) 28.8 cm²
D) 57 cm²

View Answer

B) 30.8 cm²

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:
$\text{Area of Sector} = \frac{1}{5} \times \frac{22 \times 49}{7}$
$\text{Area of Sector} = \frac{1}{5} \times \frac{1078}{7}$
$\text{Area of Sector} = \frac{1}{5} \times 154$
$\text{Area of Sector} = 30.8 \, \text{cm}^2$
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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