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

Polycet Mock Test

56) A chord of a circle of radius 4 cm is making an angle 60° at the centre, then the length of the chord is 4 సెం.మీ. వ్యాసార్ధం కలిగిన వృత్తంలో ఒక జ్యా కేంద్రం వద్ద 60° కోణం చేస్తుంది. అప్పుడు, ఆ జ్యా పొడవు

A) 1 cm
B) 2 cm
C) 3 cm
D) 4 cm

View Answer

D) 4 cm

Explanation:To find the length of the chord of a circle given the radius and the angle at the center, we can use the following formula:
$\text{Length of the chord} = 2r \sin\left(\frac{\theta}{2}\right)$
Where:
– $r$ is the radius of the circle.
– $\theta$ is the angle at the center in degrees.
Given:
– Radius $r = 4$ cm
– Angle $\theta = 60^\circ$
Now, using the formula:
$\text{Length of the chord} = 2 \times 4 \times \sin\left(\frac{60^\circ}{2}\right)$
$\text{Length of the chord} = 8 \times \sin(30^\circ)$
Since $\sin(30^\circ) = \frac{1}{2}$ :
$\text{Length of the chord} = 8 \times \frac{1}{2} = 4 \, \text{cm}$
Final Answer:
The length of the chord is 4 cm.
So, the correct answer is:4. 4 cm.

57) The discriminant of the quadratic equation $3x^2-2x+\frac13=0$ $3x^2-2x+\frac13=0$ అను వర్గ సమీకరణము యొక్క విచక్షణి

A) 32
B) 16
C) 0
D) 1

View Answer

C) 0

Explanation:To find the discriminant ( $\Delta$ ) of a quadratic equation, we use the formula:
$\Delta = b^2 - 4ac$
For the quadratic equation $ax^2 + bx + c = 0$ , the coefficients are:
– $a = 3$
– $b = -2$
– $c = \frac{1}{3}$
Step 1: Calculate the discriminant
Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula:
$\Delta = (-2)^2 - 4 \times 3 \times \frac{1}{3}$
Step 2: Simplify the expression
$\Delta = 4 - 4 \times 1$
$\Delta = 4 - 4 = 0$
Final Answer:
The discriminant is 0.
So, the correct answer is:3. 0.

58) If −1, −2 are two zeros of a polynomial $2x^3\;+\;ax^2\;+\;bx\;-\;2$ , then the values of a and b are $2x^3\;+\;ax^2\;+\;bx\;-\;2$ అను బహుపది యొక్క రెండు శూన్యాలు -1, -2 అయిన, a మరియు b యొక్క విలువలు

A) 2, -1
B) -5, -1
C) 5, 1
D) -2, -1

View Answer

C) 5, 1

Explanation:We are given the polynomial equation $2x^3 + ax^2 + bx - 2$ , and the two zeros of the polynomial are $x = -1$ and $x = -2$ .
Step 1: Use the fact that the polynomial is zero at these points
We can substitute $x = -1$ and $x = -2$ into the polynomial equation. Since these values are roots of the polynomial, the equation should equal zero at these points.
For x = -1:
Substitute $x = -1$ into the equation $2x^3 + ax^2 + bx - 2 = 0$ :
$2(-1)^3 + a(-1)^2 + b(-1) - 2 = 0$
$2(-1) + a(1) + b(-1) - 2 = 0$
$-2 + a - b - 2 = 0$
$a - b - 4 = 0$
$a - b = 4 \quad \text{(Equation 1)}$
For $x = -2$ :
Substitute $x = -2$ into the equation $2x^3 + ax^2 + bx - 2 = 0$ :
$2(-2)^3 + a(-2)^2 + b(-2) - 2 = 0$
$2(-8) + a(4) + b(-2) - 2 = 0$
$-16 + 4a - 2b - 2 = 0$
$4a - 2b - 18 = 0$
$2a - b = 9 \quad \text{(Equation 2)}$
Step 2: Solve the system of equations
We now have the system of equations:
– 1. a – b = 4
– 2. 2a – b = 9
Subtract Equation 1 from Equation 2:
(2a – b) – (a – b) = 9 – 4
2a – b – a + b = 5
a = 5
Substitute a = 5 into Equation 1:
5 – b = 4
b = 1
Final Answer:
The values of a and b are:a = 5, b = 1
So, the correct answer is 3. 5, 1.

59) If α, β are the zeros of the polynomial P(x) = 3x² – x – 4, then α . β = P(x) = 3x² – x – 4 అను బహుపది యొక్క శూన్యాలు α, β అయిన, α . β =

A) $\frac{-4}3$
B) $\frac43$
C) $\frac{-1}3$
D) $\frac13$

View Answer

A) $\frac{-4}3$

Explanation:We are given the quadratic polynomial $P(x) = 3x^2 - x - 4$ and are asked to find the product of its zeros, $\alpha \cdot \beta$ .
Step 1: Use the relationship between the coefficients and the zeros
For a quadratic equation of the form $ax^2 + bx + c = 0$ , the sum and product of the roots (zeros) are given by:
– Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
– Product of the roots: $\alpha \cdot \beta = \frac{c}{a}$
Here, the quadratic equation is $3x^2 - x - 4 = 0$ , so:
– a = 3
– b = -1
– c = -4
Step 2: Find the product of the roots
The product of the roots $\alpha \cdot \beta$ is given by:
$\alpha \cdot \beta = \frac{c}{a} = \frac{-4}{3}$
Final Answer:
The value of $\alpha \cdot \beta$ is $\frac{-4}{3}$ .
So, the correct answer is 1. $\frac{-4}{3}$ .

60) Which term of the A.P.: 20, 18, 16, is ‘-80’? 20, 18, 16, …. అనే అంకశ్రేఢిలో ‘-80’ ఎన్నవ పదము?

A) 50
B) 51
C) 52
D) 53

View Answer

B) 51

Explanation:We are given the arithmetic progression (A.P.): 20, 18, 16, … and are asked to find which term of this A.P. is equal to -80.
Step 1: Write the general formula for the nth term of an A.P.
The nth term of an arithmetic progression is given by:
$T_n = a + (n - 1) \cdot d$
where:
– $T_n$ is the nth term,
– a is the first term,
– d is the common difference,
– n is the position of the term.
Step 2: Find the common difference d
The common difference d is the difference between consecutive terms:
d = 18 – 20 = -2
Step 3: Substitute values into the formula
We need to find n when $T_n = -80$ . Using the formula:
$T_n = a + (n - 1) \cdot d$
Substitute a = 20, d = -2, and $T_n = -80$ :
$-80 = 20 + (n - 1) \cdot (-2)$
Step 4: Solve for n
Simplify the equation:
-80 = 20 – 2(n – 1)
-80 – 20 = -2(n – 1)
-100 = -2(n – 1)
Now, divide both sides by -2:
50 = n – 1
Add 1 to both sides:
n = 51
Final Answer:
The term of the A.P. that is -80 is the 51st term.

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