AP Polycet (Polytechnic) 2021 Previous Question Paper with Answers And Model Papers With Complete Analysis

31) The product of the zeroes of x + 2x² + 1 is

A) -1
B) 2
C) 1
D) ½

View Answer

D) ½

Explanation:We are given the quadratic polynomial:
$f(x) = x + 2x^2 + 1$
Rewriting it in standard form:
$f(x) = 2x^2 + x + 1$
Shortcut to find product of zeroes of a quadratic:
For a quadratic $ax^2 + bx + c$ ,
Product of zeroes = $\frac{c}{a}$
Here:
$a = 2$ $c = 1$
$\text{Product} = \frac{1}{2}$
Final Answer: D. ½

32) The zeroes of the polynomial x³ – x² are

A) 0,0,1
B) 0,1,1
C) 1,1,1
D) 0,0,0

View Answer

A) 0,0,1

Explanation:We are given the cubic polynomial:
$f(x) = x^3 - x^2$
Step 1: Factor the polynomial
Factor out the common term:
$f(x) = x^2(x - 1)$
Step 2: Find the zeroes
Set $f(x) = 0$ :
$x^2(x - 1) = 0 \Rightarrow x = 0,\ 0,\ 1$
So the zeroes are 0, 0, and 1.
Final Answer: 0, 0, 1

33) The quadratic polynomial whose zeroes are α,β is

A) x²-(α+β)x+αβ
B) x²+(α+β)x
C) x²-α-βx+αβ²
D) None of these

View Answer

A) x²-(α+β)x+αβ

Explanation:To form a quadratic polynomial with given zeroes $\alpha$ and $\beta$ , use the standard form:
$\boxed{x^2 - (\alpha + \beta)x + \alpha\beta}$
Final Answer: $x^2 - (\alpha + \beta)x + \alpha\beta$

34) The equation x-4y=5 has

A) no solution
B) unique solution
C) two solution
D) infinitely many solutions

View Answer

D) infinitely many solutions

Explanation:To determine the nature of the solutions for the equation $x - 4y = 5$ , let’s analyze it step-by-step.
Step 1: Understand the Equation
The given equation is:
x – 4y = 5
This is a linear equation in two variables ( $x$ and $y$ ).
Step 2: Express in Slope-Intercept Form
Rewriting the equation to solve for $y$ :
x – 4y = 5
-4y = -x + 5
$y = \frac{1}{4}x - \frac{5}{4}$
This is now in the slope-intercept form y = mx + b , where:
– Slope (m): $\frac{1}{4}$
– Y-intercept (b): $-\frac{5}{4}$
Step 3: Analyze the Solutions
A linear equation in two variables represents a straight line on the Cartesian plane. For such equations:
– Every point (x, y) on the line is a solution to the equation.
– Since a line extends infinitely in both directions, there are infinitely many solutions.
Step 4: Verify with Examples
Let’s find a few solutions to confirm:
→1. If $x = 1$ :
$y = \frac{1}{4}(1) - \frac{5}{4} = -1$
So, $(1, -1)$ is a solution.
→2. If $x = 5$ :
$y = \frac{1}{4}(5) - \frac{5}{4} = 0$
So, $(5, 0)$ is another solution.
→3. If $x = 9$ :
$y = \frac{1}{4}(9) - \frac{5}{4} = 1$
So, $(9, 1)$ is yet another solution.
This pattern continues infinitely, demonstrating that there are infinitely many solutions.
Step 5: Conclusion
The equation $x - 4y = 5$ represents a straight line with infinitely many points (solutions).
Final Answer: Infinitely many solutions