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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

35) If ax+b=0, then x=

A) -a

B) a

C) b/a

D) -b/a

View Answer

D) -b/a
Explanation:We are given the equation:
ax + b = 0
To solve for $x$ :
$ax = -b \Rightarrow x = \frac{-b}{a}$
Final Answer: $-\frac{b}{a}$

36) Which of the following in not a linear equation

A) 3x-2y = y+x

B) x+y = 1

C) 1+2x = y-5

D) 3-y = x²+4

View Answer

D) 3-y = x²+4
Explanation:To determine which is not a linear equation, we check for variables raised to powers other than 1 (i.e., squares, roots, etc.).
Definitions:
A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable, and no variable is raised to a power other than 1.
Options:
A. 3x – 2y = y + x
Simplifies to 2x – 3y = 0 — Linear
B. x + y = 1
Already linear
C. 1 + 2x = y – 5
Simplifies to $2x - y = -6$ — Linear
D. $3 - y = x^2 + 4$
Has $x^2$ , which is non-linear
Correct Answer: $3 - y = x^2 + 4$

37) Which of the following represents the situation where Siri bought 5 apples and 6 oranges and Laxmi bought 2 apples and 15 oranges for same amount of total money

A) 5x+6y=2x+15y

B) 5x+15y=6x+2y

C) 5x-6y=2x-15y

D) 5x-15y=6x-2y

View Answer

A) 5x+6y=2x+15y
Explanation:We are given:
Siri bought 5 apples and 6 oranges → Cost: 5x + 6y
Laxmi bought 2 apples and 15 oranges → Cost: 2x + 15y
Total cost is same for both → So, equate both expressions:
5x + 6y = 2x + 15y
Correct Answer: 5x + 6y = 2x + 15y

38) Which of the following is a quadratic equation

A) x(x+4)=12

B) x(x+4)=x²+2x+1

C) x(x+4)-x(x-2)=0

D) x(x+4)=x(x+5)-x

View Answer

A) x(x+4)=12
Explanation:Let’s check each option to see which represents a quadratic equation (i.e., an equation of the form $ax^2 + bx + c = 0$ ):
A. x(x + 4) = 12
→ Expand LHS: $x^2 + 4x = 12$
→ Rearranged: $x^2 + 4x - 12 = 0$ Quadratic!
B. $x(x + 4) = x^2 + 2x + 1$
→ LHS: $x^2 + 4x$ , RHS: $x^2 + 2x + 1$
→ Subtract RHS: $x^2 + 4x - x^2 - 2x -1 = 0 \Rightarrow 2x - 1 = 0$
Linear, not quadratic
C. x(x + 4) – x(x – 2) = 0
→ Expand: $x^2 + 4x - (x^2 - 2x) = 0$
→ Simplify: $x^2 + 4x - x^2 + 2x = 0 \Rightarrow 6x = 0$
Linear
D. x(x + 4) = x(x + 5) – x
→ LHS: $x^2 + 4x$ , RHS: $x^2 + 5x - x = x^2 + 4x$
→ Both sides equal ⇒ 0 = 0
Identically true, not a quadratic equation
Correct Answer: $x(x+4) = 12$

39) Any equation of the form p(x)=0, where p(x) is a polynomial of degree 2 is called

A) linear equation in one variable

B) linear equation in two variables

C) quadratic equation

D) none of these

View Answer

C) quadratic equation

40) The equation x²+x-306 = 0 represents that the

A) sum of two consecutive positive integers is 306

B) product of two consecutive positive integers is 306

C) sum of squares of two consecutive positive integers is 306

D) product of squares of two consecutive positive integers is 306

View Answer

B) product of two consecutive positive integers is 306

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