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

11) If α,β are the roots of a quadratic equation ax²+bx+c=0, a≠0 then $\frac1\alpha+\frac1\beta$ = ……..
A) $-\frac ba$
B) $\frac ca$
C) $-\frac bc$
D) $\frac bc$

A) A

B) B

C) C

D) D

View Answer

C) C
Explanation:We are given that α and β are roots of the quadratic equation:
ax² + bx + c = 0, a ≠ 0
We need to find the value of:
$\frac{1}{α} + \frac{1}{β}$
Shortcut Trick:
$\frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$
From the standard formulas for a quadratic:
$α + β = -\frac{b}{a}$
$α β = \frac{c}{a}$
So,
$\frac{1}{α} + \frac{1}{β} = \frac{-\frac{b}{a}}{\frac{c}{a}} = -\frac{b}{c}$
Final Answer: $-\frac{b}{c}$
Want a similar shortcut for $α^2 + β^2$ ?

12) 10th term of an arithmetic progression 2,-1,-4,…… is

A) -21

B) -23

C) -25

D) -27

View Answer

C) -25
Explanation:We are given the arithmetic progression:
2,-1,-4,…
Step 1: Identify first term and common difference
First term a = 2
Common difference d = -1 – 2 = -3
Step 2: Use formula for nth term of AP:
T_n = a + (n-1)d
For the 10th term:
TT₁₀ = 2 + (10-1)(-3) = 2 + 9(-3) = 2 – 27 = -25

13) How many two digit numbers are divisible by 7?

A) 10

B) 11

C) 12

D) 13

View Answer

D) 13
Explanation:To find how many two-digit numbers are divisible by 7, follow this shortcut method:
Step 1: Two-digit numbers range from 10 to 99.
Step 2: Find the first two-digit number divisible by 7:
$\lceil \frac{10}{7} \rceil = 2 ⇒ 2 × 7 = 14$
Step 3: Find the last two-digit number divisible by 7:
$\lfloor \frac{99}{7} \rfloor = 14 ⇒ 14 × 7 = 98$
Step 4: Use AP count formula:
$\text{Count} = \frac{(98 - 14)}{7} + 1 = \frac{84}{7} + 1 = 12 + 1 = 13$

14) The sum of 15 terms of A.P. 3, 6, 9……

A) 315

B) 360

C) 415

D) 460

View Answer

B) 360
Explanation:We are given an A.P.:3, 6, 9, …
This is a common A.P. where:
First term $a = 3$ Common difference d = 6 – 3 = 3
Number of terms n = 15
Use shortcut formula for sum of n terms of A.P.:
$S_n = \frac{n}{2} [2a + (n - 1)d]$
Plug in values:
$S_{15} = \frac{15}{2} [2(3) + (15 - 1)(3)] = \frac{15}{2} [6 + 42] = \frac{15}{2} × 48 = 15 × 24 = 360$

15) The value of x which satisfies the equation 2x-(4-x)=5-x is

A) 4.5

B) 3

C) 2.25

D) 0.5

View Answer

C) 2.25
Explanation:Solve:
2x – (4 – x) = 5 – x
Simplify LHS:
2x – 4 + x = 5 – x ⇒ 3x – 4 = 5 – x
Bring all terms to one side:
3x + x = 5 + 4 ⇒ 4x = 9 ⇒ x = $\frac{9}{4}$ = 2.25

16) Solution of the equations 3x-4y=7 and 2x+3y=-1 is not equal to ……
A) $\frac{22}{22},\frac{33}{33}$
B) $\frac{33}{33},-\frac{44}{44}$
C) $\frac{44}{44},-\frac{77}{77}$
D) $\frac{77}{77},-\frac{11}{11}$

A) A

B) B

C) C

D) D

View Answer

A) A
Explanation:Eliminate incorrect solution:
You are given equations:
⇒1. 3x – 4y = 7
⇒2. 2x + 3y = -1
Now test the options by substituting values.
⇒A. $x = \frac{22}{22} = 1, y = \frac{33}{33} = 1$
1st: $3(1) - 4(1) = 3 - 4 = -1 ≠ 7$
So, this is incorrect.
Correct Answer: $\frac{22}{22}, \frac{33}{33}$
(does notsatisfy the equations)

17) If Σn = 45, then n= ……

A) 9

B) 10

C) 11

D) 12

View Answer

A) 9
Explanation:If Σn = 45, then n = ?
We assume this means the sum of first n natural numbers:
$\sum n = \frac{n(n+1)}{2} = 45 ⇒ n(n+1) = 90$
Try n = 9:
9 × 10 = 90

18) The centre of a circle with (1, 2) and (7,-4) as end points of the diameter is

A) (-4,1)

B) (4,-1)

C) (-4,-1)

D) (4,1)

View Answer

B) (4,-1)
Explanation:Midpoint of diameter endpoints (1, 2) and (7, -4)
Use midpoint formula:
$\left( \frac{1 + 7}{2}, \frac{2 + (-4)}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1)$
Correct Answer: (4, -1)

19) Area of a triangle formed by the line xcosα+ysinα=p with the coordinate axes
A) $\frac{P^2}{2\sin\left(\alpha\right)\cos\left(\alpha\right)}$
B) $\frac{P^2}{\sin\left(\alpha\right)\cos\left(\alpha\right)}$
C) $\frac P{2\sin\left(\alpha\right)\cos\left(\alpha\right)}$
D) $\frac P{\sin\left(\alpha\right)\cos\left(\alpha\right)}$

A) A

B) B

C) C

D) D

View Answer

A) A
Explanation:Area of triangle formed by line with axes
Equation: xcosα + ysinα = p
Find x- and y-intercepts:
Put y = 0:
$x = \frac{p}{\cosα}$
Put x = 0:
$y = \frac{p}{\sinα}$
Area of triangle =
$\frac{1}{2} \cdot \frac{p}{\cosα} \cdot \frac{p}{\sinα} = \frac{p^2}{2\sinα\cosα}$
Correct Answer: $\frac{P^2}{2\sinα\cosα}$

20) If x+7y=7 and 7x-3y=-3, then y =

A) 1

B) 7

C) -3

D) 0

View Answer

A) 1
Explanation:Solve:
1) } x + 7y = 7
2) } 7x – 3y = -3
From (1):x = 7 – 7y
Substitute into (2):
7(7 – 7y) – 3y = -3
49 – 49y – 3y = -3 ⇒ -52y = -52 ⇒ y = 1

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