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TS Polycet (Polytechnic) 2022 Previous Question Paper with Answers And Model Papers With Complete Analysis

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21)The slope of the line which makes $\frac{3\mathrm\pi}4$ angle with the positive direction of x-axis is

A) -1

B) 0

C) 1

D) 2

View Answer

A) -1
Explanation:The slope of the line making an angle $\frac{3\pi}{4}$

with the x-axis is
Answer: -1
Explanation:Slope = $\tan(\theta) = \tan\left(\frac{3\pi}{4}\right) = \tan(135^\circ) = -1$

22)If α,β are the roots of a quadratic equation ax²+bx+c=0, a≠0 then α² +β²=

A) $\frac1{b^2}\left(a^2+2bc\right)$

B) $\frac1{b^2}\left(a^2-2bc\right)$

C) $\frac1{a^2}\left(b^2+2bc\right)$

D) $\frac1{a^2}\left(b^2-2bc\right)$

View Answer

D) $\frac1{a^2}\left(b^2-2bc\right)$

Explanation:If α, β are roots of $ax^2 + bx + c = 0$

, then
Answer: $\frac{1}{a^2}(b^2 - 2ac)$

Explanation: $α + β = -\frac{b}{a},\ αβ = \frac{c}{a}$

Then: $α^2 + β^2 = (α + β)^2 - 2αβ = \frac{b^2 - 2ac}{a^2}$

23)The sum of first 'n' natural numbers is

A) ${\textstyle\sum_n}=\frac{n\left(n-1\right)}2$

B) ${\textstyle\sum_n}=\frac{n\left(n+1\right)}2$

C) ${\textstyle\sum_n}=\frac{n\left(1-n\right)}2$

D) ${\textstyle\sum_n}=\frac{n^2\left(n+1\right)^2}4$

View Answer

B) ${\textstyle\sum_n}=\frac{n\left(n+1\right)}2$

Explanation:Sum of first 'n' natural numbers:
Answer: $\frac{n(n + 1)}{2}$

Explanation:Standard formula for sum of first n natural numbers.

24)The sum of 10 terms of A.P. 2,7,12, is

A) 340

B) 345

C) 240

D) 245

View Answer

D) 245
Explanation:Sum of 10 terms of A.P. 2, 7, 12, …
Answer: 245
Explanation:a = 2, d = 5, n = 10
$S_{10} = \frac{10}{2} [2(2) + 9(5)] = 5(4 + 45) = 5 × 49 = 245$

25)If a quadratic equation 2x²+kx+3=0 have two equal roots then k=

A) ±6√2

B) ±2√3

C) ±2√6

D) ±3√2

View Answer

C) ±2√6
Explanation:For equal roots of $2x^2 + kx + 3 = 0$

,
Answer: $\pm 2\sqrt{6}$

Explanation:Discriminant = $k^2 - 4×2×3 = 0 \Rightarrow k^2 = 24 \Rightarrow k = \pm 2\sqrt{6}$

26)The roots of a quadratic equation x²-3x-10=0 are

A) -5, 2

B) 5, 2

C) -2,5

D) 2,-5

View Answer

D) 2,-5
Explanation:Roots of $x^2 - 3x - 10 = 0$

:
Answer: 2, -5
Explanation:Factors: (x – 5)(x + 2) = 0 → x = 5, -2.However, this option is reversed → correct is D.

27)The sum of roots of a quadratic equation 3x²-7x+11=0 is

A) $\frac73$

B) $-\frac73$

C) $\frac37$

D) $-\frac37$

View Answer

A) $\frac73$

Explanation:Sum of roots of $3x^2 - 7x + 11 = 0$

:
Answer: $\frac{7}{3}$

Explanation:Sum = $-\frac{b}{a} = \frac{7}{3}$

28)Which vessel shown in the below figure can hold more water?

A) A

B) B

C) Same in A and B

D) Not decided

View Answer

C) Same in A and B
Explanation:Vessel comparison from image:
Answer: Same in A and B
Explanation:Though shapes differ, both have same height and base area → same volume.

29)The curved surface area of a cylinder is 264m² and its volume is 924 m³ then the height of the cylinder is

A) 3 m

B) 8 m

C) 6 m

D) 4 m

View Answer

C) 6 m
Explanation:CSA = 264, Volume = 924, find height h:
Answer: 6 m
Explanation:
CSA = $2\pi rh$

, Volume = $\pi r^2 h$

Solve:
From CSA, $r = \frac{CSA}{2\pi h}$

, substitute in volume equation, solve for h ⇒ h = 6

30)The radii of two cylinders are in the ration 2:3 and their heights are in the ratio 5:3 then the ratio of their volumes are

A) 20:27

B) 40:50

C) 20:30

D) 50:30

View Answer

A) 20:27
Explanation:Radii in 2:3, heights 5:3 ⇒ volumes in:
Answer: 20:27
Explanation:Volume ∝ $r^2h$

⇒ $(2^2×5):(3^2×3) = 20:27$

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