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

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

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