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

11) $\frac{\sin\left(18^\circ\right)}{\cos\left(72^\circ\right)}$

A) 1

B) 1/4

C) 0

D) ∞

View Answer

A) 1
Explanation:We are asked to evaluate:
$\frac{\sin(18^\circ)}{\cos(72^\circ)}$
Shortcut Observation:
Use the identity:
$\sin(x) = \cos(90^\circ - x)$
So,
$\cos(72^\circ) = \sin(18^\circ)$
Therefore:
$\frac{\sin(18^\circ)}{\cos(72^\circ)} = \frac{\sin(18^\circ)}{\sin(18^\circ)} = 1$
Final Answer: 1

12) If sinθ = $\frac12$ and θ is acute, then the value of sin2θ is
A)1
B) $\frac{\sqrt3}2$
C) $\frac12$
D) $-\frac{\sqrt3}2$

A) A

B) B

C) C

D) D

View Answer

B) B
Explanation:We are given:
$\sin\theta = \frac{1}{2}, \quad \theta \text{ is acute}$
Shortcut Step 1: Recall standard values
$\sin\theta = \frac{1}{2} \Rightarrow \theta = 30^\circ \quad \text{(since acute)}$
Step 2: Find $\sin 2\theta$
$2\theta = 60^\circ \Rightarrow \sin(60^\circ) = \frac{\sqrt{3}}{2}$
Final Answer: $\frac{\sqrt{3}}{2}$

13) If sinα = cosα, then the value of α is

A) 30°

B) 45°

C) 60°

D) 90°

View Answer

B) 45°
Explanation:We are given:
$\sin\alpha = \cos\alpha$
Shortcut Method: Use standard angles
$\sin\alpha = \cos\alpha \Rightarrow \tan\alpha = 1 \Rightarrow \alpha = 45^\circ \quad \text{(since in acute angles, only at 45° this holds)}$
Final Answer: 45°

14) The angle of elevation of the sun, when shadow of a pole of ‘h’ meter height is √3h metre long is

A) 60°

B) 30°

C) 45°

D) 50°

View Answer

B) 30°
Explanation:We are given:
Height of the pole = h
Length of the shadow = $\sqrt{3}h$
Shortcut Method: Use basic trigonometry
From the triangle formed by the pole and its shadow:
$\tan(\theta) = \frac{\text{height}}{\text{shadow}} = \frac{h}{\sqrt{3}h} = \frac{1}{\sqrt{3}} \Rightarrow \theta = 30^\circ$
Final Answer: 30°

15) The probability that a non leap year will have 53 Thursdays is

A) 1/121

B) 1/7

C) 6/7

D) 9/13

View Answer

B) 1/7
Explanation:To solve this quickly, let’s apply a shortcut method using the concept of distribution of days in a non-leap year:
Step 1: Total days in a non-leap year = 365
That’s 52 weeks + 1 extra day
So, every non-leap year has 52 Thursdays for sure, and whether it has 53 Thursdays depends on that 1 extra day.
Step 2: What can the extra day be?
The extra day could be: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday (7 possibilities).
Out of these 7, only if the extra day is a Thursday, then we will get 53 Thursdays.
Probability =
$\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{7}$
Final Answer: $\frac{1}{7}$

16) A bag contains 4 black balls and 6 red balls, if one ball is drawn at random, then the probability of getting a red ball is

A) 5/8

B) 3/5

C) 1/2

D) 1/56

View Answer

B) 3/5
Explanation:Let’s solve this using the shortcut method:
Step 1: Total number of balls
\= 4 (black) + 6 (red) = 10 balls
Step 2: Number of favorable outcomes (red ball)
\= 6 red balls
Probability =
$\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6}{10} = \frac{3}{5}$
Final Answer: $\frac{3}{5}$

17) E₁ and E₂ are mutually exclusive, then E₁∩E₂=

A) 1

B) 5

C) Φ

D) none of these

View Answer

C) Φ
Explanation:If events E₁ and E₂ are mutually exclusive, it means they cannot happen at the same time.
Shortcut Concept:
Mutually exclusive ⇒ No overlap ⇒
$E_1 \cap E_2 = \varnothing \ (\text{null set})$
Final Answer: Φ (null set)

18) If three coins are tossed, then the total number of outcomes are

A) 2

B) 4

C) 6

D) 8

View Answer

D) 8
Explanation:Shortcut Method:
When n coins are tossed, the total number of outcomes =
$2^n$
Here, n = 3 ⇒
$2^3 = 8$
Final Answer: 8

19) The formula for median in a grouped data is (with usual notation)
A) $L+\frac{{\displaystyle\frac N2}-F}f\times C$
B) $L-\frac{{\displaystyle\frac N2}-F}f$
C) $\frac{{\displaystyle\frac N2}-F}f\times L$
D) $L+\frac{{\displaystyle\frac N2}+F}f\times C$

A) A

B) B

C) C

D) D

View Answer

A) A

20) Mode of 1, 2, 3, 8, 10, 11, 16 is

A) 11

B) 1

C) 16

D) none of these

View Answer

D) none of these
Explanation:Shortcut for Mode:
Mode = Number that appears most frequently
Given numbers:
1, 2, 3, 8, 10, 11, 16
Each number appears once → no repetition.
Final Answer: none of these
(Since no number repeats, there’s no mode)

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