Embibe Experts Solutions for Chapter: UPSC CDS (I) Previous Year Paper 2022, Exercise 1: Elementary Mathematics

If $\frac{\cos \theta }{\cos ec+1}+\frac{\cos \theta }{\cos ec \theta -1}=2$ where $0<\theta <{90}^o$, then what is the value of ${\sin }^4 \theta +{\cos }^2 \theta$?

Consider the following data for the question given below:

The table below gives the age-wise population percentage of a city:

The number of persons below the age of $40$ years is $10.5$ lakhs.

What is the total population of the city (in lakhs)?

Consider the following for the question that follows:

$500$ candidates appeared in an examination comprising tests in English, Hindi and Mathematics. $30$ candidates failed in English only; $75$ failed in Hindi only; $50$ failed in Mathematics only; $15$ failed in both English and Hindi; $17$ failed in both Hindi and Mathematics; $17$ failed in both Mathematics and English; $5$ failed in all three tests.

Consider the following for the question that follows:

$500$ candidates appeared in an examination comprising tests in English, Hindi and Mathematics. $30$ candidates failed in English only; $75$ failed in Hindi only; $50$ failed in Mathematics only; $15$ failed in both English and Hindi; $17$ failed in both Hindi and Mathematics; $17$ failed in both Mathematics and English; $5$ failed in all three tests.

What is the ratio of interior angle to exterior angle of a regular polygon of $n$ sides?

The arithmetic mean and the geometric mean of two positive numbers $p$ and $q(p>q)$ are $A$ and $G$ respectively. Which one of the following is correct?

Consider a question and two statements :

Question:

Is ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2-\mathrm{ab}-\mathrm{bc}-\mathrm{ca} (\mathrm{a}, \mathrm{b}, \mathrm{c}$ are distinct real numbers) always positive?

Statement-I : $\mathrm{a}>\mathrm{b}>\mathrm{c}$

Statement-II : $\mathrm{a}+\mathrm{b}+\mathrm{c}=0$

Which one of the following is correct in respect of the question and the statements?

Consider a question and two statements :

Question:

Is $\frac{x^6+y^6}{x^4+y^4}$ always greater than $\frac{x^4+y^4}{x^2+y^2}(x\ne y\ne 0)$?

Statement-I : $\mathrm{x}>\mathrm{y}$

Statement-II : $x^2+y^2>2xy$

Which one of the following is correct in respect of the question and the statements?