Embibe Experts Solutions for Chapter: UPSC CDS (II) Previous Year Paper 2019, Exercise 1: Elementary Mathematics

Let $a \text{and} b$ be two positive real numbers such that $a\sqrt{a}+b\sqrt{b}=32\text{ and} a\sqrt{b}+b\sqrt{a}=31$. What is the value of $\frac{5\left(a+b\right)}{7}$?

Consider the following statements:

$1)$ Unit digit in ${17}^{174}$ is $7$.

$2)$ Difference of the squares of any two odd numbers is always divisible by $8$.

$3)$ Adding $1$ to the product of two consecutive odd numbers makes it a perfect square.

Which of the above statements are correct?

For what value of $k$ can the expression $x^3+k{x}^2– 7x+6$ be resolved into three linear factors?

Three parallel lines $x, y \text{and} z$ are cut by two transversals$m \text{and} n$. Transversal $m$ cuts the lines$, x, y, z \text{at} P, Q, R$ respectively, and transversal $n$ cuts the lines$x, y, z\text{ at} L, M, N$respectively. If $PQ = 3 \mathrm{cm}, QR = 9 \mathrm{cm} \text{and} MN = 10.5 \mathrm{cm}$, then what is the length of $LM$?

Let the bisector of the angle $BAC$ of a triangle $ABC$ meet $BC$ in $X$. Which one of the following is correct?

The angles of elevation of the tops of two pillars of heights $h \text{and} 2h$ from a point $P$ on the line joining the feet of the two pillars are complementary. If the distances of the foot of the pillars from the point $P$ are $x \text{and} y$ respectively, then which one of the following is correct?

The perimeter of a triangle is $22 \mathrm{cm}$. Through each vertex of the triangle, a straight line parallel to the opposite side is drawn. What is the perimeter of triangle formed by these lines?

The sides $AD, BC$ of a trapezium. $ABCD$ are parallel and the diagonals $AC\text{ and} BD$ meet at $O$. If the area of triangle $AOB \text{is} 3 {\mathrm{cm}}^2$ and the area of triangle$BDC$ is $8 {\mathrm{cm}}^2$, then what is the area of triangle $AOD$?