I'm Up and Down, and Round and Round MCQs for UPSC Prelims
134 practice questions covering I'm Up and Down, and Round and Round, organised into 1 topics.
134 questions include a written explanation.
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Assertion (A): If $AB$ subtends equal angles at two points $P$ and $Q$ lying on opposite sides of $AB$, then $A$, $B$, $P$ and $Q$ are concyclic. Reason (R): Angles subtended by a segment at two points on the same side of it are equal when all four points lie on one circle.
ABoth A and R are true, and R is the correct explanation of A
BBoth A and R are true, but R is not the correct explanation of A
CA is true, but R is false
DA is false, but R is true
Show answer and explanation
Correct answer: D - A is false, but R is true
(R) states the rule correctly, with the two points on the same side of the segment. (A) moves them to opposite sides, and there the test is different: concyclic points on opposite sides of $AB$ see it under angles that add to $180^\circ$, not under equal angles. So equal angles on opposite sides do not make the four points concyclic, and (A) is false while (R) is true.
Q2
medium
The point $M$ is the midpoint of the chord $AB$ of a circle with centre $O$, with $OM = 8\ \text{cm}$ and $MA = 6\ \text{cm}$. What is the radius of this circle?
A$12\ \text{cm}$
B$14\ \text{cm}$
C$2\ \text{cm}$
D$10\ \text{cm}$
Show answer and explanation
Correct answer: D - $10\ \text{cm}$
Since $M$ is the midpoint of the chord, $OM$ is perpendicular to $AB$, so triangle $OMA$ has a right angle at $M$ and the radius $OA$ is its hypotenuse. Then $OA = \sqrt{8^2 + 6^2} = \sqrt{100} = 10\ \text{cm}$. Adding the two legs gives $14\ \text{cm}$ and subtracting them gives $2\ \text{cm}$, neither of which is how a hypotenuse is found, and $12\ \text{cm}$ is just double $MA$, which is the whole chord.
Q3
medium
Statement 1: Two points $A$ and $B$ on a circle determine exactly one arc. Statement 2: Two points $A$ and $B$ on a circle determine two arcs, and those arcs are unequal unless $AB$ is a diameter. Which is correct?
ABoth statements are true
BNeither statement is true
COnly Statement 1 is true
DOnly Statement 2 is true
Show answer and explanation
Correct answer: D - Only Statement 2 is true
Two points of a circle always split it into two arcs, one on each side of the chord $AB$, so Statement 1 counts one too few. Those two arcs have the same size only when $AB$ is a diameter, and otherwise one is minor and the other major, which is exactly what Statement 2 states.
Q4
hard
For which kind of parallelogram do all four vertices lie on one circle?
AOnly for a rhombus
BOnly for a parallelogram whose diagonals are perpendicular
CFor no parallelogram at all
DOnly for a rectangle
Show answer and explanation
Correct answer: D - Only for a rectangle
In a parallelogram the opposite angles are equal, while in a cyclic quadrilateral they add to $180^\circ$, so each of them must be $\frac{180^\circ}{2} = 90^\circ$, and a parallelogram with four right angles is a rectangle. A rhombus has four equal sides but need not have right angles, and perpendicular diagonals again point to a rhombus. Rectangles are parallelograms that are cyclic, so it is wrong to say no parallelogram ever is.
Q5
hard
Four points $A$, $B$, $C$ and $D$ lie in this order on a circle with centre $O$. The arcs $AB$, $BC$ and $CD$ subtend $60^\circ$, $85^\circ$ and $95^\circ$ at the centre. What angle does the arc from $D$ to $A$ that contains neither $B$ nor $C$ subtend at the centre?
A$60^\circ$
B$120^\circ$
C$180^\circ$
D$240^\circ$
Show answer and explanation
Correct answer: B - $120^\circ$
Travelling once round the circle, the four arcs $AB$, $BC$, $CD$ and $DA$ follow one another, so their central angles fill a complete turn of $360^\circ$. The three given angles add to $60^\circ + 85^\circ + 95^\circ = 240^\circ$, leaving $360^\circ - 240^\circ = 120^\circ$ for the last arc. The value $240^\circ$ is the sum already used, $60^\circ$ copies the first arc, and $180^\circ$ would require $DA$ to be a diameter.
Q6
medium
In a cyclic quadrilateral $PQRS$, $\angle P = 3x$ and $\angle R = 2x$. What is the value of $x$?
A$36^\circ$
B$45^\circ$
C$72^\circ$
D$30^\circ$
Show answer and explanation
Correct answer: A - $36^\circ$
$\angle P$ and $\angle R$ are opposite angles of a cyclic quadrilateral, so $3x + 2x = 180^\circ$, giving $5x = 180^\circ$ and $x = 36^\circ$. The value $72^\circ$ is $\angle R$ itself rather than $x$. Taking $45^\circ$ divides $180^\circ$ by $4$ and $30^\circ$ divides $360^\circ$ by $12$, and neither uses the opposite-angle rule.
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