Tag: maths

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Questions Related to maths

Multiple choice maths introduction to set different sets de morgan's law de morgan's law for set theory

If the universal set is U = $ \displaystyle \left { 1^{2},2^{2},3^{2},4^{2},5^{2},6^{2} \right }  $   What is the complement of the intersection of set A = $ \displaystyle \left { 2^{2},4^{2},6^{2} \right }  $ and set B=$ \displaystyle \left { 2^{2},3^{2},4^{2} \right }  $ ?  

  1. $ \displaystyle \left \{ 2^{2},4^{2} \right \} $
  2. $ \displaystyle \left \{ 1^{2},5^{2} \right \} $
  3. $ \displaystyle \left \{ 1^{2},5^{2},6 ^{2} \right \} $
  4. $ \displaystyle \left \{ 1^{2},3^{2},5^{2},6^{2} \right \} $

  5. Answer required
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$A\cap B={2^2,4^2}$
$\bar{A\cap B}=U-(A\cap B)={1^2,3^2,5^2,6^2}$
Option D is correct.

Multiple choice maths introduction to set different sets de morgan's law de morgan's law for set theory

In a battle $70\% $ of the combatants lost one eye, $80\% $ an ear, $75\% $ an arm, $85\% $ a leg and $x\% $ lost all the four limbs the minimum value of $x$ is 

  1. $10$
  2. $12$
  3. $15$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the total number of combatants be 100. The number of soldiers who did NOT lose an eye is 30, an ear is 20, an arm is 25, and a leg is 15. The maximum number of soldiers who lost none of these is the sum of those escaping each injury, which is 30 + 20 + 25 + 15 = 90. Therefore, at least 100 - 90 = 10 percent lost all four limbs.

Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

Statement-I: The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$.
Statement-2: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.

  1. (1 ) StatementI is true. Statement-1 is true: Statement--2 is a correct explanation for Statement-1.

  2. (2) StatementI is true, Statement-2 is true: Statement-9 is not a correct explanation for statement-1.

  3. (3) Statement--I is true, Statement-2 is false.

  4. (4) StatementI is false. Statement-2 is true.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mid-point of AB=$\begin{array}{l} = \left( {\dfrac{{3 + 1}}{2},\dfrac{{1 + 3}}{2},\dfrac{{4 + 6}}{2}} \right)\ = (2,2,5)\end{array}$
lies on the plane as it satisfies the equation of the plane
and DR s of AB $ = (2, - 2,2)$
DR s of normal to the plane $= (1, - 1,1)$
AB is the perpendicular bisector.
Hence, A is the image of 2 
 

Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

If the points $(1,2,3)$ and $(2,-1,0)$ lie on the opposite sides of the plane $2x+3y-2z=k$, then

  1. $k< 1$
  2. $k> 2$
  3. $k< 1$ or $k> 2$
  4. $1< k< 2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

 Given plane equation is $2{x}+3{y}-2{z}-k=0$

$(1,2,3)$ and $(2,-1,0)$ lies on the opposite sides of the plane
$(2(1)+3(2)-2(3)-k)(2(2)+3(-1)-2(0)-k)<0$
$(2-k)(1-k)<0\implies (k-1)(k-2)<0$
$\implies 1<k<2$

Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

If the planes $x - cy - bz = 0,cx - y + az = 0\,$ and $bx + ay - z = 0$ pass through a stright line,then the value of ${a^2} + {b^2} + {c^2} + 2abc\,$ is:

  1. $1$
  2. $2$
  3. $3$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
If the planes $(x - cy - bz = 0), (cx - y + az = 0)$ and $(bx + ay - z = 0)$ are in same line.
$\therefore$ They must be collinear.
$\begin{vmatrix}1 & -c & -b\\ c & -1 & a\\ b & a & -1\end{vmatrix} = 0$
$\Rightarrow 1(1 - a^2) + c(-c - ab) -b(ac + b) = 0$
$\Rightarrow 1 - a^2 - c^2 - abc - abc - b^2 = 0$
$\therefore a^2 + b^2 + c^2 + 2abc = 1$
Option A is correct