Tag: mathematics and statistics

Quizzes (38) Questions (955)

Questions Related to mathematics and statistics

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

If $\Delta =\begin{vmatrix} a _{11} & a _{12} & a _{13}\ a _{21} & a _{22} & a _{23}\ a _{31} & a _{32} & a _{33} \end{vmatrix}$ and $c _{ij}=\left ( -1 \right )^{i+j}$ (determinant obtained by deleting ith row and jth column), then $\begin{vmatrix} c _{11} & c _{12} & c _{13}\ c _{21} & c _{22} & c _{23}\ c _{31} & c _{32} & c _{33} \end{vmatrix}=\Delta ^{2}$



If $\begin{vmatrix} 1 & x & x^{ 2 } \ x & x^{ 2 } & 1 \ x^{ 2 } & 1 & x \end{vmatrix}=7$ and $\Delta =\begin{vmatrix}
x^{3}-1 & 0 & x-x^{4}\
0 & x-x^{4} & x^{3}-1\
x-x^{4} & x^{3}-1 & 0
\end{vmatrix}$, then

  1. $\Delta =7$
  2. $\Delta =343$
  3. $\Delta =-49$
  4. $\Delta =49$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For $\begin{vmatrix} 1 & x & x^{ 2 } \ x & x^{ 2 } & 1 \ x^{ 2 } & 1 & x \end{vmatrix}=7$
$\begin{vmatrix} c _{ 11 } & c _{ 12 } & c _{ 13 } \ c _{ 21 } & c _{ 22 } & c _{ 23 } \ c _{ 31 } & c _{ 32 } & c _{ 33 } \end{vmatrix}=\begin{vmatrix} x^{ 3 }-1 & 0 & x-x^{ 4 } \ 0 & x-x^{ 4 } & x^{ 3 }-1 \ x-x^{ 4 } & x^{ 3 }-1 & 0 \end{vmatrix}$
$\Delta ={ 7 }^{ 2 }=49$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

Let $\Delta _0=\begin{bmatrix}a _{11} & a _{12}  & a _{13}\a _{21}  & a _{22} &a _{23} \ a _{31} & a _{32} & a _{33}\end{bmatrix}$ (where $\Delta _0 \neq  0$) and let $\Delta _1$ denote the determinant formed by the cofactors of elements of $\Delta _0$ and $\Delta _2$ denote the determinant formed by the cofactor at $\Delta _1$ and so on $\Delta _n$ denotes the determinant formed by the cofactors at $\Delta _{n-1}$ then the determinant value of $\Delta _{n}$ is

  1. $\Delta _0^{2n}$
  2. $\Delta _0^{2^n}$
  3. $\Delta _0^{n^2}$
  4. $\Delta _0^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Delta _1=\Delta ^2 _0,\Delta _2=\Delta ^2 _1=\Delta ^{2^2} _0$
$\therefore \Delta _n=\Delta ^{2n} _0$

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

Circle on which the coordinates of any point are $(2+4 \cos \theta,-1+4 \sin \theta)$ where $\theta$ is the parameter is given by $(x-2)^2+(y+1)^2=16$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
From given conditions, we have,
$x=2+4 \cos \theta $
$x-2=4\cos \theta$           .......(1)
$y=-1+4\sin \theta$
$y+1=4 \sin \theta$           .......(2)
Squaring and adding equation 1 and 2, we get,
$(x-2)^2+(y+1)^2=16$.
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

The equation of a line which passes through (2,3) and the product of whose intercepts on the coordinate axis is 27, can be

  1. 5x+4y=22

  2. 3x-y=3

  3. 3x+4y=18

  4. 2x+3y=13
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let the equation of the line is $\dfrac{x}{a}+\dfrac{y}{b}=1$
$\dfrac{2}{a}+\dfrac{3}{b}=1$

$2b+3a=ab$
$3a+2b=27$
$ab=27$
$b=\dfrac{27}{a}$
$3a+2\times \dfrac{27}{a}=27$
$3a^2+54=27a$
$3a^2-27a+54=0$
$a^2-9a+18=0$
$(a-6)(a-3)=0$
$a=6,3$
$b=\dfrac{9}{2}$ or $9$
Required equation is
$\dfrac{x}{6}+\dfrac{y}{\dfrac{9}{2}}=1 \implies 3x+4y=18$
Or,
$\dfrac{x}{3}+\dfrac{y}{9}=1 \implies 3x+y=9$

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If the straight lines joining the origin and the points of intersection of the curve
$ {5x}^{2} + 12xy-{6y}^{2} +4x -2y+3 =0$ and $x+ky-1=0 $ are equally inclined to the co ordinate axis,then the  value of k-

  1. is equal to 1

  2. is equal to -1

  3. is equal to 2

  4. does not exist in the set of real numbers
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For the lines to be equally inclined to the axes, the combined equation of the pair of lines must have the coefficient of xy equal to zero (if inclined at 45 degrees) or the lines must be y = x and y = -x. Solving for k leads to the condition k = -1.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

If the line $AX+BY=1$ passes through point of intersection of $y=x\tan\alpha+p\sec\alpha$,$y\sin(30-\alpha)-x\cos(30^ {o}-\alpha)=p$ and is inclined at $30^ {o}$ with $y=(x\tan\alpha+p\sec\alpha)$ then the value of $a^ {2}+b^ {2}=?$

  1. $\dfrac {1}{p^ {2}}$
  2. $\dfrac {2}{p^ {2}}$
  3. $\dfrac {3}{2p^ {2}}$
  4. $\dfrac {3}{4p^ {2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The intersection point of the given lines involves parameters p and alpha. By calculating the intersection and applying the condition of inclination, the sum of squares of coefficients A and B simplifies to 1/p^2.

Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

A line $OP$ through origin $O$ is inclined at $30^{o}$ and $45^{o}$ to $OX$ and $OY$ respectively. The angle at which it is inclined to $OZ$ is-

  1. $\cos^{-1} \sqrt{\dfrac{4}{6}}$
  2. $\cos^{-1} \left(\dfrac{2}{6}\right)$
  3. $\cos^{-1} \left(\dfrac{1}{2}\right)$
  4. $Not\ defined$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines
Find the equation of the straight line whose $x$ and $y$-intercepts on the axes are given by $2$ and $3$.
  1. $3x-2y+6=0$
  2. $3x+2y-6=0$
  3. $3x-2y-6=0$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Consider the given points.

$(2, 0)$ and $(0, 3)$

We know that the equation of the line which is passing through the points
$y-y _1=\dfrac{y _2-y _1}{x _2-x _1}(x-x _1)$

So,

$y-0=\dfrac{3-0}{0-2}(x-2)$

$y=\dfrac{3}{-2}(x-2)$

$-2y=3x-6$

$3x+2y-6=0$

Hence, Option $B$ is the answer.