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By using
properties of determinants, show that:

What is the relation between time and number of items sold in the given graph?

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad one will be approved for sale.

A knight is placed somewhere on the chess board. If you have 2 consecutive chances then taking the initial position of knight as the origin which of these position can be the maximum displaced position of the knight?

Let $f\left(x\right)=\{\begin{array}{c}ax,x<2\\ a{x}^2-bx+3,x\ge 2\end{array}$
If f(x) is differentiable for all x, then

If sum upto $(n+1)$ terms for series $\frac{C_0}{1\cdot 2}+\frac{C_1}{2\cdot 3}+\frac{C_2}{3\cdot 4}+\cdots$ is represented by $S_{n+1},$ then the value of $S_{2021}$ is

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$($Here, $C_r={}^n{C}_r)$

If $A$ is an $3\times 3$ non–singular matrix such that $AA\text{'}=A\text{'}A$ and $B=A^{-1}A\text{'}$, then $BB\text{'}$ equals:

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

If matrix $A=\left[\begin{array}{cc}2x& -4\\ -2& x\end{array}\right]$ is singular, then the value(s) of $x$ can be

Let $a,b,c,d$ be distinct real numbers such that $a,b$ are roots of $x^2-5cx-6d=0,$ and $c,d$ are roots of $x^2-5ax-6b=0.$ Then $b+d$ is

Slope at point 'B' is

The mean of discrete observations $y_1,y_2......,y_n$ is given by

If $I$ is the greatest of the definite integrals
$I_1={\int }_0^1{e}^{-x}co{s}^{2}xdx,I_2{\int }_0^1{e}^{-x^2}co{s}^{2}xdx$
$I_3={\int }_0^1{e}^{-x^2}dx,I_4={\int }_0^1{e}^{-x^2/2}dx,$ then

A ray eminating from the point $(-3,0)$ is incident on the ellipse $16x^2+25y^2=400$ at the point $P$ with ordinate $4.$ Equation of the reflected ray is

Let $f\left(x\right)=x^5+a{x}^4+b{x}^3+c{x}^2+dx-420$ where $a,b,c,d$ are real parameters, be a polynomial. If all zeros of the polynomial $f\left(x\right)$ are integers larger than $1,$ and $f\left(4\right)$ is equal to $k,$ then $k$ is divisible by

A curve is such that the length of the intercept on the $x-$axis of the tangent at a point is twice the abscissa and passes through the point $(1,2).$ Then the equation of the curve is:

If the angle between the lines whose direction ratios are 2,-1 , 2 and a, 3, 5 be ${45}^{\circ }$, then a =

Evaluate
the determinants in Exercises 1 and 2.

(i) (ii)

If $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }+\frac{1}{d+\omega }=\frac{2}{\omega },$ where $a,b,c$ are real and $\omega$ is non real cube root of unity, then:

All $x$ satisfying the inequality $({\cot }^{-1}x{)}^2-7({\cot }^{-1}x)+10>0$, lie in the interval :

Equation of the tangent to the conic $x^2-y^2$ - 8x+2y+11=0 at (2,1) is

The area (in sq. units) of the part of the circle $x^2+y^2=36$, which is outside the parabola $y^2=9x,$ is :

Evaluate : $\int \frac{x^3}{(x-1)(x^2+1)}dx.$

OR

Evaluate $\int \frac{sinx-xcosx}{x(x+sinx)}$ dx.

The number of ordered pairs $(x,y),$ satisfying $|x|+|y|=3$ and $\sin \left(\frac{\pi x^2}{3}\right)=1$ is equal to

If the vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ form the sides BC, CA and AB respectively of a $\Delta ABC$, then

The maximum value of $3\cos \theta +5\sin (\theta -\frac{\pi }{6})$ for any real value of $\theta$ is:

The sum of the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$ is , while the product of the roots of the quadratic equation is .