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The domain of the function $f\left(x\right)=\frac{1}{[x{]}^2-7[x]+10}$ is
(where $[.]$ denotes the greatest integer function)

If $\alpha$ and ${\beta }^2$ are the roots of the equation $8x^2-10x+3=0,$ where ${\beta }^2>\frac{1}{2},$ then an equation whose roots are $(\alpha +i\beta {)}^{100}$ and $(\alpha -i\beta {)}^{100}$ is

Let $A=(\sin \theta +1,\cos \theta )$ and $B=(1-\cos \theta ,-\sin \theta )$. Then the maximum value of $AB$ is

Select the correct approach for solving a pair of linear equations in 2 variables by elimination method.

i) Add or subtract one equation from the other so that one variable gets eliminated.

ii) Multiply both the equations by any non-zero constant to make the coefficients of one variable (either x or y) numerically equal.

iii) Solve the equation in one variable (x or y) to get its value.

iv) Substitute the value of x (or y) in either of the original equations to get the value of the other variable.

Let $z_1=10+6i$ and $z_2=4+6i,$ where $i=\sqrt{-1}.$ If $z$ is any complex number such that $\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4},$ then the value of $|z-7-9i|$ is

The Boolean expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is equivalent to

The differential equation for all the straight lines which are at a unit distance from the origin is

Let $P(1,1)$ be a point inside the circle $x^2+y^2+2x+2y-8=0.$ The chord $AB$ is drawn passing through the point $P.$ If $\frac{PA}{PB}=\frac{\sqrt{5}-2}{\sqrt{5}+2},$ then equation of chord $AB$ is

A two-digit number $\overline{ab}$ is called almost prime if one obtains a two-digit prime number by changing at most one of its digit $a$ and $b.$ (For example, $18$ is an almost prime number because $13$ is a prime number). Then the number of almost prime two-digit numbers is

For what values of $x\in (2\pi ,8\pi ),\sin x\le 0?$

The total number of terms in the expansion of ${\left((1+x^{2/3})(1+x^{4/3}-x^{2/3})\right)}^{2021}$ is

Out of 10 students there are 6 girls and 4 boys.A team of 4 students is selected at random.Find the probability that there are at least 2 girls.

If the distance of the point $P(1,-2,1)$ from the plane $x+2y-2z=\alpha$, where $\alpha >0$, is $5$, then the foot of the perpendicular from $P$ to the plane is

Sketch the graph of the following functions:
y= 2 cot 2x

If the point (2, k) lies outside the circles
$x^2+y^2+x-2y-14=0$ and $x^2+y^2=13$
then k lies in the interval

Which of the following is true about multiplication of a vector by a scalar.

1) Scalar multiplication by a positive number other than 1 changes its magnitude but not direction.

2) Scalar multiplication always lead to change in magnitude and direction.

3) Scalar multiplication by -1 will not change the magnitude of the vector but will change its direction.

4) Scalar multiplication by a negative number other than -1 will reverse its direction and change its magnitude as well.

For the given graph of $f\left(x\right),$ select the correct graph of $f\left(|x|\right).$

How many terms of the G.P. 3, $\frac{3}{2},\frac{3}{4},....$ be taken together to make $\frac{3069}{512}$ ?

Which of the following is/are equal to $\int {\sin }^{5}xdx$

(where $C$ is integration constant)

Eleven animals of a circus have to be placed in eleven cages one in each cage. If 4 of the cages are too small for 6 of the animals, find the number of ways of caging the animals.___

The value of $\sum _{r=0}^{20}{}^{50-r}{C}_6$ is equal to

The error in ${\begin{array}{c}\frac{d}{dx}f\left(x\right)\end{array}|}_{x=x_0}$ for a continuous function estimated with h = 0.03 using the central difference formula



${\begin{array}{c}\frac{d}{dx}f\left(x\right)\end{array}|}_{x=x_0}=\frac{f(x_0+h)-f(x_0-h)}{2h}$ is $2\times {10}^{-3}$. The value of $x_0$ and $f\left(x_0\right)$ are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately.

The range of $f\left(x\right)=\frac{x^2}{x^4+1}$ is

If the function $f\left(x\right)=\sqrt{{0.25}^{4-3x}-4^{2x}}$ is defined, then the possible of $x$ is/are

If $a={\sin }^4(\frac{3\pi }{2}-\alpha )+{\sin }^4(3\pi +\alpha )$ and $b={\sin }^6(\frac{\pi }{2}+\alpha )+{\sin }^6(5\pi -\alpha )$, then the value of $3a-2b$ is

The equation of the circle inscribed in the triangle formed by the straight line $4x+3y=6$ and both the coordinate axes is

The complete set of values of $k$ for which the equation $4^x-(k+2)2^x+2k=0$, has exactly one positive root is

Let $x,y$ be the length and breadth of a rectangular field and they are prime numbers. If the area of the rectangular field is an odd number less than $35$ and the perimeter is a number less than $24,$ then the value(s) of $x-y$ is/are

If all the letters of the word $RANK$ are rearranged to form $4$ letter words and arranged in ascending order as in a dictionary, then the rank of the word $RANK$ is