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For the complex number $z=\frac{3+\sqrt{-1}}{2-\sqrt{-1}}$, the correct option(s) is/are

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equdistant from the two axes, then

If from a point $P$, $3$ normals are drawn, to parabola $y^2=4ax$. Then the locus of $P$ such that three normals intersect the $x-$ axis at points whose distances from vertex are in $A.P.$ is

If the line x-y+k=0 is a normal to $y^2=4ax$ then the value of k is

If a+b+c=0 and |a|=3, |b|=4 and $|c|=\sqrt{37}$, the angle between a and b is

Let $f\left(x\right)=x^4+a{x}^3+b{x}^2+ax+1$ be a polynomial, where $a,b\in R$. If $b=-1$, then the range of $a$ for which $f\left(x\right)=0$ does not have real roots is

If the sets $A$ and $B$ are defined as



$A=\left\{\right(x,y):y=\frac{1}{x},x\in R-\{0\left\}\right\}$

$B=\left\{\right(x,y):y=-x,x\in R\}$, then

If $\int \frac{2dx}{\left(\right(x-5)+(x-7\left)\right)\sqrt{(x-5)(x-7)}}=f\left(g\right(x\left)\right)+c$, then

If $D\ne 0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

Let $\frac{\sin x}{a}=\frac{\cos x}{b}=\frac{\tan x}{c}=2,$ where $0<x<\frac{\pi }{2}$ and $R=bc+\frac{1}{2c}+\frac{2a}{1+2b}.$ Then the minimum value of $R$ is

The point of concurrency of the altitudes drawn from the vertices $A(a{t}_1{t}_2,a(t_1+t_2\left)\right),B(a{t}_2{t}_3,a(t_2+t_3\left)\right)$ and $C(a{t}_3{t}_1,a(t_3+t_1\left)\right)$ of the triangle $ABC$ (where $t_1\ne t_2\ne t_3)$ is

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

In a class of $140$ students numbered $1$ to $140$, all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

A woman has $11$ close friends. The number of ways in which she can invite $5$ of them to dinner, if two particular of them are not on speaking terms and will not attend together, is

If, getting a number greater than $4$ on a fair die is considered a success, then the variance of the distribution of success on tossing a die five times is

If f(x) satisfies f(x+y) = f(x) + f(y) and f(5) = 15 then find f(3)?

If $(1,1)$ and $(-3,5)$ are vertices of a diagonal of a square, then the equations of its sides through $(1,1)$ are

The roots of the equation $i{x}^2-4x-4i$ = 0 are

The transverse axis of a hyperbola is double the conjugate axes. Whats the eccentricity of the hyperbola

The equation of the curve passing through the origin if the middle point of the segment of its normal form any point of the curve to the x-axis, lies on the parabola $2y^2=x$

The domain of the function

$f\left(x\right)=x\cdot \frac{1+2(x+4{)}^{-0.5}}{2-(x+6{)}^{0.5}}+(x+5{)}^{0.5}+4(x+10{)}^{-0.5}$ is

If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle 2 $\theta$ such that $|\overrightarrow{a}-\overrightarrow{b}|$ <1 and 0 $\le \theta \le \pi$, then $\theta$ lies in the interval

Of $\alpha ,\beta$ are the roots of $a{x}^2+c=bx,$ then the equation $(a+cy{)}^2=b^{2}y$ in y has the roots

The least positive integer value of $a$ for which both roots of the quadratic equation $(a^2-6a+5)x^2+\left(\sqrt{a^2+2a}\right)x+(6a-a^2-8)=0$ lie on either side of origin, is -

Equation of the normal to $y^2=4x$ which is perpendicular to $x+3y+1=0$ is

The number of ways in which $3$ scholarships of unequal value be awarded to $17$ candidates, such that no candidate gets more than one scholarship is

The odds in favour of A solving a problem are 3 to 4 and the odds against B solving the same problem are 5 to 7. If they both try the problem, the probability that the problem is solved is:

What is the transpose conjugate for the matrix. $\left[\begin{array}{cc}2+3i& -i\\ 5& 5-i\end{array}\right]$

A straight line, makes an angle of ${60}^{\circ }$ with each of y and z-axis,then the inclination of the line with x-axis is

If A is $3\times 4$ matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type

In a hyperbola the latusrectum equals to semitransverse axis,then its eccentricity is

Equation of the hyperbola with foci (0,± 5) and e = $\frac{5}{3}$ is

If $\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln (4e^x+5e^{-x})+c$, then

Find the set of solution in ${\log }_{\frac{1}{2}}\left(x^2-6x+12\right)\ge -2$

If $k{x}^2+k{y}^2-4x-6y-2k=0$ represents a real circle, then the set of values of $k$ is

If $\text{cosec}\theta =\frac{13}{5}$ and $\theta \notin 1^{\text{st}}\text{quadrant}$. Then the value of $\sec \theta$ is

The number of real solution of:$|2x-x^2-3|=1$ is

The absolute value of ${\int }_{10}^{19}\frac{\sin x}{1+x^8}dx$ is

What is the equation of chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose middle point is $(x_1,y_1)$ ? You are given. $T=\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}-1and{S}_1\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}-1$

If $\sec \theta =max(x+\frac{1}{x}),x\in R,$ where $x<0$, then the value of $\theta$

A maze with following pattern is given. An insect has to reach point R.It can only land only on R if it lands on the adjacent cubes. What is the probability of it reaching R.

If $||2x-x^2+8|-|x^2+5||=|2x+13|$, then $x$ lies in

If $\varnothing$ is an empty set, then ${\varnothing }^{\prime }=$

The sum of the product of the integers 1,2,3,...., n taken two at a time is