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If a+b+c=0, then the solution of the equation $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$ is

All possible values of expression $x^2-4x+9$ is

The set of points on the axis of the parabola $y^2-2y-4x+5=0$ from which all the three normals to the parabola are real is :

If the angle of intersection of the circles $x^2+y^2+x+y=0$ and $x^2+y^2+x-y=0$ is $\theta$, then equation of the line passing through $(1,2)$ and making an angle $\theta$ with the y − axis is

The volume of the tetrahedron with vertices at (1,2,3), (4,3,2), (5,2,7), (6,4,8) is

In throwing a pair of dice, find the probability of getting an odd number on the first die and a total of 7 on both the sides.

The locus of mid points of the chords of the parabola $y^2=4(x+1)$ which are parallel to $3x=4y$ is

A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching $x$-axis, the equation of the reflected ray is

For the three events A, B and C, P (exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C or A occurs)=p and P(all the three events occur simultaneously)=$p^2$, where 0<p<1/2. Then the probability of at least one of the three events A, B and C occuring is

Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin, then $n$ must be of the form

If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$. Then $k$ is equal to

The locus of point of intersection of pair of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ if the sum of ordinates of their point of contact is half the length of minor axis, is

If $A=\{1,3,5\},B=\{2,4,6\},C=\{1,4\}$ then which of the following is universal set?

If tangents are drawn to the parabola $(x-3{)}^2+(y+4{)}^2=\frac{(3x-4y-6{)}^2}{25}$ at the extremities of the chord $2x-3y-18=0$, then angle between tangents is

The number of ways in which 5 identical balls can be kept in 10 identical boxes, if not more than one can go into a box, is

Among the given polynomials equations, select the biquadratic polynomial equation(s).

If $A$ and $B$ are two events such that $P\left(A\right)=\frac{3}{4}$ and $P\left(B\right)=\frac{5}{8}$, then

If the lines $a_{1}x+b_{1}y+c_1=0\text{and}{a}_{2}x+b_{2}y+c_2=0$ cut the coordinates axes in concyclic points, then

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected and $\frac{(1+x{)}^{3/2}-{(1+\frac{1}{2}x)}^3}{(1-x{)}^{1/2}}$ may be approximated as $a+bx+c{x}^2$, then

If $f\left(x\right)=\sqrt{x+3}$ and $g\left(x\right)=1+x^2,$ then $fog\left(x\right)=$ ____.

The lengths of the transverse axis and the conjugate axis of the hyperbola $9x^2-y^2=1$ are and respectively.

If the normal to the parabola $y^2=4ax$ at the point $(a{t}^2,2at)$ cuts the parabola again at $(a{T}^2,2aT)$, then

The number of onto functions $f$ from $\{1,2,3,....,20\}$ to $\{1,2,3,...,20\}$ such that $f\left(k\right)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is :

If the mean deviation of the numbers $1,1+d,1+2d,\dots ,1+100d$ from their mean is $255$, then $|d|$ is equal to

One or more options can be correct:

Find the coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

Which of the following equation(s) ($t$ being the parameter) represents a hyperbola?

In $△ABC,R,r,r_1,r_2,r_3$ denote the circumradius, inradius, the exradii opposite to the vertices $A,B,C$ respectively. Given that $r_1:r_2:r_3=1:2:3$.

The value of $R:r$ is

In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x+y=5$ and $x=4$. Then area of $△ABC$ (in sq. units) is :

If matrix $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$ where a, b, c are real positive numbers, abc = 1 and $A^{T}A=I,$ then the value of $a^3+b^3+c^3$ is ___

If $f\left(x\right)={\sin }^{6}x+{\cos }^{6}x,x\in R,$ then $f\left(x\right)$ lies in the interval

If $\left|\frac{x+1}{x}\right|+|x+1|=\frac{(x+1{)}^2}{|x|},$ then $x\in$

If the distance between the points $(5,-2)$ and $(1,a)$ is $5$ units, then the value of $a$ can be

In a hyperbola e=2 and the length of semitransverse axis is 3 and the length of conjugate axis is

Let $f\left(x\right)=\{\begin{array}{cc}|x^2-3x|+a,0\le x<\frac{3}{2}& \\ -2x+3x\ge \frac{3}{2}& \end{array}$ If f(x) has a local maximum at x =.

The triangle formed by the points (0, 7, 10), (–1, 6, 6),(– 4, 9, 6) is [RPET 2001]

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c, respectively. Of these subjects, the students has a 75% chance of passing in atleast one, a 50% chance of passing in atleast two and a 40% chance of passing in exactly two. Which of the following relations are true?

Match the columns by referring to the definition given below.

"A conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line”.

Which of the following quadratic equations does not have both of its roots lying in the range of $y=3\sin x$?

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0$. If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$, then ${\alpha }_1+{\beta }_2$ equals:

If two adjacent vertices of a regular hexagon are $(0,0)$ and $(1,2),$ then equation of the circumcircle of the hexagon is

A variable circle passes through the point $P(1,2)$ and touches the $x-$axis. The locus of the other end of the diameter through $P$ is

The range of the function $f\left(x\right)=4-\sqrt{x^2-10x+25}$ is

The fourth term in the expansion of $(1-2x{)}^{\frac{3}{2}}$ will be

If $A,B,C,D$ are the angles of a cyclic quadrilateral, then $\cos A+\cos B+\cos C+\cos D$ is equal to

If the vertex of parabola y = $x^2$-8x + c lies on x - axis, then the value of c is

If P is a point on the rectangular hyperbola $x^2-y^2=a^2,$ C is its centre and $S,S^{\prime }$ are the two foci, the $SP\cdot S^{\prime }P=$

The slope intercept form of the line $\frac{x}{2}+\frac{y}{4}=1$ is