List of top Questions asked in TS EAMCET

The foci of the ellipse \[ \frac{x^2}{25}+\frac{y^2}{16}=1 \] and that of the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] are same. The greatest length of the transverse axis of the hyperbola such that the difference of the squares of their eccentricities is at least one is

If \[ \frac{x+3}{a}=\frac{y-1}{b}=\frac{z}{c} \] is the common line of the two perpendicular planes \[ 2x+3y+4z+d=0 \] and \[ x+py+z+5=0, \] then \(p+d=\)

The radical axis of the circles \[ x^2+y^2+4x+6y+7=0 \] and \[ 4x^2+4y^2+8x+12y-24=0 \] is a tangent to the circle \[ x^2+y^2=13 \] at a point \((\alpha,\beta)\), then \(\alpha+\beta=\)

If the latus rectum of the ellipse \[ \frac{x^2}{16}+\frac{y^2}{b^2}=1 \] subtends an angle \[ \frac{\pi}{3} \] at the centre of the ellipse, then \(b^2=\)

If a normal drawn at the point \(t=1\) on the parabola \[ x^2+4y+2x-8=0 \] intersects the parabola again at a point \(A(\alpha,\beta)\), then \(4\alpha\beta=\)

If \[ P\equiv y^2+2y-x+6=0 \] is a parabola. \(S=0\) is another parabola such that the axes of \(S=0\) and \(P=0\) are same and the distance between their foci is \[ \frac54. \] If the foci of these parabolas are on either side of the vertex of \(S=0\) and the vertex of \(S=0\) is on the directrix of \(P=0\), then the length of latus rectum of \(S=0\) is

If the locus of midpoints of the chords of the circle \[ S\equiv x^2+y^2-6x-8y-11=0 \] which subtend a right angle at \(A(1,2)\) is another circle \(S'=0\), then the centre of \(S'=0\)

Let \[ S\equiv x^2+y^2-6x+4y+c=0,\qquad S'\equiv x^2+y^2-4x+6y+9=0, \] \[ S''\equiv x^2+y^2+5x+3y+k=0 \] be three circles. If the angles of intersection of the circle \(S'=0\) with the circles \(S=0\) and \(S''=0\) are respectively \[ \frac{\pi}{4} \quad\text{and}\quad \frac{\pi}{2}, \] then \(c+k=\)

Let \(L=0\) be the chord of contact of \((5,1)\) with respect to the circle \[ S\equiv x^2+y^2+8x+10y-8=0. \] If the pole of \(L=0\) with respect to the circle \[ S'\equiv x^2+y^2+4y-21=0 \] is \((-k,-h)\), then \(k+h=\)

The locus of midpoints of the chords of the circle \[ x^2+y^2-2x-2y+1=0 \] which are parallel to the line \[ x+y+2=0 \] is

If the equation of the line joining the points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is \[ ax+by=c \] and the distance between \(A\) and \(B\) is \(\sqrt{a^2+b^2}\), then \(c^2=\)

Equation of the image of the pair of lines \[ ax^2+2hxy+by^2=0 \] with respect to \(y=0\) is

Two diagonal sides of a parallelogram \(ABCD\) are \[ x-y+1=0 \] and \[ 3x-y=9. \] If one of its diagonals is \[ 7x-5y+3=0, \] then the equation of the other diagonal is

The line common to the two families of concurrent lines \[ x+y+\lambda(4x+3y-1)=0 \] and \[ 2x-3y+k(x+y-5)=0 \] is along the base of a triangle. If \((1,1)\) is the vertex of the triangle, then the perpendicular distance from that vertex to its base is

The locus of the centre of a circle touching the lines \[ x+2y=0 \] and \[ x-2y=0 \] is

Let \(X\) be a discrete random variable. If \[ P(X=r)=\frac{3}{7}a^r,\qquad r=0,1,2,\ldots,\infty,\quad 0<a<1, \] then \(aP(X=2)=\)

Let \(A(a,0)\) be a point on the \(X\)-axis. The locus of the midpoints of the line segments joining \(A\) and any point \(P\) on the parabola \[ y=16-x^2 \] is

Which of the following is not the correct data of a binomial distribution?

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spade cards. The probability that the missing card is not a spade is

Let \(E\) and \(F\) be two events such that \[ P(\bar E)=\frac13,\qquad P(E\cap F)=\frac13,\qquad P(\overline{E\cup F})=\frac16. \] Which of the following is correct?

The transformed equation of \[ 4x^2-4xy+7y^2-24=0 \] in the new coordinate system when the axes are rotated through an angle \(\theta\) about the origin in the positive direction is \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] If \[ 0<\theta<\frac{\pi}{4}, \] then \[ 1+b^2\sin^2\theta= \]

If \[ \vec r=2\hat i-\hat j+2\hat k,\qquad \vec s=3\hat i-3\hat j+3\hat k,\qquad \vec t=\hat i+2\hat j+\hat k \] are three vectors and \(\vec a\) is a vector such that \[ \vec s\times\vec a=\vec r\times\vec a \] and \[ \left|\vec t\times\vec a\right|=\sqrt{128}, \] then \(|\vec t-\vec a|=\)

If \[ \vec r=\vec a+t\vec b \quad\text{and}\quad \vec r=\vec p+l\vec b \] are two lines, where \[ \vec a=\hat i+2\hat j,\qquad \vec b=\hat i+\hat j+\hat k,\qquad \vec p=2\hat i+\hat k. \] The position vector of a point \(A\) is \[ 5\hat i+\hat j-3\hat k. \] Let \(B,C\) be the feet of the perpendiculars drawn from \(A\) to the given two lines respectively, then \[ \overrightarrow{BA}+\overrightarrow{AC}= \]

If \(P(x,y,z)\) is the intersection of the lines \[ \vec r=(2\hat i-2\hat j+3\hat k)+t(\hat i-3\hat j+\hat k) \] and \[ \vec r=(2\hat j-\hat k)+s(\hat i-\hat j+3\hat k), \] then \(x+y+z=\)

If \(\alpha,\beta,\gamma\;(\alpha>\beta>\gamma)\) are roots of the equation \[ x^3-x^2-4x+4=0, \] the volume of the parallelepiped whose coterminous edges are \[ \alpha\hat i+\beta\hat j+\gamma\hat k,\; \beta\hat i+\gamma\hat j+\alpha\hat k,\; \gamma\hat i+\alpha\hat j+\beta\hat k \] is