List of practice Questions

If the angle \(\theta\) between the line \[ \frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2} \] and the plane \[ 2x-y+\sqrt{\lambda}z+4=0 \] is such that \(\sin\theta=\frac13\), then the value of \(\lambda\) is:

Evaluate: \[ \lim_{x\to e}\frac{\log x-1}{x-e} \]

Evaluate: \[ \lim_{x\to \frac{\pi}{3}} \frac{\tan^3x-3\tan x} {\cos\left(x+\frac{\pi}{6}\right)} \]

If \[ f(x)= \begin{cases} \dfrac{\sqrt{2+\cos x}-1}{(\pi-x)^2}, & x\neq \pi\\ k, & x=\pi \end{cases} \] is continuous at \(x=\pi\), then \(k=\)

The set of all the points at which \[ f(x)=|2-|x|| \] is continuous but not differentiable is:

If \[ f(x)=\sqrt{2^{2x}\log(3x-2)} \] then \(f'(2)=\)

A chord is drawn through the focus of the parabola \(y^2=6x\) such that its perpendicular distance from the vertex is \(\frac{\sqrt5}{2}\). Then its slope can be:

A perpendicular is drawn through the vertex \(O\) of the parabola \(y^2=8x\) to any non-vertical tangent meeting the parabola at \(P\). Then \(OP.OQ=\)

The tangents drawn at the points \(P_1\) and \(P_2\) lying on the ellipse \(\frac{x^2}{4}+y^2=1\) are parallel to the chord joining the points \((0,1)\) and \((2,0)\), then the distance between \(P_1\) and \(P_2\) is:

An ellipse intersects the hyperbola \(2x^2-2y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then the equation of the ellipse is:

If \(A(1,1,1)\), \(B(2,3,4)\) and \(C(2,5,7)\) are the vertices of \(\triangle ABC\), then the length of the altitude drawn through the vertex \(A\) is:

If \((K,3,5)\), \((2,-1,2)\) are direction ratios of two lines and the angle between them is \(45^\circ\), then a value of \(K\) is:

The circumcenter of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points \((a^2+1,a^2+1)\) and \((2a,-2a)\), where \(a\neq0\). Then the equation of the parabola passing through the orthocentre is

If the chord \(y=mx+1\) of the circle \(x^2+y^2=1\) subtends an angle \(45^\circ\) at the major segment of the circle, then the value of \(m\) is:

Equation of the circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length \(3\) is:

The tangent to the circle \(C_1:x^2+y^2-2x-1=0\) at the point \((2,1)\) cuts off a chord of length \(4\) units from a circle \(C_2\) whose centre is \((3,-2)\). The radius of circle \(C_2\) is:

If the radical axis of the circles \(x^2+y^2+2gx+2fy+c=0\) and \(2x^2+2y^2+3x+8y+2c=0\) touches the circle \(x^2+y^2+2x+2y+1=0\), then:

The radical axis of two orthogonal circles is \(x+1=0\). If one of those circles is \(x^2+y^2=4\), then the equation of the other circle is:

The following table gives the probability distribution of a random variable \(X\): \[ \begin{array}{|c|c|c|c|c|} \hline x_i & 0 & 1 & 2 & 3 \\ \hline P(X=x_i) & 0.1 & 0.3 & 0.5 & 0.1 \\ \hline \end{array} \] The standard deviation of \(X\) is:

If \(O\) is the origin and \(P\) is a point moving on the straight line \(lx + my + n = 0 \; (n \neq 0)\). If \(Q\) is a point on the segment \(OP\) such that \(OP \cdot OQ = k^2\), where \(k \neq 0\), then the locus of \(Q\) is

If the axes are rotated about the origin in positive direction through an angle of \(30^\circ\), then the transformed equation of \[ x^2+2\sqrt{3}xy-y^2=2a^2 \] is

If a non-horizontal line \(L\) passes through the point \((4,-2)\) and the distance of \(L\) from the origin is \(2\) units, then the equation of the line \(L\) is

If the equation of the line passing through the orthocentre and circumcentre of triangle \(ABC\), whose vertices are \(A(3,1)\), \(B(3,3)\), \(C(6,1)\), is \(2x+by+c=0\), then \(b+c=\)

The equations of the pairs of opposite sides of a parallelogram are \[ x^2-5x+6=0 \] and \[ y^2-6y+5=0. \] Then the equations of its diagonals are

If the mean of the set of values \( (3,6,9,10,n) \) is \( 9 \), then the variance of the given set of values is: