List of practice Questions

If \(a,b\) are arbitrary constants, then the differential equation corresponding to family \[ y=ax^2-2abx+ab^2 \] is

If \(\theta\) is angle between parabolas \[ y^2=108x \] and \[ x^2=32y \] then \[ \cos2\theta= \]

If the real valued function \[ f(x)=\log(2x-3)-2x^2+6x-4 \] then the interval in which \(f(x)\) is increasing is

If \(m\) and \(M\) are respectively the absolute minimum and absolute maximum values of the function \[ f(x)=|2x^2-x-6|+2x-3 \] in the interval \([-2,4]\), then \(2M+8m=\)

Evaluate \[ \int \frac{\cos x+\sin2x}{1-\sin x-2\sin^2x}\,dx \]

Evaluate \[ \int\frac{\cos x}{\sqrt{16\cos^2x+9}}dx \]

If \[ f(x)=\pi-\cos^{-1}\left(\frac{x^2+4x+3}{x^2+4x+5}\right) \] then \(f'(1)=\)

If \[ x=\sin2\theta+\sin3\theta,\qquad y=\cos2\theta-\cos3\theta \] then \[ \frac{d^2y}{dx^2} = \]

If \[ f(x)=(1+x^3)(1+x^6)(1+x^{12})(1+x^{24}) \] then \(f'(-1)=\)

By application of derivatives, approximate value of \[ \sqrt[5]{242} \] is

If displacement of particle moving in straight line is \[ S=t^3-3t^2+3t-4 \] then time interval in which \(S\) is increasing is

If \((1,-2,2)\) and \((2,6,-3)\) are the direction ratios of two straight lines, then the direction cosines of the line bisecting an angle between these two lines are

If the image of point \((1,-1,1)\) in the plane \[ x-2y+3z=4 \] is \((x_1,y_1,z_1)\), then \(x_1-y_1-z_1=\)

Let \[ f(x)=\frac{([x]+|x|-x)x}{\sin|x|} \] be a real valued function. If \[ \alpha=\lim_{x\to0^-}f(x),\qquad \beta=\lim_{x\to0^+}f(x) \] then

If the function \[ f(x)= \begin{cases} \dfrac{p(1+\sin3x)}{(\pi+6x)^2}, & -\frac{\pi}{2}[4mm]\\ z, & x=-\frac{\pi}{6}[4mm]\\ \dfrac{q(\sin12x+2\sin6x)}{\cos^3\left(\frac{\pi+12x}{2}\right)}, & -\frac{\pi}{6}& lt;x& lt;0 \end{cases} \] is continuous at \[ x=-\frac{\pi}{6} \] then \(p+2q=\)

The domain of derivative of real valued function \[ f(x)=(x^2-x-2)|x^2+x-6| \] is

If a straight line \[ y=mx+c \] touches the circle \[ x^2+y^2=4 \] and parabola \[ y^2=4x \] then \(2m^2=\)

Let \(S,S'\) be foci and \(B\) one end of minor axis of an ellipse. If \[ \angle SBS'=120^\circ \] and area of triangle \(SBS'\) is \(\sqrt3\), then length of latus rectum is

If \(\theta\) is acute angle between tangents drawn from point \((3,4)\) to ellipse \[ \frac{x^2}{25}+\frac{y^2}{9}=1 \] then \(\theta=\)

If both foci of a hyperbola having eccentricity \(\sqrt3\) lie on x-axis and x coordinates of foci are roots of equation \[ x^2-4x+1=0 \] then length of chord passing through focus and perpendicular to transverse axis is

Let D be harmonic conjugate of point C with respect to points \[ A(1,-3,5),\qquad B(5,-3,1) \] If C divides AB in ratio \(3:5\), then point dividing CD in ratio \(1:2\) is

Let \(L_{1}\equiv3x+4y-1=0,\; L_{2}\equiv8x-6y+1=0,\; L_{3}\equiv12x+9y-1=0\) be three tangents drawn to the circle \[ x^2+y^2+2gx+2fy+c=0 \] and \(L_1& gt;0,\;L_2& gt;0,\;L_3& gt;0\) at the centre \((-g,-f)\). Then \(g+2f=\)

If \[ l_1x+m_1y+n_1=0 \] and \[ l_2x+m_2y+n_2=0 \] are tangents drawn from point \((2,-1)\) to circle \[ x^2+y^2=4 \] then \(n_1+n_2=\)

The external centre of similitude for circles \[ x^2+y^2+10x-16y-11=0 \] and \[ x^2+y^2-2x+4y-4=0 \] is

A circle \(S\) passing through origin cuts another circle \[ x^2+y^2-6x+8y+16=0 \] orthogonally and makes a chord of maximum length on line \[ x-y-2=0 \] then one diameter of circle \(S\) is