If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

z₁, z₂ and z₃ are complex number such that z₁ + z₂ + z₃ = 0 and |z₁| = |z₂| = |z₃| = 1 then z₁² + z₂² + z₃³ is

If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

The polynomial x³ - kx² + 9x has three real zeros if and only if, k satisfies

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

The polynomial x³ + 2x + 3 has

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to

The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x² + y² − 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x − 3)² + (y + 2)² = r² , then the value of r² is

If x + y = k is a normal to the parabola y² = 12x, then the value of k is

The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.

Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.

Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

Prove that the function f(x)=2x²+3x is strictly increasing on \(\left[ -\frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right] \)

Prove that the function f(x)=e^-x is strictky increasing on [0,1]

IF u(x, y) = x² + 3xy + y², x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx } \)

Find the area of the region bounded by the curve \(\sqrt { x } +\sqrt { y } =\sqrt { a } \) (x,y>0) and the co-ordinate axes.

Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that A X B = C.

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \) 

Solve the following system of linear equations by matrix inversion method :
2x  −  y  =  8 ,   3x  +  2y  =  −2.

If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that 
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)

Obtain the Cartesian equation for the locus of z = x + iy in each of the following cases:
|z - 4|²- |z -1 |² = 16

Find solution, if any, of the equation 2cos²x - 9cosx + 4 = 0

Find the domain of sin⁻¹(2−3x²)

Find the value of
\(sin\left( { tan }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) \right) \)

Find the value of  \(tan\left[ \frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 2a }{ 1+{ a }^{ 2 } } \right) +\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-{ a }^{ 2 } }{ 1+{ a }^{ 2 } } \right) \right] \)

Find the equation of the hyperbola in each of the cases given below:
passing through (5, −2) and length of the transverse axis along x axis and of length 8 units.

A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40 cm wide from rim to rim and 30 cm deep. The bulb is located at the focus.
(1) What is the equation of the parabola used for reflector?
(2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

If the straight line joining the points (2, 1, 4) and (a−1, 4, −1) is parallel to the line joining the points (0, 2, b −1) and (5, 3,  −2), find the values of a and b.

Find the coordinates of the foot of the perpendicular drawn from the point (-1, 2, 3) to the straight line \(\vec { r } =(\hat { i } -4\hat { j } +3\hat { k } )+t(2\hat { i } +3\hat { j } +\hat { k } )\). Also, find the shortest distance from the point to the straight line.

Find the parametric form of vector equation and Cartesian equations of a straight line passing through (5, 2,8) and is perpendicular to the straight lines 
\(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )+s(2\hat { i } -2\hat { j } +\hat { k } )\)
\(\vec { r } =(\hat { 2i } -\hat { j } -3\hat { k } )+t(\hat { i } +2\hat { j } +2\hat { k } )\).

Find the values in the interval \((\frac{1}{2},2)\) satisfied by the Rolle's theorem for the function \(f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]\)

Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).

Find the area bounded by the curves y=|x|-1 and y=-|x|+1

Find the area of the region bounded by a²y²=a²(a²-x²)

AOB is the positive quadrant of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

A population grows at the rate of 2% per year. How long does it take for the population to double?

Solve : \({ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5\)

Solve :(x²+xy)dy=(x²+y²)dx

Solve :\(x\frac { dy }{ dx } sin\left( \frac { y }{ x } \right) +x-ysin\left( y\frac { y }{ x } \right) =,y(1)=\frac { \pi }{ 2 } \)

Verify (p ∧ ~p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.