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

If A^TA⁻¹ is symmetric, then A² =

If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

Equation of tangent at (-4, -4) on x² = -4y is _____________

The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is __________

The length of major and minor axes of 4x² + 3y² = 12 are ____________

The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is _____________

If the vectors \(a\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \), \(\overset { \wedge }{ i } +b\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +c\overset { \wedge }{ k } \) (a ≠ b ≠ c ≠ 1) are coplaner, then \(\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } =\) _____________

The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________

If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________

The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is __________

The I.F of y log y \(\frac{dx}{dy}+x-log\ y=0\) is __________

If a random variable X has the p.d.f.\(f(x)=\frac { k }{ { x }^{ 2 }+1 }\) ,0 then k is _____________

The sum of the mean and variance of a binomial distribution for 6 total is 2.16. Then the probability of success p =__________

The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________

Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z is ________

Which of the following is a statement?

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) find \(\frac { \lambda }{ c } \) such that \(\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b } \) is perpendicular to \(\overset { \rightarrow }{ c } \)

Evaluate  \(\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})\).

Find df for f(x) = x² + 3x and evaluate it for
x = 3 and dx = 0.02

If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)

Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²

Find the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).

Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)

Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?

Solve : ydx+(x-y²)dy=0

Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

The girder of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.

Gas is escaping from a spherical balloon at the rate of 900 cm³/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?

Find the intervals for which the function f(x)=2x²-9x²-12x+1 is increasing or decfreasing and find the local extermems.

Find the local maximum and local minimum values for f(x)=12x²-2x²-x⁴.

Let X be a random variable denoting the life time of an electrical equipment having probability density function
\(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\) 
Find
(i) the value of k
(ii) Distribution function 
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time 
(v) P(X = 3)

Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.