If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

If A = [2 0 1] then the rank of AA^T is ______

If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_o g)(x), then the degree of h is

If ax² + bx + c = 0, a, b, c \(\in\) R has no real zeros, and if a + b + c < 0, then __________

The equation \(\tan ^{-1} x-\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)has

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is

The angle between the lines \(\frac{x-2}{3}=\frac{y+1}{-2}, z=2 \text { and } \frac{x-1}{1}=\frac{2 y+3}{3}=\frac{z+5}{2}\) is

The point of inflection of the curve y = (x - 1)³ is

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) 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 \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is

The value of \(\int _{ 0 }^{ \frac { \pi }{ 3 } } { tan } x \ dx\) __________

The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

The differential equation associated with the family of concentric circles having their centres at the origin is _________.

If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

Determine the truth value of each of the following statements:
(a) 4 + 2 = 5 and 6 + 3 = 9
(b) 3 + 2 = 5 and 6 + 1 = 7
(c) 4 + 5 = 9 and 1 + 2 = 4
(d) 3 + 2 = 5 and 4 + 7 = 11

The identity element in the group {R - {1},x} where a * b = a + b - ab is __________

Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.

Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.

Find the acute angle between the following lines
2x = 3y = −z and 6x = − y = −4z.

Determine the domain of convexity of the function y=e^x

If w(x, y, z) = x² y + y²z + z²x, x, y, z∈R, find the differential dw .

Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.

Solve the Linear differential equation:
\(\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y\)

A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
\(f(x)= \begin{cases}\frac{1}{30} & 0

For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?

Simplify the following:
i ¹⁷²⁹

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

Solve: (x-1)⁴+(x-5)⁴ = 82

Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)

Identify the type of the conic for the following equations:
(1) 16y² = −4x²+64
(2) x²+y² = −4x−y+4
(3) x²−2y = x+3
(4) 4x²−9y²−16x+18y−29 = 0

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

Compute the value of 'c' satisfied by Rolle’s theorem for the function \(f(x)=log(\frac{x^{2}+6}{5x})\) in the interval [2, 3]

Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form

The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3

Using the l’Hôpital Rule prove that, \(\underset{x\rightarrow 0^{+}}{lim}(1+x)^{\frac{1}{x}}=e\)

Prove that f(x, y) = x³ - 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.

Find the area of the region common to the circle x²+y²=16 and the parabola y²=6x.

The equation of electromotive force for an electric circuit containing resistance and self inductance is E = Ri + L\(\frac{di}{dt},\) Where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0.

Sovle \(\left( x+2 \right) \frac { dy }{ dx } =x2+4x-9\) .Also find the domain of the function.