Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

Solve the following system of linear equations by matrix inversion method:
2x + 5y = −2, x + 2y = −3

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

If z₁ = 3, z₂ = -7i, and z₃ = 5 + 4i, show that z₁(z₂ + z₃) = z₁ z₂ + z₁ z₃

The complex numbers u, v, and w are related by \(\frac { 1 }{ u } =\frac { 1 }{ v } +\frac { 1 }{ w } \) If v = 3−4i and w = 4+3i, find u in rectangular form.

Explain the falacy:

Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².

Solve the equation x³-3x²- 33x + 35 = 0.

Find the number of real solutions of sin (e^x) -5^x + 5^-x

Solve: (x-1)⁴+(x-5)⁴ = 82

Find the domain of the following
 \(f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right) \)

Solve \({ cot }^{ -1 }x-{ cot }^{ -1 }\left( x+2 \right) =\frac { \pi }{ 12 } ,x>0\)

Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)

Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)

Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
\(\frac { { \left( x-3 \right) }^{ 2 } }{ 225 } +\frac { { \left( y-4 \right) }^{ 2 } }{ 289 } =1\)

The equation of the ellipse is \(\frac { { \left( x-11 \right) }^{ 2 } }{ 484 } +\frac { { y }^{ 2 } }{ 64 } =1\). ( x and y are measured in centimeters) where to the nearest centimeter, should the patient’s kidney stone be placed so that the reflected sound hits the kidney stone?

Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0

For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.

Find the altitude of a parallelepiped determined by the vectors \(\vec { a } =-2\hat { i } +5\hat { j } +3\hat { k } \), \(\hat { b } =\hat { i } +3\hat { j } -2\hat { k } \) and \(\vec { c } =-3\vec { i } +\vec { j } +4\vec { k } \) if the base is taken as the parallelogram determined by \(\vec { b } \) and \(\vec { c } \)

Find the equation of the plane passing through the intersection of the planes 2x + 3y −z + 7 = 0 and and x +y −2z + 5 = 0 and is perpendicular to the plane x +y −3z −5 = 0.

Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)

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s, t

Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

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1

Find the equations of tangent and normal to the curve y = x² + 3x − 2 at the point (1, 2)

Using mean value theorem prove that for, a > 0, b > 0, le^-a - e^-bl < la - bl.

The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

Evaluate the following limits, if necessary use L’Hopitals rule
(i) \(\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ sinx }\)
(ii) \(\underset { x\rightarrow 0 }{ lim } \cfrac { cotx }{ cot2x } \) 
(iii) \(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }\)

If w(x, y) = x³ − 3xy + 2y², x, y ∊ R, find the linear approximation for w at (1,−1)

Let g( x, y) = x² - yx + sin(x+y), x(t) = e^3t, y(t) = t², t ∈ R. Find \(\frac { dg }{ dt } \) 

If f = \(\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } \) then show that = \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) = -f

Find the approximate value of \(\left( \frac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }\) using linear approximation.

Evaluate \(\\ \int _{ 0 }^{ 1 }{ { e }^{ -2x }(1+x-{ 2x }^{ 3 })dx } \)

Evaluate \(\int _{ 0 }^{ 2a }{ { x }^{ 2 }\sqrt { 2ax-{ x }^{ 2 } } } dx\)

If \(f(x)=\left| \begin{matrix} x+1 & 2x+1 & 3x+1 \\ 2x+1 & 3x+1 & x+1 \\ 3x+1 & x+1 & 2x+1 \end{matrix} \right| \) ,then find \(\int _{ 0 }^{ 1 }{ f(x)dx } \) 
[Hint:R₂➝R₂➝R₁;R₂➝R₃➝R₁]

Evaluate \(\int _{ 0 }^{ 1 }{ x(1-x) } ^{ n }dx\)

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)

Solve \({ y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } =xy\frac { dy }{ dx } \)

Verify that y=-x-1 is a solution of the D.E (y-x)dy-(y²-x²)dx=0

Solve :(1+e²x)dy+(1+y²)e^xdx=0

Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.

For the random variable X with the given probability mass function as below, find the mean and variance.
\(f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)

Establish the equivalence property p ➝ q ≡ ㄱp ν q

Verify whether the following compound propositions are tautologies or contradictions or contingency
(p ∧ q) ∧ ¬ (p ∨ q)

Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.