If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹, then the value of x is

The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

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 x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

\(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}+\sec ^{-1} \frac{5}{3}-\operatorname{cosec}^{-1} \frac{13}{12}\) is equal to

If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________

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 volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

The maximum product of two positive numbers, when their sum of the squares is 200, is

The angle made by any tangent to the curve y = x⁵ + 8x + 1 with the X-axis is a __________

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

If f(x, y) = 2x² - 3xy + 5y + 7 then f(0, 0) and f(1, 1) is _____________

For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

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

The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is

Using y = vx, the differential equation \(\frac { dy }{ dx } =\frac { y }{ x+\sqrt { xy } } \) is reduced to ________.

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

The proposition p ∧ (¬p ∨ q) is

Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z 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 number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2

Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis

The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \), \(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

A particle moves in a line so that x =\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

Evaluate \(\begin{gathered} \text { lim } \\ (x, y) \rightarrow(1,2) \end{gathered}\)g(x, y), if the limit exists, where g\((x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 } \)

Evaluate \(\int { \sum _{ r=0 }^{ \infty }{ \cfrac { { x }^{ r }{ 2 }^{ r } }{ r! } } dx } \)

Determine the order and degree (if exists) of the following differential equations: 
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function \(f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
Find the expected life of this electronic equipment.

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?

If \(\frac { z+3 }{ z-5i } =\frac { 1+4i }{ 2 } \), find the complex number z in the rectangular form

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: 2x+2^x-1+2^x-2 = 7x+7^x-1+7^x-2

If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

Find the equations of the tangent and normal to hyperbola 12x²−9y² = 108 at \(\theta =\frac { \pi }{ 3 } \) (Hint: use parametric form)

Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

Find the local extremum of the function f (x) = x⁴ + 32x

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+{ tan }^{ 3 }x } } \)

Construct the truth table for (-p) v (q ∧ r)

The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1

Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.

Find the principal value of
sec⁻¹(−2).

Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)

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.

Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

Evaluate: \(\underset{x \rightarrow1}{lim} \ x^{\frac{1}{1-x}}\)

Let f(x, y) = sin(xy²) + \(e^{{x^3}+5y}\) for all ∈ R². Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3