Mathematics Moc...

What is the principal solutions of the equation \(\tan x=-\frac{1}{\sqrt{3}}\)?

If \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\matrix{ {0,\ x = 0} \cr {x-3,\ x>0} \cr } } \right.\)The function f (x) is,

If U = [2 -3 4], X = [0 2 3], V = \(\left[ {\begin{array}{*{20}{c}} 3\\ 2\\ 1 \end{array}} \right]\) and Y = \(\left[ {\begin{array}{*{20}{c}} 2\\ 2\\ 4 \end{array}} \right]\), then UV + XY =

What is the degree of the differential equation \(\rm \left(\frac {d^3y}{dx^3}\right)^{3/2} + \left(\frac {d^2y}{dx^2}\right)^{2} = 0\) ?

If x takes a non-positive permissible value, then sin^-1 x will be-

Let f(x) = e^x, g(x) = sin^-1 x and h(x) = f(g(x)), then h'(x) / h(x) -

Solve:

\(\int_{\rm{0}}^{\frac{{\rm{\pi }}}{{\rm{2}}}} {\frac{{{\rm{f(x)}}}}{{{\rm{f(x) \,+\, f}}\left( {\frac{{\rm{\pi }}}{{\rm{2}}}\,{\rm{ - }}\,{\rm{x}}} \right)}}{\rm{dx}}} \) = ?

If \(A\, = \,\left[ {\begin{array}{*{20}{c}} 2&0&0\\ 0&2&0\\ 0&0&2 \end{array}} \right]\,and\,B\, = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 0&1&3\\ 0&0&2 \end{array}} \right]\,,then\,\left| {AB} \right|\,\) is equal to

The value of c in Lagrange's theorem for the function |x| in the interval [-1, 1] is - 

For a certain curve y = f(x) satisfying \(\frac{{{d^2}y}}{{d{x^2}}}\) = 6x – 4, f(x) has a local minimum value 5 when x = 1. The global maximum value of f(x) if 0 ≤ x ≤ 2, is 

If k is a scalar and I is a unit matrix of order 3, then adj (k I)=

Find the angle between the pair of lines

\(\frac{{x - 5}}{3} = \frac{{y + 2}}{5} = \frac{{z + 2}}{4}\)

And \(\frac{{x - 1}}{1} = \frac{{y - 3}}{1} = \frac{{z - 3}}{2}\)

Find the quantity of methods of picking 4 cards from a set of 52 playing cards if four cards are of the same unit?

If \(\overrightarrow a = 2\widehat i + 2\widehat j + 3\widehat k,\overrightarrow b = - \widehat i + 2\widehat j + \hat k \rm \ and\overrightarrow c = 3\widehat i + \widehat j\) are such that  \(\overrightarrow a + γ \overrightarrow b \) is perpendicular to \(\overrightarrow c \), then determine the value of  γ?

Find the vector and Cartesian equations of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1?

Determine the co-ordinates of the foot of the perpendicular drawn from the origin to the plane 4x - 2y + 3z - 6 = 0

Let a, b, and c be distinct non-negative numbers. If the vectors \(a\hat i + a\hat j +c\hat k\), i\(\hat i +\hat k\) and \(c \hat i +c \hat j + b \hat k\) lie in a plane c is,

f(x) = x + √x2 is a function from R → R. Then f(x) is

In a job placement event, there are 1000 aspirants attending a company interview, out of which 430 are females. It is familiar that out of 430, 10 % of the females are graduate. Determine the probability that an aspirant selected randomly is graduate given that the selected aspirant is a female?

The vectors \(\vec c\), \(\vec a = x\hat i + y\hat j + z\hat k\) and \(\vec b = \hat j \)are such that, \(\vec a, \vec c, \ and \ \vec b\) from a right-handed system, then c is

If \(\mathop \smallint \nolimits_0^{2\pi } \left| {x\;sin\;x} \right|dx = k\pi ,\) then the value of k is equal to ______.

If \(\frac{{x - 1}}{l} = \frac{{y - 2}}{m} = \frac{{z + 1}}{n}\) is the equation of the line passing through (1, 2, -1) and (-1, 0, 1), then (l, m, n) is:

Let g (x) be the inverse of an invertible function f(x) which is differentiable at x = c, then g'(f(c)) equals

If the line ax + by + c = 0 is a tangent to the curve xy = 4, then 

Urn A consists 3 blue and 4 green balls while another urn B consists 5 blue and 6 green balls. One ball is drawn at random from one of the urns and it is found to be blue. Determine the probability that it was drawn from urn B?

For what value of λ, the equations 7x - 5y + 3 = 0, 2x + 3y = 8 and 4x - 6y + λ = 0 are consistent?

If the vertices A,B,C of a triangle ABC are (1, 1, 3), (-1, 0, 0),(0, 1, 2) respectively, then determine ∠ABC. (∠ABC is the angle between the vectors \(\overrightarrow {BA} \) and \(\overrightarrow {BC} \))

Which of the following is the correct inequality for extracting all  possible values of b for which the function \(f(x)=\left\{\begin{matrix} 7x-x^3+\log(b^2-4b+4) &0\leq x<3 \\ x-9&x\geq3 \end{matrix}\right.\) has local maxima at x = 3 are - 

The number of real values of α for which the system of equations

x + 3y + 5z = αx

5x + y + 3z = αy

3x + 5y + z = αz

has infinite number of solution is

Let \(A = \left[ {\begin{array}{*{20}{c}} 2&b&1\\ b&{{b^2} + 1}&b\\ 1&b&2 \end{array}} \right]\) where b > 0. Then the minimum value of \(\frac{{det\left( {\rm{A}} \right)}}{{\rm{b}}}\)is:

The number of solution of the following equations x₂ - x₃ = 1,- x₁ + 2x₃ = - 2, x₁ - 2x₂ = 3 is

If x is a positive integer, then 

\(\Delta = \left| {\begin{array}{*{20}{c}} {x!}&{(x + 1)!}&{(x + 2)!}\\ {(x + 1)!}&{(x + 2)!}&{(x + 3)!}\\ {(x + 2)!}&{(x + 3)!}&{(x + 4)!} \end{array}} \right|\) is equal to

If \(a = \frac{2sinθ}{1 + cosθ + sinθ}\) then \(\frac{1+ \sin \theta - \cos \theta}{1+ \sin \theta}\) is

In a triangle ABC,  \(\overrightarrow {\left| {BC} \right|} = 8,\,\,\overrightarrow {\left| {CA} \right|} \, = 7,\,\,\overrightarrow {\left| {AB} \right|} = 10,\) then the projection of the  \(\overrightarrow {AB\,\,} \) on  \(\overrightarrow {AC\,\,} \) is equal to

If x & y are Random variables such that expectation and variances of X and Y are E(x) = 10, V(x) = 25 and E(Y) = 0, V(Y) = 1, The value of a, b such that y = ax - b will be?

If \(\vec a = 4\hat i + 6 \hat j\) and \(\vec b = 3 \hat j + 4 \hat k\) then the component of \(\vec a\) along \(\vec b\) is

A manufacturer produces two types of products, 1 and 2, at production levels of x1 and x2 respectively. The profit given is 2x1+ 5x2. The production constraints are
x1 + 3x2 ≤ 40
x1 + x2 ≤ 10

x1 ≥ 0, x2 ≥ 0
The maximum profit which can meet the constraints is?

The vectors \(\overrightarrow {AB} = 3 \hat i + 4 \hat k\) and \(\overrightarrow {AC} = 5\hat i - 2\hat j + 4\hat k\) are the sides of a triangle ABC. The length of the median through A is,

If α, β are the roots of the equation 2x² + 3x + 5 = 0, then the value of the determinant \(\left| {\begin{array}{*{20}{c}} 0&{ β}&{ β}\\ α&0&α\\ β&a&0 \end{array}} \right|\) is

Let X be a discrete random variable and f be a function given by \(P(X=x)=f(x)=\frac{1}{2^x}\) for \(x=1,2,3,...\). Then the expected value of X are 

If the slope of the tangent to curve y = x³ at a point is equal to ordinate of point, then point is,

Area bounded by the region R ≡ {(x, y) : y² ≤ x ≤ | y |} is

Which of the following is NOT a known factor to uniquely determine a plane?

Determine the co-ordinates of the point where the line through the points A (2, 3, 2) and B (5, 1, 6) crosses the XY- plane?

The minimum area bounded by circle x²+ y² = 9 and straight line x + y = 3 is

Consider the following probability distribution function defined as,

\({f_x}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\left| x \right|,\;\left| x \right| < 1}\\ {0,\;otherwise} \end{array}} \right.\)

Which of the following is true regarding this PDF?

Which of the following is true for binomial distribution?

A unit vector a is making angle π/3 with î, π/4 with ĵ and an acute angle θ with k̂.

I. The value of θ is π/3.

II. The components of a are \(\frac{1}{2}, \frac{1}{\sqrt 2}, \frac{1}{\sqrt 2}\)

Which of the above statement(s) is/are correct?

The minimum value of z = 5x - 7y, subject to the constraints x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0, y ≥ 0, is given by 

Let A = {x ∈  R | x ≥ 0}. A function f : A → R⁺ is defined by f(x) = x². Which one of the following is correct?