Mathematics Moc...

The number of 2 × 2 matrices A, with each element as a real number and satisfying A + A^T = I and A^TA = I, is 

The maximum value of the function f(x) = (x - 1) (x - 2) (x - 3) is -

Below is a figure indicating a sample space S and the probability of occurrences of events A and B.

Which colored section of the portion will indicate the probability of events B and A occurring simultaneously?

Let R be the relation on the set R of all real numbers defined by aRb if |a - b| ≤ 1. Then R is 

The value of \(\sin [2{\cos ^{ - 1}}\frac{{\sqrt 5 }}{3}] \) is,

If y = (1 + x) (1 + x²) (1 + x⁴) _____ (1 + x²ⁿ), then the value of \(\frac{{dy}}{{dx}}\) at x = 0 is 

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0 and x + 4cy + cz = 0 has a non - zero solution then

The inverse of the function \(f\left( x \right) = \frac{{{{10}^x} - {{10}^{ - x}}}}{{{{10}^x} + {{10}^{ - x}}}}\) is,

The value of 'a' so that f(x) = sin x - cos x - ax + b decreases for all real values of x is given by

If f(x) = log_x (log x) then f'(x) at x = e is 

If \(f(α ) = \left[ \begin{array}{l} \cos α \,\sin α \\ - \sin α \,\cos α \end{array} \right]\) and if α, β and γ are angles of a triangle, then xⁿ equal to? ( where |f(α). f(β). f(γ)| = x and n = 3)

The integral \(\int\limits_0^{\pi /2} {\left| {\sin x - \cos x} \right|dx} \) is equal to 

The area in the first quadrant between x² + y² = π² and y = sin x is,

The solution of \(\frac{{dy}}{{dx}} + 2y\,\tan x = \sin x,\) is

A unit vector is coplanar to the vector \(\vec{a}\) = l̂ + ĵ  + 2k̂   and \(\vec{b}\) = l̂ + 2ĵ  + k̂ and perpendicular to the vector \(\vec{c}\) = l̂  + ĵ  + k̂, is 

\(I = \int {\frac{{{\mathop{\rm sin}\nolimits} x}}{{{\mathop{\rm sin}\nolimits} x - {\mathop{\rm cos}\nolimits} x}}} \,dx\) is equal to,

The triangle formed by the points A(0, 7, 10), B(-1, 6, 6) and C(-4, 9, 6) is 

Let the line \(\frac{{x - 1}}{2} = \frac{{y + 2}}{{ - 3}} = \frac{{z - 3}}{4}\) lie in the plane x - y + az + b = 0 The value of b - a is 

The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x - y + z = 3 and at a distance \(\frac{2}{{\sqrt 3 }}\)from the point (3, 1, -1) is ax + by + cz -17 = 0 The value of a + b + c -17 is

For any two events A and B, P(A) = P(A|B) = \(\frac{1}{4}\) and P(B|A) = \(\frac{1}{2}\). Then which of the following is NOT correct?

A and B are two independent witnesses in a case (that means, there is no collision between them). The probability that A will speak the truth is \(\frac{1}{3}\) and the probability that B will speak the truth is \(\frac{3}{4}\). A and B agree on a certain statement then the probability that the statement is true is?

Complete solution set of \(\left| {\frac{{{x^2}}}{{x - 1}}} \right| \le 1\) is given by 

A function f: R → R satisfies the equation f(x + y) = f(x).f(y) for all x, y ∈ R; f(x) ≠ 0. Suppose that the function f(x) is differentiable at x = 0 and f'(0) = 2. If f'(x) = λ. f(x), then the value of λ is 

If y = |cos x - sin x|, then \(\frac{{dy}}{{dx}}\) at \(x = \frac{\pi }{4}\) is

Vikas is paying 50% of EMI under which he pays Rs 800 per month under the flat rate method which will be continued for a period of 5 years. For 10% interest, calculate the total principal amount taken under which EMI is being paid?  

The normal to the curve 5x⁵ - 10x³ + x + 2y + 6 = 0 at A (0, -3) meets the curve again at two points B and C. The length of BC is 

The lines x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other, if 

If \(\int {\frac{{dx}}{{x - {x^3}}} = A.} \) \(ln\left| {\frac{{{x^2}}}{{1 - {x^2}}}} \right| + c\), then the value of A is given by 

Let f(x) = x-[x]. for every real number x, where [x] is the greatest integer less than or equal to x. Then the value of \(\int\limits_{ - 1}^1 {f\left( x \right).dx} \) is 

The curve f(x) = sin x and g(x) = cos x intersects infinitely many times giving the bounded regions of equal areas. The area of one such region is 

A fair dice is thrown six times. The probability of getting factor of 6 four times is equal to,

The solution of the differential equation \(\frac{{dy}}{{dx}} = \frac{y}{x} + \frac{{f\left( {\frac{y}{x}} \right)}}{{f'\left( {\frac{y}{x}} \right)}}\)is (where C is an arbitrary constant)

For all real x, the vectors \(\vec{a}\) = cxî - 6ĵ + 3k̂ and \(\vec{b}\) = xî + 2ĵ +2cxk̂ makes an obtuse angle with each other. Then, the value of c must satisfy which one of the following conditions?

Match List I with List II

Choose the correct answer from the options given below:

Durgesh is working in a restaurant in which he prepares two types of dishes A and B. Dish A takes 20 minutes to be prepared and Dish B takes 30 minutes for the same. He earns Rs 50 to make one packet of dish A, while Rs 70 for one packet of dish B. He works for 12 hours a day. He needs 1 hour in those 12 hours for his personal activities. What will be the objective function, if this example is formulated as a linear programming problem? Assume x and y be the packets of dishes A and B respectively.

The ratio in which the line segment joining the points A(4, 8, 10) and B(6, 10, -8) is divided by the yz - plane is given by 

If a matrix \(A=\begin{pmatrix} 1 &-2 &1 \\ 1&1 &-1 \\ 3 &6 &-5 \end{pmatrix}\)is defined such that \(adj A=\begin{pmatrix} 1 &-4 &1 \\ 2&8 &2 \\ 3&-12 &3 \end{pmatrix}\).Then system of equations,

x - 2y + z = 5,

x + y - z = 6,

3x + 6y - 5z = 7

has-

There is a time-series data which is needed to be represented in the form of the line, y = a + bx, where y is the dependent variable and x is the independent variable time. a and b are the parameters needed to be determined. It has been found that the summation of the product of x and y is equal to 6, while the summation of the square of x is equal to 5. What will be the value of 'b' for this particular example?

The maximum value of \(f(x) = \sin \left( {x + \frac{\pi }{6}} \right) + \cos \left( {x + \frac{\pi }{6}} \right) \) lies in the interval\(\left[ {0,\frac{\pi }{2}} \right] \) if the value of x is?

For x ≥ 1, the value of f(x) = 2 tan^-1 x + sin^-1 \(\frac{{2x}}{{1 + {x^2}}}\)is 

The function f(x) is decreasing in the interval -

f(x) has -

The critical points of f(x) are -

The area formed by the tangent to the curve f(x) at P(2, \(\frac{4}{3}\)) with the co-ordinate axes is -

The local maximum value of the function at one of the critical points is -

Consider the objective function z = 3x + 4y. The constraints for the given objective function are given below :

4x + 3y ≤ 24,

3x + 4y ≤ 24,

and x, y ≥ 0. The above objective function can be maximised -

A continuous random variable x exists for an event such that

\(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - \;e^x,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;if\ \;-1\leq x < 0}\\ {\frac{1}{1 +x^2},\;\;\;\;if\ \;0 \leq x < 1}\\ {e^{-1}-\frac{\pi}{4},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;else} \end{array}} \right.\)

P(-1 < x < 1) is equal to,

Payment of perpetuity is done on a quarterly basis at the end of each period. It is known that the annual interest rate is equal to 100r% and the present value of the perpetuity is Rs 10,00,000. If the regular perpetuity payment is equal to 40,000 Rs then what will be the value of r? 

The cost of a cricket ball is Rs 500. Its marginal revenue in Rs will be,

If X= 48, Y=15, then the value of k such that X mod Y = (X + kY) mod Y