Mathematics Moc...

Which of the following is true?

Evaluate 

\(\mathop \smallint \limits_1^e \sqrt x \ln \left( x \right)dx\)

If  \(f(x) = \left[ {\begin{array}{*{20}{c}} x&λ \\ {2λ }&x \end{array}} \right]\;\) then |f(λx) - f(x)| is equal to ? 

Determine \(\left| {\overrightarrow a } \right| \rm and\left| {\overrightarrow b } \right|,if(\overrightarrow a + \overrightarrow b ).(\overrightarrow a - \overrightarrow b ) = 8\ and\left| {\overrightarrow a } \right| = 8\left| {\overrightarrow b } \right|\)?

if \({\cos ^{ - 1}}\frac{3}{5} - {\sin ^{ - 1}}\frac{4}{5} = {\cos ^{ - 1}}x\), then x =

The minors of -4 and 9 and the co-factors of - 4 and 9 in determinant \(\left| {\begin{array}{*{20}{c}} { - 1}&{ - 2}&3\\ { - 4}&-5&-6\\ { - 7}&{ - 8}&{ 9} \end{array}} \right|\) are respectively

If n (U) = 1000, n (X) = 300, n (Y) = 400 and n (X ∩ Y) = 200 then find n (P (X‘∩ Y’))

The real value of k for which the system of equations 2kx - 2y + 3z = 0, x + ky + 2z = 0, 2x + kz = 0, has non trivial solution is

The values of x for which the given matrix\(\left[ {\begin{array}{*{20}{c}} {{\rm{ - x}}}&{\rm{x}}&{\rm{2}}\\ {\rm{2}}&{\rm{x}}&{{\rm{ - x}}}\\ {\rm{x}}&{{\rm{ - 2}}}&{{\rm{ - x}}} \end{array}} \right]\) will be non-singular are

If f (1) = 1, f' (1) = 3, then the derivative of f(f(f(x))) + (f(x))² at x = 1 is 

If \(\displaystyle \int_{-3}^2 f(x)\:dx=\frac{7}{3}\:and\:\displaystyle \int_{-3}^9 f(x)\:dx=-\frac{5}{6}\), then what is the value of \(\displaystyle \int_2^9f(x)\:dx?\)

Determine the vector equation for the line, given the cartesian equation of a line is \(\frac{{x + 5}}{3} = \frac{{y - 7}}{2} = \frac{{z + 3}}{2}\)?

Determine the vector equation of the plane passing through the intersection of the planes \(\vec{r} \cdot(\hat{\imath}+\hat{\jmath}+\hat{k})=6\) and \(\vec{r}. (2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k})=-5\), and the point (1, 1, 1)?

Determine the direction cosines of the unit vector perpendicular to the plane \(\vec{r} \cdot(2 \hat{\imath}-6 \hat{\jmath}-3 \hat{k})+1=0\) passing through the origin?

If the plane 2x - y + z = 0 is parallel to the line \(\frac{{2x - 1}}{2} = \frac{{2 - y}}{2} = \frac{{z + 1}}{a}\) , then value of a is

If from a point P(a, b, c) perpendiculars PA and PB are drawn to yz and zx planes, then the equation of the plane OAB is

The ratio in which the plane x - 2y + 3z = 17 divides the line joining the points (-2, 4, 7) and (3, -5, 8) is

If 73 ≡ x mod 13, what is the value of x if 50 < x < 65? 

Solve the following Linear Programming Problem Graphically.

Maximize Z = 5x + 8y,

Subject to constraints

2x + 3y ≥ 10 

5x + 2y ≥ 14

x, y ≥ 0

For the LPP, 

A die is thrown 3 times. Events A and B stated below:

4 on the 3rd throw

6 on the 1st and 5 on the 2nd throw

What will be the probability of A knowing that B has already taken place?

If \(g\left( x \right) = x \times \left[ {\frac{1}{x}} \right]\), where [.] is the greatest integer function. Then find the value of \(\mathop \smallint \nolimits_{\frac{1}{3}}^1 g\left( x \right)\;dx\)

The marginal cost is very less than as compared to the marginal revenue for a product. The company selling the product should

If an event E has only one sample point of a sample space, it is called a ______

A player tosses three coin. If the first toss is head. Find the probability that consecutively two head on three coin.

A 10 km race is organized at 800 m circular racecourse. P and Q are the contestants of this race. If the ratio of the speeds of P and Q is 5 : 4, the difference between their speed is 1 m/sec, how many times will the winner overtake the loser?

In (- 4, 4) the function  \(f(x)={\int_{ - 10}^x {({t^4} - 4)e} ^{ - 4t}}dt\) has

A family has two children if the first child is a boy then find prob that other also a boy?

Let P(h, k) be a point on the curve y = x² + 7x +2, nearest to the line y = 3x - 3. Then the equation of the normal to the curve at P is 

Which of the following method is used to analyze the simple trend of the time series data?

If sin^-1 a + sin ^-1 b + sin^-1 c = π, then the value of a \(a\sqrt {(1-a^2)} +b\sqrt {(1-b^2)}+c\sqrt {(1-c^2)}\) will be

A water filling Pipe P is 5 times fast as second Pipe Q. If P fills a cistern in 30 minutes. How long will it take to fill the tank, when both the pipes are kept in operation simultaneous?

The number of vectors of unit length prependicular to vectors a = (1, 1.0) and b = (0, 1.1) is

If \(A = \left[ {\begin{array}{*{20}{c}} 1&{ - 2}&2\\ 0&2&{ - 3}\\ 3&{ - 2}&4 \end{array}} \right]\), then A. adj (A) is equal to

Thea area bounded by curve y = |x| - 1 and y = -|x| + 1 is:

The real values of x satisfying the inequalities:

\(\log _{0.5}(x^{2}-25)-\log _{0.5}(x-5)<\log _{0.5}3\)

In what ratio must tea at Rs. 50 per kg be mixed with tea at Rs. 75 per kg so the mixture must be worth Rs. 60 per kg?

If X & Y two independent Binomial Random variables such that X ∼ B (2, p) Y ∼ B (4, p) if p (x ≥ 1) = 5/9

i. p (y = 4)

ii. V(2x - y) respectively,

Rajesh comes across a person who tells him about a bond having Rs 3,500 as its par value, and redeemable at 7% at the end of 8 years. What will be the present value of the bond if the annual yield rate is 8%? Take (1.08)-8    = 0.54

If \(y\, = \,1\, + \,\frac{1}{x} + \,\frac{1}{{{x^2}}} + \,\frac{1}{{{x^3}}} + \,...\)to ∞ with |x| > 1 then \(\frac{{dy}}{{dx}}\, =\)

Shyam wants to have a minimum of 150 units of carbohydrates and 120 units of proteins. Product A provides 16 units of carbohydrates and 11 units of proteins, while product B provides 10 units of carbohydrates and 20 units of proteins. The cost of product A is Rs 50 per unit while the cost of product B is Rs 60 per unit. Formulate this given situation in a linear programming problem. Consider x and y as units of products A and B consumed respectively. 

The solution of \(\rm {{dy}\over {dx}} = {{x+y}\over {x-y}}\) is 

The minimum value of \({e^{\left( {2{x^2} - 2x + 1} \right)sin^2x}}\) is

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f'(2) = 5, f'(x) ≥ 1 and f''(x) ≥ 4, for all x ∈ (1, 6) then

The angle between the curves y = sin x and y = cos x is

If \(\cos x = \frac{1}{{\sqrt {1 + {t^2}} }}\) and \(\sin y = \frac{t}{{\sqrt {1 + {t^2}} }}\), then \(\frac{{dy}}{{dx}}\, = \)

Let\(f (x) = \sqrt {x - 1} + \sqrt {x + 24 - 10\sqrt {x - 1;} } 1 < x < 26\ \) be real valued function, them f'(x) for 1 <  x < 26 is,

If X = \(\left[ {\begin{array}{*{20}{c}} {\rm{3}}&{{\rm{ - 4}}}\\ 1&{{\rm{ - 1}}} \end{array}} \right]\), then the value of Xⁿ is

The function \(y =\frac{1}{1 +x^2} \)is decreasing in the interval

If  \({D_p} = \left| {\begin{array}{*{20}{c}} p&{15}&8\\ {{p^2}}&{35}&9\\ {{p^3}}&{25}&{10} \end{array}} \right|,then\) D₁ + D₂ + D₃ + D₄ + D₅ =

The differential form of the equation (y - a)² = 2bx is: