The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is_________.

If any three rows or columns of a determinant are identical then the value of the determinant is ________.

The value of n, when nP₂ = 20 is _______.

Thirteen guests has participated in a dinner. The number of handshakes happened in the dinner is __________.

The eccentricity of the parabola is _______.

The distance between directrix and focus of a parabola y² = 4ax is _______.

The value of sec A sin(270^o + A) is ______.

\(\sin\left(\cos^{-1}\frac{3}{5}\right)\) is _____.

If f(x) = x² and g(x) = 2x + 1 then (fg)(0) is ________.

\(\lim _{ x\rightarrow \infty }{ \frac { \tan { \theta } }{ \theta } } =\)________.

For the cost function C =\(\frac { 1 }{ 25 } { e }^{ 5x }\), the marginal cost is _________.

Average cost is minimum when _______.

What is the amount relalised on selling 8% stock 200 shares of face value Rs. 100 at Rs. 50.

The present value of the perpetual annuity of Rs. 2000 paid monthly at 10 % compound interest is _______.

When an observation in the data is zero, then its geometric mean is

The first quartile is also known as _________.

If the values of two variables move in opposite direction then the correlation is said to be ______.

If X and Y are two variates, there can be atmost ________.

Maximize: z = 3x₁ + 4x₂ subject to 2x₁ + x₂ ≤ 40, 2x₁+ 5x₂ ≤ 180, x₁, x₂ ≥ 0. In the LPP, which one of the following is feasible corner point?

The maximum value of the objective function Z = 3x + 5y subject to the constraints x > 0 , y > 0 and 2x + 5y ≤ 10 is ______.

Find adj A for \(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)

If nP_r = 360, find n and r.

If the lines 3x - 5y - 11 = 0, 5x + 3y - 7 = 0 and x + ky = 0 are concurrent, find the value of k.

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

If y = A sin x + B cos x, then prove that, y₂ + y = 0

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

A limited company wants to create a fund to help their employees in critical circumstances. The estimated expenses per month is Rs. 18,000. Find the amount to be deposited by the company if the rate of compound interest is 15%.

A man travelled by car for 3 days. He covered 480 km each day. On the first day he drove for 10 hours at 48 km an hour. On the second day, he drove for 12 hours at 40 km an hour and for the last day he drove for 15 hours at 32 km. What is his average speed?

The following information is given

Coefficient of correlation between X and Y is \(\frac{8}{15}\) . Find (i) The regression Coefficient of Y on X (ii) The most likely value of Y when X = Rs.100.

A fruit grower can use two types of fertilizers in his garden, brand P and brand Q. The amounts (in Kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs atIeast 240 kgs of phosphoric acid, at least 270 kg of potash and atmost 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, formulate the above as mathematical LPP.

Find the inverse of each of the following matrices.\(\left[\begin{array}{rr} 1 & -1 \\ 2 & 3 \end{array}\right]\)

Resolve into partial fractions for the following : \(\frac{4 x+1}{(x-2)(x+1)}\)

Find the separate equations of the pair of lines given by 3x² + 7xy + 2y² + 5x + 5y + 2 = 0.

If tanA =\(\frac{1}{7}\) and tanB =\(\frac{1}{3}\), show that cos2A = sin4B

Differentiate the following with respect to x. \(\sqrt { 1+{ x }^{ 2 } } \)

The total cost of x units of output of a firm is given by c = \(\frac { 2 }{ 3 } x+\frac { 35 }{ 2 } \) find the
(i) cost, when output is 4 units
(ii) average cost, when output  is 10 units
(iii) marginal cost, when output is 3 units

A man wishes to pay back his depts of Rs.3783 due after 3 years by 3 equal yearly instalments. Find the amount of each instalments,money being worth 5% p.a. compounded annually

Compute the mean deviation about mean from the following data:

Two phychologist ranked 12 candidates in the selection list as below:

Find the rank correlation co-efficient.

Solve the following LPP graphically. Maximize Z =−x₁ + 2x₂
Subject to the constraints −x₁ + 3x₂ ≤ 10, x₁ + x₂ ≤ 6,x₁ − x₂ ≤ 2 and x₁,x₂ ≥ 0

An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other’s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that the system is viable? 2 and 5 labour days are required to produce a tonnes of coal and steel respectively. If economy needs 100 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required.

By the principle of mathematical induction, prove the following.
5²ⁿ - 1 is divisible by 24, for all \(n\in N\) .

If m parallel lines in a plane are intersected by a family of n parallel lines. Find the number of parallelogram formed?

Show that the point (7, –5) lies on the circle x² + y² - 6x + 4y - 12 = 0 and find the coordinates of the other end of the diameter through this point.

If \(\sin { \theta \frac { 3 }{ 5 } } \), \(\tan { \phi } =\frac { 1 }{ 2 } \)and \(\frac { \pi }{ 2 } <\theta <\pi<\varphi <\frac { 3\pi }{ 2 } \) , then find the value of \(8\tan { \theta } -\sqrt { 5 } \sec {\varphi } \)

Prove that cos 4x = 1 - 8 sin²x cos²x.

Draw the graph of the following function f(x) = |x - 2|

 The demand for a commodity x is q = 5-2p₁+ P₂ -\({ p }_{ 1 }^{ 2 }{ p }_{ 2 }\). Find the partial elasticities \(\frac { Eq }{ { EP }_{ 1 } } \) and \(\frac { Eq }{ { EP }_{ 2 } } \) when p₁= 3 and p₂ = 7

The demand for a quantity A is q = 16- 3P_I - 2P₂². Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

Kamal sold Rs.9000 worth 7% stock at 80 and invested the proceeds in 15% stock at 120. Find the change in his income?

Calculate the Mean deviation about median and its relative measure for the following data.

Calculate correlation coefficient for the following data.

A dietician wishes to mix two types of food F₁ and F₂ in such a way that the vitamin contents of the mixture contains atleast 6 units of vitamin A and 9 units of vitamin B. Food F₁ costs Rs.50 per kg and F₂ costs Rs 70 per kg. Food F₁ contains 4 units per kg of vitamin A and 6 units per kg of vitamin B while food F₂ contains 5 units per kg of vitamin A and 3 units per kg of vitamin B. Formulate the above problem as a linear programming problem to minimize the cost of mixture.

The following table use the activities in a building project.

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.