Evaluate: \(\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}\)

Show that \(\begin{vmatrix}x+a &b&c \\a &x+b&c\\a&b&x+c \end{vmatrix}=x^2(x+a+b+c)\)

Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \) = 0

If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A ^-1.

How many numbers lesser than 1000 can be formed using the digits 5, 6, 7, 8 and 9 if no digit is repeated?

Resolve into Partial Fractions : \(\frac { 5x+7 }{ (x+1)(x+3) } \)

In an examination, Yamini has to select 4 questions from each part. There are 6, 7 and 8 questions is Part I, Part II and Part III respectively. What is the number of possible combinations in which she can choose the questions?

Let p(n) be the statement "n² + n is even". If P(k) is true, then show that P(k+1) is true.

Find the vertex, focus, axis, directrix and the length of latus rectum of the parabola y² - 8y - 8x + 24 = 0

A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16, find its locus.

Prove that the lines (b - c) x + (c - a) y + (a - b) = 0, (c - a) x + (a - b) y + (b - c) = 0, and (a - b) x + (b - c) y+c-a=0 are concurrent.

Find the value of k so that the line 3x + 4y - k = 0 is a tangent to x² + y² - 64 = 0

Prove that  \(\sec { \left( \frac { 3\pi }{ 2 } -\theta \right) } \sec { \left( \theta -\frac { 5\pi }{ 2 } \right) } +\tan { \left( \frac { 5\pi }{ 2 } +\theta \right) } \tan { \left( \theta -\frac { 5\pi }{ 2 } \right) } =-1\)

Find all other trigonometrical ratios if \(\sin x=\frac{-2\sqrt6}{5}\) and x lies in III quadrant?

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

If \(\sin { A } =\frac { 12 }{ 13 } \), find sin 3A

If \(y=\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) show that \(2x\frac { dy }{ dx } +y=2\sqrt { x } \).

Differentiate: \(\sin ^{ -1 }{ \left( \sqrt { \cos { x } } \right) } \)

Evaluate: \(\underset { x\rightarrow a }{ lim } \frac { { x }^{ \frac { 3 }{ 5 } }-{ a }^{ \frac { 3 }{ 5 } } }{ { x }^{ \frac { 1 }{ 5 } }-{ a }^{ \frac { 1 }{ 5 } } } \)

Differentiate the following with respect to x.
(i) x^x
(ii) (log x)^{cos x}

For the given demand function p = 40 – x, find the value of the output when η_d = 1

The average cost function associated with producing and marketing x units of an item is given by AC = 2x - 11+\(\frac { 50 }{ x } \). Find the range of values of the output x, for which AC is increasing.

Separate the intervals in which the function x³ + 8x² + 5x - 2 is increasing or decreasing.

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

Find the annual rate of interest, to get a perpetuity of Rs. 675 for every half yearly from the present value of Rs. 30,000

Which is better investment: 12% Rs. 20 shares at Rs. 16 (or) 15% Rs. 20 shares at Rs. 24.

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I. 5% per year, find out the amount I will receive at the end of 10 years.

Ram bought at 9% stock for Rs.5400 at a discount of 11%. If the paid 1% brokerage, find the percentage of his income.

Calculate the value of Q₁, Q₃, D₆ and P₅₀ from the following data

Roll No	1	2	3	4	5	6	7
Marks	20	28	40	12	30	15	50

Calculate the Harmonic Mean of the following values:
1, 0.5, 10, 45.0, 175.0, 0.01, 4.0, 11.2

A card from pack 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart? 

Calculate the harmonic mean for the following data:

Size of items	50-60	60-70	70-80	80-90	90-100
No.of items	12	15	22	18	10

Calculate the correlation co-efficient for the following data.

X	5	10	5	11	12	4	3	2	7	1
Y	1	6	2	8	5	1	4	6	5	2

There are two series of index numbers P for price index and S for stock of the commodity. The mean and standard deviation of P are 100 and 8 and of S are 103 and 4 respectively. The correlation coefficient between the two series is 0.4. With these data obtain the regression lines of P on S and S on P.

prove that the correlation co-efficient is the geometric mean of regression co-efficients.

Find the co-variance and co-efficient of correlation for the following data:
n=10, \(\sum\)x=50, \(\sum\)y=-30, \(\sum\)x²=290, \(\sum\)y²=300 and \(\sum\)xy=-115.

A company is producing three products P₁, P₂ and P₃, with profit contribution of Rs.20, Rs.25 and Rs.15 per unit respectively. The resource requirements per unit of each of the products and total availability are given below.

Product	P1	P2	P3	Total availability
Man hours/unit	6	3	12	200
Machine hours/unit	2	5	4	350
Material/unit	1kg	2kg	1kg	100kg

Formulate the above as a linear programming model.

A firm manufactures two products A and B on which the profits earned per unit are Rs. 3 and Rs. 4 respectively. Each product is processed on two machines M₁ and M₂. Product A requires one minute of processing time on M₁ and two minutes on M₂, While B requires one minute on M₁ and one minute on M₂. Machine M₁ is available for not more than 7 hrs 30 minutes while M₂ is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:

Immediate Predecessor	A	B	C	D	E	F	G	H	I
Activity	B	C	D,E,F	G	I	H	J	K	L