Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

Using matrix method, solve x + 2y + z = 7, x + 3z = 11 and 2x - 3y =1.

if A =\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

Solve: 2x + 5y = 1 and 3x + 2y = 7 using matrix method.

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 = \(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)^-1 = B^-1A^-1

Resolve into partial factors:\(\frac { x+4 }{ ({ x }^{ 2 }-4)(x+1) } \)

How may different numbers between 100 and 1000 can be formed using the digits 0, 1,2,3,4, 5, 6 assuming that in any number, the digits are not repeated.

In how many ways can n prizes be given to n boys, when a boy may receive any number of prizes?

In how many ways can 12 things be equally divided among 4 persons?

If tan \(\alpha={{1}\over{7}},\sin\beta{{1}\over{\sqrt{10}}},\) Prove that \(\alpha+2\beta{{\pi}\over4{}}\) where \(0<\alpha<{{\pi}\over{2}}\) and \(0<\beta<{{\pi}\over{}2}.\)

A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). Find its locus.

Find the equation of a circle of radius 5 whose centre lies on X-axis and passes through the point (2, 3).

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

Find the condition that the straight lines y=m₁x+C_1, y=m₂x+C₂, and y=m₃x+C₃ may meet at a point?

Find the slope of the lines which make an angle of 45° with the line 3x - y + 5 = 0.

Find the combined equation of the straight line through the origin, one of which is parallel to and the other is perpendicular to the straight line 2x + y + 1 = 0

Find the equation of the tangent lines to the circle x² + y² = 9 which are parallel to 2x + y - 3 = 0

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

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

Prove that \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}\)

Prove that \(\frac{\sin5x-2\sin3x+sinx}{\cos5x-\cos x}=\tan x\)

Show that \(\cos^{-1}\left(\frac{3}{5}\cos x+\frac45\sin x\right)=x-\tan^{-1}\left(\frac43\right)\)

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

Is the function defined by f(x) = x² -sin x + 5 is continuous at x =\(\pi\)?

Differentiate: sin² x + cos² y = 1.

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Find the stationary points and stationary values of the function f(x) = x³ - 3x² - 9x + 5.

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]

A person borrows Rs.5000 at 5% p.a.interest compounded half yearly and agrees to pay both the principal and interest at 10 equal instalments at the end of each six months.Find the amount of these instalments.

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.

Find the present value of an annuity due of Rs.200 p.a. payable annually for 2 years at 4%p.a

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

Find the yield on 20% stock at 80.

Find the amount of an ordinary annuity of 12 monthly payments of Rs.1000 that earn interset at 12% per year compounded monthly.

What is the present value of an annuity that pays 250 per month at the end of each month for 5 years assuming money to be worth 6% compounded monthly?

Which is the better investment?7% stock at 80(or) stock at 96.

A fair die is rolled. A = {1, 3, 5} B = {2, 3} and C = {2, 3, 4, 5}. Find (i) P(A/B) and P(B/A) (ii) P(A/C) and p(C/A).

Events A and B are such that P(A)=\(\frac { 1 }{ 2 } \), P(B)=\(\frac { 7 }{ 12 }\), and P(not A or not B) = \(\frac { 1 }{ 4 }\), state whether A and B are independent?

Find the geometric mean of 3, 6, 24, 48

Find Q₂ for 37, 32, 45, 36, 39, 37, 46, 57, 27, 34, 28, 30, 21

Find the geometric mean for the following data

Value	10	12	15	20	50
Frequency	2	3	10	8	2

A bag contains 6 black and 5 red balls. Two balls are drawn at random. What is the probability that they are of the same colour?

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 mean deviation about median for the following data:

Class	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	15	16	6

Calculate the correlation co-efficient from the below data:

X	1	2	3	4	5	6	7	8	9
Y	9	8	10	12	11	13	14	16	5

Calculate the correlation co-efficient from the following data:

X	12	9	8	10	11	13	7
Y	14	8	6	9	11	12	3

Calculate the covariance of the following pairs of observation of two variates X and Y. (1, 5)(2, 4)(3, 3)(4, 2)(5, 1)

Calulate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)\(\sum\)x=10, \(\sum\)x²=250, \(\sum\)y=70, \(\sum\)y²=300, \(\sum\)xy=75 and n =20

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

For the following observations, find the regression co-efficients b_yx and b_xy and hence find the correlation co-efficient between x and y.(4,2) (2, 3)(3, 2)(4, 4)(2, 4)

Calculate Co-efficient of correlation for the following data:

X	-3	-2	-2	0	1	2	3
Y	9	4	1	0	1	4	9

If two regression co-efficient are 2 and 0.45, what will be the co-efficient of correlation?

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.

Ten students got the following percentage of marks in maths and physics in their second term examinations.

Maths	28	36	99	30	78	85	95	65	68	38
Physics	87	54	94	63	71	65	89	61	38	52

Find the co-efficient of rank correlation.

Solve the following LPP graphically. Maximize Z = 6x₁ + 5x₂ Subject to the constraints 3x₁ + 5x₂ ≤ 15, 5x₁ + 2x₂ ≤ 10 and x₁,x₂ ≥ 0

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

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

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 following:

Activity	A	B	C	D	E	F
Immediate Predecessor	-	-	-	A	B	C

Develop a network based on the following information.

Activity	A	B	C	D	B	E
Immediate Predecessor	-	-	A	C	E	F