If \(\triangle=\begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{vmatrix}\) then \(\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix}\) is ________.

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

The number of ways to arrange the letters of the word "CHEESE" is ______.

The number of permutation of n different things taken r at a time, when the repetition is allowed is ________.

The x-intercept of the straight line 3x + 2y - 1 = 0 is _______.

Combined equation of co-ordinate axes is _______.

The equation of the circle with centre (3,-4) and touches the x - axis is _______.

The value of \(\sin(-420^o)\) is _______.

The value of \(cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)\) is ________.

\(\tan\left(\frac{\pi}{4}-x\right)\) is _______.

The graph of the line y = 3 is _______.

If y = x and z = \(\frac{1}{x}\) then \(\frac{dy}{dz}=\)________.

If u = 4x² + 4xy + y² + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to ________.

If u = \({ e }^{ { x }^{ 2 } }\) then \(\frac { \partial u }{ \partial x } \) is equal to _______.

A man purchases a stock of Rs. 20,000 of face value Rs. 100 at a premium of 20%, then investment is ________.

The annual income on 500 shares of face value Rs.100 at 15% is _______.

Example of contingent annuity is _______.

When calculating the average growth of economy, the correct mean to use is?

If two events A and B are dependent then the conditional probability of P(B/A) is _________.

Probability that at least one of the events A, B occur is _________.

In the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 6 }\), find the third term.

Find the center and radius of the circle 5x² + 5y² + 4x - 8y - 16 = 0

Show that \(\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{2}{11}\right)=\tan^{-1}\left(\frac{3}{4}\right)\)

If \(f\left( x \right) =\frac { 1 }{ 2x+1 } ,x> -\frac { 1 }{ 2 } \) then show that \(f\left( f\left( x \right) \right) =\frac { 2x+1 }{ 2x+3 } \)

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

Gopal invested Rs. 8,000 in 7% of Rs. 100 shares at Rs. 80. After a year he sold these shares at Rs. 75 each and invested the proceeds (including his dividend) in 18% for Rs. 25 shares at Rs. 41. Find
(i) his dividend for the first year
(ii) his annual income in the second year
(iii) The percentage increase in his return on his original investment

An investor buys Rs. 1,500 worth of shares in a company each month. During the first four months he bought the shares at a price of Rs. 10, 15, 20 and 30 per share. What is the average price paid for the shares bought during these four months? Verify your result.

Show that \(\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.\)

Show that the middle term in the expansion of (1 + x)²ⁿ is \(\frac { 1.3.5....(2n-1){ 2 }^{ n }.{ x }^{ n } }{ n! } \)

Find the locus of the point which is equidistant from (2, –3) and (3, –4).

Prove that \(\frac { sin(-\theta )tan({ 90 }^{ o }-\theta )sec\left( { 180 }^{ o }-\theta \right) }{ sin(180+\theta )cot(360-\theta )cosec({ 90 }^{ o }-\theta ) } =1\)

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

Verify Euler’s theorem for the function \(u=\frac{1}{\sqrt{x^2+y^2}}\)

Find the amount of an ordinary annuity of Rs 500 payable at the end of each year for 7 years at 7% per year compounded annually.

Prove that the term independent of x in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 2n }is\quad \frac { 1.3.5.....,(2n-1){ 2 }^{ n } }{ n! } \)

The average variable cost of a monthly output of x tonnes of a firm producing a valuable metal is  Rs. \(\frac { 1 }{ 5 } { x }^{ 2 }-6x+100\). Show that the average variable cost curve is a parabola. Also find the output and the  average cost at the vertex of the parabola.

Prove that \({ cot }^{ -1 }\left[ \frac { \sqrt { 1+sinx } +\sqrt { 1-sinx } }{ \sqrt { 1+sinx } -\sqrt { 1-sinx } } \right] =\frac { x }{ 2 } \) where \(x\in \left( 0,\frac { \pi }{ 4 } \right) \)

Calculate the geometric mean of the data given below giving the number of families and the income per head of different classes of people in a village of Kancheepuram District.

In a Shooting test, the probabilities of hitting the target are \(\frac { 1 }{ 2 } \) for A, \(\frac { 2 }{ 3 } \) for B and \(\frac { 3 }{ 4 } \)  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

Examine the following functions for continuity at indicated points 
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-9}{x-3}, & \text { if } x \neq 3 \\ 6, & \text { if } x=3 \end{array}\right.\) at x = 3