If ^(n-1)P₃ :ⁿ P₄ = 1 : 10, find n

Find the sum of all 4-digit numbers that can be formed using digits 0, 2, 5, 7, 8 without repetition?

If ⁽ⁿ⁺²⁾C₇ : ^(n-1)P₄ = 13 : 24 find n.

Five boys and 5 girls form a line. Find the number of ways of making the seating arrangement under the following condition.

	C1		C2
(a)	Boys and girls sit alternate	(i)	5! \(\times\) 6!
(b)	No two girls sit together	(ii)	10! - 5! 6!
(c)	All the girls sit together	(iii)	(5 !)2 + (5!)2
(d)	All the girls are never together	(iv)	2! 5! 5!

Write the n^th term of the following sequences
2,2,4,4,6,6

For what value of n, the nth term of the series "3 + 10 + 17 +..+ and 63 + 65 + 67 +... are equal

Expand (1+ x)^{\(2\over 3\)} up to four terms for |x| < 1.

Find the 18^th and 25^th terms of the sequence defined by
a_n = {\(n(n+2),\quad if\quad n\quad is\quad even\quad natural\quad number\\ \frac { 4n }{ { n }^{ 2 }+1 } ,\ if\ n\ is\ odd\ natural\ number\\ \)

Sum up to n terms the series: 
7 + 77 + 777 + 7777 + ...

A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total length of the spring was measured each time shown in the following table.

Weight, (kg)	2	4	5	8
Length, (cm)	3	4	4.5	6

(i) Draw a graph showing the results.
(ii) Find the equation relating the length of the spring to the weight on it.
(iii) What is the actual length of the spring.
(iv) If the spring has to stretch to 9 cm long, how much weight should be added?
(v) How long will the spring be when 6 kilograms of weight on it?

A family is using Liquefied petroleum gas (LPG) of weight 14.2 kg for consumption. (Full weight 29.5kg includes the empty cylinders tare weight of 15.3kg.). If it is use with constant rate then it lasts for 24 days. Then the new cylinder is replaced.
(i) Find the equation relating the quantity of gas in the cylinder to the days.
(ii) Draw the graph for first 96 days.

If θ is a parameter, find the equation of the locus of a moving point, whose coordinates are (a(θ - sin θ), a(1 - cos θ)).

The seventh term of an arithmetic progression is 30 and tenth term is 21.
(i) Find the first three terms of an A.P.
(ii) Which term of the A.P. is zero (if exists)
(iii) Find the relationship between Slope of the straight line and common difference of A.P.

A fruit shop keeper prepares 3 different varieties of gift packages. Pack-I contains 6 apples, 3 oranges, and 3 pomegranates. Pack-II contains 5 apples, 4 oranges and 4 pomegranates and Pack –III contains 6 apples, 6 oranges and 6 pomegranates. The cost of an apple, an orange and a pomegranate respectively are Rs. 30, Rs. 15 and Rs. 45. What is the cost of preparing each package of fruits?

Find the matrix A which satisfies the matrix relation A\(\begin{bmatrix} 1 & 2&3 \\4 & 5&6 \end{bmatrix}\)=\(\begin{bmatrix} -7 & -8&-9 \\2 & 4&6 \end{bmatrix}\)

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
\(\begin{bmatrix} 4 & -2 \\ 3& -5 \end{bmatrix}\)

Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

In a triangle ABC, if \(\begin{vmatrix} 1& 1 &1 \\1+sin A &1+sin B &1+sin C \\ sinA(1+sin A) &sin B(1+sin B) &sin C(1+sin C) \end{vmatrix}=0,\)
prove that \(\triangle\)ABC is an isosceles triangle.

Determine the values of a so that the following matrices are singular: A =\(\begin{bmatrix} 7& 3 \\ -2 & a \end{bmatrix}\)