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 ________.

(a)

\(\triangle\)

(b)

-\(\triangle\)

(c)

3\(\triangle\)

(d)

-3\(\triangle\)

The inverse matrix of \(\begin{pmatrix} \frac { 1 }{ 5 } & \frac { 5 }{ 25 } \\ \frac { 2 }{ 5 } & \frac { 1 }{ 2 } \end{pmatrix}\) is ________.

(a)

\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

(b)

\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 1 }{ 5 } \end{pmatrix}\)

(c)

\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

(d)

\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

The value of \(\begin{vmatrix} 5 & 5 & 5 \\ 4x & 4y & 4z \\ -3x & -3y & -3z \end{vmatrix}\)is ________.

(a)

5

(b)

4

(c)

0

(d)

-3

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

(a)

0

(b)

2

(c)

1

(d)

3

The number of ways selecting 4 players out of 5 is _______.

(a)

4!

(b)

20

(c)

25

(d)

5

The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for n \(\in\) N is ________.

(a)

2

(b)

6

(c)

20

(d)

24

If \(\frac { kx }{ (x+4)(2x-1) } =\frac { 4 }{ x+4 } +\frac { 1 }{ 2x-1 } \) then k is equal to _______.

(a)

9

(b)

11

(c)

5

(d)

7

The value of (5C₀ + 5C₁) + (5C₁ + 5C₂) + (5C₂ + 5C₃) + (5C₃ + 5C₄) + (5C₄ + 5C₅ ) is ________.

(a)

2⁶-2

(b)

2⁵-1

(c)

2⁸

(d)

2⁷

Sum of the binomial coefficients is ________.

(a)

2ⁿ

(b)

n²

(c)

2n

(d)

n + 17

The locus of the point P which moves such that P is always at equidistance from the line x + 2y+ 7 = 0 is _______.

(a)

x+2y+2 = 0

(b)

x - 2y + 1 = 0

(c)

2x - y + 2 = 0

(d)

3x + y + 1 = 0

If the perimeter of the circle is 8π units and centre is (2, 2) then the equation of the circle is _______.

(a)

(x - 2)² + (y - 2)² = 4

(b)

(x - 2)² + (y - 2)² = 16

(c)

(x - 4)² + (y - 4)² = 2

(d)

x² + y² =4

The double ordinate passing through the focus is _______.

(a)

focal chord

(b)

latus rectum

(c)

directrix

(d)

axis

The equation of directrix of the parabola y² = - x is _______.

(a)

4x+ 1 =0

(b)

4x - 1 = 0

(c)

x - 4=0

(d)

x + 4 = 0

If \(\tan\theta=\frac{1}{\sqrt5}\) and \(\theta\) lies in the first quadrant, then \(\cos\theta\) is _______.

(a)

\(\frac{1}{\sqrt6}\)

(b)

\(\frac{-1}{\sqrt6}\)

(c)

\(\frac{\sqrt5}{\sqrt6}\)

(d)

\(\frac{-\sqrt5}{\sqrt6}\)

The value 4cos³40^o - 3cos40^o is ________.

(a)

\(\frac{\sqrt3}{2}\)

(b)

\(\frac{-1}{2}\)

(c)

\(\frac{1}{2}\)

(d)

\(\frac{1}{\sqrt2}\)

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

(a)

\(\left(\frac{1+\tan x}{1-\tan x}\right)\)

(b)

\(\left(\frac{1-\tan x}{1+\tan x}\right)\)

(c)

1-tan x

(d)

1+tan x

If \(f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}\), then f(5) is _______.

(a)

-1

(b)

2

(c)

5

(d)

7

The graph of f(x) = e^x is identical to that of ________.

(a)

f(x) = a^x, a > 1

(b)

f(x) = a^x, a < 1

(c)

f(x) = a^x, 0 < a < 1

(d)

y = ax +b, a \(\ne\) 0

\(\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =\)________.

(a)

e

(b)

nx^(n-1)

(c)

1

(d)

0

If y = log x then y₂ = ________.

(a)

\(\frac{1}{x}\)

(b)

\(-\frac{1}{x^2}\)

(c)

\(-\frac{2}{x^2}\)

(d)

e²

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.6 & 0.9 \\ 0.20 & 0.80 \end{bmatrix}\) .Test whether the system is viable as per Hawkins-Simon conditions.

Find the minors and cofactors of all the elements of the following determinants \(\begin{vmatrix}5&20\\ 0&-1 \end{vmatrix}\)

How many triangles can be formed by joining the vertices of a hexagon?

In how many ways can a cricket team of 11 players be chosen out of a batch of 15 players?
(i) There is no restriction on the selection.
(ii) A particular player is always chosen.
(iii) A particular player is never chosen.

Find a point on x axis which is equidistant from the points (7, -6) and (3,4)

Convert the parabola y²=4x+4y into standard form.

Find the principal value of the following cosec^-1(2)

Solve : \(\tan^{-1}(x+1)+\tan^{-1}(x-1)=\tan^{-1}\left(\frac{4}{7}\right)\)

Determine whether the following functions are odd or even?
f(x) = sin x + cos x

if y = 2 sin x + 3 cos x, then show that y₂ + y = 0

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

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\)  and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y =  X^-1

Find the Co-efficient of x¹¹ in the expansion of \({ \left( x+\frac { 2 }{ { x }^{ 2 } } \right) }^{ 17 }\)

Find the distance of the point (4,1) from the line 3x - 4y + 12 = 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.

Find the values of the following  cot 75°

If \(\sin { A } =\frac { 3 }{ 5 } \) 0 < A < \(\frac{\pi}{2}\)  and \(\cos { B } =\frac { -12 }{ 13 } \) , π < B < \(\frac{3\pi}{2}\) find the values of the following \(\tan(A-B)\)

Find  \(\frac{dy}{dx}\)  of the following functions: x = acos³θ, y = asin³θ

Evaluate the following \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 3 }+2 }{ x+1 } } \)

1. Solve by matrix inversion method : 2x - z = 0; 5x + y = 4; y + 3z = 5.

2. Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using matrix inversion method.

   Week	Number of employees			Total weekly Salary (in rupees)
   	A	B	C
   1st week	4	2	3	4900
   2nd week	3	3	2	4500
   3rd week	4	3	4	5800

1.  Find the middle terms in the expansion of \({ \left( { 2x }^{ 2 }-\frac { 3 }{ { x }^{ 3 } } \right) }^{ 10 }\)

2. Find the term independent of x in the expansion of \({ \left( { 2 }x^{ 2 }+\frac { 1 }{ x } \right) }^{ 12 }\)

1. If \(y={ \left( x+\sqrt { 1+{ x }^{ 2 } } \right) }^{ m }\), then show that (1 + x)² y₂ + xy₁ - m² = 0.

2. Verify the continuity and differentiability of \(f(x)= \begin{cases}1-x & \text { if } x<1 \\ (1-x)(2-x) & \text { if } 1 \leq x \leq 2 \\ 3-x & \text { if } x>2\end{cases}\) at x = 1 and x = 2

1. Find the rank of the word 'CHAT' in dictionary.

2. Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.

1. 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.

2. As the number of units manufactured increases from 6000 to 8000, the total cost of production increases from Rs. 33,000 to Rs. 40,000. Find the relationship between the cost (y) and the number of units made (x) if the relationship is linear.

1. Prove that : cos20°cos40°cos80° = \(\frac { 1 }{ 8 } \)

2. Prove that : \(2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13} = 0\) .

1. If cosec A + sec A = cosec B + sec B, prove that cot\(\left( \frac { A+B }{ 2 } \right) \) = tanA tanB

2. If y = 500 e^7x + 600 e^-7x then show that y₂ - 49y = 0.