## Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

Let *a, b, c* be any real numbers. Suppose that there are real numbers *x, y, z* not all zero such that *x = cy + bz, y = az + cx and z = bx + ay*. Then *a ^{2} + b^{2} + c^{2} + 2abc* is equal to

Let *A* be a 2 × 2 matrix with real entries. Let* I* be the 2 × 2 identity matrix. Denote by tr(*A*), the sum of diagonal entries of *A*. Assume that *A ^{2} = I*.

**Statement1 :** If *A I* and *A – I*, then det *A = –1*.

**Statement2 :** If *A I* and *A – I*, then tr(*A*) 0.

Let A = .

If *|A ^{2}| = 25* the |

*a|*quuals to

If D = for x 0, y 0 then D is

Let A = and B = , *a, b N*. Then

If A and B are square matrices of size n × n such that A^{2} – B^{2} = (A – B)(A + B), then what will be always true?

The system of equations *ax + y + z = a – 1, x + ay + z = a – 1, x + y + az = a – 1* has no solutions, if *a* is

If *a ^{2} + b^{2} + c^{2} = – 2* and

then what is *f(x)* is a polynomial of degree?

If A = and I = , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

If *A ^{2} – A + I = 0*, then the inverse of

*A*is

If a_{1}, a_{2}, a_{3}, ...., a_{n}, ... are G.P., then the value of the determinant

is

Let A = and 10(B) = .

If B is the inverse of matrix A, then what is the ?

Let A = ̄

The only correct statement about the matrix A is

If A = and A^{2} = , then

A motor boat going down stream over came a raft at a point A, t = 60 min later it turned back and after some time passed the raft at a distance l = 6.0 km from the point A. Find the flow velocity assuming the duty of the engine to be constant?

A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as

Where is a constant with appropiate dimension with is a constant with dimension of temperature .

The heat capacity of the metal is?

Divide 64 into two parts such that sum of the cubes of two parts is minimum.

What is maximum acceleration so that so that two block move together?

Two particles of masses m1 and m2 are placed 'd' distance apart. Due to gravitational attraction they move towards each other.What is speed of m1 when their separation reduces to d/2.

What is the Value of ?

If *w = *[*z / z - *(1/3)*i*] -and *|w| = 1*, then *z* lies on

If *z _{1}* and

*z*are two nonzero complex numbers such that |

_{2}*z*+

_{1}*z*| = |

_{2}*z*| + |

_{1}*z*|, then argz

_{2}_{1}– argz

_{2}is equal to

If *|z ^{2} – 1| = |z|^{2} + 1,* then

*z*lies on

If *z = x – iy* and *z ^{1/3} = p + iq*, then when will value of ?

Let *z, w* be complex numbers such that *z + = 0* and *zw = * Then arg z equals

Let *z _{1}* and

*z*be two roots of the equation

_{2}*z*,

^{2}+ az + b = 0*z*being complex further, assume that the origin,

*z*and

_{1}*z*form an equilateral triangle, then what will be

_{2}*a*?

^{2}

If z and w are two non-zero complex numbers such that *|zw| = 1,* and Arg(*z*) – Arg(*w*) = then *zw* is equal to

If [(1 + i) / (1 - i)]^{x} = 1, then what will be x?

What will be the locus of the centre of a circle which touches the circle *| z - z _{1} | = a* and

*| z - z*externally?

_{2}| = bNote: *z, z _{1} & z_{2}* are complex numbers.

Two particle of masses m1 and m2 are 'd' distance apart. Due to gravitational attraction they move towards each other. what is speed of m1 when their separation reduces to d/2.

Prove that every first degree equation in x, y represents a straight line?

Prove that the co-ordinates of the vertices of an equilateral triangle can not all be rational?

If the vertices of a triangle are (1, 2), (4, –6) and (3, 5) then its area is

Find the hyperbola whose asymptotes are 2x – y = 3 and 3x + y – 7 = 0 and which passes through the point (1, 1).

Let *z, w* be complex numbers such that *z + = 0* and *zw = * Then arg z equals

Let *z _{1}* and

*z*be two roots of the equation

_{2}*z*,

^{2}+ az + b = 0*z*being complex further, assume that the origin,

*z*and

_{1}*z*form an equilateral triangle, then what will be

_{2}*a*?

^{2}

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle x2 +y2 –2x–6y+6=0

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is-

Consider a family of circles which are passing through the point (–1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval-

Three distinct points A, B and C are given in the 2–dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (–1, 0) is equal to 1/3 . Then the circumcentre of the triangle ABC is at the point :-

The point diametrically opposite to the point (1, 0) on the circle x2 + y2 + 2x + 4y – 3 = 0 is-

A and B are two fixed points and P moves such that PA = nPB where n != 1. Show that locus of P is a circle and for different values of n all the circles have a common radical axis.

The circle x2 + y2 – 6x – 10y + k = 0 does not touch or intersect the coordinate axes and the point (1, 4) is inside the circle. Find the range of the value of k.

Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0?

A real valued function f (x) satisfies the functional equation *f(x – y) = f (x)f (y) – f(a – x)f(a + y)* where *a* is a given constant and *f(0) = 1, f(2a – x)* is equal to

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

Let *f* : (–1, 1) B, be a function defined by f(x) = , then *f* is both one-one and onto when B is the interval

Let *R =* {(3, 3) (6, 6) (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set *A* = {3, 6, 9, 12}. What is the relation then?

What is the domain of the function f(x) = ?