Question(s) from Search: IIT

If a, b, c be positive and not all equal, show that the value of the determinant  is negative.

Match the following

Normals are drawn at the points P, Q and R lying on the parabola  which intersect at (3, 0) then

If a polynomial of degree 3, then  equals

a)

b)

c)

d) a constant

If ε then  is always greater than or equal to

a) 2 tan

b) 1

c) 2

d)

If the expression  is real then the set of all possible values of x is .  .  .  .

a) x = 2nπ or mπ + π/4

b) x = nπ or mπ + π/4

c) x = 2nπ or 2mπ + π/4

d) x = nπ or 2mπ + π/4

(Assertion and reason)

The question contains statement – 1 (assertion) and statement 2 (reason). Of these statements mark correct choice if

a) Statement 1 and 2 are true. Statement 2 is a correct explanation for statement 1.

b) Statement 1 and 2 are true. Statement 2 is not a correct explanation for statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true

Statement 1 – The curve  is symmetric with respect to the line x = 1

Statement 2 – The parabola is symmetric about its axis.

If  then

a)

b)

c)

d)

Let P = (x, y) be any point on  with focii  and  equals

a) 8

b) 6

c) 10

d) 12

Let α, β be roots of the equation are the roots of the equation  then the value of r is equal to

a)

b)

c)

d)

Show that square of  is a rational number.

The determinants   are.

a) Identical

b) Not identical

c) Identical if a = b = c

d) None of the above

Given that x = −9 is a root of  = 0

.

a) {2, 7}

b) {−2, −7}

c) {2, 0}

d) {0, 7}

An ellipse has OB as a semi-minor axis. F, F’ are its focii and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is . . . . .

If x = 9 is the chord of contact of the hyperbola  then the equation of the corresponding pair of tangents is

a)

b)

c)

d)

Solve for x

The rational number which equals the numbers with recurring decimals is

a)

b)

c)

d)

(Fill in the blanks) The function y = 2x² – ln|x| is monotonically increasing for values of x (≠0) satisfying the inequalities .  .  .  . and monotonically decreasing for values of x satisfying the inequalities .  .  .  .

a)

b)

c)

d)

Find  

a) 0

b) 1

c) 2

d) 4

The probability that an event A happens in one of the experiments is 0.4 Three independent trials of these experiments are performed. The probability that the event A happens at least once is

a) 0.936

b) 0.784

c) 0.904

d) None of these

Let  be roots of the equations and  respectively. If the system of equations  and  have non-trivial solutions then prove that

If  are in Arithmetic Progression
then

a) a, b, c are in Arithmetic Progression

b)  are in  Arithmetic Progression

c) a, b, c are in Geometric Progression

d) a, b, c are in Harmonic Progression

Let f(x) = ∫e^x (x – 1) (x − 2) dx, then f(x) decreases in the interval

a) (−∞, −2)

b) (−2, −1)

c) (1, 2)

d) (2, ∞)

The harmonic means of the roots of the equation
 is

a) 2

b) 4

c) 6

d) 8

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

Consider the two curves  then

a)  touch each other at only one point

b)  touch each other exactly at two points

c)  intersect(but not touch) at exactly two points

d)  neither intersect nor touch each other