Question(s) from Search: IIT

Let  and  where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = .  .  .  .  .

If M is a 3 x 3 matrix where det (M) = 1 and MM^T = I, then prove that det (M – I) = 0.

Let A =

If U₁, U_2, U₃ are column matrices satisfying
AU₁ =  , AU₂ =  and AU₃ =

and U is a 3 x 3 matrix whose columns are U₁, U_2, U₃  then the value of [ 3  2  0 ] U  is

a)

b)

c)

d)

Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations

 
 
 
Then show that the roots of the equation
 
 
and  are reciprocal of each other.

The interior angles of a polygon are in Arithmetic Progression. The smallest angle is 120° and the common difference is 5. Find the number of sides of the polygon.

Let a₁, a₂, … a_n be positive real numbers in Geometric Progression. For each n let A_n, G_n, H_n be respectively the arithmetic mean, geometric mean and harmonic mean of a₁, a₂, .  .  .  ., a_n. Find the expressions for the Geometric mean of G₁, G₂, .  .  .  .G_n in terms of A₁, A₂, .  .  .  .,A_n; H₁, H₂, .  .  .  .H_n

If total number of runs scored in n matches is
 where n > 1 and the runs scored in the k^th match are given by k.2^{n + 1 – k}  where 1 ≤ k ≤ n. Find n.

In a triangle ABC if cotA, cotB, cotC are in Arithmetic Progression then a, b, c are in .  .  .  .  . Progression.

For any odd integer n ≥ 1,
n³ – (n – 1)³ + .  .  . + (−)^{n – 1} 1³ = .  .  .

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  such that . Then the volume of the parallelepiped is

a)

b)

c)

d)

Consider three planes
P₁ : x – y + z = 1

P₂ : x + y – z = −1

P₃  : x – 3y + 3z = 2

Let L₁, L₂, L₃ be lines of intersection of planes P₂ and P₃, P₃ and P_1, and P₁ and P₂ respectively.

Statement 1 – At least two of the lines L₁, L₂, L₃ are non parallel

Statement 2 – The three planes do not have a common point.

a) Statement 1 is true. Statement 2 is true. Statement 2 is a correct explanation of statement 1.

b) Statement 1 is true. Statement 2 is true. Statement 2 is not a correct explanation of statement 1.

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

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

The solution of primitive equation
 is . If  and  then is

a)

b)

c)

d)

If  then prove that

Solution of the differential equation is

The parameter on which the value of the determinant
Δ =
does not depend upon is

a) a

b) p

c) d

d) x

For what value of m does the system of equations 3x + my = m, 2x − 5y = 20 have a solution satisfying the condition x > 0, y > 0.

a) m  (−∞, ∞)

b) m  (−∞, −15) ∪ (30, ∞)

c)  

d)  

If α is a repeated root of a quadratic equation f(x) = 0 and A(x), B(x), C(x) be polynomials of degree 3, 4, 5 respectively, Then show that
 

is divisible by f(x) where prime denotes the derivatives.

If , for every real number x, then the minimum value of f

a) does not exist because f is unbounded

b) is not attained even though f is bounded

c) is equal to 1

d) is equal to –1

The function  is defined by then  is

a)

b)

c)

d) None of these

Suppose  for x ≥ . If g(x) is the function whose graph is the reflection of f(x) with respect to the line y = x then g(x) equals

a)

b)

c)

d)

Domain of definition of the function   for real values of x is

a)

b)

c)

d)

Let f be a one–one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly one of the following statements is true and remaining statements are false f (1) = 1, f (y) ≠ 1, f (z) ≠ 2. Determine  

Let {x} and [x] denote the fractional and integral part of a real number x respectively. Solve 4{x} = x + [x]

a) True

b) False

Find the range of values of t for which  

a) (−, −)

b) ( ,  )

c) (− , −  ) U ( ,  )

d) (−,  )