List of practice Questions

Consider the following statements in respect of the function f(x)=x³-1, x∈[-1,1]. I. f(x) is increasing in [-1,1] II. f(x) has no root in (-1,1). Which of the statements given above is/are correct?

If a,b,c are real numbers then the roots of the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are always

Evaluate limₙtₒᵢₙfty(aⁿ+bⁿ)/(aⁿ-bⁿ), where a>b>1

The number of points at which the function f(x)=(1)/(log|x|) is discontinuous is

If f(x)= begincases (xlog(cos x))/(log(1+x²)), & x≠0
0, & x=0 endcases then f(x) is

For any differentiable function y of x, (d²x)/(dy²)((dy)/(dx))³+(d²y)/(dx²)=

The set of all values of a for which the function f(x)=(a²-3a+2)(cos²x-4sin²x/4)+(a-1)x+sin x does not possess critical points is

Evaluate (x+\frac1x)²+(x²+\frac1x²)²+(x³+\frac1x³)² up to n terms is

If z₁=\sqrt3+i\sqrt3 and z₂=\sqrt3+i, then the complex number ((z₁)/(z₂))⁵0 lies in the

If the matrix beginbmatrix 1 & 3 & λ+2
2 & 4 & 8
3 & 5 & 10 endbmatrix is singular, then λ=

Let alpha₁,alpha₂ and beta₁,beta₂ be the roots of ax²+bx+c=0 and px²+qx+r=0 respectively. If the system alpha₁ y+alpha₂ z=0, beta₁ y+beta₂ z=0 has a non-trivial solution, then

If [x] denotes the greatest integer ≤ x and -1≤ x<0, 0≤ y<1, 1≤ z<2, then the value of the determinant beginvmatrix [x]+1 & [y] & [z]
[x] & [y]+1 & [z]
[x] & [y] & [z]+1 endvmatrix is

If α,β are the roots of x²-2x-1=0, then the value of α²β²-α²-β² is

In a △ ABC, if (cos A)/(a)=(cos B)/(b)=(cos C)/(c), and the side a=2, then area of the triangle is:

If sin⁻1((2a)/(1+a²)) -cos⁻1((1-b²)/(1+b²)) =tan⁻1((2x)/(1-x²)), then what is the value of x?

The arithmetic mean of numbers a,b,c,d,e is M. What is the value of (a-M)+(b-M)+(c-M)+(d-M)+(e-M)?

The sum to n terms of the series \frac12+\frac34+\frac78+(15)/(16)+⋯ is

Let f(x)=dfracax+bcx+d, then f∘ f(x)=x, provided that:

If A and B are positive acute angles satisfying 3cos²A+2cos²B=4 and (3sin A)/(sin B)=(2cos B)/(cos A), then the value of A+2B is equal to: