List of practice Questions

Let \(S=\){\(n∈NIn^3+3n^2+5n+3\) is not divisible by \(3\)}.Then, which of the following statements is true about \(S\)

If f(z)=zn+an-1zn-1 +......+a1zn+a0 ∈ R[z] is a polynomial in z with no root over R,then deg(f)is
If \(x+iy=\dfrac{1}{(1+cosθ)+isinθ}\),then the value of \(x^2+1\) is ?
Suppose \(A=\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{bmatrix}\) is an adjoint of the matrix \(\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}\). Then the value of \(\frac{a_!+b_2+c_3}{b_1a_2}\) is

If \(A =[5a-b 32]\) and A.adjA=AA^T,then which of the following statements is true

The determinant of the matrix\(\begin{vmatrix}      1 & 4 & 8\\      1 & 9 & 27\\      1 & 16 & 64  \end{vmatrix}\)  is?
The contrapositive of the statement "If the number is not divisible by 3, then it is not divisible by 15" is
Let A be (2n+1)×(2n+1) matrix with integer entries and positive determinant,where n∈N.If AAT = I = ATA, then which of the following statements always holds?
\(\int^{1}_{0}\frac{2e^x}{1+e^{2x}}dx=\)
The equation of the plane through the points (2, 1, 0), (3, 2, -2) and (3, 1, 7) is
The sides of a right-angled triangle ae in an arithmetic progression. If the area of the triangle is 54, then the length of the longest side is
Let \(a_1,a_2,a_3,..\). be an increasing sequence of natural numbers, which are in an arithmetic progression with common difference d. Suppose \(a_1+a_2+a_3=27\) and \(a_1^2+a_2^2+a_3^2=275\). Then the value of \(a_1,d\) are 

Let a,b be two real numbers between \(3\) and \(81 \)such that the resulting sequence \(3,a,b,81\) is in a geometric progression. The value of \(a+b\) is 

Let \(f(x)=\dfrac{x-1}{x+1}\) ,Let \(S ={x∈R \text{ Iff } -1(x)=x \text{  does not hold} }\).The cardinality of S is 

The plane passing through the points(2,1,0), (5,0,1)and (4,1,1)intersects the x axis at
The line of intersection of the planes \(3x-6y-2z-15=0\) and \(2x=y-2z-5=0\) is ?
The plane that is perpendicular to the planes \(x-y+2z-4=0\) and  \(2x-2y+z=0\) and passes through \( (1,-2,1)\) is 
The angle between the lines, whose direction cosines are proportional to 4,√3-1,-√3-1 and 4,-√3-1,√3-1, is

If \(θ1,θ2, θ3\) are the angles made by a line with the positive directions of the \(x,y,z\) axes, then the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is

If the \(XZ\)-plane divides the straight line joining the points \((2,4,7)\) and \((3,-5,8)\)  in the ratio \(α:1\),then the value of \(α\) is ? 
Given the points \(A(6,-7,0),B(16,-19,-4),C(0,3,-6)\) and \(D(2,-5,10)\),the point of intersection of the lines \(AB\) and \(CD\) is 
Suppose a and b are lengths of major and minor axes of an ellipse that passes through the points (4,3) and (-1,4). If the major axis of the ellipse lies along the x-axis , then the value of \(\frac{1}{a^2}+\frac{16}{b^2}\) is
For the circle \(C=x^2+y^2-6x+2y=0\),which of the following is incorrect:
The equation of the circle that can be inscribed in the square formed by \(x^2-8x+12=0\) and \(y^2-14y+45=0\) is ?
Suppose the point \(P(1,1)\) is translated to \(Q\) in the direction of y = 2x. If \(PQ = 1\),then Qis