List of practice Questions

The number of common tangents that can be drawn to the circle x²+y²-4x-6y-3=0 and x²+y²+2x+2y+1=0 is

The area of region bounded by the curve y²=x, the y-axis and between y = 2 and y = 4 is

The area of the region bounded by the curve x²=4y and the straight line x = 4y - 2 is

If log₃2,log₃(2^x-5) and log₃(2^x-7/2) are in A.P., then x is equal to

A single 6-sided dice is rolled, then the probability of getting an odd number is

In a class of 49 students, the ratio of girls to boys is 4:3. If 4 girls leave the class, the ratio of girls to boys would be

If income of A is 20% more than that of B, then income of B is how much percent less than that of A?

The average of 5 consecutive odd positive integers is 9. Then sum of smallest and greatest number is

If 2.5x=0.05 y, then find the value of \((\frac{y-x}{y+x}).\)

If P(x) is a polynomial of degree 10 with leading coefficient as 11 and having (x-1), (x-2), (x-3),...(x-10) as factor, then the coefficient of x⁹ in P(x) is

In the neighborhood of z = 1, the function f(z) has a power series expansion of the form f(z) = 1+(1-z)+(1-z)² + .... then f(z) is

A rectangular box open at the top is to have volume of 32 cubic feets. The minimum outer surface area of the box is

The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4 × 4 symmetric positive definite matrix is

The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?

If the matrices \(\left(\begin{matrix} 1 & 0 \\   0 & 0  \end{matrix}\right),\left(\begin{matrix}   -1 & 0 \\   0 & 0  \end{matrix}\right),\left(\begin{matrix}   i & 0 \\   0 & 0  \end{matrix}\right) and \left(\begin{matrix}   -i & 0 \\   0 & 0  \end{matrix}\right)\) form a group with respect to matrix multiplication, then which one of the following statements about the groups, thus formed is correct?

Which of the following are generators of the multiplicative group {(1,2,3,4,5,6), x₇} where x₇ denotes multiplication moduls 7?

The general solution of differential equation \(\frac{d^2y}{dx^2}+9y=sin^3x\) is
(given that c₁ and c₂ are arbitrary constants)

The solution of (x² -√2y) dx + (y² - √2x) dy = 0 is given by

Consider the linear mapping F: R² →R² defined by F(x, y) = (3x+4y, 2x-5y) and following bases of R²: E= {e₁, e₂} = {(1, 0), (0, 1)} and S = {u₁, u₂} = {(1, 2), (2, 3)}. Then the matrix A representing F relative to the basis E is:

In a group G, if a⁵ = e, aba^-1 = b² for a, b ∈ G then o(b) is equal to:

Which one of the following statements is wrong.

Which one of the following is wrong?

If f(x) satisfies the conditions of Rolle's theorem in [1, 2] and f(x) is continuous in [1, 2], then \(\int_1^2f'(x)dx\) is equal to

If f is twice differentiable function such that f''(x) = - f(x) and f'(x) = g(x), h(x) = [f(x)]²+[g(x)]² and h(5)=11, then h(10) =

Let f: R→R such that f(1) =3 and f'(1) = 6. Then \(\lim\limits_{x\rightarrow0}\left(\frac{f(1+x)}{f(1)}\right)^{1/x}\)equals