List of practice Questions

\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)

Then

Let \(S=2+\frac{6}{7}+\frac{12}{7 ^2}+\frac{20}{7 ^3}+\frac{30}{7 ^4}+…\) Then \(4S\) is equal to

The area bounded by the curves y = |x² – 1| and y = 1 is

If α, β are the roots of the equation
\(x^2-(5+3^{\sqrt{log_35}}-5^{\sqrt{log_53}})+3(3^{(log_35)^{\frac{1}{3}}}-5^{(log_53)^{\frac{2}{3}}}-1) = 0\)
then the equation, whose roots are α + 1/β and β + 1/α , is

If \( A = \{3, 5, 7\} \) and \( B = \{1, 2, 3, 5\} \), then \( A \times B \cap B \times A \) is equal to:

Let \(x(t)=2\sqrt2costsin\sqrt2t \) and \(y(t)=2\sqrt2sintsin\sqrt2t\) ,\(t ∈(0,\frac{π}{2})\)

Then \(\frac{1+(\frac{dy}{dx})^2}{\frac{d^2y}{dx^2}}\) at \(t=\frac{π}{4}\) is equal to

If the function
\(f(x) = \begin{cases} \frac {log_e(1−x+x^2)+log_e(1+x+x_2)}{sec⁡\ x−cos⁡\ x}, & \quad x∈(−\frac \pi2,\frac \pi2)−{0}\\ k, & \quad x=0 \end{cases}\)
is continuous at \(x = 0\), then k is equal to

The sum of the absolute maximum and absolute minimum values of the function \(\begin{array}{l} f\left(x\right)=\tan^{-1}\left(\sin x-\cos x\right) \end{array}\)in the interval\( [0, π]\) is

1L aqueous solution of \(H_2SO_4\) contains 0.02 m mol \(H_2SO_4\). 50% of this solution is diluted with deionized water to give 1L solution (A). In solution (A), 0.01 m mol of \(H_2SO_4\) are added. Total m mols of \(H_2SO_4\) in the final solution is ______ \(× 10^3\) m mols.

1 mole of HI is heated in a closed container of capacity of 2 L. At equilibrium half a mole of HI is dissociated. The equilibrium constant of the reaction is

Let the normal at the point P on the parabola y² = 6x pass through the point (5, –8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is

Beriberi is the disease caused by the deficiency of.............

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x – y = 3, respectively. If its orthocentre is
\((2, a),−\frac{1}{2}<a<2 \)
then p is equal to _______.

Let a, b, c be distinct non-negative numbers. If the vectors \( a\hat{i} + a\hat{j} + c\hat{k} \), \( \hat{i} + \hat{k} \) and \( c\hat{i} + c\hat{j} + b\hat{k} \) lie in a plane, then c is

Suppose that the temperature at a point (x, y) on a metal plate is \( T(x, y) = 4x^2 - 4xy + y^2 \). An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What is the highest temperature encountered by the ant?

The 10th and 50th percentiles of the observation 32, 49, 23, 29, 118 respectively are

If \( \csc\theta - \cot\theta = 2 \), then the value of \( \csc\theta \) is

The first three moments of a distribution about 2 are 1, 16, -40 respectively. The mean and variance of the distribution are

If \( \alpha, \beta \) are the roots of \( x^2 - x - 1 = 0 \), and \( A_n = \alpha^n + \beta^n \), then Arithmetic mean of \( A_{n-1} \) and \( A_n \) is

If \( x^m y^n = (x+y)^{m+n} \), then \( \frac{dy}{dx} \) is

The value of \( 3^{3-\log_3 5} \) is

There are two sets A and B with \( |A| = m \) and \( |B| = n \). If \( |P(A)| - |P(B)| = 112 \) then choose the wrong option (where \( |A| \) denotes the cardinality of A, and P(A) denotes the power set of A)

If \( (\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c}) \), then

In the following figure, find the total number of squares.