List of practice Questions

The integral \( \int \frac{\cos x - \sin x}{10 + \sin 2x} dx \) is equal to:

The integral \( \int_{0}^{1} x^{5/2} (1-x)^{3/2} dx \) is equal to:

The integral \( \int_{\sqrt{2}}^{2} \frac{x}{(x^3 - x^2 + x - 1)(x+1)} dx \) is equal to:

The integral \( \int_{0}^{\pi} |x \cos 2x| dx \) is equal to:

A circle passes through the ends of the latus rectum of parabola \( y^2=12x \) and has its centre at the vertex. The area inside the circle and outside the parabola in the \( 1^{st} \) quadrant is:

Consider the differential equations \(\frac{dy}{dx}(x+y+1)=1\) and \(\frac{dx}{dy}=3y+2x^2\).
Which of the following is correct regarding these two differential equations?

The surface area of a cube is 150 sq. cm. If it is increased by 0.025 sq. cm, then the approximate increase in its volume (in c.c.) is:

The length of the tangent drawn at the point \(P(1, 3\sqrt{3})\) on the curve is \(\frac{x^2}{3} + \frac{y^2}{27} = 4\):

The value of the integral \(\int \frac{dx}{\cos 2x + \sin 2x + 2\sin^2 x}\) is:

If \(x > 0\), then \(\int \frac{1}{\sqrt{x^4 + 2x^3 + 2x^2}} dx =\)

The function \(f(x) = \frac{x}{2} + \frac{2}{x}\) has a local minimum at

The integral \( \int \frac{\cos^3 x}{(1+\sin x)^4} dx \) is equal to:

If \(f(x)=\frac{\cos^{4}x-1}{x^{2}}\), \(x\ne0\) and \(f(0)=2\) is a real valued function, then:

If \(f(x)=\sqrt{-(1+x)}\sec^{-1}x\) is a real valued function, then \(f'(x)=\)

If \(x \in [-1,1]\) and \(y=(\cot^{-1}x)^{\cot^{-1}x}\), then \(\left(\frac{dy}{dx}\right)_{x=0}=\)

If \(x=\sinh^{-1}t+\log(t^{2}+1)\) and \(y=\tan^{-1}t+\log|t|\), then \(\frac{dy}{dx}=\)

If \(f(x)=ax^{3}+bx^{2}+cx+1\) attains an extreme value \(2\) at \(x=1\) and another extreme value at \(x=\frac{2}{3}\), then \(2b+3c\) is equal to

If a cylindrical tank of radius 3 m is filled with water at the rate of \(\frac{3}{2} \, m^3/sec \), then the rate of change of its water level in (m/sec) is:

If \(d_1\) and \(d_2\) are the distances of the foci of the hyperbola \(4x^{2}-9y^{2}-16x+54y-101=0\) from the point (2,-3), then \(d_1+d_2=\)

\(A(-4,9,k)\), \(B(-1,k,k)\), \(C(0,7,10)\) form an isosceles right-angled triangle. If \(AB=BC\) and \(AC\) is an integer then the perimeter of \(\Delta ABC\) is

If N is the foot of the perpendicular drawn from the point \(P(5,-1,3)\) to the line passing through the points \(A(1,3,-5)\) and \(B(3,-1,5)\) then the ratio in which N divides AB is

If l, m, n are the direction cosines of a normal drawn to the plane \(2x-3y+6z-7=0\) and d is the length of the perpendicular drawn from origin to this plane then \(7d|l+m+n|=\)

Evaluate \(\lim_{x\rightarrow2}(x^{2}-3x+3)^{\frac{1}{x^{2}-4}}\).

If \[ f(x)= \begin{cases} \dfrac{x^{2}-4}{\sqrt{2-x}}, & x<2\\ a, & x=2\\ \log(x-2), & x>2 \end{cases} \] is a real valued function, then:

Two circles, each of radius 5, touch at $(1,2)$. If the common tangent at the point of contact is $4x+3y=10$, then the equation of one of the circles is: