List of practice Questions

If a particle is moving in a straight line so that after \(t\) seconds its distance \(S\) (in cms) from a fixed point on the line is given by \(S = f(t) = t^3 - 5t^2 + 8t\) then the acceleration of the particle at \(t=5\) sec is (in cm/sec\(^2\))

If \(f:[a,b] \to [c,d]\) is a continuous and strictly increasing function, then \(\frac{d-c}{b-a}\) is

\(\int \left( \frac{1}{x^2} + \frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} \right) dx =\)

If the points \((1, 1, \lambda)\) and \((-3, 0, 1)\) are equidistant from the plane \(3x + 4y - 12z + 13 = 0\), then the values of \(\lambda\) are

If \(f(x) = \frac{x(a^x - 1)}{1 - \cos x}\) and \(g(x) = \frac{x(1 - a^x)}{a^x \left(\sqrt{1 - x^2} - \sqrt{1 + x^2}\right)}\), then \(\lim_{x \to 0} (f(x) - g(x)) =\)

Consider the following statements
Assertion (A): For \(x \in \mathbb{R} - \{1\}\), \(\frac{d}{dx}\left(\tan^{-1}\left(\frac{1+x}{1-x}\right)\right) = \frac{d}{dx}(\tan^{-1}x)\)
Reason (R): For \(x<1\), \(\tan^{-1}\left(\frac{1+x}{1-x}\right) = \frac{\pi}{4} + \tan^{-1}x\),
for \(x>1\), \(\tan^{-1}\left(\frac{1+x}{1-x}\right) = -\frac{3\pi}{4} + \tan^{-1}x\)
The correct answer is

If \(\frac{d}{dx}\left\{ \frac{x-1}{x-\sqrt{x}} e^{2x+1} \right\} = \frac{x-1}{x-\sqrt{x}} e^{2x+1} f(x)\), then \(f(4) =\)

The locus of a point which divides the line segment joining the focus and any point on the parabola \(y^2 = 12x\) in the ratio \(m:n\) (\(m+n \ne 0\)) is a parabola. Then the length of the latus rectum of that parabola is

The curve represented by \(\frac{x^2}{12-\alpha} + \frac{y^2}{\alpha-10} = 1\) is

If any tangent drawn to the ellipse \(\frac{x^2}{16} + \frac{y^2}{9} = 1\) touches one of the circles \(x^2 + y^2 = \alpha^2\), then the range of \(\alpha\) is

Let \(x\) be the eccentricity of a hyperbola whose transverse axis is twice its conjugate axis. Let \(y\) be the eccentricity of another hyperbola for which the distance between the foci is 3 times the distance between its directrices. Then \(y^2 - x^2 =\)

O(0,0,0), A(3,1,4), B(1,3,2) and C(0,4,-2) are the vertices of a tetrahedron. If G is the centroid of the tetrahedron and \(G_1\) is the centroid of its face ABC, then the point which divides \(GG_1\) in the ratio 1:2 is

If L is a line common to the planes \(3x + 4y + 7z = 1\), \(x - y + z = 5\) then the direction ratios of the line L are

The locus of the centre of the circle touching the x-axis and passing through the point \((-1,1)\) is

The centres of all circles passing through the points of intersection of the circles \(x^2 + y^2 + 2x - 2y + 1 = 0\) and \(x^2 + y^2 - 2x + 2y - 2 = 0\) and having radius \(\sqrt{14}\) lie on the curve

For the parabola \(y = x^2 - 3x + 2\), match the items in list-1 to that of the items in list-2.
S is a focus, Z is intersection of axis and directrix, P is one end point of latus rectum, Q is the point on the parabola at which tangent is parallel to X-axis

O(0,0), B(-3,-1), C(-1,-3) are vertices of a triangle OBC. D is a point on OC and E is a point on OB. If the equation of DE is \(2x + 2y + \sqrt{2} = 0\), then the ratio in which the line DE divides the altitude of the triangle OBC is

Every point on the curve \(3x + 2y - 3xy = 0\) is the centroid of a triangle formed by the coordinate axes and a line (L) intersecting both the coordinate axes. Then all such lines (L)

The value of 'a' for which the equation \((a^2-3)x^2 + 16xy - 2ay^2 + 4x - 8y - 2 = 0\) represents a pair of perpendicular lines is

The slope of a common tangent to the circles \(x^2 + y^2 = 16\) and \((x-9)^2 + y^2 = 16\) is

The equation of the circle whose radius is 3 and which touches the circle \(x^2 + y^2 - 4x - 6y - 12 = 0\) internally at \((-1, -1)\) is

Suppose C1 and C2 are two circles having no common points, then

A, B are the events in a random experiment. If \( P(A)=\frac{1}{2}, P(B)=\frac{1}{3}, P(A \cap B)=\frac{1}{4} \), then \( P(A^c | B^c) + P(A | B) = \)

Two persons A and B play a game by throwing two dice. If the sum of the numbers appeared on the two dice is even, A will get \( \frac{1}{2} \) point and B will get \( \frac{1}{2} \) point. If the sum is odd, A will get one point and B will get no point. The arithmetic mean of the random variable of the number of points of A is

A typist claims that he prepares a typed page with typo errors of 1 per 10 pages. In a typing assignment of 40 pages, if the probability that the typo errors are at most 2 is p, then \(e^2 p =\)