List of practice Questions

If \[ I = \int_0^{\frac{\pi}{2}} \frac{\sin^2 \frac{3}{2}x}{\sin^2 x + \cos^2 x} \, dx, \] then \[ \int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals:

The equation of the chord of the ellipse $ \frac{x^2}{25} + \frac{y^2}{16} = 1 $, whose mid-point is $ (3, 1) $ is:

The value of $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k+3)!}$ is:

The number of solutions of the equation $ 2x + 3\tan x = \pi $, $ x \in [-2\pi, 2\pi] - \left\{ \pm \frac{\pi}{2}, \pm \frac{3\pi}{2} \right\} $ is

Let $ [x] $ denote the greatest integer less than or equal to $ x $. Then the domain of $ f(x) = \sec^{-1} (2[x] + 1) $ is:

If the components of $ \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} $ along and perpendicular to $ \vec{b} = 3\hat{i} + \hat{j} - \hat{k} $ respectively are $ \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) $ and $ \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

Let the area of a triangle $ \triangle PQR $ with vertices $ P(5, 4) $, $ Q(-2, 4) $, and $ R(a, b) $ be 35 square units. If its orthocenter and centroid are $ O(2, \frac{14}{5}) $ and $ C(c, d) $ respectively, then $ c + 2d $ is equal to:

Let \[ \int x^3 \sin x \, dx = g(x) + C, \quad \text{where $ C $ is the constant of integration.} \] If \[ g\left( \frac{\pi}{2} \right) + g\left( \frac{\pi}{2} \right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \quad \alpha, \beta, \gamma \in {Z}, \] then \[ \alpha + \beta - \gamma \text{ equals:} \]

Let $ f: \mathbb{R} \to \mathbb{R} $ be a function defined by $ f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 $. If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of $ 28 \sum_{i=1}^5 f(i) $ is:

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ be a function defined by $ f(x) = ||x+2| - 2|x|| $. If m is the number of points of local maxima of f and n is the number of points of local minima of f, then m + n is

Let $ \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} $ and a vector $ \vec{c} $ be such that \[ (\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] and \[ \vec{a} \cdot \vec{c} = 3. \] If $ \vec{b} \times \vec{c} = \vec{d} $, then find $ |\vec{a} \cdot \vec{d}| $.

If the image of the point $ (4, 4, 3) $ in the line $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} $ is $ (a, \beta, \gamma) $, then $ a + \beta + \gamma $ is equal to:

Let $ (2, 3) $ be the largest open interval in which the function $ f(x) = 2 \log_e (x - 2) - x^2 + ax + 1 $ is strictly increasing, and $ (b, c) $ be the largest open interval, in which the function $ g(x) = (x - 1)^3 (x + 2 - a)^2 $ is strictly decreasing. Then $ 100(a + b - c) $ is equal to:

Suppose that the number of terms in an A.P. is $ 2k, k \in \mathbb{N} $. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then $ k $ is equal to:

The sum of the squares of all the roots of the equation $ x^2 + [2x - 3] - 4 = 0 $ is:

Let $ \langle a_n \rangle $ be a sequence such that $ a_0 = 0 $, $ a_1 = \frac{1}{2} $, and $ 2a_{n+2} = 5a_{n+1} - 3a_n $.n= 0,1,2,3.... Then $ \sum_{k=1}^{100} a_k $ is equal to:

Let the ellipse $ E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, $ a > b $ and $ E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 $, $ A < B $, have the same eccentricity $ \sqrt{\frac{1}{3}} $. Let the product of their lengths of latus rectums be $ \sqrt{\frac{32}{3}} $ and the distance between the foci of $ E_1 $ be 4. If $ E_1 $ and $ E_2 $ meet at $ A $, $ B $, $ C $, and $ D $, then the area of the quadrilateral ABCD equals:

The area of the region enclosed by the curves $ y = x^2 - 4x + 4 $ and $ y^2 = 16 - 8x $ is:

Suppose A and B are the coefficients of the 30th and 12th terms respectively in the binomial expansion of $ (1 + x)^{2n - 1} $. If $ 2A = 5B $, then $ n $ is equal to:

The value of \[ \int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left( (\log_e x)^2 +1 \right)^{-1}}}{e^{\left( (\log_e x)^2 +1 \right)^{-1}} + e^{\left( (6-\log_e x)^2 +1 \right)^{-1}}} \right) dx \] is:

What is the sum of the infinite series $ S = \sum_{n=0}^{\infty} \frac{1}{3^n} $?

The total number of 10-digit sequences formed by only $ \{0, 1, 2\} $, where 1 should be used at least 5 times and 2 should be used exactly three times, is:

Let a circle $ C $ with radius $ r $ passes through four distinct points $ (0, 0), (k, 3k), (2, 3), (-1, 5) $, such that $ k \neq 0 $, then $ (10k + 2r^2) $ is equal to:

Let (a, b) be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_{a}^{b} \frac{9x^2}{1+5^{x}} \, dx$ is equal to:

If $ A $ and $ B $ are binomial coefficients of the 30$^\text{th}$ and 12$^\text{th}$ terms of the binomial expansion $ (1 + x)^{2n-1} $, and $ 2A = 5B $, then the value of $ n $ is