The Set of Intelligent Students in a class is

An equilateral triangle is inscribed in the parabola $y^{2} = 4ax$, such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:

A chain of video stores sells three different brands of DVD players. Of its DVD player sales, 50% are brand 1, 30% are brand 2 and 20% are brand 3. Each manufacturer offers one year warranty on parts and labor. It is known that 25% of brand 1 DVD players require warranty repair work whereas the corresponding percentage for brands 2 and 3 are 20% and 10% respectively. The probability that a randomly selected purchaser has a DVD player that will need repair while under warranty, is:

The locus of the intersection of the two lines $\sqrt{3} x-y=4k\sqrt{3}$ and $k(\sqrt{3}x+y)=4\sqrt{3}$, for different values of k, is a hyperbola. The eccentricity of the hyperbola is:

Constant forces $\vec{P}= 2\hat{i} - 5\hat{j} + 6\hat{k}$ and $\vec{Q}= -\hat{i} + 2\hat{j}- \hat{k}$  act on a particle. The work done when the particle is displaced from A whose position vector is $4\hat{i} - 3\hat{j} - 2\hat{k}$, to B whose position vector is $6\hat{i} + \hat{j} - 3k\hat{k}$ , is:

The value of $\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to:

For the vectors $\vec{a}=-4\hat{i}+2\hat{j}, \vec{b}=2\hat{i}+\hat{j}$ and $\vec{c}=2\hat{i}+3\hat{j}$, if $\vec{c}=m\vec{a}+n\vec{b}$ then the value of m + n is

The value of $\int_{0}^{\pi/4} log(1+tanx)dx$ is equal to:

The number of ways in which 5 days can be chosen in each of the 12 months of a non-leap year, is:

If [x] represents the greatest integer not exceeding x, then $\int_{0}^{9} [x] dx$ is

In a group of 200 students, the mean and the standard deviation of scores were found to be 40 and 15, respectively. Later on it was found that the two scores 43 and 35 were misread as 34 and 53, respectively. The corrected mean of scores is:

If the matrix has an inverse matrix, then the value of K is:

The mean deviation from the mean of the AP a, a + d, a + 2d, ..., a + 2nd, is:

If (x₀, y₀) is the solution of the equations (2x)^ln2 = (3y)^ln3 and 3^lnx = 2^lny, then x₀ is:

The value of tan 1° tan 2° tan 3° ... tan 89° is:

If $\alpha$ and $\beta$ are the roots of the equation $2x^{2}+ 2px + p^{2} = 0$, where $p$ is a non-zero real number, and $\alpha^{4}$ and $\beta^{4}$ are the roots of $x^{2} - rx + s = 0$, then the roots of $2x^{2} - 4p^{2}x + 4p^{4} - 2r = 0$ are:

The number of ways to arrange the letters of the English alphabet, so that there are exactly 5 letters between a and b, is:

Suppose, the system of linear equations 

If $\vec{A}=4\hat{i}+3\hat{j}+\hat{k}$ and $\vec{B}=2\hat{i}-\hat{j}+2\hat{k}$ , then the unit vector $\hat{N}$ perpendicular to the vectors $\vec{A}$ and $\vec{B}$ ,such that $\vec{A}, \vec{B}$ , and $\hat{N}$ form a right handed system, is:

The value of $\int \frac{(x+1)}{x(xe^{x}+1)} dx$ is equal to

The sum of two vectors $\vec{a}$ and $\vec{b}$ is a vector $\vec{c}$ such that $|\vec{a}|=|\vec{b}|=|\vec{c}|=2$. Then, the magnitude of $\vec{a}-\vec{b}$ is equal to:

If x and y are positive real numbers satisfying the system of equations $x^{2}+y\sqrt{xy}=336$ and $y^{2}+x\sqrt{xy}=112$, then x + y is:

From three collinear points A, B and C on a level ground, which are on the same side of a tower, the angles of elevation of the top of the tower are 30°, 45° and 60° respectively. If BC = 60 m, then AB is:

If $x = 1$ is the directrix of the parabola $y^{2} = kx - 8$, then k is:

If $sin x + a cos x = b$, then $|a sin x - cos x|$ is:

A condition that $x^{3} + ax^{2} + bx + c$ may have no extremum is

If n and r are integers such that 1 ≤ r ≤ n, then the value of n ^n-1C_r-1is

If the foci of the ellipse $b^{2}x^{2}+16y^{2}=16b^{2}$ and the hyperbola $81x^{2}-144y^{2}=\frac{81 \times 144}{25}$ coincide, then the value of $b$, is

There are 8 students appearing in an examination of which 3 have to appear in Mathematics paper and the remaining 5 in different subjects. Then, the number of ways they can be made to sit in a row, if the candidates in Mathematics cannot sit next to each other is

If $x$ is so small that $x^{2}$ and higher powers of $x$ can be neglected, then $\frac{(9+2x)^{1/2}(3+4x)}{(1-x)^{1/5}}$ is approximately equal to

If the sets A and B are defined as A = {(x, y) | y = 1 / x, 0 ≠ x ∈ R}, B = {(x, y)|y = -x ∈ R} then

If A, B and C is three angles of a ΔABC, whose area is Δ. Let a, b and c be the sides opposite to the angles A, B and C respectively. Is $s=\frac{a+b+c}{2}=6$, then the product $\frac{1}{3} s^{2} (s-a)(s-b)(s-c)$ is equal to

A normal to the curve $x^{2} = 4y$ passes through the point (1, 2). The distance of the origin from the normal is

Suppose r integers, 0 < r < 10, are chosen from (0, 1, 2, ...,9) at random and with replacement. The probability that no two are equal, is

If $x^{2} + 2ax + 10 - 3a > 0$ for all x ∈ R, then

A box contains 3 coins, one coin is fair, one coin is two headed and one coin is weighted, so that the probability of heads appearing is $\frac{1}{3}$ . A coin is selected at random and tossed, then the probability that head appears is

If a vector $\vec{a}$ makes an equal angle with the coordinate axes and has magnitude 3, then the angle between $\vec{a}$ and each of the three coordinate axes is

If $f(x)=\left\{\begin{matrix} \frac{sin[x]}{[x]} &, [x]\ne0 \\ 0 &, [x]=0 \end{matrix}\right.$ , where [x] is the largest integer but not larger than x, then $\lim_{x\to0}f(x)$ is

If tan A - tan B = x and cot B - cot A = y, then cot (A - B) is equal to

If $a = log_{12}^{18}$, $b = log_{24}^{54}$, then $ab + 5(a - b)$ is

A student takes a quiz consisting of 5 multiple choice questions. Each question has 4 possible answers. If a student is guessing the answer at random and answer to different are independent, then the probability of atleast one correct answer is

The condition that the line lx + my + n = 0 becomes a tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ , is

The value of sin 20° sin 40° sin 80° is

Two non-negative numbers whose sum is 9 and the product of the one number and square of the other number is maximum, are

The median AD of ΔABC is bisected at E and BE is produced to meet the side AC at F. Then, AF ∶ FC is

If PQ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ such that OPQ is an equilateral triangle, where O is the centre of the hyperbola, then which of the following is true?

In ΔABC, if a = 2, b = 4 and ∠C = 60°, then A and B are respectively equal to

If $\int \frac{xe^{x}}{\sqrt{1+e^{x}}}=f(x)\sqrt{1+e^{x}}-2log \frac{\sqrt{1+e^{x}}-1}{\sqrt{1+e^{x}}+1}+C$ then $f(x)$ is

The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

How many even integers between 4000 and 7000 have four different digits?