Recent questions in JEEMAIN PAST PAPERS

If, for a positive integer n, the quadratic equation, $x(x + 1) + (x + 1)(x + 2) +.....+ (x + \overline{n – 1})(x + n) = 10n$ has two consecutive integral solutions, then n is equal to

asked Dec 11, 2018 by pady_1

1 answer

The radius of a circle, having minimum area, which touches the curve $y = 4 - x^2$ and the lines, $y = |x|$ is

asked Dec 11, 2018 by pady_1

1 answer

Let $a, \; b, \; c \in R$ . If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x+y) = f(x) + f(y) + xy, \forall x, y \in R$ , then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to

asked Dec 11, 2018 by pady_1

1 answer

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is

asked Dec 11, 2018 by pady_1

1 answer

The value of $(^{21} C_1 - ^{10}C_1) + (^{21}C_2 - ^{19}C_2) + (^{21}C_3 -^{10}C_3) + (^{21}C_4 - ^{10}C_4) +...+(^{21}C_{10} - ^{10}C_{10})$ is

asked Dec 11, 2018 by pady_1

1 answer

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is

asked Dec 11, 2018 by pady_1

1 answer

It two different numbers are taken from the set $\{0,1,2,3,....., 10\}$ ; then the probability that their sum as well as absolute difference are both multiple of 4, is

asked Dec 11, 2018 by pady_1

1 answer

The normal to the curve $y(x-2)(x-3) = x+6$ at the point where the curve intersects the y-axis passes through the point :

asked Dec 11, 2018 by pady_1

1 answer

Let $\overrightarrow{a} = 2 \hat{i} +\hat{j} - 2 \hat{k}$ and $\overrightarrow{b} = \hat{i} + \hat{j}$ . Let $\overrightarrow{c}$ be a vector such that $| \overrightarrow{c} - \overrightarrow{a} | = 3, \; |(\overrightarrow{a} \times \overrightarrow{b} ) \times \overrightarrow{c} |=3$ and the angle between $\overrightarrow{c}$ and $\overrightarrow{a} \times \overrightarrow{b}$ be $30^{\circ}$ . The $\overrightarrow{a}.\overrightarrow{c}$ is equal to

asked Dec 11, 2018 by pady_1

1 answer

$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2 x)^3}$ equals

asked Dec 11, 2018 by pady_1

1 answer

The function $f : R \to \begin{bmatrix}-\frac{1}{2}, \frac{1}{2} \end{bmatrix}$ defined as $f(x) = \frac{x}{1+x^2}$ , is :

asked Dec 11, 2018 by pady_1

1 answer

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$ . Then the tangent to this hyperbola at $P$ also passes through the point :

asked Dec 11, 2018 by pady_1

1 answer

The eccentriciy of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directrices is $x = -4$ , then the equation of the normal to it at $(1, \frac{3}{2})$ is

asked Dec 11, 2018 by pady_1

1 answer

Let $I_n = \int \tan^n x\; dx, \; (n > 1)$ . If $I_4 + I_6 = a \tan^5 x + bx^5 + C$ , where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to

asked Dec 11, 2018 by pady_1

1 answer

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$ , having normal perpendicular to both the lines $\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-4}{3}$ and $\frac{x-2}{2} = \frac{y+1}{-1} = \frac{z+7}{-1}$ , is

asked Dec 11, 2018 by pady_1

1 answer

For any three positive real numbers $a, \;b$ and $c,\; 9(25a^2 + b^2 ) + 25(c^2 – 3ac) = 15b(3a + c)$ , Then

asked Dec 11, 2018 by pady_1

1 answer

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$ , then $adj (3A^2 + 12A)$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

A vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that $AP = 2 AB$ . If $\angle{BPC} = \beta$ , then $\tan \beta$ is equal to

asked Dec 11, 2018 by pady_1

1 answer

If $(2+\sin x) \frac{dy}{dx} +(y+1) \cos x = 0$ and $y(0)=1$ , then $y(\frac{\pi}{2})$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

If for $x \in (0, \frac{1}{4})$ , the derivative of $\tan^{-1} \begin{pmatrix} \frac{6x \sqrt{x}}{1-9x^3} \end{pmatrix}$ is $\sqrt{x}. g(x)$ , then $g(x)$ equals :

asked Dec 11, 2018 by pady_1

1 answer

If the image of the point P(1, -2, 3) in the plane $2x + 3y - 4z + 22 =0$ measured parallel to the line, $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$ is $Q$ , then $PQ$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

The area (in sq. units) of the region $\{ (x, y) : x \geq 0, \; x + y \leq 3, \; x^2 \leq 4y$ and $y \leq 1 + \sqrt{x} \}$ is :

asked Dec 11, 2018 by pady_1

2 answers

Twenty meters of wires is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

asked Dec 11, 2018 by pady_1

1 answer

Let $k$ be an integer such that the triangle with vertices $(k, -3k),\; (5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point :

asked Dec 11, 2018 by pady_1

1 answer

Let $\omega$ be a complex number such that $2 \omega + 1 = z$ where $z = \sqrt{-3}$ . If $\begin{vmatrix} 1 & 1& 1 \\ 1 & -\omega^2-1& \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$ , then $k$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\frac{1}{4}$ and P (All the three events occur simultaneously) = $\frac{1}{6}$ . Then the probability that at least one of the events occurs, is :

asked Dec 11, 2018 by pady_1

1 answer

If $5(\tan^2 x – \cos^2 x) = 2 \cos2x + 9$ , then the value of $\cos 4 x$ is :

asked Dec 11, 2018 by pady_1

1 answer

The following statement $(p \to q)\; [(\sim p \to q) \to q]$ is :

asked Dec 11, 2018 by pady_1

1 answer

If S is the set of distinct values of 'b' for which of the following system of linear equations $x + y + z = 1$ $x + ay + z = 1$ $ax + by + z = 0$ has no solution, then S is :

asked Dec 11, 2018 by pady_1

1 answer

The Boolean Expression $(p \wedge \sim q) \vee q \vee(\sim p \wedge q)$ is equivalent to :

asked Dec 11, 2018 by pady_1

1 answer

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is $30^{\circ}$ . After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is $60^{\circ}$ . Then the time taken (in minutes) by him, from B to reach the pillar, is :

asked Dec 11, 2018 by pady_1

1 answer

If $0 \leq x \leq 2\pi$ , then the number of real values of $x$ , which satisfy the equation $\cos x+\cos 2x+\cos 3x+\cos 4x=0$ , is :

asked Dec 11, 2018 by pady_1

1 answer

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ?

asked Dec 11, 2018 by pady_1

1 answer

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true ?

asked Dec 11, 2018 by pady_1

1 answer

Let $\overrightarrow{a}, \; \overrightarrow{b}$ and $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a} \times (\overrightarrow{b} \times \overrightarrow{c} ) = \frac{\sqrt 3}{2} (\overrightarrow{b} + \overrightarrow{c})$ . If $\overrightarrow{b}$ is not parallel to $\overrightarrow{c}$ , then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is :

asked Dec 11, 2018 by pady_1

1 answer

If the line, $\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$ lies in the plane, $lx + my - z =9$ , then $l^2 + m^2$ is equal to

asked Dec 11, 2018 by pady_1

1 answer

The distance of the point $(1, −5, 9)$ from the plane $x−y+z=5$ measured along the line $x=y=z$ is :

asked Dec 11, 2018 by pady_1

1 answer

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

asked Dec 11, 2018 by pady_1

1 answer

Let P be the point on the parabola, $y^2=8x$ which is at a minimum distance from the centre C of the circle, $x^2+(y+6)^2=1$ . Then the equation of the circle, passing through C and having its centre at P is :

asked Dec 11, 2018 by pady_1

1 answer

If one of the diameters of the circle, given by the equation, $x^2+y^2−4x+6y−12=0$ , is a chord of a circle S, whose centre is at (−3, 2), then the radius of S is :

asked Dec 11, 2018 by pady_1

1 answer

The centres of those circles which touch the circle, $x^2+y^2−8x−8y−4=0$ , externally and also touch the x-axis, lie on :

asked Dec 11, 2018 by pady_1

1 answer

Two sides of a rhombus are along the lines, $x−y+1=0$ and $7x−y−5=0$ . If its diagonals intersect at $(−1, −2)$ , then which one of the following is a vertex of this rhombus ?

asked Dec 11, 2018 by pady_1

1 answer

If a curve $y = f(x)$ passes through the point $(1, -1)$ and satifies the differential equation, $y(1+xy) dx = x ;dy$ , then $f(-\frac{1}{2})$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

The area (in sq. units) of the region $\{ (x,y) : y^2 \geq 2x$ and $x^2 + y^2 \leq 4x, \; x \geq 0, \; y \geq 0 \}$ is :

asked Dec 11, 2018 by pady_1

1 answer

$\begin{align} \displaystyle{\lim_{n \to \infty} \begin{pmatrix} \frac{(n+1) (n+2) ...3n}{n^{2n}} \end{pmatrix} }^{1/n} \end{align}$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

The integral $\begin{align} \int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx \end{align}$ is equal to : where C is an arbitrary constant.

asked Dec 11, 2018 by pady_1

1 answer

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then :

asked Dec 11, 2018 by pady_1

1 answer

Consider $f(x) = \tan^{-1} \begin{pmatrix} \sqrt{\frac{1+\sin x}{1 - \sin x}} \end{pmatrix}, x \in (0, \frac{\pi}{2})$ . A normal to $y = f(x)$ at $x = \frac{\pi}{6}$ also passes through the point :

asked Dec 11, 2018 by pady_1

2 answers

For $x \in R, \; f(x) = | \log 2 - \sin x |$ and $g(x) = f( f(x))$ , then :

asked Dec 11, 2018 by pady_1

1 answer

Let $p = \displaystyle{\lim_{x \to 0^+} (1 + \tan^2 \sqrt{x} )^{1/2x} }$ then $\log p$ is equal to :

asked Dec 11, 2018 by pady_1

1 answer

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