Recent questions tagged maths

Consider the following two statements : <br> P : If 7 is an odd number, then 7 is divisible by 2. <br> Q : If 7 is a prime number, then 7 is an odd number. <br> If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of contrapositive of Q, then the ordered pair $(V_1, V_2)$ equals :

asked May 29, 2017 by meena.p

1 answer

If $m$ and $M$ are the minimum and the maximum values of $4 +\large\frac{1}{2} $$\sin ^2 2x - 2 \cos ^4 x , x \in R$ then $M−m$ is equal to :

asked May 29, 2017 by meena.p

1 answer

The number of $x \in [0, 2 \pi]$ for which $| \sqrt{2 \sin ^4 x +18 \cos ^2 x} - \sqrt {2 \cos ^4 x +18 \sin ^2 x}|=1$ is :

asked May 29, 2017 by meena.p

1 answer

If $A$ and $B$ are any two events such that $P(A)= \large\frac{2}{5}$ and $P(A \cap B)=\large\frac{3}{20}$,then the conditional probability, $P(A |(A' \cup B')),$ where $A'$ denotes the complement of A, is equal to :

asked May 29, 2017 by meena.p

1 answer

If the mean deviation of the numbers $1, 1+ d, ..., 1+100d$ from their mean is $255$, then a value of $d$ is :

asked May 29, 2017 by meena.p

1 answer

In a triangle $ABC$, right angled at the vertex $A$, if the position vectors of A, B and C are respectively $3 \hat i + \hat j - \hat k , \hat i+3 \hat j +p \hat k $ and $5 \hat i+q \hat j -4 \hat k$ then the point $(p, q)$ lies on a line :

asked May 29, 2017 by meena.p

1 answer

The distance of the point $(1, −2, 4)$ from the plane passing through the point $(1, 2, 2)$ and perpendicular to the planes $x−y+2z=3$ and $2x−2y+z+12=0,$ is :

asked May 29, 2017 by meena.p

1 answer

The shortest distance between the lines $\large\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\large\frac{x+2}{-1} = \frac{y-4}{8} =\frac{z-5}{4}$ lies in the interval

asked May 29, 2017 by meena.p

1 answer

Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation $9e^2−18e+5=0$. If $S(5, 0)$ is a focus and $5x=9$ is the corresponding directrix of this hyperbola, then $a^2−b^2$ is equal to

asked May 29, 2017 by meena.p

1 answer

A circle passes through $(−2, 4)$ and touches the y-axis at $(0, 2)$. Which one of the following equations can represent a diameter of this circle ?

asked May 26, 2017 by meena.p

1 answer

The point $(2, 1)$ is translated parallel to the line $L : x−y=4$ by $2 \sqrt 3$ units. If the new point Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is :

asked May 26, 2017 by meena.p

1 answer

If a variable line drawn through the intersection of the lines $\large\frac{x}{3}+\frac{y}{4}$$=1$ and $\large\frac{x}{4}+\frac{y}{3}$$=1$ , meets the coordinate axes at $A$ and $B,(A \neq B),$ then the locus of the midpoint of AB is :

asked May 26, 2017 by meena.p

1 answer

If $f(x)$ is a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim \limits_{t \to x} \large\frac{t^2f(x)-x^2f(t)}{t-x} $$=1$ fopr each $x >0$ then $f( \large\frac{3}{2})$ is equal to :

asked May 26, 2017 by meena.p

1 answer

The area (in sq. units) of the region described by $A=\{(x, y)|y ≥ x2−5x+4, x+y ≥ 1, y ≤ 0\}$ is :

asked May 26, 2017 by meena.p

1 answer

If $2 \int \limits_0^1 \tan ^{-1} x dx = \int \limits _0^1 \cot ^{-1} (1-x+x^2)dx$ then $\int \limits_0^1 \tan ^{-1} (1-x+x^2)dx$ is equal to :

asked May 26, 2017 by meena.p

1 answer

If $\int \Large\frac{dx}{\cos ^3 x \sqrt {2 \sin 2x}}$$=(\tan x)^A+C(\tan x)^B +k$ where k is a constant of integration, then $A+B+C$ equals :

asked May 26, 2017 by meena.p

1 answer

The minimum distance of a point on the curve $y=x^2−4$ from the origin is :

asked May 26, 2017 by meena.p

1 answer

If the tangent at a point $P,$ with parameter $t$, on the curve $x=4t^2+3, y=8t^3−1, t \in R,$ meets the curve again at a point Q, then the coordinates of Q are :

asked May 26, 2017 by meena.p

1 answer

If the function $f(x) = \left\{ \begin{array}{l l} -x, & \quad x < 1 \\ a+\cos ^{-1} (x+b), & \quad 1 \leq x \leq 2 \end{array} \right.$ is differentiable at $x=1$, then $\large\frac{a}{b}$ is equal to :

asked May 26, 2017 by meena.p

1 answer

If $\lim \limits_{x \to \infty}$ \bigg( 1+ \large\frac{a}{x}-\frac{4]{x^2} \bigg)^{2x} =e^3$, then 'a' is equal to :

asked May 26, 2017 by meena.p

1 answer

The value of $\sum \limits_{r=1}^{15} r^2 \bigg( \large\frac{^{15} C_r}{^{15}C_{r-1}}\bigg)$ is equal to :

asked May 26, 2017 by meena.p

1 answer

Let $x, y, z$ be positive real numbers such that $x+y+z=12$ and $x^3y^4z^5=(0.1) (600)^3$. Then $x^3+y^3+z^3$ is equal to :

asked May 26, 2017 by meena.p

1 answer

For $x \in R,x \neq -1$, if $(1+x)^{2016} +x(1=x)^{2015} +x^2(1+x)^{2014}+.......+x^{2016}=\sum \limits_{i=0} ^{2016} a_i x^i $, then $a_{17}$ is equal to :

asked May 25, 2017 by meena.p

1 answer

If the four letter words (need not be meaningful ) are to be formed using the letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is :

asked May 25, 2017 by meena.p

1 answer

The number of distinct real roots of the equation $P=\begin{vmatrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{vmatrix} = 0 $ in the interval $\bigg[ \large\frac{-\pi}{4} ,\frac{\pi}{4} \bigg]$ is :

asked May 25, 2017 by meena.p

1 answer

If $P=\begin{bmatrix} \large\frac{\sqrt 3}{2} & \frac{1}{2} \\ \large\frac{-1}{2} & \large\frac{\sqrt 3}{2} \end{bmatrix} $ , $A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and $Q= PAP^{T}$, then $P^Tq^{2015}P$ is :

asked May 25, 2017 by meena.p

1 answer

If the equations $x^2+bx−1=0$ and $x^2+x+b=0$ have a common root different from $−1$, then $|b|$ is equal to :

asked May 25, 2017 by meena.p

1 answer

The point represented by $2+i$ in the Argand plane moves $1 \;unit $ eastwards, then $2 \;units$ northwards and finally from there $2 \sqrt 2$ units in the south-westwards direction. Then its new position in the Argand plane is at the point represented by :

asked May 25, 2017 by meena.p

1 answer

For $ x \in R, x \neq 0,x \neq 1 ,$ let $f_0(x) =\large\frac{1}{1-x} $ and $f_{n+1}(x) =f_0 (f_n(x)),n =0,1,2$.... Then the value of $f_{100} (3)+f_1\bigg( \large\frac{2}{3}\bigg) +f_2 \bigg( \large\frac{3}{2}\bigg)$ is equal to :

asked May 25, 2017 by meena.p

1 answer

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

asked Mar 22, 2017 by meena.p

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 Mar 22, 2017 by meena.p

1 answer

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

asked Mar 22, 2017 by meena.p

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 Mar 21, 2017 by meena.p

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 Mar 21, 2017 by meena.p

1 answer

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

asked Mar 21, 2017 by meena.p

1 answer

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

asked Mar 21, 2017 by meena.p

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 the distance between its foci, is:

asked Mar 21, 2017 by meena.p

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 Mar 21, 2017 by meena.p

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 Mar 21, 2017 by meena.p

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 Mar 21, 2017 by meena.p

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 Mar 21, 2017 by meena.p

1 answer

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

asked Mar 20, 2017 by meena.p

1 answer

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

asked Mar 20, 2017 by meena.p

1 answer

$\lim_\limits{n \to \infty} \bigg( \large\frac{(n-1)(n+2)....3n}{n^{2n}}\bigg)^{1/n}$ is eaual to

asked Mar 20, 2017 by meena.p

1 answer

The integral $\int \large\frac{2x^{12} +5x^{9}}{(x^2+x^3+1)^3}$$dx$ is equal to

asked Mar 17, 2017 by meena.p

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 Mar 16, 2017 by meena.p

1 answer

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

asked Mar 16, 2017 by meena.p

1 answer

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

asked Mar 16, 2017 by meena.p

1 answer

Let $p = \lim \limits_{x \to 0+} (1+ \tan ^2 \sqrt x )^{1/2x}$ then long p is equal to

asked Mar 16, 2017 by meena.p

1 answer

If the sum of the first ten terms of the series $\bigg(1 \large\frac{3}{5}\bigg) \normalsize +\bigg(2 \large\frac{2}{5} \bigg)^2 \normalsize +\bigg(3 \large\frac{1}{5}\bigg)^2 \normalsize +4^2 +\bigg(4 \large\frac{4}{5} \bigg)^2 \normalsize +.....is\;\large\frac{16}{5}$$\;m$ then m is equal to

asked Mar 16, 2017 by meena.p

1 answer

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