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Study Stuff for MCA Examinations

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The expression $\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}$ can be written as

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If $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$, then $\begin{bmatrix}{\vec{a}} & {\vec{b}} & {\vec{c}}\end{bmatrix}$ depends on

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Angle of elevation of the top of the tower from 3
points (collinear) A, B and C on a road leading to the
foot of the tower are 30°, 45° and 60°, respectively.
The ratio of AB and BC is

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Let the height of tower = h

If $\vec{a}, \vec{b}$ are unit vectors such that $2\vec{a}+\vec{b} =3$ then which of the following statement is true?

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The area enclosed between the curves y^{2} = x and
y = |x| is

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we get, y = 0, x = 0, y = 1, x = 1 Therefore,

$\int f(x)\mathrm{d}x=g(x)$, then $\int {x}^5f({x}^3)\mathrm{d}x$

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Test the continuity of the function at x = 2

$f(x)= \begin{cases} \frac{5}{2}-x & \text{ if } x<2 \\ 1 & \text{ if } x=2 \\ x-\frac{3}{2}& \text{ if } x>2 \end{cases}$

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$\lim _{{x}\rightarrow1}\frac{{x}^4-1}{x-1}=\lim _{{x}\rightarrow k}\frac{{x}^3-{k}^2}{{x}^2-{k}^2}=$, then find k

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The value of

2tan^{-1}[cosec(tan^{-1}x) - tan(cot^{-1}x)]

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The graph of function $f(x)=\log _e({x}^3+\sqrt[]{{x}^6+1})$ is symmetric about:

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If $3 sin x + 4 cos x = 5$, then $6tan\frac{x}{2}-9tan^2\frac{x}{2}$

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If the equation $|x^2 – 6x + 8| = a$ has four real solution then find the value of $a$?

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If A is a subset of B and B is a subset of C, then
cardinality of A ∪ B ∪ C is equal to

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Largest value of $cos^2\theta -6sin\theta cos\theta+3sin^2\theta+2 $ is

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Given to events A and B such that odd in favour A are 2 : 1 and odd in favour of $A \cup B$ are 3 : 1. Consistent with this information the smallest and largest value for the probability of event B are given by

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If A and B are square matrices such that $B=-A^{-1} BA$, then $(A + B)^2$ is

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A bag contain different kind of balls in which 5 yellow, 4 black & 3 green balls. If 3 balls are drawn at random then find the probability that no black ball is chosen

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Between any two real roots of the equation $e^x sin x = 1$, the equation $e^x cos x = –1$ has

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If f(x) is a polynomial of degree 4, f(n) = n + 1 & f(0) = 25, then find f(5) = ?

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The maximum value of $f(x) = (x – 1)^2 (x + 1)^3$ is equal to $\frac{2^p3^q}{3125}$
then the ordered pair of (p, q) will be

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The coefficient of $x^{50}$ in the expression of ${(1 + x)^{1000} + 2x(1 + x)^{999} + 3x^2(1 + x)^{998} + ...... + 1001x^{1000}}$

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If ${{x}}_k=\cos \Bigg{(}\frac{2\pi k}{n}\Bigg{)}+i\sin \Bigg{(}\frac{2\pi k}{n}\Bigg{)}$ , then $\sum ^n_{k=1}({{x}}_k)=?$

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Number of point of which f(x) is not differentiable $f(x)=|cosx|+3$ in $[-\pi, \pi]$

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If $n_1$ and $n_2$ are the number of real valued solutions $x = | sin^{–1} x |$ & $x = sin (x)$ respectively, then the value of $n_2– n_1$ is

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The negation of $\sim S\vee(\sim R\wedge S)$ is equivalent to

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A point P in the first quadrant, lies on $y^2 = 4ax$, a > 0, and keeps a distance of 5a units from its focus. Which of the following points lies on the locus of P?

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If $\int x\, \sin x\, sec^3x\, dx=\frac{1}{2}\Bigg{[}f(x){se}c^2x+g(x)\Bigg{(}\frac{\tan x}{x}\Bigg{)}\Bigg{]}+C$, then which of the following is true?

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Let a, b, c, d be no zero numbers. If the point of intersection of the line 4ax + 2ay + c = 0 & 5bx + 2by + d=0 lies in the fourth quadrant and is equidistance from the two are then

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$\theta={\cos }^{-1}\Bigg{(}\frac{3}{\sqrt[]{10}}\Bigg{)}$ is the angle between $\vec{a}=\hat{i}-2x\hat{j}+2y\hat{k}$ & $\vec{b}=x\hat{i}+\hat{j}+y\hat{k}$ then possible values of (x,y) that lie on the locus

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Let R be reflexive relation on the finite set a having 10 elements and if m is the number of ordered pair in R, then

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If $| x - 6|= | x - 4x | -| x^2- 5x +6 |$ , where x is a real variable

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The range of values of $\theta$ in the interval $(0,\pi)$ such that the points (3, 2) and $(cos\theta ,sin\theta)$ lie on the samesides of the line x + y – 1 = 0, is

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Which of the following number is the coefficient of $x^{100}$ in the expansion of $\log _e\Bigg{(}\frac{1+x}{1+{x}^2}\Bigg{)},\, |x|{\lt}1$ ?

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A real valued function f is defined as $f(x)=\begin{cases}{-1} & {-2\leq x\leq0} \\ {x-1} & {0\leq x\leq2}\end{cases}$.
Which of the following statement is FALSE?

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A line segment AB of length 10 meters is passing through the foot of the perpendicular of a pillar, which is standing at right angle to the ground. Top of the pillar subtends angles $tan^{–1}$ 3 and $tan^{–1} 2$ at A and B respectively. Which of the following choice represents the height of the pillar?

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If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes,then the sum of the magnitude of the projections on each of the axis is

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Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ballsis transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred is red, is:

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Let $f(x)=\frac{x^2-1}{|x|-1}$. Then the value of $lim_{x\to-1} f(x)$ is

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The mean of 5 observation is 5 and their variance is 12.4. If three of the observations are 1,2 and 6; then the mean deviation from the mean of the data is:

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In a beauty contest, half the number of experts voted Mr. A and two thirds voted for Mr. B 10 voted for both and 6 did not for either. How may experts were there in all.

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Let the total number of experts be .

is the set of experts who voted for miss .

is the set of experts who voted for miss .

Since did not vote for either, n.

and

.

So,

Solving the above equation gives

is the set of experts who voted for miss .

is the set of experts who voted for miss .

Since did not vote for either, n.

and

.

So,

Solving the above equation gives

A circle touches the x–axis and also touches the circle with centre (0, 3) and radius 2. The locus of the centre of the circle is

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The value of non-zero scalars α and β such that for all vectors and such that is

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A force of 78 grams acts at the point (2,3,5). The direction ratios of the line of action being 2,2,1 . The magnitude of its moment about the line joining the origin to the point (12,3,4) is

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Number of real solutions of the equation is

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The sum of infinite terms of a decreasing GP is equal to the greatest value of the function in the interval [-2,3] and the difference between the first two terms is f'(0). Then the common ratio of GP is

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Number of onto (surjective) functions from A to B if n(A)=6 and n(B)=3, is

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If $|z|<\sqrt{3}-1$, then $|z^{2}+2z cos \alpha|$ is

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A computer producing factory has only two plants T_{1} and T_{2} produces 20% and plant T_{2} produces 80% of the total computers produced. 7% of the computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant T_{1} 10P(computer turns out to be defective given that it is produced in plant T_{2} ). A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T_{2} is

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If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $tanA + tanB$

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x= tanA + tan(π/6-A)

we get :

dx/dA = sec²A-sec²(π/6-A)

now putting

dx/dA=0

we get

cos²(A) = cos²(π/6-A) so 0≤A≤π/6

therefore

A=π/6-A from here we get A = π/12 = B

so minimum value of that function is

2tanπ/12 which is equal to 2(2-√3)

Inverse of the function $f(x)=\frac{10^x-10^{-x}}{10^{x}+10^{-x}}$ is

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⇒ $\frac{10^x-10^-x}{10^x+10^-x}=y$

⇒ $\frac{10^{2x}-1}{10^{2x}+1}=y$

⇒ $10^{2x}=\frac{1+y}{1-y}$ By Componendo Dividendo Rule

⇒ $x=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(y)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(x)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+x}{1-x}\Bigg{)}$

The tangent at the point (2, -2) to the curve $x^2 y^2-2x=4(1-y)$ does not passes through the point

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The integral $\int \sqrt{1+2 cot x(cosec x+cotx)} dx$ , $(0<x<\frac{\pi}{2})$ (where C is a
constant of integration) is equal to

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If all the words, with or without meaning, are written using the letters of the word QUEEN add are arranged as in English Dictionary, then the position of the word QUEEN is

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Letters of the word QUEEN are E,E,N,Q,U

Words beginning with E (4!) = 24

Words beginning with N (4!/2!)=12

Words beginning with QE (3!) = 6

Words beginning with QN (3!/2!)= 3

Total words = 24+12+6+9=45

QUEEN is the next word and has rank 46th.

The curve satisfying the differential equation and passing through the point (1,1) also passes through the point __________

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If S and S' are foci of the ellipse , B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is

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The equation of the circle passing through the point (4,6) and whose diameters are along x + 2y - 5 =0 and 3x - y - 1=0 is

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In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q. Then AQ : QC =

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The median AD of ΔABC is bisected at E and BE is extended to meet the side AC in F. The AF : FC =

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Let $X_i, i = 1,2,.. , n$ be n observations and $w_i = px_i +k, i = 1,2,
,n$ where p and k are constants. If the mean of $x_i 's$ is 48 and the standard deviation is 12, whereas the mean of $w_i 's$ is 55 and the standard deviation is 15, then the value of p and k should be

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If , then the values of A_{1}, A_{2}, A_{3}, A_{4} are

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Equation of the tangent from the point (3,−1) to the ellipse 2x^{2} + 9y^{2} = 3 is

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The position vectors of the vertices

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The $sin^2 x tanx + cos^2 x cot x-sin2x=1+tanx+cotx $, $x \in (0 , \pi)$, then x

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Not Available right now

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In a chess tournament, n men and 2 women players participated. Each player plays 2 games against every other player. Also, the total number of games played by the men among themselves exceeded by 66 the number of games that the men played against the women. Then the total number of players in the tournament is

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Suppose A_{1} , A_{2} , A_{3} , …..A_{30} are thirty sets each having 5 elements with no common elements across the sets and B_{1} , B_{2} , B_{3} , ..... , B_{n} are

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Let ,where [x]denotes the greatest integer

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Let U and V be two events of a sample space S and P(A) denote the probability of an event A. Which of the following statements is true?

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If a man purchases a raffle ticket, he can win a first prize of Rs.5,000 or a second prize of Rs.2,000 with probabilities 0.001 and 0.003 respectively. What should be a fair price to pay for the ticket?

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If the mean deviation 1, 1+d, 1+2d, … , 1+100d from their mean is 255, then d is equal to

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If and , then a possible value of n is among the following is

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Let S be the set $\{a\in Z^+:a\leq100\}$.If the equation
$[tan^2 x]-tan x - a = 0$ has real roots (where [ . ] is the greatest
integer function), then the number of elements is S is

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The solution set of the inequality is

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If $a, b, c$ are in GP and $log a - log 2b$, $log 2b - log 3c$ and $log 3c - log a$ are in AP, then $a, b, c$are the lengths of the sides of a triangle which
is

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If (1 + x – 2x^{2})^{6 }= 1 + a_{1}x + a_{2}x^{2 }+ ... + a_{12}x^{12},
then the value a_{2 }+ a_{4 }+ a_{6 }+ ... + a_{12}

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A man takes a step forward with probability 0.4 and backward with probability 0.6. The probability that at the end of eleven steps, he is one step away from the starting point is

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If x, y, z are distinct real numbers then = 0, then xyz=

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If $a, a, a_2, ., a_{2n-1},b$ are in AP, $a, b_1, b_2,...b_{2n-1}, b $are in GP and $a, c_1, c_2,... c_{2n-1}, b $ are in HP, where a, b are positive, then the
equation $a_n x^2-b_n+c_n$ has its roots

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For the two circles $x^2+y^2=16$ and $x^2+y^2-2y=0$, there is/are

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Let $\vec{a}, \vec{b}, \vec{c} $ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$ , $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then c is

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$\Rightarrow\begin{vmatrix}{a} & {a} & {c} \\ {1} & {0} & {1} \\ {c} & {c} & {b}\end{vmatrix}=0$

$\Rightarrow-ac-ab+ac+{c}^2=0$

$\Rightarrow{c}^2=ab$

A particle P starts from the point

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The correct expression for $cos^{-1} (-x)$ is

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If $\Delta=a^2-(b-c)^2$, where $\Delta$ is the are of the triangle ABC, then $tanA=$

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Two numbers $a$ and $b$ are chosen are random from a set of the first 30
natural numbers, then the probability that $a^2 - b^2$ is divisible by
3 is

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