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Using the properties of determinants, solve the following for x:

$\left|\begin{array}{ccc}x+2& x+6& x-1\\ x+6& x-1& x+2\\ x-1& x+2& x+6\end{array}\right|=0$

Two numbers a and b are chosen at random from the set of integers {1, 2, 3, ......, 15}. The probability that equation 2a – 3b = 0 is satisfied, is -

Find the middle term in the expansion of:

$\left(i\right){(\frac{2}{3}x-\frac{3}{2x})}^{20}$

$\left(ii\right){(\frac{a}{x}+bx)}^{12}$

$\left(iii\right){(x^2-\frac{2}{x})}^{10}$

$\left(iv\right){(\frac{x}{a}-\frac{a}{x})}^{10}$

Find the slope of the tangent to the curve $y=3x^4-4x$ at x=4.

The solution set of $\frac{1-\sqrt{1-4x^2}}{x}<3$ is

Solution of the differential eqaution $(1+\ln 2-\frac{dy}{dx})2^x=\frac{dy}{dx}-1$ if the curve passes through $(0,\ln 2)$ is

If V₁,V₂,V₃ are unit vectors such that V₁+V₂+V₃=0 then

Chose the correct answer.

The slope of the normal to the curve $y=2x^2+3$sinx at x=0 is

(a) 3 (b) $\frac{1}{3}$ (c)-3 (d)- $\frac{1}{3}$

The total costC(x) associated with the production of x units of an item is given by C(x) =$0.005x^3-0.02x^2+30x+5000$. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

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

Sin40/sin80 + sin80/sin20 - sin20/sin40

A 1

B 2

C 3

D 4

The equation of the circle whose radius is 5 and which touches the circle $x^2+y^2-2x-4y-20=0$ externally at the point (5, 5), is

The sides of triangle are in the ratio $1:\sqrt{3}:2,$ then the angles of the triangle are in ratio

Derivative of $y=(sinx{)}^2$ with respect to x will be

The number of different signals can be given by using any number of flags from $4$ flags of different colours is

Magnetic field at the center of regular hexagon of side ${}^{\prime }{a}^{\prime }$ carrying current ${}^{\prime }{i}^{\prime }$ is

Let $a,b,c,d$ be real positive numbers. Then the maximum number of roots of the equation $a|x{|}^3+b{x}^2+c|x|+d=0$ is

Let $y=f\left(x\right)$ be a parabola, having its axis parallel to $y-$axis, which is touched by the line $y=x$ at $x=1$, then

In a culture the bacteria count is 100000. The number is increases by 10% in 2h. In how many hours will the count reach 200000, if the ratio of growth of bacteria is proportional to the number present?

The number of all three element subsets of the set $\{a_1,a_2,a_3\cdots ,a_n\}$ which contain $a_3$ is:

How many two digit positive integers N have the property that the sum of N and number obtained by reversing the order of the digits of N is a perfect square.

Identify the functions in the graph and match the columns

If c₁, c₂,……, c_n are constants and x₁, x₂,….., x_n are variables, then the linear function Z = c₁x₁+ c₂x₂ + …..c_nx_n which is to be maximized or minimized is called the

The equation of locus of a point where distance from the y-axis is equal to its distance from the point A(2,1,-1) is-

The value of $\left\{\sin \frac{25\pi }{3}\right\}$ is
(where $\{.\}$ is fractional part function)

Total number of $4$ letter words that can be formed using the letters of the word ${}^{\prime }FLOWE{R}^{\prime }$, such that the word starts with $F$ and ends with $R$ is

The size of the Variable Number Tandem Repeats (VNTR) vary from

Find the value of y, when x = 60${}^0$ in the figure below.

If $y=4e^{2x}+3e^{3x},$ then the value of $\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y$ is

A man alternately tosses a coin and throws a die beginning with the coin. The probability that he gets a head in the coin he gets a 5 or 6 in the dice is

Given the following frequency distribution with some missing frequencies
$\begin{array}{cccccccc}\text{Class}& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ \text{Frequency}& 180& & 34& 180& 136& & 50\end{array}$
If the total frequency is $685$ and approximate value of median is $42.6$, then the approximate values for missing frequencies are

Let $\alpha ,\beta$ are the angle of inclination of the tangents to the axis of the parabola $y^2=4ax$ drawn from the point $P.$

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\cot \alpha \cot \beta =k,\text{then locus of}P\text{is}& \text{(P)}& kx=a\\ \text{(B)}& \text{If}\tan \alpha +\tan \beta =k,\text{then locus of}P\text{is}& \text{(Q)}& y=k(x-a)\\ \text{(C)}& \text{If}\tan (\alpha +\beta )=k,\text{then locus of}P\text{is}& \text{(R)}& kx=y\\ \text{(D)}& \text{If}\tan \alpha \tan \beta =k,\text{then locus of}P\text{is}& \text{(S)}& xy=k\\ & & \text{(T)}& x=ka\end{array}$

Which of the following is the only CORRECT combination?

Write the solution set of $|x+\frac{1}{x}|>2$

Write down different notations used in mathematics.

A man has $7$ letters for his $7$ friends. The letter are kept in the envelopes at random. The number of ways in which exactly $3$ letters are going to correct envelope and rest $4$ letters are going to the wrong envelopes is

Find the absolue maximum value and the absolute minimum value of the following function in the given intervals:

$f\left(x\right)=4x-\frac{1}{2}{x}^2,xϵ[-2,\frac{9}{2}]$

Number of solutions of the equation $[y+\left[y\right]]=2$ = 2 cosx is not equal to, where $y=\frac{1}{3}[sinx+[sinx+\left[sinx\right]\left]\right]$ and [.] denotes the greatest integer function