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Match the following for system of linear equations

2x -3y + 5z =12

$3x+y+\lambda z=\mu$

x -7y + 8z =17

$\begin{array}{cccc}& \text{Column - I}& & \text{Column - I}\\ \left(P\right)& \text{Unique solution}& \left(1\right)& \lambda =2,\mu =7\\ \left(Q\right)& \text{Infinite solution}& \left(2\right)& \lambda \ne 2,\mu =7\\ \left(R\right)& \text{No solution}& \left(3\right)& \lambda \ne 2,\mu \ne 7\\ \left(S\right)& \text{Consistent system}& \left(4\right)& \lambda \in R,\mu \ne 7\\ & \text{equation}& & \\ & & \left(5\right)& \lambda =2,\mu \ne 7\end{array}$

There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?

Consider the parabola $x^2+4y=0.$ Let $P(a,b)$ be any fixed point inside the parabola and let $S$ be the focus of parabola. Then the minimum value of $SQ+PQ$ as point $Q$ moves on parabola is:

Which of the following function is differentiable for all $x\in R$ (where $[.]$ represents the greatest integer function and ${}^{\prime }{\text{sgn}}^{\prime }$ represents signum function.)

A variable chord of circle $x^2+y^2=4$ is drawn from the point $P(3,5)$ meeting the circle at the points $A$ and

$B$. A point $Q$ is taken on this chord such that $2PQ=PA+PB$. Locus of ${}^{\prime \prime }{Q}^{\prime \prime }$ is

Find the mean and variance for each of the data in Exercise 1 to 5:

$\begin{array}{cccccccc}{x}_i& 6& 10& 14& 18& 24& 28& 30\\ f_i& 2& 4& 7& 12& 8& 4& 3\end{array}$

Find the equation of the circle passing through the points (1, -2) and (4, -3) and whose centre lies on the line 3x + 4y = 7.

For a positive integer n, let

$f_n\left(\theta \right)=\left(tan\frac{\theta }{2}\right)(1+sec\theta )(1+sec2\theta )(1+sec4\theta )\dots (1+sec{2}^n\theta )$.

Then

Find the greater number in 300! and $\sqrt{{300}^{300}}$

IF $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r},$ then $\frac{t_n}{S_n}$ is equal to

The average weight of 35 students in a class is 40kg. If the weight of the teacher is also included, the average rises by 0.5 kg. The weight of the teacher is

The real value of $\lambda$ for which the image of the point $(\lambda ,\lambda -1)$ with respect to the line $3x+y=6\lambda$ is the point $({\lambda }^2+1,\lambda )$ is

Match each of the set on the left in the roster form with the same set on the right described in set-builder form :

(i) {1, 2, 3, 6} (a) {x : x is a prime number and a divisor of 6}

(ii) {2, 3} (b) {x : x is an odd natural number less than 10}

(iii) {M, A, T, H, E, I, C, S} (c) {x : x is natural number and divisor of 6}

(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}

What is the equation of the normal to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ at the point $(5\sqrt{3},2\sqrt{2})$

If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x-\frac{1}{2}\sin 2x,$ then the range of $f\left(x\right)$ is

Harikiran purchased a house in Rs. 15000 and paid Rs. 5000 at once. Rest money he promised to pay in annual instalment of Rs. 1000 with 10% per annum interest. How much money is to be paid by him

Let p and q be real numbers such that $p\ne 0$, $p^3\ne 2q$ and $p^3\ne -q$. If $\alpha$ and $\beta$ are non zero complex numbers satisfying $\alpha +\beta =-p$ and ${\alpha }^3+{\beta }^3=q,$ then a quadratic equation having $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ as its roots is

Find all possible values of $\sqrt{x^2+25}$

Two complex numbers are given as,

$Z_1$ = $a_1$ + i$b_1$

$Z_2$ = $a_2$ + i$b_2$

For the complex numbers $Z_1$ and $Z_2$ to be equal, the necessary conditions are

If x $\in [\frac{\pi }{2},\frac{5\pi }{2}]$, the greatest positive solution of 1 + $si{n}^{4}x=co{s}^{2}3x$ is

In a survey of 25 students, it was found that 12 have taken physics, 11 have taken chemistry and 15 have taken mathematics; 4 have taken physics and chemistry; 9 have taken physics and mathematics; 5 have taken chemistry and mathematics while 3 have taken all the three subjects.
Find the number of students who have taken

(i) physics only;
(ii) chemistry only;
(iii) mathematics only;
(iv) physics and chemistry but not mathematics;
(v) physics and mathematics but not chemistry;
(vi) only one of the subjects;
(vii) at least one of the three subjects;
(viii) none of the three subjects.

The maximum distance from the origin of coordinates to the point z satisfying the equation $|z+\frac{1}{z}|$=a is

Let z be a complex number such that the imaginary part of z is non-zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

Find the $n^{th}$ term of the series. 1 + 4 +13 +40 + 121 +.................

$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\cdots +\frac{1}{(3n-1)(3n+2)}=\frac{n}{6n+4}$

If $\cot A+\text{cosec}A=3$ and $A$ is an acute angle, then the value of $\cos A$ is

Prove by using the principle of mathematical induction that $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{n(n+1)}=\frac{n}{n+1}$

The total number of ways in which $6$ person can be seated at a round table, so that all person shall not have the same neighbors in any two arrangements

Which of the following gives the equation of director circle of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$?

If x = 1 + ${\log }_a$ bc. Find the value of ${\log }_{abc}$ a in terms of x.

Values of $x$ satisfying the inequality $x^{\left(\frac{1}{{\log }_{10}x}\right)}.{\log }_{10}x<1$ is -

Two statements p and q are given below.

p: 2 plus 3 is 5
q: Delhi is the capital of India

Then the statement "Delhi is the capital of India and it is not that 2 plus 3 is five" is

The graph of $y=1+{\log }_{4}x$ is

If the three consecutive coefficient in the

expansion of $(1+x{)}^n$ are 28, 56 and 70, then the

value of n is

The first negetive term in the sequence of $56,55\frac{1}{5},54\frac{2}{5},\cdots$

If x ∈ ($\pi$, 2$\pi$) and $\frac{\sqrt{1+cosx}+\sqrt{1-cosx}}{\sqrt{1+cosx}-\sqrt{1-cosx}}$ =$cot(a+\frac{x}{2})$, then a is equal to