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Solve for $x$:
$\frac{1}{x-2}+\frac{2}{x-1}=\frac{6}{x}$

Solve

$\frac{a}{b}x-\frac{b}{a}y=a+b$

$ax-by=2ab$

TA and TB are tangents to a circle with centre O from an external point T. OT intersects the circle at point P. Then prove that AP bisects $\angle TAB$.

Find the values of a and b for whic the following system of linear equations has infinite number of solutions:

2x - 3y = 7

(a + b) x - (a + b - 3) y = 4a + b

If $d,a$ and $l$ denote the common difference, first and last terms respectively, then $n^{th}$ term from the end is ___________ .

In the given figure, PQ = QR, $\angle RQP={68}^{\circ }$, PC and CQ are tangents to the circle with center O. Calculate the values of:



$\left(i\right)\angle QOP$



$\left(ii\right)\angle QCP$ [4 MARKS]

There are 2 ships on either side of a lighthouse. If the angles of elevation of the lighthouse from both the ships are ${30}^{\circ }$ and ${45}^{\circ }$ respectively. If the height of lighthouse is h and distance between the two ships is d, the ratio of d to h up to 1 decimal place is - ___

Kapil told his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.

For what values of k are the roots of the quadratic equation $3x^2+2kx+27=0$ real and equal?

Which term of the AP 8, 14, 20, 26,.. will be 72 more than its 41st term?

The value of $\cos$ ${60}^{\circ }$ is

The point that divides AB in the ratio a: 1 and lies on the line x + y = 10, find 'a'

Find the value of 'k', for which the points (8, 1), (k, -4), (2, -5) are collinear.

Each of letters of the word PILOTS is on separate cards, face down on the table. If you pick a card at random, what is the probability that the letter will be a T or a L?

The Following data shows the goals scored by 19 clubs in a Premiere League Season :

$\begin{array}{cc}\text{Goals Scored}& \text{No. of Clubs}\\ 0-10& 2\\ 10-20& 3\\ 20-30& 2\\ 30-40& 4\\ 40-50& 3\\ 50-60& 5\end{array}$

Find the Median number of Goals scored in the season.

If the numbers 3, 5, 7, 4 have frequencies $x,x+4,x-3,x+8$ with their arithmetic mean of 4, then $x$ will be:

Find the median of the following cumulative frequency distribution:

$\begin{array}{ccc}x& frequency& Cumulativefrequency\\ 22& 1& 1\\ 34& 2& 3\\ 39& 2& 5\\ 45& 8& 13\\ 54& 8& 21\\ 56& 9& 30\\ 68& 4& 34\\ 78& 4& 38\\ 84& 2& 40\end{array}$

Express the given pattern as a sequence.

The side of one square is $3cm$ longer than the twice of the second square the difference in their areas is $144c{m}^2.$ Find the sides of the two squares.

Find the value of k if the point (2, k) lies on the line 2x - y = 3

How to learn trigonomentery table easily?

Over the past 200 working days, the number of defective parts produced by a machine is given in the following table.

$\begin{array}{ccccccccccccccc}\text{Number of}& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ \text{defective parts}& & & & & & & & & & & & & & \\ \text{Days}& 50& 32& 22& 18& 12& 12& 10& 10& 10& 10& 8& 6& 6& 2\end{array}$

Determine the probability that tomorrow’s output will have:
(i) No defective part.
(ii) Atleast one defective part.
(iii) Not more than 5 defective parts.
(iv) More than 13 defective parts.

Question 1 (v)
State whether the pair of triangles in the below figure is similar. Write the similarity criterion used by you for answering the question and also write the pair of similar triangles in the symbolic form:

Find the area of a rectangle, if the length of the rectangle is x and the breadth is twice the length the area of the rectangle?

The radii of the circular ends of a solid frustum of a cone are 18 cm and 12 cm and its height is 8 cm. Find its total surface area. $[Use\pi =\mathrm{3.14.}]$

Two circles touch each other internally at a single point. Their radii are 2 cm and 3 cm. The biggest chord of outer circle which is a tangent to the inner circle, is of length ____.

The following data shows monthly savings of 100 families . Calculate the mode of the given frequency distribution.

$$\begin{array}{cc}Monthlysavings\left(Rs\right)& Numberoffamilies\\ 1000-2000& 14\\ 2000-3000& 15\\ 3000-4000& 21\\ 4000-5000& 27\\ 5000-6000& 25\end{array}$$

An AP consists of 50 terms of which the third term is 12 and the last term is 106. Find the ${29}^{th}$ term.