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Evaluate each of the following

$si{n}^2{30}^o+si{n}^2{45}^o+4ta{n}^2{30}^o+\frac{1}{2}si{n}^2{90}^o-2co{s}^2{90}^o+\frac{1}{24}co{s}^2{0}^o$

The pair of equations 2x+3y=5 and 4x+6y=15 has

(a) a unique solution (b) exactly two solution

(c) infinitely many solutions (d) no solution

66 cubic cm of silver is drawn into a wire 1 mm in diameter. Calculate the length of the wire in metres.

In making 1000 revolutions, a wheel covers 88 km. The diameter of the wheel is

(a) 14 m (b) 24 m (c) 28 m (d) 40 m

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ration 2:3. Determine the fraction.

The total surface area of a right circular cylinder of height 9m is 440 sq. meter.Find the radius of the cylinder. (Take~$\pi =\frac{22}{7}$)

The area of the sector of a circle of radius 10.5 cm is 69.3cm^2. Find the central angle of the sector.

1. 27

If the area of the base of a right circular cone is 3850 $c{m}^2$ and its height is 84 cm, find the slant height of the cone.

Match the following to their roots.

If x = sin (${90}^{\circ }$- a) cot (${90}^{\circ }$- a), then x is equal to ............................

Find the value of k for which each of the following systems of equations have infinitely many solution: (9 - 19)

kx - 2y + 6 = 0

4x - 3y + 9 = 0

Five coins were simultaneously tossed 1000 times, and at each toss the number of heads was observed. The number of tosses during which 0, 1, 1 3, 4 and 5 heads were obtained are shown in the table below. Find the mean number of heads per toss
No. of heads per toss (x) : 0 1 2 3 4 5
No. of tosses (f) : 38 144 342 287 164 25

Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is $50$.

$\begin{array}{cc}{x}_i& f_i\\ 10& 17\\ 30& f_1\\ 50& 32\\ 70& f_2\\ 90& 19\\ \text{Total}& 120\end{array}$

A copper sphere of radius 3 cm is melted and recast into a right circular cone of height 3 cm. Find the radius of the base of the cone.

If the product of the zeros of the quadratic polynomial $x^2-4x+k$ is 3 then write the value of k.

An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250km away, in time, it had to increase its speed by 250km/hr from its usual speed. Find its usual speed.

The area of a rectangle plot is 462 $m^2$ and its length is 28 m.Find its perimeter.

Which of the following numbers has both 7 and 5 as factors?

From a thin metallic piece in the shape of a trapezium ABCD in which AB || CD and $\angle$ BCD = ${90}^{\circ }$, a quarter circle BFEC is removed. Given, AB = BC = 3.5 cm and DE = 2 cm, calculate the area of remaining (shaded) part of metal sheet.

In a cyclic quadrilateral ABCD, it is given that $\angle A=(2x+4{)}^{\circ },\angle B=(y+3{)}^{\circ },\angle C=(2y+10{)}^{\circ },and\angle D=(4x-5{)}^{\circ }$.Find the four angles.

$a{x}^2$+ bx + c = 0, where a, b, c are real, has real roots if:

A chord of a circle of radius 12 cm subtends an angle of ${120}^{\circ }$ at the centre. Find the area in $c{m}^2$ of the corresponding segment of the circle.

$\text{Use}\pi =3.14\text{and}\sqrt{3}=1.73$

The sum of n terms of the series: 13+105+1007+10009.... is

In fig, $\frac{PS}{SQ}$ = $\frac{PT}{TR}$ and $\angle$PQR. Prove that PQR is an isosceles triangle.

A 1.8 m tall man is standing at some distance from a building. The angle of elevation from his eyes to the top of the building increases from ${30}^{\circ }$ to ${60}^{\circ }$ as he walks 20 m towards the building. Find the height of the building (in m).

Find the greatest number of 6digits which is exactly divisible by 24,15 and 36

Let $N={55}^3+{17}^3-{72}^3.$ N is divisible by : (CAT 2000)

Ashish deposits a certain sum of money every month in a Recurring Deposit Account for a period of 12 months. If the bank pays interest at the rate of 11% p.a. and Ashish gets ₹ 12,715 as the maturity value of this account, what sum of money did he pay every month?

The ratio of sum of m and n terms of an AP is m²:n². Show that the ratio of m^th and n^th term is (2m-1):(2n-1).

prove (sin a ÷1+cos) +(1+cos a÷sin a)=2 cosec a

It takes 14 hr to travel 30km in upstream and 44 km in downstream. If the distance covered in upstream is doubled and distance covered in downstream is increased by 11km then total time taken is 11hr more than earlier. Find the speed of stream and speed of boat.

Which of the following is not a linear polynomial?

Given $M=\left(\begin{array}{ccc}9& 7& -8\\ -6& 5& 4\\ -3& 2& -1\end{array}\right)$, Find a matrix N such that M + N is a null matrix.

If $(x^3-3x^2-x+3)$ is divided by (x-1) we get $a{x}^2-bx+c$, then value of (a+b+c)

From a solid right circular cylinder with height 10 cm and radius of the base 6 cm, a right circular cone of the same height and same base is removed. Find the volume of the remaining solid.

A man travels 600 km to his home ,partly by train and partly by bus.If he travels 120km by train and rest by bus it would take 8hours.If he travels 200km by train and rest by bus it will take 20 minutes more .find speed of train and bus saparetly.

(Use a graph paper for this question) The daily pocket expenses of 200 students in a school are given below: [4 MARKS]

Draw a histogram representing the above distribution and estimate the mode from the graph.

Solve equation, given below, using factorisation method:

$(x+3{)}^2-4(x+3)-5=0$

Identify the minor segment in the given image.
(O is the centre of the circle)

Which of the following equation has infinitely many solutions?

Solve the following:
Radha takes some flowers in a basket and visits three temples one-by-one. At each temple, she offers one half of the flowers from the basket. If she is left with $3$ flowers at the end, then find the number of flowers she had in the beginning.

A square matrix is having p number of rows. What is ‘j’ if $a_{ij}$ is situated in the last row and second last column

Rs 50 shares of a company are quoted at a discount of 10%. Find the rate of dividend given by the company. the return on the investment on these shares being 20 percent.