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What is the sum of the first 100 natural numbers?

The sum of the roots of the equation $x^2–8x+2=0$ is

If sin$\theta$=a,and cos$\theta$=b, then the value of $\frac{(1-a^2)}{b}$ = __

Find the value of 'c' if the distance between the point (c, 4) and the origin is 5 units.

For what value(s) of k will the equation kx²-5x+k=0 have a repeated root?

In the ___ method of calculating mean we divide the deviations by the same number to simplify calculations.

If sin4A = cos (A - 30°), then find the value of A which satisfies the given equation.

Assertion (A) in the given figure, O is the centre of the circle with D, E and F as mid-points of AB, BO and OA respectively. If $\angle DEF$ is ${30}^{\circ }$, then $\angle ACB$ is ${60}^{\circ }$

Reason (R) Angle subtended by an arc at the centre is twice the angle subtended by it on the remaining part of the circle.

A sector of ${56}^{\circ }$, cut out from a circle, contains 17.6 $c{m}^2$. Find the radius of the circle.

Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles. Find the radius of the third circle correct to one decimal place.

Consider coordinate pointes A, B and C. Match the following.

$$\begin{array}{cc}I.IfAB+BC=AC,A,B,Care& A.Rightangledtriangle\\ II.IfAB+BC>AC,A,B,Care& B.Collinearpoints\\ III.IfA{B}^2+B{C}^2=A{C}^{2}A,B,Care& C.Pointsofatriangle\end{array}$$

A hemospherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are necessary to empty the bowl ?

The sum of the radius of the base and the height of a solid cylinder is 37 metres. If the total surface area of the cylinder be 1628 sq metres, find its volume.

Two vertices of an isosceles triangle (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

In the given figure, a semicircle is drawn on the hypotenuse of the right-angled triangle. Another arc of radius 30 cm is drawn passing through A and C with O as the center. What is the area of the shaded region? (Take $\pi$ as 3.14 and $\sqrt{3}$ = 1.732)

Find the area of a ring whose outer and inner radii are respectively 23 cm and 12 cm.

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 $m^2$.If the pipe is made of 99 $c{m}^3$ of metal,find the outer and inner radii of the pipe.

A cricket ball of radius R is exactly fitted in a cylindrical tin. Find the ratio of surface areas of the cricket ball to the tin.

In a shower, 5 cm of rain falls. The volume of the water that falls on 2 hectares of ground, is

(a) 100 $m^3$ (b) 10 $m^3$ (c) 1000 $m^3$ (d) 10000 $m^3$

The length of a rectangular field is 23 m more than its breadth. If the perimeter of the field is 206 m, then its area is

(a) 2420 $m^2$ (b) 2520 $m^2$ (c) 2480 $m^2$ (d) 2620 $m^2$

Question 8
The number of planks of dimensions $(4m\times 50cm\times 20cm)$ that can be stored in a pit which is 16 m long, 12 m wide and 4 m deep is:

A) 1900
B) 1920
C) 1800
D) 1840

In the figure, D and E trisect BC, Prove that $8{AE}^2=3{AC}^2+5{AD}^2$ [4 MARKS]

Which of the following is the reduced form of the equation $(2x-1{)}^2=x^2+2$?

With a ruler and compass which of the following angles cannot be constructed?

${60}^{\circ }$ ,${80}^{\circ }$, ${90}^{\circ }$ ,${105}^{\circ }$

In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then the radius of the circle is

(a) 10 cm (b) 12 cm (c) 13 cm (d) 15 cm

The length of a string between a kite and a point on the roof of a building 10 m high is 180 m. If the string makes an angle A with the level ground such that tan A =$\frac{4}{3}$. How high is the kite from the ground?

In a circus, a motorcyclist performs his stunts in a death-well which is in the form of a hollow cylinder. The thickness of the cylinder is negligible. If the diameter of the hollow cylinder is 14 m, and height is 20 m, then find the area available for the motorcyclist to perform stunts. (Take $\pi =\frac{22}{7}$)

A class contains 12 boys and 8 girls. The teacher calls on a student at random to go to the black board. What is the probability that the first student called is a boy? Give the answer up to 1 decimal place.

__

1. 18

The co-ordinates of the point which divides the line joining of (–1, 7) and (4, –3) in the ratio 2:3 is

A sum of Rs 700 is used to give seven cash prizes to students of a school for their overall academic performance. If, each prize is Rs 20 less than its preceding term, find the value of each of the prizes.

Which among the following are triangular matrices?

If you take a random 4 digit number, what is the probability that
(i)the ten's place digit is greater than the unit's place digit
(ii) that the leftmost digit is a multiple of 3 (iii) the rightmost digit is a multiple of 3. [4 MARKS]

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion : $\frac{129}{2^2{.5}^7{.7}^5}$

Dhoni purchased a bat for Rs 1100 inclusive of the sales tax. If the rate of sales tax is 10 %, then what is the selling price of the bat?

In the given figure 'O' is the center of the circle. SP and TP are the two tangents at S and T respectively. $\angle$SPT is ${50}^{\circ }$, the value of $\angle$SQT is:

Given below are pairs of statements. In each case, combine them using 'if and only if'.

(i) p: In $\Delta ABC,\angle B=\angle C.$

q: In $\Delta ABC,AC=AB.$

(ii) p: A and B are two sets such that $A\subseteq B$ and $B\subseteq A.$

q: $A=B.$

(iii) p: $\Delta ABC$ is equilateral.

q: $\Delta ABC$ is equiangular.

(iv) p: $\{a\in Rsuchthat\left|a\right|<2\}.$

q: $\{a\in Rsuchthat(a>-2anda<2\left)\right\}.$

A frequency distribution table is given for the daily wages and the number of workers getting the said wages. Find the median for the given data.

$\begin{array}{cc}\text{Daily Wages}& \text{No. of Workers}\\ 225& 4\\ 250& 5\\ 270& 7\\ 300& 9\\ 350& 2\\ 400& 3\end{array}$

The distance (in units) between the points (1, 3), (5, 6) is _____.

Question 48

State whether the following statement is true (T) or false (False)
Two numbers differ by 40.If both number is increased by 8 then larger number is equal to three times the smaller number, then the number is 12 and 52.

Find the common factors of:

$35$ and $50$

Evaluate : $\frac{sin{30}^{\circ }+tan{45}^{\circ }-cosec{60}^{\circ }}{sec{30}^{\circ }+cos{60}^{\circ }+cot{45}^{\circ }}$

Question 61

The value of a car, bought for Rs 440000 depreciates each year by 10% of its value at the beginning of that year. So, its value becomes Rs 308000 after three years.

1. 1

Choose the correct options: the equation (m being real), $m{x}^2+2x+m=0$ has two distinct roots if -