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A cylindrical pencil sharpened at one edge is the combination of

(a) a cylinder and a cone

(b) a cylinder and frustum of a cone

(c) a cylinder and a hemisphere

(d) two cylinders

If A = ${30}^o$ and B = ${60}^o$,verify that

(i) sin (A + B) = sin A cos B + cos A sin B

(ii) cos (A + B) = cos A cos B - sin A sin B

If (2, p) is the midpoint of the line segment joining the points A(6, -5) and B(-2, 11), find the value of p.

$\text{If}3cos\theta =1,\text{find the value of}\frac{6si{n}^2\theta +ta{n}^2\theta }{4cos\theta }$

A solid cuboid of iron with dimensions $53cm\times 40cm\times 15cm$ is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are 8 cm and 7 cm respectively. Find the length of pipe.

If the vertices of a triangle are (a, 1), (b, 3) and (4, c), then the centroid of the triangle will lie on the x-axis, if

State the co-ordinates of the following points under reflection in x-axis :

(i) (3, 2) (ii) (-5, 4) (ii) (0, 0)

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion : $\frac{35}{50}$

A solid metallic hemisphere of diameter 28 cm is melted and recast into a number of identical solid cones, each of diameter 14 cm and height 8 cm. Find the number of cones so formed.

In the given figure, $\Delta$ ABC ~ $\Delta$ ADE. If AE : EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'.

When each side of a square is reduced by 3 cm, the area becomes 49 sq cm. The side of the original square is ___ cm.

Show that A (3, 2), B (6, -2) and C (2, -5) can be the vertices of a square.

(i) Find the co-ordinates of its fourth vertex D, if ABCD is a square.

(ii) Without using the co-ordinates of vertex D, find the equation of side AD of the square and also the equation of diagonal BD.

If
$cosec\theta -cot\theta =\frac{1}{2},0<\theta <{90}^{\circ }$,
then $cos\theta$
is equal to

Express sin A, sec A and tan A in terms of cot A.

Find the value of each given polynomial at the given value of its variable :
(i) $5x^2–7x+2$ at x = 2
(ii) $x^2+15x–4$ at x = -3
(iii) $3y+8–2y^2$ at y = –3
(v) $x^2+3x+1$ at x = -1/3
(v) $x^3–3x^2+5x+2$ at x = – 1

$Inthefigure,givenalongside,PBandQAareperpendicularstothelinesegmentAB.IfPO=3cm,QO=4.5cmandareaof△POB=60c{m}^2,findtheareaof△QOA.$

Find the 11th term of the last term ( towards the first term) of the arithmetic progression 10, 7, 4, .......-62

A line segment joining A(-1,$\frac{5}{3}$ ) and B(a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects the y-axis.
(i) Calculate the value of 'a'.
(ii) Calculate the co-ordinates of 'P'.

Given that HCF (2520, 6600) = 120, LCM (2520, 6600) = 252 k, then the value of k is:

A bag contains 3 white and 5 red balls. If a ball is drawn at random, the probability that the drawn ball is red is

Find the mean, median & mode of the following distribution:

8, 10, 7, 6, 10, 11, 6, 13, 10.

If P(E) and P(E’) are the probabilities of two complementary events, then which of the following is always true?

Express $\frac{6x^2-21x-12}{54x^2+45x+9}$ in its lowest form.

From a pack of 52 playing cards all cards whose numbers are multiples of 3 are removed. A card is now drawn at random. What is the probability that the card is

Solve the equation 4$x^2$+ 7x = 1 by the method of completing the square. In one of the options, one of the roots and the converted form of the equation after completing the square is given. Select this option.

In a survey of 200 students from 7 different schools, 50 people do not play only NFS, 40 people do not play only Dota and 10 people play no online game. Then find the no. of people out of 200 people who do not play both the games.

Arrange the following equations based on the descending order of their root values.(Use quadratic formula )

The volume of a sphere (in cubic cm) is equal to its surface area (in sq. cm). The diameter of the sphere (in cm) is

If $x=\sqrt{a}tan\alpha$ and $y=\sqrt{a}sec\alpha$. Find the value of $y^2–x^2$

From the top of a cliff 25m high, the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. The height of the tower is:

$\frac{PS}{SQ}$ = $\frac{PT}{TR}$ and $\angle$PST = $\angle$PRQ. Prove that $\Delta$ PQR is an isosceles triangle.

Two circles with centers A and B, and radii 5 cm and 3 cm, touch each other internally, If the perpendicular bisector of the segment AB meets the bigger circle in P and Q; find the length of PQ.

The sequence of the area of the equilateral triangles having sides 1 cm, 2 cm, 3 cm ___is

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be $\alpha and\beta$. Show that the height in miles of aeroplane above the road is given by $\frac{tan\alpha tan\beta }{tan\beta +tan\alpha }$

The side of a square is 10 cm. Find the area of circumscribed and inscribed circles.

Solve each of the following systems of eqautions by the method of cross-multiplication:

2x+y=35

3x+4y=65

Fint the roots of the following quadratic equations (if they exist) by the method of completing the square.
$4x^2+4\sqrt{3}x+3=0$

Find the coordinates of a point A , where AB is a diameter of the circle whose centre is (2, -3) and B is (1, 4).

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

Question 96

In a survey, 200 students were asked what influenced them most to buy their latest CD. The results are shown in the circle graph

(a) How many students said radio influenced them most?
(b) How many students were influenced by radio than by a music video channel?
(c) How many said a friend or relative influenced them or they heard the CD in a Shop?

In the given figure find
(i) position of $A$ with respect to $B$
(ii) position of $O$ with respect to $B$

Question 10
Obtain the mean of the following distribution.

$\begin{array}{cc}\text{Frequency}\left(f_i\right)& \text{Variable}\left(x_i\right)\\ 4& 4\\ 8& 6\\ 14& 8\\ 11& 10\\ 3& 12\end{array}$

(i) $\frac{sec\theta -1}{sec\theta +1}=\frac{si{n}^2\theta }{(1+cos\theta {)}^2}$

(ii) $\frac{sec\theta -tan\theta }{sec\theta +tan\theta }=\frac{co{s}^2\theta }{(1+sin\theta {)}^2}$

Question 113

A sugar factory has annual sales of 3 billion 720 million kilograms sugar. Express this number in the standard form.

Amit deposited Rs 150 per month in a month in a bank for 8 months under the Recurring Deposit Scheme. What will be the maturity value of his deposits, if the rate of interest is 8% per annum and interest is calculated at the end of every month ?

The formula for finding mean by direct method is $\frac{\sum (A\times B)}{\sum A}$

Where B and A are respectively: