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One of the two events must occur. If the chance of one is $\frac{2}{3}$ of the other, then odds in favour of the other are

Find the sum of the squares of the first n natural numbers.

If the slant height of a pyramid is 5 m and its base edge is 6 m long, then find the height of the pyramid.

Adding 6 to numerator of a fraction , the numerator becomes twice the denominator. The quadratic equation satisfying the statement is

Find the point where 2x-3y = -9 intersects the y axis. [1 MARK]

You have budgeted $\$66$ a month to spend on yoga classes. Your yoga studio charges $\$22$ per month membership fee, plus $\$5$ per class attended. Find the possible number of classes you can attend each month.

The angles of depression of two objects from the top of a 100 m hill lying to its east are found to be ${45}^{\circ }and{30}^{\circ }$. Find the distance between the two objects. (Take $\sqrt{3}=1.732$)

Find the sum : 34 + 32 + 30 + .... + 10

Find the value of $\frac{sec\theta }{tan\theta }$.

Find the area of the rhombus, the lengths of whose diagonals are 30 cm and 16 cm. Also, find the perimeter of the rhombus.

Find the value of k for which each of the following systems of equations has no solution:
$3x-y-5=0,6x-2y+k=0.(k\ne 0).$

Find the value of k for which each of the following systems of linear equations has an infinite number of solutions:

$kx+3y=(2k+1)2(k+1)x+9y=(7k+1).$

While computing the mean of the grouped data,we assume that the frequencies are:

(a) evenly distributed over the classes

(b) centred at the class marks of the classes

(c) centred at the lower limits of the classes

(d) centred at the upper limits of the classes

The given figure represents a solid consisting of a cylinder surmounted by a cone at one end and a hemisphere at the other. Find the volume of the solid.

The angles of a quadrilateral are $x^{\circ },(x-10{)}^{\circ },(x+30{)}^{\circ }$ and $(2x{)}^{\circ }$. Find the value of the greatest angle.

Question 72

In the following figure, each division represents 1 cm

Express numerically the ratios of the following distances

(i) AC : AF
(ii) AG : AD
(iii) BF : AI
(iv) CE : DI

The radius of sphere is r cm. It is divided into two equal parts. Find the whole surface of two parts.

If the occurrence of an event precludes the occurrence of a second event, and vice versa, then the events are

A conical tomb is made up of bricks whose surface area is 5 m${}^2$. The slant height and radius of the tomb are measured to be 25 m and 7 m respectively. The approximate number of bricks used to construct the tomb are ____ (Take $\pi =\frac{22}{7}$)

The perimeter of a sector of circle with radius 5.7m is 27.2m. The length of the arc of sector (in m) is _______

AB and BC are the tangents to the circle from the points B. D is the centre of the circle. BD = 5 cm and CD = 3 cm. Find the value of AB - BC.

If LM $\parallel$ AB, AL= x-3, AC = 2x, BM = x-2, BC = 2x+3. The value of AC is ___.

$△$ABC is a right angled triangle, right angled at B. BD is perpendicular to AC. What is AC . DC?

The angle bisector of $\angle A$ divides the side BC of $△ABC$ in the ratio 2 :3. The area of $△ABD$ = 20 $c{m}^2$. Area of $△$ ADC =

Question 11

Find the area of the quadrilateral ABCD.
Here, AC = 22 cm, BM = 3 cm, DN = 3 cm and BM $\perp$ AC, DN $\perp$ AC.

A Ltd issued 10,000, 10% Debentures of Rs.100 each at a premium of 5% payable as follows:

Rs.10 on application, Rs.20 along with premium on allotment and balance on First and Final Call. Record necessary journal entries.

Question 120

Madhu's room measures $6m\times 3m$. Her carpet covers 8 $m^2$. What per cent of floor is covered.by the carpet?

A girl is twice as old as sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

Describe the locus for questions 1 to 13 given below:

The locus of the centres of all circles passing through two fixed points.

Find the middle term of the AP 10, 7, 4, ... , (-62).

Which one of the following is a compound surd?

Show that the points A(3, 1), B(0, -2),C(1,1) and D(4, 4) are the vertices of a parallelogram ABCD.

If $xcos\theta -ysin\theta =a$ and $xsin\theta +ycos\theta =b,$ then $\frac{x^2+y^2}{a^2+b^2}$ = ___.

Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

Find the roots of the quadratic equation $2x^2+x–6=0$ by factorisation.

A die is thrown once. The probability of getting an even number is
(a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{6}$ (d) $\frac{5}{6}$

The sum of the ages of a boy and his brother is 25 years, and the product of their ages in years is 126. Find their ages.

Find the equation of the circle passing through the points (4, 1) and (6, 5) whose centre is on the line 4x + y = 16.

or

A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of locus of point P on the rod, which is 3 cm from the end in contact with the X-axis.

Show that the points A(7 ,10), B(-2, 5) and C(3, -4) are the vertices of an isosceles right triangle.

A solid metal sphere is cut through its centre into 2 equal parts.If the diameter of the sphere is 3$\frac{1}{2}$ cm,find the total surface area of each part correct to two decimal places.

Question 1 (iii)

Find the roots of the quadratic equations by using the quadratic formula in the following question.

$-3x^2+5x+12=0$

If the frequencies of the observations are doubled, then the mode will:

D, E, F are the mid-points of the sides BC, CA and AB respectively of a $\Delta$. Then the ratio of the areas of $\Delta DEF$ and $\Delta ABC$ is