If the sum of the roots of the equation 3x² – (3k – 2) x – (k – 6) = 0 is equal to the product of its roots then k = ?

The number of all 2-digit numbers divisible by 6 is

A fair die is thrown once. The Probability of getting a composite number is

Which of the following statements is true?

In the given figure, PA and PB are tangents to a circle such that PA = 8 cm and ∠ APB = 60˚. The length of the chord AB is

The angle of depression of an object from a 60-m-high tower is 30˚. The Distance of the object from the tower is

In what ratio does the point P (2, – 5) divide the line segment joining A (– 3,5) and B(4, – 9)?

Three solid spheres of radii 6 cm, 8 cm and 10 cm are melted to form a sphere. The radius of the sphere so formed is

Find the value of p for which the quadratic equation

x² – 2px + 1 = 0 has no real roots.

Find the 10th term form the end of the AP 4, 9, 14, .. , 254.

Which term of the AP 24, 21 ,18, 15, … is the first negative term?

A circle is touching the side BC of a Δ ABC at P and is touching AB and AC when produced at Q and R respectively.

Prove that AQ = 1/2 (perimeter of Δ ABC).

Two vertical of a ΔABC are given by A(6, 4) and B (– 2, 2) and its centroid is G(3, 4). Find the coordinates of its third vertex C.

A box contain 150 orange is taken out from the box at random and the probability of its being rotten is 0.06 then find the number of good orange in the box.

A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.

Solve: a²b²x² – (4b^{4 –} 3a⁴) x – 12a²b² = 0.

If the 8^th term of an AP is 31 and its 15^th term is 16 more than the 11^th term, find the AP.

Find the sum of all two-digit odd positive numbers.

In the adjoining figure, PA and PB are tangents drawn from an external point P to a circle with centre O. Prove that

∠ APB = 2 ∠ OAB.

In the adjoining figure, quadrilateral ABCD is circumscribed. If the radius of the in circle with centre O is 10 cm and AD ⊥ DC, find the value of x.

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct a pair of tangents to the circle. Measure the length of each of the tangent segments.

The three vertices of a parallelogram ABCD, taken in order are A (1, – 2), B (3, 6) and C (5, 10). Find the coordinates of the fourth vertex D.

Find the third vertex of a Δ ABC if two of its vertices are

B (– 3, 1) and C (0, – 2) and its centroid is at the origin.

Cards marked with all 2-digit numbers are placed in a box and are mixed thoroughly. One card is drawn at random. Find the probability that the number on the card is

(a) divisible by 10

(b) a perfect square number

(c) a prime number less than 25

A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the area of the road. [Take π = 22/7.]

A round table cover shown in the adjoining figure has six equal designs. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹ 0.50 per cm². [Use √ 3 = 1.73.]

In an equilateral triangle of side 12 cm, a circle is inscribed touching its sides. Find the area of the portion of the portion of the triangle not included in the circle. [Take √ 3 = 1.73 and π = 3.14.]

If a sphere has the same surface area as the total surface area of a circular cone of height 40 cm and radius 30 cm, find the radius of the sphere.

A two–digit number is such that the product of its digits is 35. If 18 is added to the number, the digit interchange their places. Find the number.

Two water taps together can fill a tank in hours. The larger tap takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact to the centre.

From the top of a 7 – m high building, the angle of elevation of the top of a cable tower is 60˚ and the angle of depression of its foot is 45˚. Find the height of the tower. [Given √ 3 = 1.73.]

Puja works in a bank and she gets a monthly salary of ₹ 35000 with annual increment of ₹ 1500. What would be her monthly salary in the 10^th year?

In the given figure ABCD represent the quadrant of a circle of radius 7 cm with centre A. Calculate the area of the shaded region. [Take π = 22/7.]

The radii of the circular end of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. Find its capacity and total surface area. [Take π = 22/7.]

From an external point p, tangents PA and PB are drawn to a circle with centre O. If CD is the tangent to the circle at the point E and PA = 14 cm, find the perimeter of ΔPCD.

Construct a ΔABC in which BC = 5.4 cm, AB = 4.5 cm and ∠ ABC = 60˚.

Construct a triangle similar to this triangle, whose side are 3/4 of the corresponding sides of ΔABC.

A bag contain 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.

In what ratio is the line segment joining the points (– 2, – 3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

The values of k for which the equation 2x² + kx + 3 = 0 has two real equal root are

How many terms are there in the AP 7, 11,15, …, 139?

One card is drawn from a well-shuffled deck of card. The probability of drawing a 10 of a black suit is

In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT = 24 cm. If O is the centre of the circle then OP =?

The ratio in which the line segment joining the points A(– 3/ 2) and B(6, 1) is divided by the y – axis is

The distance of the point P (6, – 6) from the origin is

A kite is flowing at a height of 75 cm from the level ground, attached to a string inclined at 60˚ to the horizontal. The length of string with no slack in it, is

A solid metal cone with radius of base 12 cm and height 24 cm is melted to form solid spherical balls of diameter 6 cm each. The number of balls formed is

If the roots of the equation (a – b) x² + (b – c) + (c – a) = 0 are equal, prove that b + c = 2a.

Find the 10^th term form the end of the AP 4, 14, … , 254.

Or, which term of the AP 3, 15, 27, 34, … will be 132 more than its 54^th term?

Which term of the AP 3, 15, 27, 39, … will be 132 more than its 54^th term.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel

From an external point P tangents PA and PB are drawn to a circle with centre at the point E and PA = 14 cm, find the perimeter of ΔPCD.

The area of the circular base of a cone is 616 cm² and its height is 48 cm. Find its whole surface area. [Take π = 22/7.]

In the adjoining figure, the area enclosed between two concentric circles is 770 cm² and the radius of the outer circle is 21 cm. Find the radius of the inner circle.

Solve for x: 12abx² – (9a² – 8b²) x – 6ab = 0.

If the 8^th term of an AP is 31 and its 15^th term is 16 more than the 11^th term. Find the AP.

Prove that the parallelogram circumscribing a circle is a rhombus.

A ΔABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of length 8 cm and 6 cm respectively. Find AB and AC.

Draw a circle of diameter 12 cm. From a point 10 cm away its centre, construct a pair of tangents to the circle. Measure the length of each tangent segment.

Show that the point A (a, a), B (– a, – a) and C(– a√ 3, a√ 3) are the vertices of an equilateral triangle.

Find the area of a rhombus if its vertices are A (3, 0), B (4, 5), C (– 1, 4) and D (– 2, – 1).

Cards marked with numbers 13, 14, 15, …, 60 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that the number on the card drawn is

(a) Divisible by 5 (b) a number which is a perfect square

A window in a building is at a height of 10 m from the ground. The angle of depression of a point P on the ground from the window is 30˚. The angle of elevation of the top of the building from the point P is 60˚. Find the height of the building.

In a violent storm, a tree got bent by the wind. The top of the tree meet the ground at an angle of 30˚, at a distance of 30 metres from the root. At what height from the bottom did the tree get bent? What was the original height of the tree?

A wire bent in the form of a circle of radius 42 cm is cut and again bent in the form of a square. Find the ratio of the areas of the regions enclosed by the circle and the square.

A metallic sphere of radius 10.5 cm is melted and then recast into smaller cones, each of radius 3.5 cm and height 3 cm. How many cones are obtained?

In the given figure ΔABC is right angled at A. Semicircles are drawn on AB, AC and BC as diameter. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region.

₹ 250 is divided equally among a certain number of children. If there were 25 more children, each would have received 50paise less. Find the number of children.

The hypotenuse of a right – angled triangle is 6 cm more than twice the shortest side. If the third side 2 cm less than the hypotenuse, find the sides of the triangle.

If the sum of first n, 2n and 3n term of an AP be S_{1 ,} S₂ and S₃ respectively, then prove that S₃ = 3 (S₂ – S₁).

The angle of elevation of a jet plane from point A on the ground is 60˚. After A flight of 15 seconds, the angle of elevation changes to 30˚. If the jet plane is flying at a constant height of 1500 √ 3 m, find the speed of the jet plane.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the given figure. Prove that: AB + CD = AD + BC

A solid is made up of a cube and a hemisphere attached on its top as shown in the figure. Each edge of the cube measures 5 cm and the hemisphere has a diameter of 4.2 cm. Find the total area to be painted.

The diameter of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm. find:

(i) The capacity of the bucket

(ii) The area of the metal sheet used to make the bucket. [Take π = 3.14.]

Find the value of k for which the point A (– 1, 3), B (2, k) and C (5, – 1) are collinear.

Two dice are thrown at the same time. Find the probability that the sum of two numbers appearing on the top of the dice is more than 9.

A circus tent is cylindrical to a height of 3 cm and conical above it. If its base radius is 52.5 m and the slant height of the conical portion is 53 m, find the area of canvas needed to make the tent. [Take π = 22/7.]