Write the smallest number which is divisible by both 306 and 657.
Find a relation between x and y if the points A(x, y), B(–4, 6) and C(–2, 3) are collinear.
Find the area of a triangle whose vertices are given as (1, –1) (–4, 6) and (–3, –5).
The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is 1/5. The probability of selecting a black marble at random from the same jar is 1/4 . If the jar contains 11 green marbles, find the total number of marbles in the jar.
Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.
The larger of two supplementary angles exceeds the smaller by 18°. Find the angles.
Sumit is 3 times as old as his son. Five years later, he shall be two and a half times as old as his son. How old is Sumit at present ?
Using Euclid’s Algorithm, find the HCF of 2048 and 960.
Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, prove that AP × PC = BP × DP.
Diagonals of a trapezium PQRS intersect each other at the point O, PQ ∥ RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS.
In Figure, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting PQ at A and RS at B. Prove that ∠AOB = 90º.
Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (–2, –5) and (6, 3). Find the coordinates of the point of intersection.
In Figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of the shaded region. (Use π = 3.14)
In Figure, ABCD is a square with side 2√2 cm and inscribed in a circle. Find the area of the shaded region. (Use π = 3.14)
A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use π = 22/7)
For what value of k, is the polynomial f(x) = 3x4 – 9x3 + x2 + 15x + k completely divisible by 3x2 – 5 ?
Find the zeroes of the quadratic polynomial 7y2 – 11y/3 – 2/3 and verify the relationship between the zeroes and the coefficients.
Write all the values of p for which the quadratic equation x2 + px + 16 = 0 has equal roots. Find the roots of the equation so obtained.
Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30°. Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45°. Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.
A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8 gm mass. (Use π = 3.14)
Which term of the Arithmetic Progression –7, –12, –17, –22, ... will be –82 ? Is –100 any term of the A.P. ? Give reason for your answer.
How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ? Explain the double answer.
In a class test, the sum of Arun’s marks in Hindi and English is 30. Had he got 2 marks more in Hindi and 3 marks less in English, the product of the marks would have been 210. Find his marks in the two subjects.
Given that √3 is an irrational number, prove that (2 + √3) is an irrational number.
X is a point on the side BC of ΔABC. XM and XN are drawn parallel to AB and AC respectively meeting AB in N and AC in M. MN produced meets CB produced at T. Prove that TX2 = TB × TC
In Figure, ABC is a triangle in which ∠B = 90º, BC = 48 cm and AB = 14 cm. A circle is inscribed in the triangle, whose centre is O. Find radius r of in-circle.
Find the linear relation between x and y such that P(x, y) is equidistant from the points A(1, 4) and B(–1, 2).
A, B, C are interior angles of ΔABC. Prove that cosec (A+B)/2 = sec C/2
A right circular cylinder and a cone have equal bases and equal heights. If their curved surface areas are in the ratio 8 : 5, show that the ratio between radius of their bases to their height is 3 : 4.
Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number.
Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.
If two positive integers p and q are written as p = a2b3 and q = a3b; a, b are prime numbers, then verify:
LCM (p, q) × HCF (p, q) = pq
Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Using Euclid’s division algorithm find the HCF of the numbers 867 and 255.
Divide 27 into two parts such that the sum of their reciprocals is 3/20.
The sum of first n terms of an AP is given by Sn = 2n2 + 3n. Find the sixteenth term of the AP.
An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.
In an A.P if sum of its first n terms is 3n2 + 5n and its kth term is 164, find the value of k.
If coordinates of two adjacent vertices of a parallelogram are (3, 2), (1, 0) and diagonals bisect each other at (2, –5), find coordinates of the other two vertices.
If the area of triangle with vertices (x, 3), (4, 4) and (3, 5) is 4 square units, find x.
In figure, AB is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at A and B intersect at P. Find the length of AP.
The side of a square is 10 cm. Find the area between inscribed and circumscribed circles of the square.
The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 48 hours.
If sin (A + 2B) = √3/2 and cos (A + 4B) = 0, A > B, and A + 4B ≤ 90º, then find A and B.