In electrical circuit theory, a circuit C(t) is called a linear circuit if it satisfies the superposition principle given by C(at₁+ bt₂) = aC(t₁) + bC(t₂)  where a,b are constants. Show that the circuit C(t) = 3t is linear.

Let A = {9, 10, 11, 12, 13, 14, 15, 16, 17} and let f: A ⟶ N be defined f(n) = the highest prime factor of n \(\in \) A. Write f as a set of ordered pairs and find the range of f.

State whether the graph represent a function. Use vertical line test.

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Letf: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as  a table .

Find the next three terms of the sequences.
1, 0.1, 0.01,...

Find the sum of the following series
51 + 52 + 53 + ....... + 92

Use Euclid's algorithm to find the HCF of 4052 and 12756.

Prove that \(\sqrt { 3 } \) is irrational

Find the LCM and GCD for the following and verify that f(x) x g(x) = LCM x GCD
21x²y, 35 xy²

Simplify
\(\frac { { x }^{ 3 } }{ x-y } +\frac { { y }^{ 3 } }{ y-x } \)

Using quadratic formula solve the following equations.
p²x² + (P² -q²) X - q² = 0

Using quadratic formula solve the following equations.9x²-9(a+b)x+(2a²+5ab+2b²)=0

QA and PB are perpendiculars to AB. If AO = 10 cm, BO = 6 cm and PB = 9 cm. Find AQ.

D and E are respectively the points on the sides AB and AC of a \(\triangle\)ABC such that AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm, show that DE || BC

In figure if PQ || RS Prove that \(\Delta POQ\sim \Delta SOQ\)

In figure the line segment xy is parallel to side AC of \(\Delta ABC\) and it divides the triangle int two parts of equal areas. Find the ratio \(\cfrac { AX }{ AB } \)

Find the value of ‘a’, if the line through (–2, 3) and (8, 5) is perpendicular to y = ax + 2

Find the slope of the following straight lines 5y − 3 = 0

Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).

Show that the points (1, 7), (4, 2), (-1,-1) and (-4,4) are the vertices of a square.

A road is flanked on either side by continuous rows of houses of height \( 4\sqrt { 3 } \)m with no space in between them. A pedestrian is standing on the median of the road facing a row house. The angle of elevation from the pedestrian to the top of the house is 30°. Find the width of the road.

calculate \(\angle \)BAC inthe given triangles ( tan 69.4° = 2.6604 )

Prove that sec A (1 - sin A) (sec A + tan A) = 1.

If 2sin²θ-cos2θ=2, then find the value θ.

The external radius and the length of a hollow wooden log are 16 cm and 13 cm respectively. If its thickness is 4 cm then find its T.S.A.

A cone of height 24 cm is made up of modeling clay. A child reshapes it in the form of a cylinder of same radius as cone. Find the height of the cylinder.

If the radii of the circular ends of a conical bucket which is 45 cm high are 28 cm and 7 cm, find the capacity of the bucket. (Use π = \(\frac{22}{7}\))

Find the depth of a cylindrical tank of radius 28 m, if its capacity is equal to that of a rectangular tank of size 28 m x 16 m x 11 m.

The range of a set of data is 13.67 and the largest value is 70.08. Find the smallest value.

If the standard deviation of a data is 4.5 and if each value of the data is decreased by 5, then find the new standard deviation.

Find the standard deviation for the following data. 5, 10, 15, 20, 25. And also find the new S.D. if three is added to each value.

The marks scored by 5 students in a test for 50 marks are 20, 25, 30, 35, 40. Find the S.D for the marks. If the marks are converted for 100 marks, find the S.D. for newly obtained marks.