TextBook Solutions
- A 1.0 m long metallic rod is rotated with an angular frequency of 400 rad s-1about an axis normal to the rod passing Hence, the emf developed between the centre and the ring is 100 V. 1
- A long solenoid with 15 turns per cm has a small loop of area 2 2.0 cm placed inside the solenoid normal to its axis Hence, the induced voltage in the loop is = 7.54x10⁻⁶ V 1
- A chemist has one solution which is 50% acid and a second which is 25% acid. How much of each should be mixed to make Let x litres and y litres be the amount of acids from 50% and 25% acid solutions respectively. As per the question 50% of x + 25% of y = 40%... 1
- Find the values of a and b if the points A(–2,1), B(a,b) and C(4,–1) are collinear and a – b = 1. Answer: a = 1, b = 0 Three points are collinear when the area of the triangle formed by them is zero: x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂... 1
- If (3,3),(6,y),(x,7) and (5,6) are the vertices of a parallelogram taken in order, find the value of x and y. Answer: x = 8, y = 4 Let A(3, 3), B(6, y), C(x, 7), D(5, 6). The diagonals of a parallelogram bisect each other, so the mid-point of AC equa... 1
- If xcosθ – ysinθ = a, xsinθ + ycos θ = b, prove that x2+y2=a2+b2. Answer: Square both equations and add; the cross terms cancel and sin²θ + cos²θ = 1 gives x² + y² = a² + b² . Squaring the first equation: a... 1
- If cosφ + sinφ = √2 cosφ, prove that cosφ - sinφ = √ 2sin φ. Answer: From the given condition, cosφ = (sqrt2 + 1)sinφ , which rearranges to cosφ - sinφ = sqrt2sinφ . Given cosφ + sinφ = sqrt2cosφ . So... 1
- Prove that one of every three consecutive integers is divisible by 3. Answer: Every integer leaves remainder 0, 1 or 2 on division by 3; in each case exactly one of n, n + 1, n + 2 is a multiple of 3. Let the t... 1
- A bag contains 5 red and some blue balls, if probability of drawing a blue ball from the bag is twice that of Answer: 10 blue balls Let the number of blue balls be x . Total balls = 5 + x . P(red) = 5/5 + x,qquad P(blue) = x/5 + x Given P(blue) = 2 ×... 1
- A card is drawn at random from a well shuffled deck of playing cards. Find the probability that the card drawn Answer: (i) 11/13 (ii) 2/13 A deck has 52 cards, with 4 kings and 4 queens, so 8 cards are kings or queens. (ii) Either a king or a queen: P... 1
- Find the point on the y-axis which is equidistant from the points (5,–2) and (–3,2). Answer: (0, −2) Any point on the y-axis has x-coordinate 0, so let the point be P(0, y) . Let A(5, -2) and B(-3, 2) . Given PA = PB , so PA²... 1
- Which term of the sequence 4, 9, 14.... is 124? 2 Answer: 124 is the 25th term. The sequence 4, 9, 14, … is an AP with first term a = 4 and common difference d = 9 - 4 = 5 . Let a_n = 124 .... 1
- Find the LCM and HCF of 120 and 144 by using Fundamental Theorem of Arithmetic. Answer: HCF = 24, LCM = 720 Write each number as a product of primes (Fundamental Theorem of Arithmetic): 120 = 2³ × 3 × 5 144 = 2⁴ × 3² HCF... 1
- Determine whether the following system of equations has a unique solution, no solutions or infinitely many solution Answer: infinitely many solutions For a₁x + b₁y = c₁ and a₂x + b₂y = c₂ , compare the ratios of coefficients. Here a₁ = 3, b₁ = -7, c₁ = 5 a... 1
- If sec A+ tan A = p, then find the value of sec A – tan A. Answer: sec A - tan A = 1/p Use the identity sec²A - tan²A = 1 . Factorising the left side: (sec A + tan A)(sec A - tan A) = 1 Given sec A +... 1
- D and E are respectively the points on the sides AB and AC of a triangle ABC such that AD = 2 cm, BD = 3 cm, BC = 7.5 cm Answer: DE = 3 cm Since DE ∥ BC, triangle ADE is similar to triangle ABC (AA similarity: ∠A is common and corresponding angles ∠ADE = ∠ABC).... 1
- The first three terms of an AP are 3y – 1, 3y +5 and 5y +1 respectively. Then find the value of ‘y’. Answer: y = 5 If a, b, c are in AP, then b - a = c - b , i.e. 2b = a + c . 2(3y + 5) = (3y - 1) + (5y + 1) 6y + 10 = 8y 2y = 10 ⇒ y = 5 Chec... 1
- The mid-point of line segment AB is the point P (0, 4). If the coordinates of B are (-2, 3) then find the coordinates Answer: A = (2, 5) Let A = (x, y) . By the mid-point formula, the mid-point of AB is P = (x + (-2)/2,\ y + 3/2) = (0, 4) Equating coordinate... 1
- For what value of k, are the roots of the quadratic equation kx (x – 2) + 6 = 0 equal? Answer: k = 6 Expand the equation: kx(x-2) + 6 = 0 ⇒ kx² - 2kx + 6 = 0 . Here a = k , b = -2k , c = 6 . Roots are equal when the discriminan... 1
- Without actual division find whether 18/30 will have a terminating decimal expansion or a non-terminating repeating Answer: 18/30 has a terminating decimal expansion. First reduce the fraction to its lowest terms: 18/30 = 3/5 A rational number p/q in lowes... 1
- My favourite sports person PV Sindhu information My favourite sports person: P. V. Sindhu My favourite sports person is the badminton champion Pusarla Venkata Sindhu, popularly known as P.... 1
- List a few examples of environmental degradation that you may have observed around you. The air quality is bad, many people in my neighbourhood suffer from respiratory problem. The river which flows from the city, looks like a d... 1
- “The Earth has enough resources to meet the needs of all but not enough to satisfy the greed of even one person”. This statement implies that human beings are overusing the natural resources of Earth in the name of development. Development is not sustain... 1
- Why is the issue of sustainability important for development? Development does not only mean securing a better present, but it also means securing a better future for the generations to come. If we use... 1