Intermediate mathematics 2nd year important questions

Maths IIA-IIB Important Questions (24-25)

📘 Maths IIA & IIB – Important Questions (2024-25)

📐 Mathematics IIA

Essay Answer Questions (7 Marks)

De Moivre's Theorem Q : 18

  1. If \(n\) is an integer then show that \((1 + i)^{2n} + (1 - i)^{2n} = 2^{n + 1}\cos \frac{n\pi}{2}\).
  2. If \(n\) is a positive integer, show that \((1 + i)^{n} + (1 - i)^{n} = 2^{\frac{n + 2}{2}}\cos \left(\frac{n\pi}{4}\right)\).
  3. If \(\alpha, \beta\) are the roots of the equation \(x^{2} - 2x + 4 = 0\) then for any \(n\in \mathbb{N}\) show that \(\alpha^{n} + \beta^{n} = 2^{n + 1}\cos \left(\frac{n\pi}{3}\right)\).
  4. If \(n\) is a positive integer, show that \((p + iQ)^{\frac{1}{n}} + (P - iQ)^{\frac{1}{n}} = 2(P^{2} + Q^{2})^{\frac{1}{2n}}\cos \left[\frac{1}{n}\tan^{-1}\frac{Q}{P}\right]\).
  5. If \(\cos \alpha +\cos \beta +\cos \gamma = 0 = \sin \alpha +\sin \beta +\sin \gamma\) then show that
    • \(\cos 3\alpha +\cos 3\beta +\cos 3\gamma = 3\cos (\alpha +\beta +\gamma)\)
    • \(\sin 3\alpha +\sin 3\beta +\sin 3\gamma = 3\sin (\alpha +\beta +\gamma)\)
  6. If \(\cos \alpha +\cos \beta +\cos \gamma = 0 = \sin \alpha +\sin \beta +\sin \gamma\) prove that \(\cos \alpha +\cos \beta +\cos \gamma = \frac{3}{2} = \sin^{2}\alpha +\sin^{2}\beta +\sin^{2}\gamma\).
  7. If \(n\) is an integer then show that \((1 + \cos \theta + i\sin \theta)^{n} + (1 + \cos \theta - i\sin \theta)^{n} = 2^{n + 1}\cos^{n}\left(\frac{\theta}{2}\right)\cos \left(\frac{n\theta}{2}\right)\).
  8. Show that one value of \(\left[1 + \sin \frac{\pi}{8} + i\cos \frac{\pi}{8}\right]^{3}\) is \(-1\).
  9. If \(n\) is an integer and \(z = \cos \theta\) \(\left(\theta \neq (2n + 1)\frac{\pi}{2}\right)\) then show that \(\frac{z^{2n} - 1}{z^{2n} + 1} = i\tan \theta\).
  10. Find all the roots of the equation
    • \(x^{11} - x^{7} + x^{4} - 1 = 0\)
    • \(x^{9} - x^{5} + x^{4} - 1 = 0\)
  11. If \((1 + x)^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{n}x^{n}\), then show that
    • \(a_{0} - a_{2} + a_{4} - \ldots = 2^{\frac{n}{2}}\cos \left(\frac{n\pi}{4}\right)\)
    • \(a_{1} - a_{3} + a_{5} - \ldots = 2^{\frac{n}{2}}\sin \left(\frac{n\pi}{4}\right)\)
  12. If \(z^{2} + z + 1 = 0\), where \(z\) is a complex number, prove that \(\left(\tau \cdot \frac{1}{z}\right)^{2} + \left(\tau^{2} \cdot \frac{1}{z^{2}}\right)^{2} + \left(\tau^{3} \cdot \frac{1}{z^{3}}\right)^{2} + \left(\tau^{4} \cdot \frac{1}{z^{4}}\right)^{2} + \left(\tau^{5} \cdot \frac{1}{z^{5}}\right)^{2} + \left(\tau^{6} \cdot \frac{1}{z^{6}}\right)^{2} = 12\)

Theory of Equations Q : 19

  1. Solve:
    • \(4x^{3} - 24x^{2} + 23x + 18 = 0\), given that the roots are in A.P.
    • \(8x^{3} - 36x^{2} - 18x + 81 = 0\), given that the roots are in A.P.
  2. Solve:
    • \(x^{3} - 7x^{2} + 14x - 8 = 0\), given that the roots are in geometric progression.
    • \(3x^{3} - 26x^{2} + 52x - 24 = 0\), given that the roots are in geometric progression.
  3. Solve:
    • \(15x^{3} - 23x^{2} + 9x - 1 = 0\), given that the roots are in H.P.
    • \(6x^{3} - 11x^{2} + 6x - 1 = 0\), given that the roots are in H.P.
  4. Solve \(18x^{3} + 81x^{2} + 121x + 60 = 0\) given that a root is equal to half the sum of the remaining roots.
  5. Solve \(x^{4} - 2x^{3} + 4x^{2} + 6x - 21 = 0\), the sum of two roots being zero.
  6. Solve:
    • \(x^{4} - 5x^{3} + 5x^{2} + 5x - 6 = 0\), the product of two roots being 3.
    • Solve the equation \(x^{4} + x^{3} - 16x^{2} - 4x + 48 = 0\), given that the product of two roots is 6.
  7. Solve \(x^{4} + 4x^{3} - 2x^{2} - 12x + 9 = 0\), if it has a pair of equal roots.
  8. Solve: \(x^{4} - 16x^{3} + 86x^{2} - 176x + 105 = 0\).
  9. Solve:
    • \(6x^{4} - 35x^{3} + 62x^{2} - 35x + 6 = 0\)
    • \(x^{4} - 10x^{3} + 26x^{2} - 10x + 1 = 0\)
  10. Solve:
    • \(2x^{5} + x^{4} - 12x^{3} - 12x^{2} + x + 2 = 0\)
    • \(x^{5} - 5x^{4} + 9x^{3} - 9x^{2} + 5x - 1 = 0\)
  11. Solve: \(6x^{6} - 25x^{5} + 31x^{4} - 31x^{2} + 25x - 6 = 0\).
  12. Solve \(x^{3} - 9x^{2} + 14x + 24 = 0\) given that two of the roots are in the ratio 3:2.
    • Find the repeated roots of the equation \(x^{5} - 3x^{4} - 5x^{3} + 27x^{2} - 32x + 12 = 0\).
    • Show that \(x^{5} - 5x^{3} + 5x^{2} - 1 = 0\) has three equal roots and find that root.
    • Solve \(x^{4} + 2x^{3} - 5x^{2} + 6x + 2 = 0\), given that one root of it is \(1 + i\).
    • Solve the equation \(x^{4} - 9x^{3} + 27x^{2} - 29x + 6 = 0\), given that one root of it is \(2 - \sqrt{3}\).
    • Given that \(- 2 + \sqrt{- 7}\) is a root of the equation \(x^{4} + 2x^{2} - 16x + 77 = 0\), solve it completely.
  15. Find the algebraic equation of degree 5 whose roots are the translates of the roots of \(x^{5} + 4x^{3} - x^{2} + 11 = 0\) by \(-3\).
  16. Transform \(x^{4} + 4x^{3} + 2x^{2} - 4x - 2 = 0\) into another equation in which the coefficient of second highest power of \(x\) is zero and find the transformed equation.
  17. If the roots of the equation \(x^{3} + 3px^{2} + 3qx + r = 0\),
    • are in A.P, then show that \(2p^{3} - 3pq + r = 0\).
    • are in G.P then show that \(p^{3}r = q^{3}\).
    • are in H.P show that \(2q^{3} = 4(3pq - r)\).

Binomial Theorem Q : 20

  1. Prove that \(\mathbf{C}_{0} + \frac{\mathbf{C}_{1}}{2} x + \frac{\mathbf{C}_{2}}{3} x^{2} + \ldots + \frac{\mathbf{C}_{n}}{n + 1} x^{n} = \frac{(1 + x)^{n + 1} - 1}{(n + 1)x}\). Deduce that \(\frac{\mathbf{C}_{1}}{2} + \frac{\mathbf{C}_{3}}{4} + \frac{\mathbf{C}_{5}}{6} + \ldots = \frac{2^{n} - 1}{n + 1}\).
  2. Prove that \(\mathbf{C}_{0}\mathbf{C}_{r} + \mathbf{C}_{1}\mathbf{C}_{r + 1} + \mathbf{C}_{2}\mathbf{C}_{r + 2} + \ldots + \mathbf{C}_{n - 1}\mathbf{C}_{n} = 2^{n}\mathbf{C}_{n + 1}\). Deduce that
    • \(\mathbf{C}_{0}^{2} + \mathbf{C}_{1}^{2} + \mathbf{C}_{2}^{2} + \ldots + \mathbf{C}_{n}^{2} = 2^{n}\mathbf{C}_{n}\)
    • \(\mathbf{C}_{0}\mathbf{C}_{1} + \mathbf{C}_{1}\mathbf{C}_{2} + \mathbf{C}_{2}\mathbf{C}_{3} + \ldots + \mathbf{C}_{n - 1}\mathbf{C}_{n} = 2^{n}\mathbf{C}_{n + 1}\)
    • If the \(2^{\text{nd}}, 3^{\text{rd}}\) and \(4^{\text{th}}\) terms in the expansion of \((a + x)^{n}\) are respectively 240, 720, 1080, find \(a, x, n\).
    • If 36, 84, 126 are three successive binomial coefficients in the expansion of \((1 + x)^{n}\), then find \(n\).
    • If \((7 + 4\sqrt{3})^{n} = I + f\) where \(I\) and \(n\) are positive integers and \(0 < f < 1\) then show that (i) \(I\) is an odd positive integer (ii) \((1 + f)(1 - f) = 1\).
    • If \(R, n\) are positive integers, \(n\) is odd, \(0 < F < 1\) and if \((5\sqrt{5} + 11)^{n} = R + F\), then prove that (i) \(R\) is an even integer (ii) \((R + F)F = 4^{n}\).
  5. If \(P\) and \(Q\) are the sum of odd terms and the sum of even terms respectively in the expansion of \((x + a)^{n}\), then prove that
    • \(P^{2} - Q^{2} = (x^{2} - a^{2})^{n}\)
    • \(4PQ = (x + a)^{2n} - (x - a)^{2n}\)
    • If the coefficients of \(r^{\text{th}}, (r + 1)^{\text{th}}, (r + 2)^{\text{nd}}\) terms in the expansion of \((1 + x)^{n}\) are in A.P, then show that \(n^{2} - (4r + 1)n + 4r^{2} - 2 = 0\).
    • If the coefficients of \(x^{3}, x^{10}, x^{11}\) in the expansion of \((1 + x)^{n}\) are in A.P. then prove that \(n^{2} - 41n + 398 = 0\).
  7. If the coefficients of 4 consecutive terms in the expansion of \((1 + x)^{n}\) are \(a_{1}, a_{2}, a_{3}, a_{4}\) respectively, then show that \(\frac{a_{1}}{a_{1} + a_{2}} + \frac{a_{3}}{a_{3} + a_{4}} = \frac{2a_{2}}{a_{2} + a_{3}}\).
  8. If \(n\) is a positive integer, prove that \(\sum_{r = 1}^{n}r^{3}\left(\frac{nC_{r}}{nC_{r - 1}}\right)^{2} = \frac{n(n + 1)^{2}(n + 2)}{12}\).
  9. If the coefficient of \(x^{10}\) in the expansion of \((ax^{2} + \frac{1}{bx})^{11}\) is equal to the coefficient of \(x^{10}\) in the expansion of \((ax - \frac{1}{bx^{2}})^{11}\), find the relation between \(a\) and \(b\), where \(a\) and \(b\) are real numbers.
  10. Prove that \((\mathbf{C}_{0} + \mathbf{C}_{1})(\mathbf{C}_{1} + \mathbf{C}_{2})(\mathbf{C}_{2} + \mathbf{C}_{3})\ldots(\mathbf{C}_{n + 1} + \mathbf{C}_{n}) = \frac{(n + 1)^{n}}{n!}\mathbf{C}_{0}\mathbf{C}_{1}\ldots\mathbf{C}_{n}\).
  11. If \(n\) is a positive integer, then prove that \(\mathbf{C}_{0} + \frac{\mathbf{C}_{1}}{2} + \frac{\mathbf{C}_{2}}{3} + \ldots + \frac{\mathbf{C}_{n}}{n + 1} = \frac{2^{n + 1} - 1}{n + 1}\).

Binomial Theorem Q : 21

  1. Find the sum of infinite series \(\frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \ldots\)
  2. If \(t = \frac{4}{5} + \frac{4 \cdot 6}{5 \cdot 10} + \frac{4 \cdot 6 \cdot 8}{5 \cdot 10 \cdot 15} + \ldots\) then prove that \(9t = 16\).
  3. If \(x = \frac{1}{5} + \frac{1 \cdot 3}{5 \cdot 10} + \frac{1 \cdot 3 \cdot 5}{5 \cdot 10 \cdot 15} + \ldots\) then find the value of \(3x^{2} + 6x\).
  4. If \(x = \frac{1 \cdot 3}{3 \cdot 6} + \frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{3 \cdot 6 \cdot 9 \cdot 12} + \ldots\) then prove that \(9x^{2} + 24x = 11\).
  5. Find the sum of the series \(\frac{3 \cdot 5}{5 \cdot 10} + \frac{3 \cdot 5 \cdot 7}{5 \cdot 10 \cdot 15} + \frac{3 \cdot 5 \cdot 7 \cdot 9}{5 \cdot 10 \cdot 15 \cdot 20} + \ldots\)
  6. Find sum of the infinite series \(\frac{3}{4 \cdot 8} - \frac{3 \cdot 5}{4 \cdot 8 \cdot 12} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12 \cdot 16} - \ldots\)
  7. If \(x = \frac{5}{(2!3)} + \frac{5 \cdot 7}{(3!3)^{2}} + \frac{5 \cdot 7 \cdot 9}{(4!3)^{3}} + \ldots\) then find the value of \(x^{2} + 4x\).
  8. Find the sum of the series \(\frac{7}{5}\left[1 + \frac{1}{10^{2}} + \frac{1 \cdot 3}{1 \cdot 2} \cdot \frac{1}{10^{4}} + \frac{1 \cdot 3 \cdot 5}{1 \cdot 2 \cdot 3} \cdot \frac{1}{10^{6}} + \ldots\right]\)
  9. Find the sum of the infinite series \(1 + \frac{2}{3} + \frac{1}{3 \cdot 6} + \frac{2 \cdot 5}{3 \cdot 6}\left(\frac{1}{3}\right)^{2} + \frac{2 \cdot 5 \cdot 8}{3 \cdot 6 \cdot 9}\left(\frac{1}{3}\right)^{3} + \ldots\)
  10. Show that for any non zero rational number \(x\), \[x + \frac{x(x - 1)}{2} + \frac{x(x - 1)(x - 2)}{2 \cdot 4} + \ldots = 1 + \frac{x}{3} + \frac{x(x + 1)}{3 \cdot 6} + \frac{x(x + 1)(x + 2)}{3 \cdot 6 \cdot 9} + \ldots\]
  11. If \(|x|\) is so small that \(x^{2}\) and higher power of \(x\) may be neglected, then find approximate value of
    • \(\left(1 - \frac{2x}{3}\right)^{\frac{3}{2}} (32 + 5x)^{\frac{1}{5}}\)
    • \(\left(\frac{y}{y + x}\right)^{\frac{3}{4}} - \left(\frac{y}{y + x}\right)^{\frac{4}{5}}\)

Measures of Dispersion Q : 22

  1. Calculate the mean deviation about the mean for the following data:
    Class interval25781035
    Frequency6810682
  2. Find the mean deviation from the mean of the following data, using the step deviation method:
    Marks0-1010-2020-3030-4040-5050-6060-70
    No. of Students65815763
  3. Find the mean deviation from the median for the following data:
    \(x_i\)6931215132122
    \(f_i\)45325443
  4. Find the mean deviation from the median of the following data:
    Age (Years)20-2525-3030-3535-4040-4545-5050-5555-60
    No. of workers (f)12012517516015014010030
  5. Calculate the variance and standard deviation for the discrete frequency distribution:
    \(x_i\)481117202432
    \(f_i\)3595431
  6. Calculate the variance and standard deviation of the following continuous frequency distribution:
    Class interval30-4040-5050-6060-7070-8080-9090-100
    Frequency371215832
  7. The following table gives the daily wages of workers in a factory. Compute the standard deviation and the coefficient of variation of the wages of the workers:
    Wages125-175175-225225-275275-325325-375375-425425-475475-525525-575
    No. of Workers222191434611
  8. The scores of two cricketers A and B in 10 innings are given below. Find who is a better run getter and who is a more consistent player:
    Scores of A (\(x\))140251980388671216676
    Scores of B (\(y\))12870310141116631254
  9. The mean of 5 observations is 4.4. Their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.

Probability Q : 23

  1. State and explain the axioms that define 'Probability function'. Prove addition theorem on probability, i.e. \(P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)\).
  2. A, B, C are three horses in a race. The probability of A to win the race is twice that of B, and probability of B is twice that of C. What are the probabilities of A, B, C to win the race? Also find the probability that A loses in the race.
  3. A, B, C are 3 newspapers from a city. 20% of the population read A, 16% read B, 14% read C, 8% read both A and B, 5% read both A and C, 4% read both B and C and 2% read all the three. Find the percentage of the population who read at least one newspaper and find the percentage of the population who read the newspaper A only.
  4. The probabilities of three events A, B, C are such that \(P(A) = 0.3\), \(P(B) = 0.4\), \(P(C) = 0.8\), \(P(A \cap B) = 0.08\), \(P(A \cap C) = 0.28\), \(P(A \cap B \cap C) = 0.09\) and \(P(A \cup B \cup C) \ge 0.75\). Show that \(P(B \cap C)\) lies in the interval \([0.23, 0.48]\).
  5. The probabilities of three mutually exclusive events are respectively given as \(\frac{1 + 3n}{2}\), \(\frac{1 - n}{1 - 2n}\). Prove that \(\frac{1}{2} \le p \le \frac{1}{2}\).
  6. A, B, C are aiming to shoot a balloon. A will succeed 4 times out of 5 attempts. The chance of B to shoot the balloon is 3 out of 4 and that C is 2 out of 3. If the three aim the balloon simultaneously, then find the probability that at least two of them hit the balloon.
  7. In a shooting test the probability of A, B, C hitting the targets are \(1/2, 2/3\) and \(3/4\) respectively. If all of them fire at the same target, find the probability that (i) only one of them hits the target, (ii) at least one of them hits the target.
  8. If \(E_1, E_2, E_3\) are three independent events such that \(P(E_1 \cap \overline{E_2} \cap \overline{E_3}) = \frac{1}{4}\), \(P(\overline{E_1} \cap E_2 \cap \overline{E_3}) = \frac{1}{8}\), \(P(\overline{E_1} \cap \overline{E_2} \cap E_3) = \frac{1}{4}\), then find \(P(E_1)\), \(P(E_2)\), \(P(E_3)\).
  9. Define conditional event and Conditional Probability. There are 3 black and 4 white balls in one bag; 4 black and 3 white balls in the second bag. A die is rolled and the first bag is selected if it is 1 or 3, and the second bag for the rest. Find the probability of drawing a black ball from the selected bag.
  10. State and prove Baye's theorem.
  11. Three boxes numbered I, II, III contain 1 white, 2 black and 3 red balls; 2 white, 1 black and 1 red ball; 4 white, 5 black and 3 red balls respectively. One box is randomly selected and a ball is drawn from it. If the ball is red then find the probability that it is from box II.
  12. Three boxes \(B_1, B_2, B_3\) contain balls with different colours as follows:
    WhiteBlackRed
    \(B_1\)212
    \(B_2\)324
    \(B_3\)432
    A die is thrown. If 1 or 2 turns up on the dice, box \(B_1\) is selected; if 3 or 4 turns up \(B_2\) is selected; if 5 or 6 turns up, then \(B_3\) is selected. If a box is selected like this, a ball is drawn from that box. If the ball is red, then find the probability that it was drawn from \(B_2\).
  13. In a certain college, 25% of the boys and 10% of the girls are studying mathematics. The girls constitute 60% of the student strength. If a student selected at random is found studying mathematics, find the probability that the student is a girl.

Random Variable and Distributions Q : 24

  1. The probability distribution of a random variable X is given below:
    \(X = x_i\)12345
    \(P(X = x)\)k2k3k4k5k
    Find the value of \(k\) and the mean, variance of X.
  2. \(X = x\)-2-10123
    \(P(X = x)\)0.1K0.22K0.3K
    is the probability distribution of a random variable X. Find the value of \(K\) and the variance of X.
  3. A random variable X has the following probability distribution:
    \(X = x\)01234567
    \(P(X = x)\)0K2k2k3kk²2k²7k² + k
    Find (i) \(k\) (ii) The mean (iii) \(P(0 < X < 5)\).
  4. A cubical die is thrown. Find the mean and variance of X, giving the number on the face that shows up.
  5. The range of a random variable X is \(\{0, 1, 2\}\). Given that \(P(X = 0) = 3C^3\), \(P(X = 1) = 4C - 10C^2\), \(P(X = 2) = 5C - 1\). Find (i) the value of \(C\) (ii) \(P(X < 1)\) (iii) \(P(1 < X \le 2)\) (iv) \(P(0 < X \le 3)\).
  6. One in nine ships is likely to be wrecked when they set on sail. When 6 ships are set on sail, find the probability for: (i) at least one will arrive safely (ii) exactly three will arrive safely.
  7. If the mean and variance of a binomial variate X are 2.4 and 1.44 respectively, find \(P(1 < X \le 4)\).
  8. In the experiment of tossing a coin \(n\) times, if the variable X denotes the number of heads and \(P(X = 4), P(X = 5), P(X = 6)\) are in A.P, then find \(n\).
  9. If the difference between the mean and variance of binomial variate is \(\frac{5}{9}\) then, find the probability for the event of 2 successes when the experiment is conducted 5 times.
  10. If \(X: S \to R\) is a discrete random variable with range \(\{x_{1}, x_{2}, x_{3}, \ldots\}\), \(\mu\) is mean and \(\sigma^2\) is variance of X then prove that \(\sigma^2 + \mu^2 = \Sigma x^2 P(X = x)\).

Short Answer Questions (4 Marks)

Complex Numbers Q : 11

  1. Show that the points in the Argand diagram represented by the complex numbers \(2 + 2i, -2 - 2i, -2\sqrt{3} + 2\sqrt{3}i\) are the vertices of an equilateral triangle.
  2. Show that the four points in the Argand plane represented by the complex numbers \(2 + i, 4 + 3i, 2 + 5i, 3i\) are the vertices of a square.
  3. Show that the points in the Argand plane represented by the complex numbers \(-2 + 7i, \frac{-3}{2} + \frac{1}{2}i, \frac{1}{2} - 2i, \frac{7}{2}(1 + i)\) are the vertices of rhombus.
  4. If \(z = 3 - 5i\), then show that \(z^3 - 10z^2 + 58z - 136 = 0\).
  5. If \((x - iy)^{1/3} = a - ib\) then show that \(\frac{x}{a} + \frac{y}{b} = 4(a^{2} - b^{2})\).
    • If \(x + iy = \frac{3}{2 + \cos \theta + i \sin \theta}\) then, show that \(x^{2} + y^{2} = 4x - 3\).
    • If \(x + iy = \frac{1}{1 + \cos \theta + i \sin \theta}\), show that \(4x^{2} - 1 = 0\).
  7. If \(z = x + iy\) and if the point P in the Argand plane represents \(z\), find the locus of \(z\) satisfying the equation \(|z - 2 - 3i| = 5\).
  8. If the point P denotes the complex number \(z = x + iy\) in the Argand plane and if \(\frac{z - i}{z - 1}\) is a purely imaginary number, find the locus of P.
  9. If the amplitude of \(\left(\frac{z - 2}{z - 6i}\right) = \frac{\pi}{2}\), find its locus.
  10. Determine the locus of \(z, z \neq 2i\), such that \(\operatorname{Re}\left(\frac{z - 4}{z - 2i}\right) = 0\).
  11. Find the real values of \(\theta\) in order that \(\frac{3 + 2i \sin \theta}{1 - 2i \sin \theta}\) is (a) a real number (b) a purely imaginary number.
  12. The points P, Q denote the complex numbers \(z_{1}, z_{2}\) in the Argand diagram. O is the origin. If \(z_{1}z_{2} + z_{1}z_{2} = 0\), then show that \(\angle POQ = 90^{\circ}\).
  13. Show that the points in the Argand diagram represented by the complex numbers \(z_{1}, z_{2}, z_{3}\) are collinear if and only if there exist three real numbers \(p, q, r\) not all zero, satisfying \(p z_{1} + q z_{2} + r z_{3} = 0\) and \(p + q + r = 0\).

Quadratic Expressions Q : 12

  1. If \(x\) is a real number, find the range of
    • \(\frac{x + 2}{2x^{2} + 3x + 6}\)
    • \(\frac{x^{2} + x + 1}{x^{2} - x + 1}\)
  2. Show that \(\frac{x}{x^{2} - 5x + 9}\) lies between \(\frac{1}{11}\) and 1.
  3. If \(x\) is real, show that the values of the expression \(\frac{x^{2} + 34x - 71}{x^{2} + 2x - 7}\) do not lie between 5 and 9.
  4. If \(x\) is real, find the maximum value of the expression \(\frac{x^{2} + 14x + 9}{x^{2} + 2x + 3}\).
  5. Prove that \(\frac{1}{3x + 1} + \frac{1}{x + 1} = \frac{1}{(3x + 1)(x + 1)}\) does not lie between 1 and 4, if \(x\) is real.
  6. If the expression \(\frac{x^{2} - 3x + 2}{x^{2} - 3x + 2}\) takes all real values for \(x \in \mathbb{R}\), then find the bounds for \(p\).
  7. If \(c^{2} \neq ab\) and the roots of \((c^{2} - ab)x^{2} - 2(a^{2} - bc)x + (b^{2} - ac) = 0\) are equal then show that \(a^{3} + b^{3} + c^{3} = 3abc\) or \(a = 0\).
  8. Solve \(4x^{4} - 3 \cdot 2x^{4} + 2 = 0\).
  9. Solve the equation \(\sqrt{\frac{3x}{x + 1}} + \sqrt{\frac{x + 1}{3x}} = 2\).
  10. If \(x_{1}, x_{2}\) are the roots of the quadratic equation \(ax^{2} + bx + c = 0\) and \(c \neq 0\). Find the value of \((ax_{1} + b)^{2} + (ax_{2} + b)^{2}\) in terms of \(a, b, c\).
  11. If the roots of \(ax^{2} + bx + c = 0\) are imaginary, show that for all \(x \in \mathbb{R}\), \(ax^{2} + bx + c\) and \(a\) have the same sign.
  12. Let \(\alpha, \beta\) be the real roots of \(ax^{2} + bx + c = 0\) where \(\alpha < \beta\), then prove that
    • for \(\alpha < x < \beta\), \(ax^{2} + bx + c\) and \(a\) have opposite signs.
    • for \(x < \alpha\) or \(x > \beta\), \(ax^{2} + bx + c\) and \(a\) have the same sign.

Permutations Q : 13

  1. Find the rank of the words
    • "MASTER"
    • "REMASTER"
    • "PRISON"
    • "EAMCET"
    • "JANATA"
  2. Find the number of 4 letter words that can be formed using the letters of the word 'MIXTURE' which
    • contain the letter X
    • do not contain the letter X
  3. Find the number of ways of arranging 6 boys and 6 girls in a row so that
    • all the girls sit together
    • no two girls sit together
    • boys and girls sit alternately
  4. Find the number of ways of permuting the letters of the word 'PICTURE' so that
    • All vowels come together
    • No two vowels come together
  5. Find the number of ways of arranging 5 different mathematics books, 4 different physics books and 3 different chemistry books such that the books of the same subject are together.
  6. Find the sum of all 4 digit numbers that can be formed using the digits 0, 2, 4, 7, 8 without repetition.
  7. Find the sum of all 4 digit numbers that can be formed using the digits 1, 3, 5, 7 and 9 (without repetition).
  8. Find the number of numbers that are greater than 4000 which can be formed using the digits 0, 2, 4, 6, 8 without repetition.
  9. Find the number of numbers less than 2000 that can be formed using the digits 1, 2, 3, 4 if repetition is allowed.
  10. Find the number of ways of arranging 7 gents and 4 ladies around a circular table if no two ladies wish to sit together.
  11. Find the number of different ways of preparing a garland using 7 distinct red roses and 4 distinct yellow roses such that no two yellow roses come together.
  12. Find the number of four digit numbers that can be formed using the digits 1, 2, 5, 6, 7. How many of them are divisible by (i) 2 (ii) 3 (iii) 4 (iv) 25?
  13. Prove that \({}^nP_r = r \cdot {}^{n-1}P_{r-1} + {}^{n-1}P_r\).
  14. Find the number of ways of arranging the letters of the word 'SINGING' so that
    • They begin and end with I
    • The two G's come together
  15. A family consists of the father, mother, 2 daughters and 2 sons. In how many different ways can they sit at a round table, if the two daughters wish to sit on either side of father?

Combinations Q : 14

  1. Find the number of ways of selecting a cricket team of 11 players from 7 batsmen and 6 bowlers such that there will be at least 5 bowlers in the team.
  2. Find the number of ways of forming a committee of 5 members out of 6 Indians and 5 Americans so that always Indians will be in majority in the committee.
  3. Find the number of ways of selecting 11 member cricket team from 7 batsmen, 6 bowlers and 2 wicket keepers so that the team contains 2 wicket keepers and at least 4 bowlers.
  4. A question paper is divided into 3 sections A, B, C containing 3, 4, 5 questions respectively. Find the number of ways of attempting 6 questions choosing at least one from each section.
  5. Simplify: \({}^{34}C_{5} + \sum_{r = 0}^{4} {}^{38-r}C_{4}\).
  6. Prove that for \(3 \le r \le n\), \({}^{n-3}C_{r} + 3{}^{n-3}C_{r-1} + 3{}^{n-3}C_{r-2} + {}^{n-3}C_{r-3} = {}^{n}C_{r}\).
  7. Show that \(\frac{{}^{4n}C_{2n}}{{}^{2n}C_{n}} = \frac{1 \cdot 3 \cdot 5 \ldots (4n - 1)}{1 \cdot 3 \ldots (2n - 1)^{2}}\).
  8. Prove that \({}^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r}\).
  9. If 5 vowels and 6 consonants are given, then how many 6 letter words can be formed with 3 vowels and 3 consonants?
  10. Find the number of subsets of A having 12 elements
    • at least 3 elements
    • at most 3 elements

Partial Fractions Q : 15

  1. Resolve into partial fractions
    • \(\frac{3x + 7}{x^2 - 3x + 2}\)
    • \(\frac{x + 4}{(x^2 - 4)(x + 1)}\)
  2. Resolve into partial fractions
    • \(\frac{x - 1}{(x - 2)^2(x + 1)}\)
    • \(\frac{x^2 + 13x + 15}{(2x + 3)(x + 3)^2}\)
    • \(\frac{3x - 18}{x^3(x + 3)}\)
    • \(\frac{2x^2 + 2x + 1}{x^3 + x^2}\)
  3. Resolve into partial fractions
    • \(\frac{3x^3 - 8x^2 + 10}{(x - 1)^4}\)
    • \(\frac{x^2 + 5x + 7}{(x - 3)^3}\)
    • \(\frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3}\)
  4. Resolve into partial fractions
    • \(\frac{x^2 - 3}{(x + 2)(x^2 + 1)}\)
    • \(\frac{2x^2 + 3x + 4}{(x - 1)(x^2 + 2)}\)
    • \(\frac{3x - 1}{(1 - x + x^2)(x + 2)}\)
  5. Resolve into partial fractions
    • \(\frac{x^3}{(x - a)(x - b)(x - c)}\)
    • \(\frac{x^3}{(2x - 1)(x + 2)(x - 3)}\)
    • \(\frac{x^4}{(x - 1)(x - 2)}\)
  6. Find the coefficient of \(x^4\) in the expansion of \(\frac{3x}{(x - 2)(x + 1)}\).
  7. Find the coefficient of \(x^n\) in the power series expansion of \(\frac{x - 4}{x^2 - 5x + 6}\) specifying the region in which the expansion is valid.

Probability Q : 16

  1. In a committee of 25 members, each member is proficient either in Mathematics or in Statistics or in both. If 19 of these are proficient in Mathematics, 16 in Statistics, find the probability that a person selected from the committee is proficient in both.
  2. Find the probability of drawing an ace or a spade from a well shuffled pack of 52 playing cards.
  3. If one ticket is randomly selected from tickets numbered 1 to 30, then find the probability that the number on the ticket is (i) a multiple of 5 or 7 (ii) Multiple of 3 or 5.
  4. In a class of 60 boys and 20 girls, half of the boys and half of the girls know cricket. Find the probability of a person selected from the class is either a boy or a girl who knows cricket.
  5. If A, B, C are three events in a sample space S, then show that \(P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)\).
  6. If two numbers are selected randomly from 20 consecutive natural numbers, find the probability that the sum of the two numbers is (i) an even number (ii) an odd number.
  7. A bag contains 12 two rupee coins, 7 one rupee coins and 4 half a rupee coins. If three coins are selected at random, then find the probability that (a) the sum of three coins is maximum (b) the sum of three coins is minimum (c) each coin is of different value.
  8. A speaks truth in 75% of the cases and B in 80% of the cases. What is the probability that their statements about an incident do not match?
  9. Two persons A and B are rolling a die on the condition that the person who gets 3 will win the game. If A starts the game, then find the probabilities of A and B respectively to win the game.
  10. In a box containing 15 bulbs, 5 are defective. If 5 bulbs are selected at random from the box, find the probability of the event, that (i) None of the defective (ii) Only one of the defective (iii) At least one of them is defective.
  11. State and prove multiplication theorem on probability.
  12. If one card is drawn from a pack of cards, then show that the events of getting an ace and getting a heart card are independent events.

Probability Q : 17

  1. If A and B are independent events with \(P(A) = 0.6\), \(P(B) = 0.7\) then compute
    • \(P(A \cap B)\)
    • \(P(A \cup B)\)
    • \(P\left(\frac{B}{A}\right)\)
    • \(P(\overline{A} \cap \overline{B})\)
  2. If A and B are independent events with \(P(A) = 0.2\), \(P(B) = 0.5\) then find
    • \(P\left(\frac{A}{B}\right)\)
    • \(P\left(\frac{B}{A}\right)\)
    • \(P(A \cap B)\)
    • \(P(A \cup B)\)
  3. If A, B are two events with \(P(A \cup B) = 0.65\), \(P(A \cap B) = 0.15\) then find \(P(\overline{A}) + P(\overline{B})\).
  4. For any two events A and B, show that \(P(\overline{A} \cap \overline{B}) = 1 + P(A \cap B) - P(A) - P(B)\).
  5. A and B are events with \(P(A) = 0.5\), \(P(B) = 0.4\), \(P(A \cap B) = 0.3\). Find the probability that (i) A does not occur (ii) neither A nor B occurs.
  6. The probability that Australia wins a match against India in a cricket game is given to be \(\frac{1}{3}\). If India and Australia play 3 matches, what is the probability that (i) Australia will lose all the three matches? (ii) Australia will win at least one match?
  7. A problem in calculus is given to two students A and B whose chances of solving it are \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively. Find the probability of the problem being solved if both of them try independently.
  8. A, B are two independent events such that the probability of both the events to occur is \(1/6\) and the probability of both the events do not occur is \(1/3\). Find \(P(A)\).
  9. A bag \(B_1\) contains 4 white and 2 black balls. Bag \(B_2\) contains 3 white and 4 black balls. A bag is drawn at random and a ball is chosen at random from it. Then what is the probability that the ball is white?
  10. Three screws are drawn at random from a lot of 50 screws, 5 of which are defective. Find the probability of the event that all 3 screws are non-defective, assuming that the drawing is (a) with replacement (b) without replacement.
  11. A number \(x\) is drawn arbitrarily from the set \(\{1, 2, 3, \ldots, 100\}\). Find the probability that \(\left(x + \frac{100}{x}\right) > 29\).
  12. Find the probability that a non leap year contains (i) 53 Sundays (ii) 52 Sundays only.

Very Short Answer Questions (2 Marks) – IIA

Complex Numbers Q1

    • If \(z_{1} = (2, -1)\), \(z_{2} = (6, 3)\) then find \(z_{1} - z_{2}\).
    • If \(z_{1} = (3, 5)\), \(z_{2} = (2, 6)\) then find \(z_{1} - z_{2}\).
    • If \(z_{1} = (6, 3)\), \(z_{2} = (2, -1)\) then find \(\frac{z_{1}}{z_{2}}\).
  2. Write the additive inverse of the complex number \((\sqrt{2}, \pi)\).
  3. Write the multiplicative inverse of (i) \((7, 24)\) (ii) \((\cos \theta, \sin \theta)\).
  4. Find the square roots of (i) \(-5 + 12i\) (ii) \(7 + 24i\) (iii) \(-8 - 6i\).
  5. If \(z = 2 - 3i\), then show that \(z^{2} - 4z + 13 = 0\).
    • If \((a + ib)^{2} = x + iy\), find \(x^{2} + y^{2}\).
    • If \((\sqrt{3} + i)^{100} = 2^{200}(a + ib)\), then show that \(a^{2} + b^{2} = 4\).
    • If \(x + iy = \cos \alpha \cdot \cos \beta\), then find the value of \(x^{2} + y^{2}\).
    • Write the conjugate of \((3 + 4i)(2 - 3i)\).
    • \(\frac{5i}{7 + i}\)
    • Show that \(\frac{2 - i}{(1 - 2i)^{2}}\) and \(\frac{-2 - 11i}{25}\) are conjugate to each other.
  8. Represent the complex number \(2 + 3i\) in Argand diagram.
  9. Find the real and imaginary parts of the complex number \(\frac{a + ib}{a - ib}\).
  10. Find the least positive integer \(n\), satisfying \(\left(\frac{1 + i}{1 - i}\right)^{n} = 1\).
  11. If \(z = (\cos \theta, \sin \theta)\), then find \(z - \frac{1}{z}\).

Complex Numbers Q2

  1. If \(z_{1} = -1\), \(z_{2} = i\), then find \(\operatorname{Arg}\left(\frac{z_{1}}{z_{2}}\right)\).
  2. If \(z_{1} = -1\), \(z_{2} = -i\), then find \(\operatorname{Arg}(z_{1} \cdot z_{2})\).
  3. If \(\operatorname{Arg} \overline{z}_{1}\) and \(\operatorname{Arg} z_{2}\) are \(\frac{\pi}{5}\) and \(\frac{\pi}{3}\) respectively, then find \((\operatorname{Arg} z_{1} + \operatorname{Arg} z_{2})\).
  4. Express \(1 + i\sqrt{2}\) in the modulus–amplitude form.
  5. Express \(-1 - \sqrt{2}\) in modulus–amplitude form.
  6. If \(\sqrt{3} + i = r(\cos \theta + \sin \theta)\), find the value of \(\theta\).
  7. If \((\cos 2\alpha + \sin 2\alpha)(\cos 2\beta + \sin 2\beta) = \cos \theta + \sin \theta\), then find the value of \(\theta\).
  8. If the amplitude of \((z - 1)\) is \(\frac{\pi}{2}\), then find the locus of \(z\).
  9. Simplify \(i^{18} - 3i^{7} + i^{2}(1 + i^{4})(-i)^{26}\).

De Moivre's Theorem Q3

  1. Find the value of (i) \((1 + i)^{16}\) (ii) \(\left(1 + i\sqrt{3}\right)^{3}\) (iii) \((1 - i)^{8}\).
  2. Find the value of \(\left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^{5} - \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^{5}\).
  3. If A, B, C are angles of a triangle such that \(x = \cos A\), \(y = \cos B\), \(z = \cos C\), then find the value of \(xyz\).
  4. If \(x = \cos \theta\), then find the value of \(\left(x^{6} + \frac{1}{x^{6}}\right)\).
  5. Find the cube root of 8.
  6. If the cube roots of unity are \(1, \omega, \omega^{2}\), then find the roots of the equation \((x - 1)^{3} + 8 = 0\).
  7. Simplify \(\frac{(\cos \alpha + \sin \alpha)^{4}}{(\sin \beta + \cos \beta)^{8}}\).
  8. If \(\alpha, \beta\) are the roots of the equation \(x^{2} + x + 1 = 0\), then prove that \(\alpha^{4} + \beta^{4} + \alpha^{-1}\beta^{-1} = 0\).
  9. If \(1, \omega, \omega^{2}\) are the cube roots of unity, show that \((1 - \omega + \omega^{2})^{6} + (1 + \omega - \omega^{2})^{6} = 128\).
  10. If \(1, \omega, \omega^{2}\) are the cube roots of unity, then prove that \(\frac{1}{2 + \omega} + \frac{1}{1 + 2\omega} = \frac{1}{1 + \omega}\).
  11. Solve the equation \(x^{4} - 1 = 0\).

Quadratic Expressions Q4

  1. If \(\alpha, \beta\) are the roots of the equation \(ax^{2} + bx + c = 0\), then find the value of
    • \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}\)
    • \(\alpha^{3} + \beta^{3}\)
    • \(\frac{1}{\alpha} + \frac{1}{\beta}\)
    • \(\alpha^{4}\beta^{7} + \alpha^{7}\beta^{4}\)
  2. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2} + x + 1 = 0\), find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\).
  3. Find the quadratic equation whose roots are (i) \(7 \pm 2\sqrt{5}\) (ii) \(\frac{p - q}{p + q}, \frac{p + q}{p - q}\).
  4. If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 3x + 6 = 0\), find the quadratic equation whose roots are \(\alpha^{3}\) and \(\beta^{3}\).
  5. Find the quadratic equation, the sum of whose roots is 7 and the sum of the squares of the roots is 25.
  6. If the equation \(x^{2} - 15 - m(2x - 8) = 0\) has equal roots, find the value of \(m\).
  7. Find the maximum or minimum value of (i) \(2x - 7 - 5x^{2}\) (ii) \(x^{2} + 5x + 6\) where \(x \in \mathbb{R}\). Also state whether it is maximum or minimum with reason.
  8. For what values of \(x\), the expression (i) \(x^{2} - 5x + 6\) (ii) \(3x^{2} + 4x + 4\) are positive?
  9. For what values of \(x\), the expression (i) \(x^{2} - 7x + 10\) (ii) \(15 + 4x - 3x^{2}\) are negative?
  10. If \(x^{2} - 6x + 5 = 0\) and \(x^{2} - 12x + p = 0\) have a common root, then find \(p\).

Theory of Equations Q5

    • Find the monic polynomial equation of degree 3 whose roots are 2, 3, 6.
    • Form a polynomial equation with rational coefficients and whose roots are \(2 \pm \sqrt{3}, 1 \pm 2i\).
    • If \(-1, 2, \alpha\) are the roots of the equation \(2x^{3} + x^{2} - 7x - 6 = 0\), then find \(\alpha\).
    • If \(1, 1, \alpha\) are the roots of \(x^{3} - 6x^{2} + 9x - 4 = 0\), then find \(\alpha\).
  3. If \(1, -2\) and \(3\) are the roots of \(x^{3} - 2x^{2} + ax + 6 = 0\), then find \(a\).
  4. If the product of roots of \(4x^{3} + 16x^{2} - 9x - a = 0\) is 9 then find \(a\).
    • If \(\alpha, \beta, 1\) are the roots of \(x^{3} - 2x^{2} - 5x + 6 = 0\), then find \(\alpha, \beta\).
    • Solve the equation \(x^{3} - 3x^{2} - 16x + 48 = 0\), one root being 3.
  6. If \(1, 2, 3\) and \(4\) are the roots of \(x^{4} + ax^{3} + bx^{2} + cx + d = 0\), then find the values of \(a, b, c\) and \(d\).
    • Find the algebraic equation whose roots are two times the roots of \(x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 4x + 3 = 0\).
    • Find the algebraic equation whose roots are 3 times the roots of \(x^{3} + 2x^{2} - 4x + 1 = 0\).
  8. If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^{3} + 2x^{2} - 4x - 3 = 0\), find the equation whose roots are \(\frac{\alpha}{\beta}, \frac{\beta}{\gamma}, \frac{\gamma}{\alpha}\).
  9. Find the polynomial equation whose roots are the reciprocals of roots of \(x^{4} - 3x^{3} + 7x^{2} + 5x - 2 = 0\).
  10. Find the transformed equation whose roots are the negatives of the roots of \(x^{7} + 3x^{5} + x^{3} - x^{2} + 7x + 2 = 0\).
  11. Form the polynomial equation whose roots are the squares of the roots of \(x^{3} + 3x^{2} - 7x + 6 = 0\).
  12. If \(\alpha, \beta, \gamma\) are the roots of \(x^{3} + px^{2} + qx + r = 0\), then find \(\alpha^{2} + \beta^{2} + \gamma^{2}\).
  13. If \(\alpha, \beta, \gamma\) are the roots of \(x^{3} + px^{2} + qx + r = 0\), then find the value of \(\alpha^{3} + \beta^{3} + \gamma^{3}\).
  14. If \(\alpha, \beta, \gamma\) are the roots of \(x^{3} - 2x^{2} + 3x - 4 = 0\), then find \(\Sigma \alpha^{2}\beta^{2}\).
  15. Find the quotient and remainder, when \(2x^{5} - 3x^{4} + 5x^{3} - 3x^{2} + 7x - 9\) is divided by \(x^{2} - x - 3\).

Permutations Q6

  1. If \({}^{n}P_{4} = 1680\), then find \(n\).
  2. If \({}^{n}P_{r} = 1320\), find \(r\).
  3. If \({}^{n}P_{2} = 42 \cdot {}^{n}P_{5}\), find \(n\).
  4. If \({}^{n}P_{5} : {}^{n}P_{3} = 3 : 2\), find \(n\).
  5. If \({}^{n}P_{5} : {}^{n}P_{3} = 9 : 7\), find \(r\).
  6. If \({}^{n}P_{r} + 5 \cdot {}^{n}P_{r} = {}^{n+1}P_{r}\), find \(r\).
  7. Find the number of ways of arranging the letters of the word
    • INTERMEDIATE
    • MATHEMATICS
    • INDEPENDENCE
  8. Find the number of ways of arranging 7 persons around a circle.
  9. Find the number of different chains that can be prepared using 7 different coloured beads.
  10. Find the number of 4 letter words that can be formed using the letters of the word PISTON in which at least one letter is repeated.
  11. Find the number of 5 letter words that can be formed using the letters of the word "NATURE" that begin with 'N' when repetition is allowed.
  12. Find the number of palindromes with 6 digits that can be formed using the digits (i) 0, 2, 4, 6, 8 (ii) 1, 3, 5, 7, 9.

Combinations Q7

  1. If \({}^{n}C_{4} = 210\), find \(n\).
  2. If \({}^{12}C_{r} = 495\), find the possible values of \(r\).
  3. If \(10 \cdot {}^{n}C_{2} = 3 \cdot {}^{n+1}C_{3}\), find \(n\).
  4. If \({}^{n}P_{4} = 5040\) and \({}^{n}C_{4} = 210\), find \(n\) and \(r\).
  5. If \({}^{n}C_{5} = {}^{n}C_{6}\), then find \({}^{13}C_{n}\).
  6. If \({}^{12}C_{5s + 1} = {}^{12}C_{25 - 5s}\), then find \(s\). (ii) If \({}^{15}C_{2r - 1} = {}^{15}C_{2r + 4}\), then find \(r\).
  7. Find the value of \({}^{10}C_{5} + 2 \cdot {}^{10}C_{4} + {}^{10}C_{3}\).
  8. Find the number of diagonals of a polygon with 12 sides.
  9. Find the number of positive divisors of 1080.
  10. If a set A has 8 elements, find the number of subsets of A, containing at least 6 elements.
  11. Find the number of ways of forming a committee of 5 members from 6 men and 3 women.
  12. Find the number of ways of selecting 4 boys and 3 girls from a group of 8 boys and 5 girls.
  13. Find the number of ways of selecting 3 vowels and 2 consonants from the letters of the word EQUATION.
  14. To pass an examination, a student has to pass in each of the three papers. In how many ways, can a student fail?
  15. In a class there are 30 students. If each student plays a chess game with each of the other students, then find the total number of chess games played by them.

Binomial Theorem Q8

  1. Find the set E of \(x\) for which the binomial expansion (i) \((3 - 4x)^{34}\) (ii) \((2 + 3x)^{23}\) is valid.
  2. Find the number of terms in the expansion of (i) \((2x + 3y + z)^7\) (ii) \((4x - 7y)^{49} + (4x + 7y)^{49}\).
  3. Write down and simplify \(6^{\text{th}}\) term in \(\left(\frac{2x}{3} + \frac{3y}{2}\right)^5\).
  4. Find the \(4^{\text{th}}\) term from the end in the expansion of \((2a + 5b)^8\).
  5. Find the middle term in the expansion of (i) \(\left(\frac{3x}{2} - 2y\right)^{10}\) (ii) \(\left(4a + \frac{3b}{2}\right)^{11}\).
  6. If the coefficients of \((2r + 4)^{\text{th}}\) term and \((3r + 4)^{\text{th}}\) term in the expansion of \((1 + x)^{21}\) are equal, then find \(r\).
  7. If \({}^{22}C_{r}\) is the largest binomial coefficient in the expansion of \((1 + x)^{22}\), find the value of \({}^{13}C_{r}\).
  8. Find the term independent of \(x\) in \(\left(\frac{3}{\sqrt[3]{x}} + 5\sqrt{x}\right)^{25}\).
  9. Find the coefficient of \(x^{6}\) in \(\left(3x - \frac{4}{x}\right)^{10}\).
  10. Prove that \(C_{0} + 2C_{1} + 4C_{2} + 8C_{3} + \ldots + 2^{n}C_{n} = 3^{n}\).
  11. Prove that \(C_{1} + 2C_{2} + 3C_{3} + \ldots + nC_{n} = n \cdot 2^{n-1}\).
  12. If \(C_{r}\) denotes \({}^{n}C_{r}\) then prove that \(aC_{0} + (a + d)C_{1} + (a + 2d)C_{2} + \ldots + (a + nd)C_{n} = 2^{n-1}(2a + nd)\).
  13. If \((1 + x + x^{2})^{n} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2n}x^{2n}\) then find the value of \(a_{0} + a_{2} + a_{4} + \ldots + a_{2n}\).
  14. Find the numerically greatest term in the expansion of \((3 + 2a)^{15}\) when \(a = 5/2\).
  15. Find the numerically greatest term in the expansion of \((3x + 5y)^{12}\) when \(x = 1/2\), \(y = 4/3\).

Measures of Dispersion Q9

  1. Find the mean deviation from the mean of the following discrete data: 3, 6, 10, 4, 9, 10.
  2. Find the mean deviation from the mean of the discrete data 6, 7, 10, 12, 13, 4, 12, 16.
  3. Compute the mean deviation about the median of the data 4, 6, 9, 3, 10, 13, 2.
  4. Compute the mean deviation about the median of the data 4, 6, 7, 10, 12, 12, 13, 16.
  5. Find the variance and standard deviation of the data 5, 12, 3, 18, 6, 8, 2, 10.
  6. The variance of 20 observations is 5. If each observation is multiplied by 2, then find the new variance of the resulting observations.
  7. The coefficient of variation of two distributions are 60 and 70 and their standard deviations are 21 and 16 respectively. Find their arithmetic means.
  8. If each of the observations \(x_{1}, x_{2}, \ldots, x_{n}\) is increased by \(k\), where \(k\) is positive or negative number, then show that the variance remains unchanged.

Random Variable & Distributions Q10

  1. Find the constant \(c\) so that \(f(x) = c\left(\frac{2}{3}\right)^{x}, x = 1, 2, 3, \ldots, \infty\) is the probability distribution function of a discrete random variable X.
  2. The range of a random variable X is \(\{1, 2, 3, \ldots, \infty\}\) and \(P(X = k) = \frac{e^{-c}c^{k}}{k!}; k = 1, 2, 3, \ldots, \infty\) find \(c\).
  3. If X is a random variable with the probability distribution \(P(X = k) = \frac{(k + 1)C}{2^{k}}\) \((k = 0, 1, 2, \ldots)\), then find \(C\).
  4. The mean and variance of a binomial distribution are 4 and 3 respectively. Fix the distribution and \(P(X \ge 1)\).
  5. For a binomial distribution with mean 6 and variance 2, find the first two terms of the distribution.
  6. If the mean and variance of a binomial variable X are 2.4 and 1.44 respectively, find \(n\).
  7. The probability that a person chosen at random is left handed (in hand writing) is 0.1. What is the probability that in a group of 10 people, there is one who is left handed?
  8. On an average, rain falls on 12 days in every 30 days. Find the probability that rain will fall on just 3 days of a given week.
  9. A Poisson variable satisfies \(P(X = 1) = P(X = 2)\), find \(P(X = 5)\).
  10. The number of persons joining a cinema ticket counter in a minute has Poisson distribution with parameter 6. Find the probability that (i) no one joins the queue in a particular minute (ii) two or more persons join the queue in a minute.

📐 Mathematics IIB

Essay Answer Questions (7 Marks)

Circle Q.No: 18

  1. Find the equation of the circle passing through the points:
    • \((3, 4), (3, 2), (1, 4)\)
    • \((1, 1), (2, -1), (3, 2)\)
    • \((5, 7), (8, 1), (1, 3)\)
    • \((1, 2), (3, -4), (5, 6)\)
    • Find the equation of the circle which passes through the vertices of the triangle formed by \(x + y + 1 = 0, 3x + y - 5 = 0, 2x + y - 5 = 0\).
  2. Show that the points are concyclic:
    • \((1, 1), (-6, 0), (-2, 2)\) and \((-2, -8)\)
    • \((1, 2), (3, -4), (5, -6)\) and \((19, 8)\)
    • \((9, 1), (7, 9), (-2, 12)\) and \((6, 10)\)
    • If \((2, 0), (0, 1), (4, 5)\) and \((0, c)\) are concyclic, then find the value of \(c\).
    • Find the equation of a circle which passes through the points \((4, 1), (6, 5)\) and whose centre lies on \(4x + 3y - 24 = 0\).
    • Find the equation of a circle which passes through the points \((4, 1), (6, 5)\) and whose centre lies on \(4x + y - 16 = 0\).
    • Find the equation of a circle which passes through the points \((2, -3), (-4, 5)\) and whose centre lies on \(4x + 3y + 1 = 0\).
    • Find the equation of the circle whose centre lies on \(x\)-axis and passing through \((-2, 3)\) and \((4, 5)\).

Circle Q.No: 19

    • Show that the circles \(x^{2} + y^{2} - 6x - 2y + 1 = 0\) and \(x^{2} + y^{2} + 2x - 8y + 13 = 0\) touch each other. Also find the point of contact and common tangent at this point of contact. (Externally)
    • Show that the circles \(x^{2} + y^{2} - 4x - 6y - 12 = 0\) and \(x^{2} + y^{2} + 6x + 18y + 26 = 0\) touch each other. Also find the point of contact and common tangent at this point of contact. (Externally)
    • Show that the circles \(x^{2} + y^{2} - 6x - 9y + 13 = 0\) and \(x^{2} + y^{2} - 2x - 16y = 0\) touch each other. Also find the point of contact and common tangent at this point of contact. (Internally)
    • Find the equation of the circle which touches the circle \(x^{2} + y^{2} - 2x - 4y - 20 = 0\) externally at \((5, 5)\) with radius 5.
    • Find the equation of the circle which touches the circle \(x^{2} + y^{2} - 4x + 6y - 12 = 0\) internally at \((-1, 1)\) with radius 2.
    • Find the transverse common tangents of the circles \(x^{2} + y^{2} - 4x - 10y + 28 = 0\) and \(x^{2} + y^{2} + 4x - 6y + 4 = 0\).
    • Find the equations of direct common tangents of the circles \(x^{2} + y^{2} + 22x - 4y - 100 = 0\) and \(x^{2} + y^{2} - 22x + 4y + 100 = 0\).
    • Show that four common tangents can be drawn for the circles given by \(x^{2} + y^{2} - 14x + 6y + 33 = 0\) and \(x^{2} + y^{2} + 30x - 2y + 1 = 0\) and find the internal and external centers of similitude.
    • Find the equation of all possible common tangents of the circles \(x^{2} + y^{2} - 2x - 6y + 6 = 0\) and \(x^{2} + y^{2} = 1\).
    • Prove that the combined equation of pair of tangents drawn from an external point \(P(x_{1}, y_{1})\) to the circle \(S = 0\) is \(S_{1}^{2} = SS_{11}\).
    • Find the pair of tangents drawn from \((1, 3)\) to the circle \(x^{2} + y^{2} - 2x + 4y - 11 = 0\) and also find the angle between them.
  5. Find the equations of the circles which touch \(2x - 3y + 1 = 0\) at \((1, 1)\) and having radius \(\sqrt{13}\).
  6. If \(\theta_{1}, \theta_{2}\) are the angles of inclination of tangents through a point P to the circle \(x^{2} + y^{2} = a^{2}\), then find the locus of P when \(\cot \theta_{1} + \cot \theta_{2} = k\).
  7. Show that the poles of the tangents to the circle \(x^{2} + y^{2} = a^{2}\) with respect to the circle \((x + a)^{2} + y^{2} = 2a^{2}\) lies on \(y^{2} + 4ax = 0\).

Parabola Q.No: 20

  1. Derive the equation of the parabola \(y^{2} = 4ax\) in the standard form.
  2. Find the coordinates of the vertex and focus, the equation of the directrix and axis of the parabola.
    • \(y^{2} + 4x + 4y - 3 = 0\)
    • \(x^{2} - 2x + 4y - 3 = 0\)
    • Find the equation of the parabola whose axis is parallel to Y-axis and which passes through the points \((4, 5), (-2, 11)\) and \((-4, 21)\).
    • Find the equation of the parabola whose axis is parallel to X-axis and which passes through the points \((-2, 1), (1, 2)\) and \((-1, 3)\).
  4. Find the equation of the parabola whose focus is \((-2, 3)\) and directrix is the line \(2x + 3y - 4 = 0\). Also find the length of the latus rectum and the equation of the axis of the parabola.
    • Show that the equations of common tangents to the circle \(x^{2} + y^{2} = 2a^{2}\) and the parabola \(y^{2} = 8ax\) are \(y = \pm (x + 2a)\).
    • Show that the common tangents to the circle \(2x^{2} + 2y^{2} = a^{2}\) and the parabola \(y^{2} = 4ax\) intersect at the focus of the parabola \(y^{2} = -4ax\).
    • If \(y_{1}, y_{2}, y_{3}\) are the y-coordinates of the vertices of the triangle inscribed in the parabola \(y^{2} = 4ax\), then show that the area of the triangle is \(\frac{1}{8a}|(y_{1} - y_{2})(y_{2} - y_{3})(y_{3} - y_{1})|\) square units.
    • Prove that the area of the triangle formed by the tangents at \((x_{1}, y_{1}), (x_{2}, y_{2})\) and \((x_{3}, y_{3})\) to the parabola \(y^{2} = 4ax (a > 0)\) is \(\frac{1}{16a}|(y_{1} - y_{2})(y_{2} - y_{3})(y_{3} - y_{1})|\) sq. units.
    • The normal at a point \(t_{1}\) on \(y^{2} = 4ax\) meets the parabola again in the point \(t_{2}\), prove that \(t_{1}t_{2} + t_{1}^{2} + 2 = 0\).
    • If a normal chord at point \(t\) on the parabola \(y^{2} = 4ax\) subtends a right angle at vertex, then prove that \(t = \pm \sqrt{2}\).
  8. Show that the equation common tangent to the parabola \(y^{2} = 4ax\) and \(x^{2} = 4by\) is \(xa^{3} + yb^{3} + a^{3}b^{3} = 0\).
  9. Prove that the two parabolas \(y^{2} = 4ax\) and \(x^{2} = 4by\) intersect (other than the origin) at an angle of \(\theta = \tan^{-1}\left|\frac{3a^{1/3}b^{1/3}}{2(a^{2/3} + b^{2/3})}\right|\).
  10. Show that the locus of point of intersection of perpendicular tangents to the parabola \(y^{2} = 4ax\) is the directrix \(x + a = 0\).
  11. From an external point P tangents are drawn to the parabola \(y^{2} = 4ax\) and these tangents make angles \(\theta_{1}, \theta_{2}\) with its axis, such that \(\cot \theta_{1} + \cot \theta_{2}\) is a constant \(d\). Then show that all such P lie on a horizontal line.

Integration Q.No: 21

  1. Evaluate \(\int \frac{x + 1}{x^{2} + 3x + 12} dx\).
    • Evaluate \(\int \frac{2x + 5}{\sqrt{x^{2} - 2x + 10}} dx\)
    • Evaluate \(\int \frac{x + 1}{\sqrt{x^{2} - x + 1}} dx\)
    • Evaluate \(\int \sqrt{\frac{5 - x}{x - 2}} dx\)
    • \(\int (6x + 5)\sqrt{6 - 2x^{2} + x} dx\)
    • \(\int (3x - 2)\sqrt{2x^{2} - x + 1} dx\)
    • \(\int x\sqrt{1 + x - x^{2}} dx\)
    • \(\int \frac{1}{(1 + x)\sqrt{3 + 2x - x^{2}}} dx\)
    • \(\int \frac{1}{(x + 1)\sqrt{2x^{2} + 3x + 1}} dx\)
    • \(\int \frac{1}{(1 - x)\sqrt{3 - 2x - x^{2}}} dx\)
    • \(\int \frac{1}{1 + \sin x + \cos x} dx\)
    • \(\int \frac{1}{3\cos x + 4\sin x + 6} dx\)
    • \(\int \frac{1}{3\sin x + 4\cos x + 5} dx\)
    • \(\int \frac{dx}{4\cos x + 3\sin x}\)
    • \(\int \frac{1}{4 + 5\sin x} dx\)
    • \(\int \frac{dx}{5 + 4\cos x}\)
    • \(\int \frac{dx}{\sin x + \sqrt{3}\cos x}\)
    • \(\int \frac{1}{5 + 4\cos 2x} dx\)
    • \(\int \frac{1}{2 - 3\cos 2x} dx\)
    • Evaluate \(\int \frac{9\cos x - \sin x}{4\sin x + 5\cos x} dx\)
    • Evaluate \(\int \frac{2\cos x + 3\sin x}{4\cos x + 5\sin x} dx\)
    • Evaluate \(\int \frac{\cos x + 3\sin x + 7}{\cos x + \sin x + 1} dx\)
    • Evaluate \(\int \frac{2\sin x + 3\cos x + 4}{\sin x + 4\cos x + 5} dx\)

Integration (Reduction Formulae) Q.No: 22

    • If \(I_{n} = \int \sin^{n}x \, dx\), then show that \(I_{n} = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}I_{n-2}\) and hence find \(I_{5}, I_{4}\).
    • If \(I_{n} = \int \cos^{n}x \, dx\), then show that \(I_{n} = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n}I_{n-2}\) and hence find \(I_{4}\).
    • If \(I_{n} = \int \sec^{n}x \, dx\), then prove that \(I_{n} = \frac{1}{n-1}\sec^{n-2}x \cdot \tan x + \frac{n-2}{n-1}I_{n-2}\); find \(\int \sec^{5}x \, dx\).
    • If \(I_{n} = \int \csc^{n}x \, dx\), then prove that \(I_{n} = -\frac{1}{n-1}\csc^{n-2}x \cdot \cot x + \frac{n-2}{n-1}I_{n-2}\); find \(\int \csc^{5}x \, dx\).
    • Find the reduction formula for \(\int \tan^{n}x \, dx\) and hence find \(\int \tan^{6}x \, dx\) and \(\int \tan^{5}x \, dx\).
    • Find the reduction formula for \(\int \cot^{n}x \, dx\) and hence find \(\int \cot^{4}x \, dx\).
    • \(\int \frac{2x + 3}{(x + 3)(x^{2} + 4)} dx\)
    • \(\int \frac{x + 3}{(x - 1)(x - 2)^{2}} dx\)
    • \(\int \frac{\sin x \cos x}{\cos^{2}x + 3\cos x + 2} dx\)
    • \(\int \frac{1}{(1 - x)(4 + x^{2})} dx\)
    • \(\int \frac{1}{x(x + 1)(x + 2)^{2}} dx\)
    • \(\int e^{ax} \cos(bx + c) \, dx\)
    • \(\int e^{ax} \sin(bx + c) \, dx\)
    • \(\int x \cos^{-1}x \, dx\)
    • \(\int x \sin^{-1}x \, dx\)
    • \(\int x \tan^{-1}x \, dx\)
    • \(\int \sqrt{a^{2} - x^{2}} \, dx\)

Definite Integrals Q.No: 23

  1. Evaluate \(\int \frac{\sin x + \cos x}{9 + 16\sin 2x} dx\).
  2. Evaluate \(\int_{0}^{1} \frac{\log(1 + x)}{1 + x^{2}} dx\).
    • Evaluate \(\int_{0}^{\pi} \frac{x}{1 + \sin x} dx\)
    • Evaluate \(\int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} dx\)
    • Evaluate \(\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^{2}x} dx\)
    • Evaluate \(\int_{0}^{\pi} \frac{x \sin^{3}x}{1 + \cos^{2}x} dx\)
    • Show that \(\int_{0}^{\pi/2} \frac{x}{\sin x + \cos x} dx = \frac{\pi}{2\sqrt{2}}\log(\sqrt{2} + 1)\)
    • Evaluate \(\int_{0}^{\pi/2} \frac{\sin^{2}x}{\cos x + \sin x} dx\)
  6. Find the value of \(\int_{-\pi/2}^{\pi/2} x \cdot \sin^{7}x \cos^{6}x \, dx\).
    • Find the area of the region bounded by the parabolas \(y^{2} = 4x\) and \(x^{2} = 4y\).
    • Find the area between the curves \(y^{2} = 4ax\) and \(x^{2} = 4by\).
    • Find the area enclosed between \(y = x^{2} - 5x\) and \(y = 4 - 2x\).
    • Find the area enclosed by the curves \(y = 3x\) and \(y = 6x - x^{2}\).
    • Find the area enclosed between \(y^{2} = 4x\) and \(y^{2} = 4(4 - x)\).
    • Find the area of the region enclosed between the curves \(y = 2 - x^{2}\) and \(y = x^{2}\).
    • Find the area of the region enclosed between the curves \(y = x^{2}\) and \(y = \sqrt{x}\).
    • Show that the area of the region bounded by \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) is \(\pi ab\). Also deduce the area of the circle \(x^{2} + y^{2} = a^{2}\).
    • Let AOB be the positive quadrant of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with OA = \(a\), OB = \(b\). Then show that the area bounded between the chord AB and the arc AB of the ellipse is \((\pi - 2)ab/4\).

Differential Equations Q.No: 24

  1. Solve the differential equation
    • \((2x + 2y - 1)dy = 0\)
    • \((2x + 2y + 3)dy/dx = (x + y + 1)\)
    • \(\frac{dy}{dx} = \frac{x - y + 3}{2x - 2y + 5}\)
    • \(\frac{dy}{dx} = \frac{4x + 6y + 5}{3y + 2x + 4}\) (Non homogeneous D.E case (ii))
    • \(\frac{dy}{dx} = \frac{2x + y + 3}{2y + x + 1}\)
    • \(\frac{dy}{dx} = \frac{3y - 7x + 7}{3x - 7y - 3}\)
    • \((x - y)dy = (x + y + 1)dy\) (Non homogeneous D.E case (iii))
    • Solve \((x^{3} - 3xy^{2})dx + (3x^{2}y - y^{3})dy = 0\)
    • \((x^{2} + y^{2})dx = 2xy \, dy\)
    • \((x^{2}y - 2xy^{2})dx + (x^{3} - 3x^{2}y)dy = 0\)
    • \((x^{2} - y^{2})dx - xy \, dy = 0\)
  4. Give the solution of \(x \sin^{2}\left(\frac{y}{x}\right) dx = y \, dx - x \, dy\) which passes through the point \(\left(1, \frac{\pi}{4}\right)\).
  5. Find the solution of the differential equation \(x(x - 2)\frac{dy}{dx} - 2(x - 1)y = x^{3}(x - 2)\) which satisfies the condition that \(y = 9\) when \(x = 3\).

Short Answer Questions (4 Marks) – IIB

Circle Q.No: 11

    • Find the length of the chord intercepted by the circle \(x^{2} + y^{2} - x + 3y - 22 = 0\) on the line \(y = x - 3\).
    • Find the length of the chord intercepted by the circle \(x^{2} + y^{2} - 8x - 2y - 8 = 0\) on the line \(x + y + 1 = 0\).
    • Find the equation of the circle with centre \((-2, 3)\) cutting a chord of length 2 units on \(3x + 4y + 4 = 0\).
    • The line \(y = mx + c\) and the circle \(x^{2} + y^{2} = a^{2}\) intersect at A and B, if AB = \(2\lambda\) then show that \(c^{2} = (1 + m^{2})(a^{2} - \lambda^{2})\).
    • Find the midpoint of the chord intercepted by \(x^{2} + y^{2} - 2x - 10y + 1 = 0\) on the line \(x - 2y + 7 = 0\).
    • Show that \(x + y + 1 = 0\) touches the circle \(x^{2} + y^{2} - 3x + 7y + 14 = 0\) and find its point of contact.
    • Show that the tangent at \((-1, 2)\) of the circle \(x^{2} + y^{2} - 4x - 8y + 7 = 0\) touches the circle \(x^{2} + y^{2} + 4x + 6y = 0\) also find its point of contact.
    • Find the pole of \(3x + 4y - 45 = 0\) with respect to \(x^{2} + y^{2} - 6x - 8y + 5 = 0\).
    • Find the value of \(k\) if \(2x + ky - 8 = 0\) and \(x + y - 5 = 0\) are conjugate lines with respect to the circle \(x^{2} + y^{2} - 2x - 2y - 1 = 0\).
    • Find the value of \(k\) if \(kx + 3y - 1 = 0\) and \(2x + y + 5 = 0\) are conjugate lines with respect to the circle \(x^{2} + y^{2} - 2x - 4y - 4 = 0\).
    • Find the equations of the tangents to the circle \(x^{2} + y^{2} - 4x + 6y - 12 = 0\) which are parallel to \(x + y - 8 = 0\).
    • Find the equations of tangents to the circle \(x^{2} + y^{2} + 2x - 2y - 3 = 0\) which are perpendicular to \(3x - y + 4 = 0\).
  5. Find the area of the triangle formed by the normal drawn at \((3, -4)\) on the circle \(x^{2} + y^{2} - 22x - 4y + 25 = 0\) with the coordinate axes.
  6. Find the condition that the tangents drawn from exterior point \((0, 0)\) to \(x^{2} + y^{2} + 2gx + 2fy + c = 0\) are perpendicular to each other.
  7. If the abscissa of points A, B are the roots of the equation \(x^{2} + 2ax - b^{2} = 0\) and ordinates of A, B are the roots of \(y^{2} + 2py - q^{2} = 0\), then find the equation of a circle for which AB is a diameter.
  8. If a point P is moving such that the lengths of tangents drawn from P to the circles \(x^{2} + y^{2} - 4x - 6y - 12 = 0\) and \(x^{2} + y^{2} + 6x + 18y + 26 = 0\) are in the ratio 2 : 3, then find the equation of the locus of P.
  9. Find the inverse point of \((-2, 3)\) with respect to the circle \(x^{2} + y^{2} - 4x - 6y + 9 = 0\).
  10. Find the equation to the pair of tangents drawn from \((0, 0)\) to \(x^{2} + y^{2} + 10x + 10y + 40 = 0\).

System of Circles Q.No: 12

  1. If the angle between the circles \(x^{2} + y^{2} - 12x - 6y + 41 = 0\) and \(x^{2} + y^{2} + kx + 6y - 59 = 0\) is \(45^{\circ}\), find \(k\).
    • Find the equation of the circle passing through the points of intersection of circles \(x^{2} + y^{2} - 8x - 6y + 21 = 0\), \(x^{2} + y^{2} - 2x - 15 = 0\) and the point \((1, 2)\).
    • If the straight line \(2x + 3y = 1\) intersects the circle \(x^{2} + y^{2} = 4\) at the points A and B, then find the equation of the circle having AB as diameter.
    • Find the equation of the circle whose diameter is the common chord of the circles \(x^{2} + y^{2} + 2x + 3y + 1 = 0\), \(x^{2} + y^{2} + 4x + 3y + 2 = 0\).
    • Find the equation and length of the common chord of the two circles \(x^{2} + y^{2} + 3x + 5y + 4 = 0\), \(x^{2} + y^{2} + 5x + 3y + 4 = 0\).
    • Show that the circles \(x^{2} + y^{2} - 2x - 4y - 20 = 0\), \(x^{2} + y^{2} + 6x + 2y - 90 = 0\) touch each other internally. Find their point of contact.
    • Show that the circles \(x^{2} + y^{2} + 2ax + c = 0\) and \(x^{2} + y^{2} + 2by + c = 0\) touch each other if \(\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{c}\).
    • Prove that the radical axis of the circles \(x^{2} + y^{2} + 2gx + 2fy + c = 0\) and \(x^{2} + y^{2} + 2g'x + 2f'y + c' = 0\) is the diameter of the latter circle if \(2g'(g - g') + 2f'(f - f') = c - c'\).
  4. Find the radical centre of the circles \(x^{2} + y^{2} - 4x - 6y + 5 = 0\), \(x^{2} + y^{2} - 2x - 4y - 1 = 0\), \(x^{2} + y^{2} - 6x - 2y = 0\).
    • Find the equation of the circle which cuts orthogonally the circle \(x^{2} + y^{2} - 4x + 2y - 7 = 0\) and having centre at \((2, 3)\).
    • Find the equation of the circle which passes through the origin, having its centre on the line \(x + y = 4\) and intersects the circle \(x^{2} + y^{2} - 4x + 2y + 4 = 0\) orthogonally.
    • Find the equation of the circle which passes through the point \((0, -3)\) and intersects the circles given by the equations \(x^{2} + y^{2} - 6x + 3y + 5 = 0\) and \(x^{2} + y^{2} - x - 7y = 0\) orthogonally.
    • Find the equation of the circle which is orthogonal to \(x^{2} + y^{2} + 2x + 17y + 4 = 0\), \(x^{2} + y^{2} + 7x + 6y + 11 = 0\) and \(x^{2} + y^{2} - x + 22y + 3 = 0\).

Ellipse Q.No: 13

  1. Find the equation of the ellipse with focus at \((1, -1)\), \(e = 2/3\) and directrix as \(x + y + 2 = 0\).
  2. Find the length of major axis, minor axis, latus rectum, eccentricity, foci, equations of directrices of the ellipse
    • \(9x^{2} + 16y^{2} = 144\)
    • \(4x^{2} + y^{2} - 8x + 2y + 1 = 0\)
    • \(9x^{2} + 16y^{2} - 36x + 32y - 92 = 0\)
    • Find the equation of the ellipse in the standard form such that the distance between the foci is 8 and the distance between the directrices is 32.
    • Find the equation of the ellipse in the standard form whose distance between foci is 2 and the length of latus rectum is \(15/2\).
    • Find the equation of the ellipse referred to its major, minor axes as the coordinate axes X, Y respectively with latus rectum of length 4, whose distance between foci is \(4/\sqrt{2}\).
  4. If \(P(x, y)\) is any point on the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with foci S and S' then prove that \(SP + S'P\) is a constant.
  5. The distance of a point on the ellipse \(x^{2} + 3y^{2} = 6\) from its centre is equal to 2. Find the eccentric angles.
  6. Show that the points of intersection of the perpendicular tangents to an ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) lies on a circle.
  7. Show that the locus of the feet of the perpendiculars drawn from either of the foci to any tangent to the ellipse is the auxiliary circle.

Ellipse Q.No: 14

  1. Find the equations of the tangents to the ellipse \(2x^{2} + y^{2} = 8\) which are
    • parallel to \(x - 2y - 4 = 0\)
    • perpendicular to \(x + y + 2 = 0\)
    • making angle \(45^{\circ}\) with x-axis.
  2. Find the equations of tangents to \(9x^{2} + 16y^{2} = 144\), which makes equal intercepts on the coordinate axes.
  3. Find the equations of the tangent and normal to the ellipse \(9x^{2} + 16y^{2} = 144\) at the end of latus rectum in the first quadrant.
  4. Find the equation of tangent and normal to the ellipse \(2x^{2} + 3y^{2} = 11\) at the point whose ordinate is one.
  5. Find the equation of tangent and normal to the ellipse \(x^{2} + 2y^{2} - 4x + 12y + 14 = 0\) at \((2, -1)\).
  6. Find the value of \(k\) if \(4x + y + k = 0\) is a tangent to the ellipse \(x^{2} + 3y^{2} = 3\).
  7. Find the equation of the ellipse in the standard form passing through \((-2, 2)\) and \((3, -1)\).
  8. If the normal at one end of a latus rectum of the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) passes through one end of the minor axis, then show that \(e^{4} + e^{2} = 1\).
  9. Find the condition for the line \(x \cos \alpha + y \sin \alpha = p\) to be a tangent to the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
  10. If the line \(y = mx + c\) touches the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), prove that \(c^{2} = a^{2}m^{2} + b^{2}\).
  11. The tangent and normal to the ellipse \(x^{2} + 4y^{2} = 4\) at a point \(P(\theta)\) on it meets the major axes in Q and R respectively. If \(0 < \theta < \frac{\pi}{2}\) and QR = 2, then show that \(\theta = \cos^{-1}(2/3)\).

Hyperbola Q.No: 15

  1. One focus of a hyperbola is \((1, -3)\) and the corresponding directrix is \(y = 2\). Find the equation of the hyperbola if its eccentricity is \(\frac{3}{2}\).
  2. Find the eccentricity, foci, equations of directrices, length of latus rectum of the hyperbola
    • \(x^{2} - 4y^{2} = 4\)
    • \(16y^{2} - 9x^{2} = 144\)
    • \(9x^{2} - 16y^{2} + 72x - 32y - 16 = 0\)
    • \(5x^{2} - 4y^{2} + 20x + 8y = 4\)
    • Find the equations of tangents to the hyperbola \(x^{2} - 4y^{2} = 4\) which are (i) parallel (ii) perpendicular to the line \(x + 2y = 0\).
    • Find the equations of tangents to the hyperbola \(3x^{2} - 4y^{2} = 12\) which are (i) parallel (ii) perpendicular to the line \(y = x - 7\).
    • Find the equation of the hyperbola whose foci are \((4, 2), (8, 2)\) and eccentricity is 2.
    • If \(P(x, y)\) is any point on the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) with foci S and S' then prove that \(|SP - S'P| = 2a\) is a constant.
  5. Show that the angle between the two asymptotes of a hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) is \(2\tan^{-1}\left(\frac{b}{a}\right)\) or \(2\sec^{-1}(e)\).
  6. Show that the condition for the line \(lx + my + n = 0\) to be a tangent to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) is \(a^{2}l^{2} - b^{2}m^{2} = n^{2}\).
  7. Show that the equation of normal at \(P(\theta)\) to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) is \(\frac{ax}{\sec \theta} + \frac{by}{\tan \theta} = a^{2} + b^{2}\).
  8. Prove that the point of intersection of two perpendicular tangents to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) lies on the circle \(x^{2} + y^{2} = a^{2} - b^{2}\).
  9. Tangents to the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) make angles \(\theta_{1}, \theta_{2}\) with transverse axis of a hyperbola. Show that the point of intersection of these tangents lies on the curve \(2xy = k(x^{2} - a^{2})\) when \(\tan \theta_{1} + \tan \theta_{2} = k\).
  10. Prove that the product of the perpendicular distances from any point on a hyperbola to its asymptotes is constant.

Definite Integrals Q.No: 16

    • Evaluate \(\int_{0}^{\pi/2} \frac{\cos^{2}x}{\sin^{2}x + \cos^{2}x} dx\)
    • Evaluate \(\int_{0}^{\pi/2} \frac{\sin^{5}x}{\sin^{5}x + \cos^{5}x} dx\)
    • Evaluate \(\int_{0}^{\pi/2} \frac{\sin^{2}x - \cos^{2}x}{\sin^{3}x + \cos^{3}x} dx\)
    • Evaluate \(\int_{0}^{\pi/2} \frac{a \sin x + b \cos x}{\sin x + \cos x} dx\)
    • Evaluate \(\int_{\pi/6}^{\pi/3} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx\)
    • Find \(\int_{-1/2}^{2} \frac{\cos x}{1 + e^{x}} dx\)
    • Evaluate \(\int_{0}^{7} \frac{7 - x}{x - 3} dx\)
    • Evaluate \(\int_{4}^{9} \frac{dx}{\sqrt{(9 - x)(x - 4)}}\)
    • Evaluate \(\int_{0}^{6} \frac{\sqrt{(x - 2)(6 - x)}}{2} dx\)
    • Evaluate \(\int_{0}^{6} \sqrt{(x - a)(b - x)} dx\)
    • Evaluate \(\int_{0}^{\pi/2} \frac{dx}{4 + 5\cos x}\)
    • Evaluate \(\int_{0}^{\pi} \frac{dx}{3 + 2\cos x}\)
  5. Evaluate \(\int_{0}^{1} x \tan^{-1}x \, dx\)
  6. Evaluate \(\int_{0}^{1} \sin^{-1}\left(\frac{2x}{1 + x^{2}}\right) dx\)
  7. Obtain a reduction formula for \(\int_{0}^{\pi/2} \sin^{n}x \, dx\).
  8. Evaluate \(\int_{0}^{1} (16 - x^{2})^{3/2} dx\)
    • Evaluate \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{i^{3}}{i^{4} + n^{4}}\)
    • Evaluate \(\lim_{n \to \infty} \frac{\sqrt{n + 1} + \sqrt{n + 2} + \ldots + \sqrt{n + n}}{n\sqrt{n}}\)
    • Evaluate \(\lim_{n \to \infty} \left[\frac{1}{n + 1} + \frac{1}{n + 2} + \ldots + \frac{1}{6n}\right]\)

Differential Equations Q.No: 17

  1. Linear differential equations in \(y\):
    • \(\frac{dy}{dx} + y \tan x = \sin x\)
    • Solve: \(\frac{dy}{dx} - y \tan x = e^{x} \sec x\)
    • \(\frac{dy}{dx} + y \sec x = \tan x\)
    • Solve: \(\frac{dy}{dx} + y \tan x = \cos^{3}x\)
    • Solve: \(\frac{dy}{dx} + \frac{4x}{1 + x^{2}} y = \frac{1}{(1 + x^{2})^{2}}\)
    • Solve: \(\cos x \frac{dy}{dx} + y \sin x = \sec^{2}x\)
    • Solve: \((1 + x^{2}) \frac{dy}{dx} + y = e^{\tan^{-1}x}\)
    • Solve: \((1 + x^{2}) \frac{dy}{dx} + y = \tan^{-1}x\)
    • Solve: \((1 + x^{2}) \frac{dy}{dx} + 2xy - 4x^{2} = 0\)
    • Solve: \(x \log x \frac{dy}{dx} + y = 2 \log x\)
  3. Linear differential equations in \(x\):
    • Solve: \((x + y + 1)\frac{dy}{dx} = 1\)
    • Solve: \((1 + y^{2})dx = (\tan^{-1}y - x)dy\)
  4. Variables separable D.E:
    • Solve: \(\frac{dy}{dx} = e^{x - y} + x^{2}e^{y}\)
    • Solve: \((xy^{2} + x)dx + (yx^{2} + y)dy = 0\)
    • Solve: \(\frac{dy}{dx} + \frac{y^{2} + y + 1}{x^{2} + x + 1} = 0\)
    • Solve: \((e^{x} + 1)y \, dy + (y + 1)dx = 0\)
    • Solve: \(y - x \frac{dy}{dx} = 5(y^{2} + \frac{dy}{dx})\)
    • Solve: \(\frac{dy}{dx} = \frac{x(2\log x + 1)}{\sin y + y \cos y}\)
    • Solve: \(\sqrt{1 + x^{2}}dx + \sqrt{1 + y^{2}}dy = 0\)
  5. Variables separable D.E (substitution):
    • Solve: \(\sin^{-1}\left(\frac{dy}{dx}\right) = x + y\)
    • Solve: \(\frac{dy}{dx} - x \tan(y - x) = 1\)
    • Solve: \(\frac{dy}{dx} + 1 = e^{x + y}\)
  6. Homogeneous D.E:
    • Solve: \((x^{2} - y^{2})dx - xy \, dy = 0\)
    • Solve: \((x^{2} - y^{2})\frac{dy}{dx} = xy\)
    • Solve: \(\frac{dy}{dx} = \frac{x - y}{x + y}\)
    • Solve: \(\frac{dy}{dx} = \frac{xy + y}{xy + x}\)
    • Solve: \((2x - y)dy = (2y - x)dx\)
    • Solve: \(x \, dy = (y + x \cos^{2}(y/x))dx\)
  7. Non-homogeneous D.E:
    • \(\frac{dy}{dx} = \frac{2x - y + 1}{x + 2y - 3}\) (case (ii))
    • Solve: \(\frac{dy}{dx} = \frac{x - y + 3}{2x - 2y + 5}\) (case (iii))

Very Short Answer Questions (2 Marks) – IIB

Circle Q1

  1. If \(ax^{2} + bxy + 3y^{2} - 5x + 2y - 3 = 0\) represents a circle, find the values of \(a\) and \(b\). Also find its radius and centre.
  2. Find \(a\) if \(2x^{2} + ay^{2} - 3x + 2y - 1 = 0\) represents a circle and also find its radius.
  3. If \(x^{2} + y^{2} + 2gx + 2fy - 12 = 0\) represents a circle with centre \((2, 3)\), find \(g, f\) and its radius.
  4. If the circle \(x^{2} + y^{2} + ax + by - 12 = 0\) has centre at \((2, 3)\) find \(a, b\) and also the radius of the circle.
  5. If \(x^{2} + y^{2} - 4x + 6y + c = 0\) represents a circle with radius 6, then find the value of \(c\).
  6. Find the centre and radius of the circle \(x^{2} + y^{2} + 6x + 8y - 96 = 0\).
  7. Find the centre and radius of the circle \(\sqrt{1 + m^{2}}(x^{2} + y^{2}) - 2cx - 2my = 0\) \((c > 0)\).
  8. Find the equation of a circle passing through \((2, -1)\) and having centre at \((2, 3)\).
  9. Find the equation of the circle with \((-4, 3), (3, -4)\) as ends of a diameter.
  10. Show that A(3, -1) lies on the circle \(x^{2} + y^{2} - 2x + 4y = 0\). Also find the other end of the diameter through A.

Circle Q2

  1. Find the equation of the circle which is concentric with \(x^{2} + y^{2} - 6x - 4y - 12 = 0\) and passing through \((-2, 14)\).
  2. Find the power of the point \((2, 3)\) with respect to the circle \(x^{2} + y^{2} - 2x + 8y - 23 = 0\).
  3. Obtain the parametric equations of the circle \((x - 3)^{2} + (y - 4)^{2} = 8^{2}\).
  4. If \(x^{2} + y^{2} - 6x + 4y - 12 = 0\) represents a circle, then find the parametric equations of the circle.
  5. Obtain the parametric equations of the circle represented by \(x^{2} + y^{2} = 4\).
  6. Find the length of the tangent from \((1, 3)\) to the circle \(x^{2} + y^{2} - 2x + 4y - 11 = 0\).
  7. If the length of the tangent from \((2, 5)\) to the circle \(x^{2} + y^{2} - 5x + 4y + k = 0\) is \(\sqrt{37}\), then find \(k\).
  8. State the necessary and sufficient condition for \(lx + my + n = 0\) to be a normal to the circle \(x^{2} + y^{2} + 2gx + 2fy + c = 0\).
  9. Find the equation of the polar of \((1, -2)\) with respect to the circle \(x^{2} + y^{2} - 10x - 10y + 25 = 0\).
  10. Show that \((4, -2)\) and \((3, -6)\) are conjugate points with respect to the circle \(x^{2} + y^{2} - 24 = 0\).
  11. Show that \((4, 2)\) and \((3, -5)\) are conjugate with respect to the circle \(x^{2} + y^{2} - 3x - 5y + 1 = 0\).
  12. Find the value of \(k\) if the points \((1, 3)\) and \((2, k)\) are conjugate with respect to the circle \(x^{2} + y^{2} = 35\).

System of Circles Q3

  1. Find the angle between the circles \(x^{2} + y^{2} - 12x - 6y + 41 = 0\) and \(x^{2} + y^{2} + 4x + 6y - 59 = 0\).
  2. Show that the angle between the circles \(x^{2} + y^{2} = a^{2}, x^{2} + y^{2} = ax + ay\) is \(\frac{3\pi}{4}\).
  3. If the angle between the circles \(x^{2} + y^{2} - 12x - 6y + 41 = 0\) and \(x^{2} + y^{2} + kx + 6y - 59 = 0\) is \(45^{\circ}\), find \(k\).
  4. Show that the circles \(x^{2} + y^{2} + 4x - 2y - 11 = 0, x^{2} + y^{2} - 4x - 8y + 11 = 0\) intersect each other orthogonally.
  5. Find \(k\) if the pair of circles \(x^{2} + y^{2} - 6x - 8y + 12 = 0, x^{2} + y^{2} - 4x + 6y + k = 0\) are orthogonal.
  6. Find the equation of radical axis of the two circles \(2x^{2} + 2y^{2} + 3x + 6y - 5 = 0, 3x^{2} + 3y^{2} - 7x + 8y - 11 = 0\).
  7. Find the equation of common chord of the circles \(x^{2} + y^{2} - 4x - 4y + 3 = 0, x^{2} + y^{2} - 5x - 6y + 4 = 0\).
  8. Find the equation of common chord of the circles \((x - a)^{2} + (y - b)^{2} = c^{2}, (x - b)^{2} + (y - a)^{2} = c^{2}\) \((a \neq b)\).
  9. Find the equation of the common tangent of the circles \(x^{2} + y^{2} + 10x - 2y + 22 = 0, x^{2} + y^{2} + 2x - 8y + 8 = 0\) at their point of contact.
  10. Find the radical centre of the circles \(x^{2} + y^{2} + 4x - 7 = 0, 2x^{2} + 2y^{2} + 3x + 5y - 9 = 0\) and \(x^{2} + y^{2} + y = 0\).

Parabola Q4

  1. Find the equation of the parabola whose focus is \(S(1, -7)\) and vertex is \(A(1, -2)\).
  2. Find the equation of the parabola whose vertex is \((3, -2)\) and focus is \((3, 1)\).
  3. Find the coordinates of the points on the parabola \(y^{2} = 2x\) whose focal distance is \(5/2\).
  4. Find the coordinates of the points on the parabola \(y^{2} = 8x\) whose focal distance is 10.
  5. If \(\left(\frac{1}{2}, 2\right)\) is one extremity of a focal chord of the parabola \(y^{2} = 8x\), find the coordinates of the other extremity.
  6. Find the value of \(k\) if the line \(2y = 5x + k\) is a tangent to the parabola \(y^{2} = 6x\).
  7. Find the equation of normal to the parabola \(y^{2} = 4x\) which is parallel to \(y - 2x + 5 = 0\).
  8. Show that the line \(2x - y + 2 = 0\) is a tangent to the parabola \(y^{2} = 16x\). Find the point of contact also.
  9. Find the equation of tangent to the parabola \(y^{2} = 16x\) inclined at an angle \(60^{\circ}\) with its axis and also find the point of contact.
  10. Find the equations of tangent and normal to the parabola \(y^{2} = 6x\) at the positive end of the latus rectum.

Hyperbola Q5

  1. Find the equation of the hyperbola whose foci are \((\pm 5, 0)\), the transverse axis is of length 8.
  2. If \(e, e_{1}\) are the eccentricities of a hyperbola and its conjugate hyperbola, prove that \(\frac{1}{e^{2}} + \frac{1}{e_{1}^{2}} = 1\).
  3. If the eccentricity of a hyperbola is \(\frac{5}{4}\), then find the eccentricity of its conjugate hyperbola.
  4. If \(3x - 4y + k = 0\) is a tangent to \(x^{2} - 4y^{2} = 5\), find the values of \(k\).
  5. If the lines \(3x - 4y = 12\) and \(3x + 4y = 12\) meet on a hyperbola \(S = 0\) then find eccentricity of hyperbola.
  6. Find the equation of hyperbola whose asymptotes are the straight lines \(x + 2y + 3 = 0\) and \(3x + 4y + 5 = 0\) and passing through the point \((1, -1)\).
  7. Find the equation of the normal at \(\theta = \frac{\pi}{3}\) to the hyperbola \(3x^{2} - 4y^{2} = 12\).
  8. Find the product of lengths of the perpendiculars from any point on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\) to its asymptotes.
  9. If the angle between the asymptotes of a hyperbola is \(30^{\circ}\) then find its eccentricity.
  10. Define rectangular hyperbola and find its eccentricity.

Integration Q6

  1. Evaluate:
    • \(\int \frac{e^{\log x}}{x} dx\)
    • \(\int \frac{e^{\tan^{-1}x}}{1 + x^{2}} dx\)
    • \(\int \left(1 - \frac{1}{x^{2}}\right) e^{x + 1/x} dx\)
  2. Evaluate:
    • \(\int \frac{\log(1 + x)}{1 + x} dx\)
    • \(\int \frac{1}{x \log x [\log(\log x)]} dx\)
    • \(\int \sec x \log(\sec x + \tan x) dx\)
  3. Evaluate:
    • \(\int e^{x} \sin e^{x} dx\)
    • \(\int \frac{\cot(\log x)}{x} dx\)
    • \(\int \frac{\cos \sqrt{x}}{\sqrt{x}} dx\)
    • \(\int \frac{\sin(\tan^{-1}x)}{1 + x^{2}} dx\)
    • \(\int \frac{e^{x}(1 + x)}{\cos^{2}(x e^{x})} dx\)
  4. Evaluate:
    • \(\int (\tan x + \sec^{2}x) e^{x} dx\)
    • \(\int (\tan^{-1}x + \frac{1}{1 + x^{2}}) e^{x} dx\)
    • \(\int e^{x}(\sec x + \sec x \tan x) dx\)
    • \(\int (\tan x + \log \sec x) e^{x} dx\)
    • \(\int e^{x}\left(\frac{1 + x \log x}{x}\right) dx\)
    • \(\int e^{x}\left(\frac{1 + x}{(2 + x)^{2}}\right) dx\)

Integration Q7

  1. Evaluate:
    • \(\int \sec^{2}x \csc^{2}x \, dx\)
    • \(\int \frac{1 + \cos^{2}x}{1 - \cos 2x} dx\)
    • \(\int \sqrt{1 - \cos 2x} \, dx\)
    • \(\int \frac{\cos x + \sin x}{\sqrt{1 + \sin 2x}} dx\)
  2. \(\int \frac{1}{1 + \cos x} dx\)
  3. \(\int \frac{\sin^{4}x}{\cos^{6}x} dx\)
  4. \(\int \sin mx \cos nx \, dx\)
  5. \(\int \cos mx \cos nx \, dx\)
  6. Evaluate:
    • \(\int \frac{2x^{3}}{1 + x^{8}} dx\)
    • \(\int \frac{x^{8}}{1 + x^{18}} dx\)
    • \(\int \frac{(a^{x} - b^{x})^{2}}{a^{x} b^{x}} dx\)
    • \(\int \frac{1}{(x + 3)\sqrt{x + 2}} dx\)
  7. \(\int \frac{x^{2} + 1}{x^{4} + 1} dx\)
  8. \(\int \frac{x^{2}}{\sqrt{1 - x^{2}}} dx\)
  9. \(\int \frac{1}{4 - 9x^{2}} dx\)
  10. \(\int \frac{1}{\sqrt{25 + 9x^{2}}} dx\)
  11. \(\int \sqrt{9 + 4x^{2}} \, dx\)
  12. \(\int \sqrt{9x^{2} - 25} \, dx\)
  13. \(\int \frac{dx}{(x + 1)(x + 2)}\)
  14. \(\int \frac{dx}{\sqrt{x^{2} + 2x + 10}}\)
  15. Evaluate:
    • \(\int \log x \, dx\)
    • \(\int \sin^{-1}x \, dx\)
    • \(\int \frac{1}{\sin^{2}x \sqrt{1 - x^{2}}} dx\)
    • \(\int \sqrt{\sin x \sqrt{x^{2} + 1}} \, dx\)
    • \(\int x \sec^{2}x \, dx\)
    • \(\int \tan^{-1}x \, dx\)
  16. \(\int \frac{x^{2}}{1 - x^{2}} dx\)
  17. \(\int \frac{1}{1 - x^{2}} dx\)
  18. \(\int \frac{5}{\sqrt{2x - 1}} dx\)
  19. \(\int \frac{1}{2 - x} dx\)
  20. \(\int |2 - x| \, dx\)
  21. \(\int \frac{\pi^{2}}{\sec^{4} \alpha} d\alpha\)

Definite Integrals Q9

  1. Evaluate \(\int_{1}^{2} \sqrt{x^{2} - 1} \, dx\).
  2. Evaluate \(\int_{0}^{1} \frac{x^{4}}{1 + x^{2}} dx\).
  3. Evaluate \(\int_{0}^{1} \frac{x^{2}}{1 + x} dx\).
  4. Evaluate:
    • \(\int_{0}^{\pi/2} \sin^{8}x \, dx\)
    • \(\int_{0}^{\pi/2} \sin^{4}x \, dx\)
    • \(\int_{0}^{\pi/2} \frac{\cos \sqrt{x}}{\sqrt{x}} dx\)
    • \(\int_{0}^{\pi/2} \sin^{11}x \, dx\)
    • \(\int_{0}^{\pi/2} \sin^{7}x \, dx\)
    • \(\int_{0}^{\pi/2} \cos^{6}x \, dx\)
  5. Evaluate:
    • \(\int_{0}^{\pi/2} \sin^{4}x \cos^{5}x \, dx\)
    • \(\int_{0}^{\pi/2} \sin^{5}x \cos^{4}x \, dx\)
    • \(\int_{0}^{\pi} \sin^{7}x \cos^{6}x \, dx\)
    • \(\int_{0}^{2\pi} \sin^{4}x \cos^{6}x \, dx\)
    • \(\int_{-\pi/2}^{\pi/2} \sin^{2}x \cos^{4}x \, dx\)
    • \(\int_{-\pi/2}^{\pi/2} \sin^{3}x \cos^{3}x \, dx\)
  6. Find the area under the curve \(f(x) = \sin x\) in \([0, 2\pi]\).
  7. Find the area under the curve \(f(x) = \cos x\) in \([0, 2\pi]\).
  8. Find the area bounded by the curves \(y = \sin x\), and \(y = \cos x\) and x-axis.
  9. Find the area bounded by the parabola \(y = x^{2}\), the X-axis and the lines \(x = -1, x = 2\).
  10. Find the area bounded between the curves \(y^{2} = 1 - 2x\) and \(x = 0\).
  11. Find the area of the region enclosed by the curves \(y = x^{3} + 3, y = 0, x = -1, x = 2\).

Differential Equations Q10

  1. Find the order and degree of the differential equation:
    • \(\left(\frac{d^{2}y}{dx^{2}}\right)^{3} - \left(\frac{dy}{dx}\right)^{5} = 6y\)
    • \(\left[\left(\frac{dy}{dx}\right)^{2} + \left(\frac{d^{2}y}{dx^{2}}\right)^{3}\right]^{1/4} = 0\)
    • \(\frac{d^{2}y}{dx^{2}} = \left[1 + \left(\frac{dy}{dx}\right)^{2}\right]^{5/3}\)
    • \(x^{2}\left(\frac{d^{2}y}{dx^{2}}\right)^{3} + x\frac{dy}{dx} + y = 0\)
    • \(\left(\frac{d^{3}y}{dx^{3}}\right)^{2} - 3\left(\frac{dy}{dx}\right)^{2} - e^{x} - 4 = 0\)
    • \(\frac{d^{2}y}{dx^{2}} + 2\left(\frac{dy}{dx}\right) + y = \log\left(\frac{dy}{dx}\right)\)
  2. Form the differential equation corresponding to \(y = ae^{3x} + be^{4x}\).
  3. Form the differential equation corresponding to \(y = A \cos 3x + B \sin 3x\), where A and B are parameters.
  4. Form the differential equation of the family of all circles with their centres at the origin and also find its order.
  5. Obtain the differential equation corresponding to family of rectangular hyperbolas which have the coordinate axes as asymptotes.
  6. Find the order of the differential equation obtained by eliminating the arbitrary constants \(a, b\) from \(xy = ae^{x} + be^{-x} + x^{2}\).
  7. Form the differential equation corresponding to \(y = cx - 2c^{2}\), \(c\) is a parameter.
  8. Find the general solution of \(\frac{dy}{dx} = e^{x + y}\).
  9. Find the integrating factor of \(\frac{dy}{dx} + y \tan x = \sin x\) by transforming it into linear form.
  10. Find the integrating factor of \(x \frac{dy}{dx} - y = 2x^{2} \sec^{2} 2x\) by transforming it into linear form.
  11. Form the differential equation corresponding to the family of curves \(y = c(x - c)^{2}\), where \(c\) is a parameter.

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