11th Grade Math Exam

• This is an 11th-grade Mathematics exam paper.
• The exam is 3 hours long and has a maximum mark of 100.

Section A: Long Answer Type Questions (5Q X 6M = 30 Marks)

• Prove Cos²x + Cos²(x + π/3) + Cos²(x - π/3) = 3/2

Alternative Question

• Find the general and principal solutions of cos 3x + cos x - cos 2x = 0.
• The coefficients of the (r-1)th, rth, and (r+1)th terms in the expansion of (x+1)^n are in the ratio 1:3:5, and it's required to find 'n' and 'r'.

Alternative Question

• Show that 9^(n+1) + 8n - 9 is divisible by 64, provided that 'n' is a positive integer.
• Find the equation of the circle that passes through the points (2,3) and (-1,1) and whose center lies on the line x - 3y - 11 = 0.

Alternative Question

• Determine the coordinates of the foci and vertices, the lengths of the major and minor axes, the eccentricity, and the length of the Latus Rectum of the ellipse x²/36 + y²/16 = 1.
• Compute the derivative of sin x using the first principle.

Alternative Question

• Find the derivative of (sin x + cos x) / (sin x - cos x).
• Find the mean deviation about the mean for the provided data.
• The data set includes values for x_i (2, 5, 6, 8, 10, 12) and their corresponding frequencies f_i (2, 8, 10, 7, 8, 5).

Alternative Question

• Requires calculation of the mean, variance, and standard deviation for a given distribution.
• The class intervals are 30-40, 40-50, 50-60, 60-70, 70-80, 80-90, and 90-100, with corresponding frequencies of 3, 7, 12, 15, 8, 3, and 2.

Section B: Short Answer Type Questions (10Q X 4M = 40 Marks)

• Given U = {1,2,3,4,5,6,7,8,9}, A = {2,4,6,8}, and B = {2,3,5,7}, verify that (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
• Determine the domain and range of the relation R defined as R = {(x, x+5) : x ∈ {1,2,3,4,5}}.
• Prove that (sin 5x + sin 3x) / (cos 5x - cos 3x) = -tan 4x.
• Prove 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² using mathematical induction principles.
• Convert the complex number Z = √3 + i into polar form.
• Find the value of 'x' if 1/6! + 1/7! = x/8!.
• Calculate the sum of the series 8 + 88 + 888 + 8888 + ... up to n terms.
• If P(a, b) is the midpoint of a line segment between axes, show that the equation of the line is x/a + y/b = 2
• Find the ratio in which the YZ-plane divides the line segment joining points (-2, 4, 7) and (3, -5, 8).
• If E and F are events where P(E) = 1/4, P(F) = 1/2, and P(E and F) = 1/8, find P(E or F) and P(Not E and Not F).
• Write all the subsets of the set A = {a, b}.
• Find the range of the function f(x) = 2 - 3x, where x ∈ R.
• Find the value of 'r' in relation to 5P_r = 6P_(r-1).
• Determine the first five terms of the sequence defined by a_n = (n² + 5) / 4.
• Reduce the equation 6x + 3y - 5 = 0.
• Determine the equation of a parabola with its focus at (6,0) and directrix x = -6.
• Evaluate the limit: lim (x->0) of (sin ax + bx) / (ax + sin bx), given that a, b, and a+b are not equal to 0.
• Write the negation of the statements "Srinagar is a city" and "√7 is a surd."
• Write the converse of the statements "If 'n' is even, then n² is even" and "If a number is divisible by 10, it is divisible by 5."
• Given that P(A) = 1/4, find the value of P(not A).

Section C: Very Short Answer Type Questions (10Q X 1M = 10 Marks)

• Determine the general solution of the equation sin x = 0.
• Options: (a) x = 2nπ, (b) x = nπ, (c) x = (2n + 1)π/2, (d) None of these
• Find the value of sin(31π/3).
• Options: (a) √3/2, (b) 1/2, (c) √3, (d) 1
• What is the set of 'x' values satisfying both 5x + 2 < 3x + 8 and (x+2)/(x-1) < 4?
• Options: (a) (2,3), (b) (-∞), (c) [(2,3), (d) (1,3)
• If AM, GM and HM are the Arithmetic Mean, Geometric Mean and Harmonic Mean respectfully of any two positive real numbers, state the relationship between them.
• Options: (a) AM ≤ GM ≤ HM, (b) AM < GM < HM, (c) AM > GM > HM, (d) AM ≥ GM ≥ HM
• Determine the coordinates of the foci of the hyperbola x²/9 - y²/16 = 1.
• Options: (a) (0 ± √17), (b) (± √17, 0), (c) (√17, 0), (d) (-√17, 0)
• A hyperbola where a = b is called an equilateral hyperbola.(True/False)
• The value of lim (as x approaches 0) of sinx/x (where 'x' is measured in radians) is = 0. (True/False)
• (VIII) The value of lim (n->∞) of (x^n - a^n) / (x - a) = na^(n-1). (True/False)
• The sum of n terms of a Geometric Progression (GP), where r > 1 and 'a' is the starting term of the series.
• Options: (a) a/(1-r), (b) n/2[2a + (n - 1)d], (c) a(r^n - 1)/(r - 1), (d) ar^(n-1)
• The least positive integral value of 'm' for which [1 + 1/x]^m = 1, is:
• Options: (a) 1, (b) 2, (c) 3, (d) 4

This is an 11th-grade mathematics exam paper designed for a 3-hour duration, with a maximum score of 100. The exam includes proving trigonometric identities and finding general solutions. It also covers binomial theorem applications and coordinate geometry problems.