MARCH EXAM 2024-25
Class: XI
Subject: MATHEMATICS
Time: 3 hours M.M.: 80
Section-A (1 mark each)
Choose the correct options in the following questions:
- Three coins are tossed. Probability of getting at most 2 heads is:
- (A) \( \frac{1}{2} \)
- (B) \( \frac{7}{8} \)
- (C) \( \frac{1}{8} \)
- (D) \( \frac{3}{8} \)
- Value of \( i^{200023} \) is equal to:
- (A) 1
- (B) -1
- (C) \( i \)
- (D) \( -i \)
- Which of the following is not a measure of dispersion?
- (A) Range
- (B) Mean deviation
- (C) Mode
- (D) Standard deviation
- Number of chords that can be drawn through 60 points on a circle are:
- (A) 400
- (B) 2000
- (C) 1770
- (D) 3600
- The derivative of \( \sqrt{x} \) at \( x = 9 \) is:
- (A) \( \frac{1}{9} \)
- (B) \( \frac{1}{6} \)
- (C) 6
- (D) \( \frac{1}{3} \)
- Radian measure of sum of angles of a rectangle:
- (A) \( \pi \)
- (B) \( 2\pi \)
- (C) \( 4\pi \)
- (D) \( \frac{3\pi}{2} \)
- sec\( \left(\frac{3\pi}{2} + x\right) \) is equal to:
- (A) cosec x
- (B) - cosec x
- (C) - sec x
- (D) sec x
- tan 1140° is equal to:
- (A) \( -\sqrt{3} \)
- (B) \( \sqrt{3} \)
- (C) \( \frac{1}{\sqrt{3}} \)
- (D) \( -\frac{1}{\sqrt{3}} \)
- If \( x - iy = 2 + 7i \), then \( x + y = \)
- (A) 5
- (B) -7
- (C) -5
- (D) 9
- Total number of ways in which "APPLE" can be arranged are:
- (A) 5
- (B) 12
- (C) 60
- (D) 120
- \(|4 + 4i|\) is:
- (A) 8
- (B) 16
- (C) \( 4\sqrt{2} \)
- (D) \( 2\sqrt{2} \)
- If \( A = \{x : x \in N, -4 < x \leq 4\} \), then which of these is correct:
- (A) -4 ∈ A
- (B) -5 ∈ A
- (C) 2 ∉ A
- (D) 4 ∈ A
- Range of \( f(x) = \lfloor x \rfloor \) is:
- (A) [0,1]
- (B) \( \mathbb{N} \)
- (C) \( \mathbb{Z} \)
- (D) \( \mathbb{R} \)
- Slope of line perpendicular to the line \( 2x - 6y = -1 \) is:
- (A) \( \frac{1}{3} \)
- (B) 3
- (C) -3
- (D) \( -\frac{1}{3} \)
- In a G.P. having third term 24 and common ratio 2, Second term is equal to:
- (A) 6
- (B) 12
- (C) 24
- (D) 48
- Number of elements in the sample space when a die and a coin is thrown:
- (A) 6
- (B) 12
- (C) 36
- (D) 216
2. Fill in the blanks from the given options:
- The distance of origin from the line 15x - 8y = -56 is ...
- Derivative of sin x w.r.t. x when x = π/4 is ...
- Distance between the points (2,1,-2) and (-2,1,3) is ...
- If the numbers -2, x, -2 are in G.P., then value of x is equal to ...
- Length of the major axis of the ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) is ...
- If A = {1,2,3,4} then the total number of subsets of A is ...
- \( \frac{d}{dx} (x^4 + a^x) = \) ...
- \( \frac{d}{dt} (3 \sin t + 4 \cos t) = \) ...
3. State true or false for the following statements:
- The point (-3,1) lies in II quadrant.
- Number of terms in the expansion of (1 - x²)⁶ is 6.
- Probability of two mutually exclusive events is always equal to zero.
- \( \frac{7!}{5!} = 2! \)
- The set of all second elements in a relation R from a set A to a set B is called Range of the relation R.
- The solution set of 4x + 3 > 8x + 7 is (-∞,-1).
- \( \lim_{x \to 0} \frac{3x}{\sin 8x} = \) ...
- If A = {x | x ∈ N, 1 < x < 3} and B = {0,1,2}, then B - A = {1,2}.
Section-B (2 marks each)
- How many words can be made from the letters of the word MISSISSIPPI?
- A die has two faces each with number ‘1’, three faces each with number ‘2’ and one face with number ‘3’. If die is rolled once, determine:
- (i) P(2)
- (ii) P(not 3)
- Which term of the sequence 2, ..., is (4096)?
- Expand \( (x + 2)^5 \), \( x \neq 0 \) using binomial theorem.
Or
Compute \( (99)^6 \) using binomial theorem. - Find the equation of the ellipse whose vertices are (±13, 0) and foci are (±5, 0).
Or
Find the coordinates of the focus, axis of the parabola, the equation of the directrix, and the length of the latus rectum for the parabola \( x^2 = -16y \). - Find the probability that when a hand of 8 cards is drawn from a well-shuffled deck of 52 cards, it contains all Kings.
Section-C (4 marks each)
- Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Or
Find \( n \) so that \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \) may be the geometric mean between \( a \) and \( b \). - If \( (a + ib) = \frac{(x + i b)^2}{2x + 1} \), prove that \( a^2 + b^2 = \frac{(x^2 + 1)^2}{(2x + 1)^2} \).
Or
Find the real numbers \( x \) and \( y \) if \( (x - iy) (3 + 5i) \) is the conjugate of \( -6 - 24i \). - (a) Find the value of \( \sin 150^\circ - \cos 180^\circ \).
(b) Show that \[ \frac{\sin x + \sin 3x + \sin 5x + \sin 7x}{\cos x + \cos 3x + \cos 5x + \cos 7x} = \tan 4x \] - Find the equation of the line passing through (-3,5) and perpendicular to the line through the points (2,5) and (-3,6).
Or
Find the equation of the circle which passes through the points (1,2), (3,-4), and (5,-6). Find its center and radius. - Calculate the mean and variance for the following distribution:
Classes 30-40 40-50 50-60 60-70 70-80 80-90 90-100 Frequency 3 7 12 15 8 3 2
Section-D (6 marks each)
- Find the derivative of \( x \cos x \) using the first principle.
Or
(a) Find \( \lim\limits_{x \to 1} \frac{x^6 - 5x + 4}{x^3 - 2x + 1} \).
(b) Differentiate \( (4x + 3) \sin x \) w.r.t. \( x \). - Find the number of permutations of the letters of the word ‘MATHEMATICS’.
Or
(a) The longest side of a triangle is 3 times the shortest side, and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
(b) How many numbers greater than 10000000 can be formed by using the digits 1, 2, 0, 2, 4, 4, 2, 4? - (a) Solve \( \frac{2x - 1}{3} \geq \frac{3x - 2}{4} - \frac{2 - x}{5} \).
- (a) Prove that \( \sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ = \frac{1}{16} \).
Or
(b) Find \( \sin \frac{x}{2}, \cos \frac{x}{2} \) and \( \tan \frac{x}{2} \) when \( \tan x = -\frac{24}{7} \), \( x \) lies in the 4th quadrant.
