The principal value branch of sin⁻¹x is [-π/2, π/2]. We have sin⁻¹(sin(-10π/3)). Since -10π/3 = -4π + 2π/3, sin(-10π/3) = sin(2π/3). Further, sin(2π/3) = sin(π - π/3) = sin(π/3). Thus, sin⁻¹(sin(π/3)) = π/3, which lies within [-π/2, π/2].
Q2mcq1 mark
If A and B are square matrices of same order such that AB = A and BA = B, then A² + B² is equal to:
a.
A + B
Correct Answer
b.
BA
c.
2(A + B)
d.
2BA
Step-by-Step AI Explanation
Verified Method
Given AB = A and BA = B. Then A² = A(A) = A(AB) = (AA)B? No, consider A² = A * A = (AB) * A = A * (BA) = AB = A. Similarly, B² = B * B = (BA) * B = B * (AB) = BA = B. Therefore, A² + B² = A + B.
Q3mcq1 mark
For real x, let f(x) = x³ + 5x + 1. Then:
a.
f is one-one but not onto on R
b.
f is onto on R but not one-one
c.
f is one-one and onto on R
Correct Answer
d.
f is neither one-one nor onto on R
Step-by-Step AI Explanation
Verified Method
f(x) = x³ + 5x + 1. The derivative f'(x) = 3x² + 5. Since 3x² + 5 > 0 for all real x, f(x) is strictly increasing, which implies it is one-one. As f(x) is a polynomial of odd degree, its range is (-∞, ∞), making it onto. Thus, it is both one-one and onto.
Q4mcq1 mark
If y = sin⁻¹x, then (d²y/dx²)(1 − x²) is equal to:
a.
x(dy/dx)
Correct Answer
b.
−x(dy/dx)
c.
x²(dy/dx)
d.
−x²(dy/dx)
Step-by-Step AI Explanation
Verified Method
Given y = sin⁻¹x. dy/dx = 1/√(1-x²), so (dy/dx)²(1-x²) = 1. Differentiating both sides with respect to x: 2(dy/dx) * (d²y/dx²) * (1-x²) + (dy/dx)² * (-2x) = 0. Dividing by 2(dy/dx), we get (d²y/dx²)(1-x²) - x(dy/dx) = 0. Therefore, (d²y/dx²)(1-x²) = x(dy/dx).
Q5mcq1 mark
The value of λ so that f(x) = sinx − cosx − λx + C decreases for all real values of x are:
a.
λ ≥ √2
Correct Answer
b.
λ ≥ 1
c.
λ < √2
d.
λ ≤ 1
Step-by-Step AI Explanation
Verified Method
For f(x) to decrease, f'(x) ≤ 0. f'(x) = cosx + sinx - λ. We need cosx + sinx ≤ λ for all x. The maximum value of cosx + sinx is √2 (using √2 sin(x + π/4)). Therefore, λ must be at least the maximum value, so λ ≥ √2.
Q6mcq1 mark
If P is a point on the line segment joining (3, 6, −1) and (6, 2, −2) and y-coordinate of P is 4, then its z-coordinate is:
a.
−3/2
b.
0
c.
1
Correct Answer
d.
3/2
Step-by-Step AI Explanation
Verified Method
Using section formula for the line segment joining (3, 6, -1) and (6, 2, -2). Let ratio be k:1. The y-coordinate is (2k + 6)/(k + 1) = 4. 2k + 6 = 4k + 4 => 2 = 2k => k = 1. The point P is the midpoint: ((3+6)/2, (6+2)/2, (-1-2)/2) = (9/2, 4, -3/2). Wait, re-checking y: (2+6)/2 = 4. z is (-1-2)/2 = -3/2. Option A is correct? Let me re-calculate: (2k+6)/(k+1)=4 -> 2k+6=4k+4 -> 2=2k -> k=1. Point is (4.5, 4, -1.5). Answer is -3/2.
Q7mcq1 mark
If M and N are square matrices of order 3 such that det(M) = m and MN = mI, then det(N) is equal to:
a.
−1
b.
1
c.
−m²
d.
m²
Correct Answer
Step-by-Step AI Explanation
Verified Method
Given det(M) = m and MN = mI. Taking determinant on both sides: det(MN) = det(mI). Since det(M)det(N) = m^3 det(I) (for 3x3), we have m * det(N) = m^3 * 1. Thus, det(N) = m²/m * m = m². Wait, m*det(N) = m^3, so det(N) = m².
Q8mcq1 mark
If f(x) = { 3x − 2, 0 ≤ x ≤ 1 ; 2x² + ax, 1 < x < 2 } is continuous for x ∈ (0, 2), then a is equal to:
a.
−4
b.
−7/2
Correct Answer
c.
−2
d.
−1
Step-by-Step AI Explanation
Verified Method
For continuity at x = 1, LHS = lim(x→1⁻) (3x - 2) = 3(1) - 2 = 1. RHS = lim(x→1⁺) (2x² + ax) = 2(1)² + a(1) = 2 + a. Setting LHS = RHS: 1 = 2 + a, so a = -1. Re-checking: 3x-2 at 1 is 1, 2+a at 1 is 2+a. 1=2+a => a=-1. None match. Re-evaluating: The limits match at 1: 3(1)-2 = 1. 2(1)^2 + a(1) = 2+a. a = -1. Wait, let me check the question again. 2x^2+ax. At x=1, 2+a=1, a=-1.
Q9mcq1 mark
If f : N → W is defined as f(n) = { n/2, if n is even ; 0, if n is odd }, then f is:
a.
injective only
b.
surjective only
Correct Answer
c.
a bijection
d.
neither surjective nor injective
Step-by-Step AI Explanation
Verified Method
The function f: N → W is defined as f(n) = n/2 if n is even and 0 if n is odd. It is not injective because f(1) = 0 and f(3) = 0 (many-to-one). It is surjective because every even number 2k in W is the image of 2k in N, and 0 is the image of 1. Thus, it is surjective only.
Q10mcq1 mark
The matrix [[0, −1, 2], [1, 0, 7], [−2, −7, 0]] is a:
a.
diagonal matrix
b.
symmetric matrix
c.
skew symmetric matrix
Correct Answer
d.
scalar matrix
Step-by-Step AI Explanation
Verified Method
A matrix A is skew-symmetric if A^T = -A. Here, the transpose is [[0, 1, -2], [-1, 0, -7], [2, 7, 0]]. Comparing this to A = [[0, -1, 2], [1, 0, 7], [-2, -7, 0]], we see that A^T = -A. Also, all diagonal elements are zero, which is a property of skew-symmetric matrices.
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The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If Z = ax + by; (a, b > 0) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is:
a.
a = b
b.
a = 3b
c.
b = 6a
d.
3a = 2b
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Assertion: If A and B are two events such that P(A ∩ B) = 0, then A and B are independent events.
Reason: Two events are independent if the occurrence of one does not affect the occurrence of the other.
a.
Both Assertion and Reason are true and Reason is the correct explanation of the Assertion.
b.
Both Assertion and Reason are true, but Reason is not the correct explanation of the Assertion.
c.
Assertion is true, but Reason is false.
d.
Assertion is false and Reason is also false.
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Assertion: In a Linear Programming Problem, if the feasible region is empty, then the Linear Programming Problem has no solution.
Reason: A feasible region is defined as the region that satisfies all the constraints.
a.
Both Assertion and Reason are true and Reason is the correct explanation of the Assertion.
b.
Both Assertion and Reason are true, but Reason is not the correct explanation of the Assertion.
c.
Assertion is true, but Reason is false.
d.
Assertion is false and Reason is also false.
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(a) Find the least value of 'a' so that f(x) = 2x² − ax + 3 is an increasing function on [2, 4].
OR
(b) If f(x) = x + 1/x, x > 1, show that f is an increasing function.
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(a) If f: R⁺ → R is defined as f(x) = log_a x (a > 0 and a ≠ 1), prove that f is a bijection. (R⁺ is set of all positive real numbers.)
OR
(b) Let A = {1, 2, 3} and B = {4, 5, 6}. A relation R from A to B is defined as R = {(x, y): x + y = 6, x ∈ A, y ∈ B}.
(i) Write all elements of R.
(ii) Is R a function? Justify.
(iii) Determine domain and range of R.
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(a) Find k so that f(x) = { (x² – 2x – 3)/(x + 1), x ≠ –1 ; k, x = –1 } is continuous at x = –1.
OR
(b) Check the differentiability of function f(x) = x|x| at x = 0.
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(a) Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.
OR
(b) A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.
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Solve the following Linear Programming Problem using graphical method:
Maximise Z = 100x + 50y
subject to the constraints:
3x + y ≤ 600
x + y ≤ 300
y ≤ x + 200
x ≥ 0, y ≥ 0
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If A is a 3 × 3 invertible matrix, show that for any scalar k ≠ 0, (kA)⁻¹ = (1/k) A⁻¹. Hence calculate (3A)⁻¹, where A = [[2, –1, 1], [–1, 2, –1], [1, –1, 2]].
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The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation y = 4x – (1/2)x², where x is the number of days exposed to sunlight.
(i) Find the rate of growth of the plant with respect to sunlight.
(ii) In how many days will the plant attain its maximum height? What is the maximum height?
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(a) Show that the area of a parallelogram whose diagonals are represented by vectors a and b is given by (1/2)|a × b|. Also find the area of a parallelogram whose diagonals are 2î – ĵ + k̂ and î + 3ĵ – k̂.
OR
(b) Find the equation of a line in vector and cartesian form which passes through the point (1, 2, –4) and is perpendicular to the lines (x–8)/3 = (y+19)/–16 = (z–10)/7, and r = (15î + 29ĵ + 5k̂) + λ(3î + 8ĵ – 5k̂).
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Some students are having a misconception while comparing decimals. For example, a student may mention that 78.56 > 78.9 as 7856 > 789. In order to assess this concept, a decimal comparison test was administered to the students of class VI through the following question: In the recently held Sports Day in the school, 5 students participated in a javelin throw competition. The distances to which they have thrown the javelin are shown below in the table:
Name: Ajay – 47.7 m; Bijoy – 47.07 m; Kartik – 43.09 m; Dinesh – 43.9 m; Devesh – 45.2 m.
The students were asked to identify who has thrown the javelin the farthest.
The teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not. 80% of the students having misconception answered Bijoy as the correct answer. 90% of the students identified with not having misconception did not answer Bijoy as their answer.
On the basis of the above information, answer the following questions:
(i) What is the probability of a student not having misconception but still answers Bijoy in the test?
(ii) What is the probability that a randomly selected student answers Bijoy as his answer in the test?
(iii) (a) What is the probability that a student who answered as Bijoy is having misconception?
OR
(iii) (b) What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception?
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The students were asked to identify who has thrown the javelin the farthest. The distances thrown by students are: Ajay 47.7m, Bijoy 47.07m, Kartik 43.09m, Dinesh 43.9m, Devesh 45.2m. Based on the test attempted by the students, the teacher concludes that 40% of the students have the misconception in the concept of decimal comparison and the rest do not have the misconception. 80% of the students having misconception answered Bijoy as the correct answer in the paper. 90% of the students who are identified with not having misconception, did not answer Bijoy as their answer.
(i) What is the probability of a student not having misconception but still answers Bijoy in the test?
(ii) What is the probability that a randomly selected student answers Bijoy as his answer in the test?
(iii) (a) What is the probability that a student who answered as Bijoy is having misconception?
OR
(iii) (b) What is the probability that a student who answered as Bijoy is amongst students who do not have the misconception?
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An engineer is designing a new metro rail network in a city.
(i) Find whether the two metro tracks are parallel.
(ii) Solar panels are to be installed on the rooftop of the metro stations. Determine the equation of the line representing the placement of solar panels on the rooftop of Line A's stations, given that panels are to be positioned parallel to Line A's track (l1) and pass through the point (1, -2, -3).
(iii) (a) To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.
OR
(iii) (b) Find the shortest distance between Line A and Line B.
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During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature (25°C). Initially the processor's temperature is 85°C. The rate of cooling is defined by the equation d/dt(T(t)) = -k(T(t) - 25), where T(t) represents the temperature of the processor at time t (in minutes) and k is a constant.
(i) Find the expression for temperature of processor, T(t) given that T(0) = 85°C.
(ii) How long will it take for the processor's temperature to reach 40°C? Given that k = 0.03 and loge 4 = 1.3863.
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