The position vectors of points A and B are i + j and j + k, respectively. If point C divides the line segment AB in the ratio 2:1, then the position vector of C is
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Correct answer: (C) (i + 3j + 2k)/3 Dividing in the ratio 2:1, C's position vector is [1·(i + j) + 2·(j + k)]/3 = (i + 3j + 2k)/3.
1 marks · 2023 · Set A
Question 102
If A = [[cos α, −sin α], [sin α, cos α]] and A + A′ = I, then the value of α is
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Correct answer: (B) π/3 A + A′ = 2cos α · I, so 2 cos α = 1 and α = π/3.
1 marks · 2023 · Set A
Question 103
If a and b are unit vectors and the angle between them is θ, then sin(θ/2) is
Which of the following partially ordered sets is a chain?
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Correct answer: (C) (R, ≤), where R is the real numbers and a ≤ b means a is less than or equal to b A chain is a partial order in which any two elements are comparable, and ≤ on the real numbers is such.
1 marks · 2023 · Set A
Question 105
If a = 2i + 5j + 3k, b = 3i + 3j + 6k and c = 2i + 7j + 4k, then |(a − b) × (c − a)| is
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Correct answer: (E) None of the above (a − b) × (c − a) works out to 8i + j − 2k, of magnitude √69, which is not among the options—so the answer is ‘None of the above’.
1 marks · 2023 · Set A
Question 106
Three vectors a, b and c are such that |a| = 3, |b| = 4, |c| = 5, and each is perpendicular to the sum of the other two. What is |a + b + c|?
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Correct answer: (A) 5√2 Each vector being perpendicular to the sum of the other two, all the dot products vanish, so |a + b + c| = √(9 + 16 + 25) = √50 = 5√2.
1 marks · 2023 · Set A
Question 107
The coordinates of the foot of the perpendicular drawn from A(1, 2, 1) to the line joining B(1, 4, 6) and C(5, 4, 4) are
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Correct answer: (A) (3, 4, 5) Taking BC's direction as (4, 0, −2) and dropping the perpendicular gives the foot (3, 4, 5).
1 marks · 2023 · Set A
Question 108
The work done by forces P = 3i − 2j + k and Q = i + 3j − 5k in displacing a particle from A, whose position vector is −2i + 5k, to B, whose position vector is 3i − 7j + 2k, is
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Correct answer: (C) 25 units The total force P + Q = 4i + j − 4k and the displacement AB = 5i − 7j − 3k; their dot product 20 − 7 + 12 = 25 units is the work done.
1 marks · 2023 · Set A
Question 109
The coordinates of the point where the line (x − 1)/2 = (y − 2)/(−3) = (z + 5)/4 cuts the plane 2x + 4y − z = 3 are
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Correct answer: (E) None of the above Putting the line's parametric form into the plane gives t = 1 and the point (3, −1, −1), which is not among the options—so the answer is ‘None of the above’.
1 marks · 2023 · Set A
Question 110
The moment about the point i + 2j − k of the force 3i + k, whose line of action passes through the point 2i − j + 3k, is
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Correct answer: (B) −3i + 11j + 9k The moment is r × F, r being the difference between the point of action and the given point; working it out gives −3i + 11j + 9k.
1 marks · 2023 · Set A
Question 111
The minimum distance between the lines (x − 3)/3 = (y − 8)/(−1) = z − 3 and (x + 3)/(−3) = (y + 7)/2 = (z − 6)/4 is
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Correct answer: (C) 3√30 Applying the shortest-distance formula for the two lines gives 3√30.
1 marks · 2023 · Set A
Question 112
Find the number of permutations of the letters of the word ‘PRAYAGRAJ’.
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Correct answer: (B) 30240 ‘PRAYAGRAJ’ has nine letters with A thrice and R twice, so the permutations number 9!/(3!2!) = 30240.
1 marks · 2023 · Set A
Question 113
The equation of the plane passing through the line of intersection of planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0, and perpendicular to the plane 5x + 3y − 6z + 8 = 0, is
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Correct answer: (B) 33x + 45y + 50z − 41 = 0 Taking the plane from the family of the two given planes and imposing perpendicularity gives 33x + 45y + 50z − 41 = 0.
1 marks · 2023 · Set A
Question 114
The term independent of x in the expansion of (3x²/2 − 1/(3x))⁶ is
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Correct answer: (A) 5/12 Setting the power of x in the general term to zero picks out the fourth term, whose value is 5/12.
1 marks · 2023 · Set A
Question 115
The mean and variance of five observations are 4.4 and 8.24, respectively. If three observations are 1, 2 and 6, then the other two observations are
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Correct answer: (D) More than one of the above The other two observations are 4 and 9; the same pair appears in reverse in another option, so the answer is ‘more than one’.
1 marks · 2023 · Set A
Question 116
If w is an imaginary cube root of unity, then the value of (1 + w − w²)(w + w² − 1)(w² + 1 − w) is
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Correct answer: (B) −8 Using 1 + ω + ω² = 0 each bracket becomes −2ω², −2 and −2ω, whose product is −8.
1 marks · 2023 · Set A
Question 117
From a group of 7 men and 4 women, a committee of 6 persons is formed. The probability that the committee consists of exactly 2 women is
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Correct answer: (A) 5/11 Exactly two women can come in C(4,2) × C(7,4) = 6 × 35 = 210 ways out of C(11,6) = 462, so the probability is 210/462 = 5/11.
1 marks · 2023 · Set A
Question 118
The sum of the series 2x/(1 + x²) + (1/3)(2x/(1 + x²))³ + (1/5)(2x/(1 + x²))⁵ + ⋯ to infinity is
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Correct answer: (B) logₑ((1 + x)/(1 − x)) The series is the expansion of tanh⁻¹; with 2x/(1 + x²) its sum is logₑ((1 + x)/(1 − x)).
1 marks · 2023 · Set A
Question 119
p ⇒ q is false if
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Correct answer: (B) p is true and q is false p ⇒ q is false only when p is true and q is false.
1 marks · 2023 · Set A
Question 120
The constraints x + 2y ≥ 10, 3x + 4y ≤ 24, x ≥ 0, y ≥ 0 determine a triangular region with vertices
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Correct answer: (A) (0, 5), (0, 6) and (4, 3) With x + 2y ≥ 10, 3x + 4y ≤ 24 and the axes, the vertices come out as (0, 5), (0, 6) and (4, 3).