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CUSAT CAT B.Tech study materials
CUSAT CAT B.Tech study resources

CUSAT CAT B.Tech practice questions

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Question 1Mathematics

If z = (1 + i)/(1 - i), then z^4 equals

  1. 1 Correct answer.
  2. -1
  3. -i
  4. i

Explanation

Multiply numerator and denominator by the conjugate 1 + i: z = (1 + i)^2/((1 - i)(1 + i)) = 2i/2 = i. Hence z^4 = i^4 = (i^2)^2 = (-1)^2 = 1.

Question 2Mathematics

The number of diagonals of a regular decagon is

  1. 35 Correct answer.
  2. 30
  3. 45
  4. 40

Explanation

A decagon has 10 vertices, so the number of line segments joining pairs of vertices is C(10,2) = 45. Of these, 10 are sides of the polygon, leaving 45 - 10 = 35 diagonals, which agrees with n(n - 3)/2 for n = 10.

Question 3Mathematics

The value of lim(x→0) sin(3x)/(5x) is

  1. 3/5 Correct answer.
  2. 5/3
  3. 1/5
  4. 3

Explanation

Write sin(3x)/(5x) as (3/5) · sin(3x)/(3x). Since sin(t)/t → 1 as t → 0, the limit is (3/5) · 1 = 3/5.

Question 4Mathematics

If A is a 3 × 3 matrix with det(A) = 5, then det(adj A) equals

  1. 25 Correct answer.
  2. 1
  3. 5
  4. 125

Explanation

For any n × n matrix A, the identity det(adj A) = (det A)^(n-1) holds. Here n = 3 and det A = 5, so det(adj A) = 5^2 = 25.

Question 5Mathematics

Two cards are drawn at random one after the other, without replacement, from a standard pack of 52 cards. The probability that both cards are kings is

  1. 2/221
  2. 1/221 Correct answer.
  3. 1/13
  4. 1/169

Explanation

The first card is a king with probability 4/52 = 1/13, and then the second is a king with probability 3/51 = 1/17. Since draws are without replacement, multiply: (1/13)(1/17) = 1/221.

Question 6Mathematics

The value of the integral ∫ from 0 to π/2 of sin x/(sin x + cos x) dx is

  1. π/3
  2. π/6
  3. π/4 Correct answer.
  4. π/2

Explanation

Let I be the integral and replace x by π/2 - x. This gives I = ∫ cos x/(sin x + cos x) dx over the same limits, so 2I = ∫ 1 dx = π/2 and therefore I = π/4.

Question 7Mathematics

If vectors a and b satisfy |a| = 3, |b| = 4 and a · b = 0, then |a + b| equals

  1. 1
  2. 7
  3. 12
  4. 5 Correct answer.

Explanation

Expand the squared magnitude: |a + b|^2 = |a|^2 + 2a·b + |b|^2 = 9 + 0 + 16 = 25. Taking the positive square root gives |a + b| = 5.

Question 8Mathematics

If α and β are the roots of x^2 - 6x + 4 = 0, then α^2 + β^2 equals

  1. 36
  2. 28 Correct answer.
  3. 20
  4. 32

Explanation

By Vieta's formulas, α + β = 6 and αβ = 4. Then α^2 + β^2 = (α + β)^2 - 2αβ = 36 - 8 = 28.

Question 9Mathematics

The sum of the first n terms of the series 1/(1·3) + 1/(3·5) + 1/(5·7) + ... is

  1. n/(n + 1)
  2. 1/(2n + 1)
  3. n/(2n + 1) Correct answer.
  4. (2n)/(2n + 1)

Explanation

The k-th term is 1/((2k - 1)(2k + 1)) = (1/2)[1/(2k - 1) - 1/(2k + 1)]. Summing telescopes to (1/2)[1 - 1/(2n + 1)] = (1/2)(2n/(2n + 1)) = n/(2n + 1).

Question 10Mathematics

The image of the point (1, 2) with respect to the line x - 3y + 4 = 0 is

  1. (6/5, 7/5) Correct answer.
  2. (6/5, -7/5)
  3. (-6/5, 7/5)
  4. (11/5, 2/5)

Explanation

For the line ax + by + c = 0, the reflection of (x1, y1) is (x1 - 2aD/(a^2 + b^2), y1 - 2bD/(a^2 + b^2)) where D = ax1 + by1 + c. Here D = 1 - 6 + 4 = -1 and a^2 + b^2 = 10, giving x = 1 + 2/10 = 6/5 and y = 2 - 6/10 = 7/5.

Question 11Mathematics

If z = 1 + i√3, then the principal value of arg(z), in radians, is

  1. π/3 Correct answer.
  2. π/2
  3. π/6
  4. π/4

Explanation

For z = x + iy with x > 0, the principal argument is arctan(y/x). Here x = 1 and y = √3, so arctan(√3) = π/3. The point lies in the first quadrant, so no quadrant adjustment is needed.

Question 12Mathematics

If A = [[0, 1], [-1, 0]], then A^2 equals

  1. -I Correct answer.
  2. 2A
  3. I
  4. 0

Explanation

Multiplying the matrix by itself gives A^2 = [[0·0 + 1·(-1), 0·1 + 1·0], [-1·0 + 0·(-1), -1·1 + 0·0]] = [[-1, 0], [0, -1]] = -I. This matrix represents a 90° rotation, whose square is a 180° rotation.

Question 13Mathematics

A fair die is rolled once. The probability that the number obtained is a prime number is

  1. 1/2 Correct answer.
  2. 2/3
  3. 1/6
  4. 1/3

Explanation

On a fair die the possible prime outcomes are 2, 3 and 5, giving three favourable outcomes among six equally likely faces. Therefore the probability is 3/6 = 1/2. The number 1 is not prime.

Question 14Mathematics

The value of lim(x→0) (1 - cos x)/x^2 is

  1. 0
  2. 1/4
  3. 1
  4. 1/2 Correct answer.

Explanation

Using the standard expansion cos x = 1 - x^2/2 + O(x^4), the numerator becomes x^2/2 plus higher-order terms. Dividing by x^2 and taking the limit gives 1/2. Equivalently, write 1 - cos x = 2 sin^2(x/2) and use lim(t→0) sin t/t = 1.

Question 15Mathematics

The distance between the parallel lines 3x + 4y = 5 and 3x + 4y = 15 is

  1. 5/2
  2. 1
  3. 2 Correct answer.
  4. 3

Explanation

For parallel lines ax + by = c1 and ax + by = c2, the distance is |c1 - c2|/√(a² + b²). Here |15 - 5|/√(3² + 4²) = 10/5 = 2. The lines are distinct because their constant terms differ.

Question 16Mathematics

For vectors a = i + j, b = j + k and c = k + i, the scalar triple product [a b c] equals

  1. 1
  2. 2 Correct answer.
  3. 4
  4. 0

Explanation

The scalar triple product is the determinant of the component rows: |1 1 0; 0 1 1; 1 0 1|. Expanding gives 1(1·1 - 1·0) - 1(0·1 - 1·1) + 0 = 1 + 1 = 2. It also equals the volume of the parallelepiped spanned by the three vectors.

Question 17Mathematics

The constant term in the expansion of (x + 1/x)^10 is

  1. 210
  2. 252 Correct answer.
  3. 120
  4. 300

Explanation

The general term is C(10, r) x^(10-r) (1/x)^r = C(10, r) x^(10-2r). For the constant term the exponent must be zero, so 10 - 2r = 0 and r = 5. The coefficient is C(10, 5) = 252.

Question 18Mathematics

The number of arrangements of the letters of the word CUSAT in which the two vowels are together is

  1. 60
  2. 36
  3. 48 Correct answer.
  4. 24

Explanation

The word CUSAT has five distinct letters, with vowels U and A. Treat UA as one block, so the objects are the block, C, S and T: 4! arrangements. The vowels inside the block can be ordered in 2! ways, giving 4! × 2! = 48.

Question 19Mathematics

If ω is a non-real cube root of unity, then (1 + ω - ω²)^3 + (1 - ω + ω²)^3 equals

  1. -16 Correct answer.
  2. -8
  3. 16
  4. -32

Explanation

Use 1 + ω + ω² = 0, so 1 + ω = -ω² and 1 + ω² = -ω. Then 1 + ω - ω² = -2ω², whose cube is -8ω^6 = -8, and 1 - ω + ω² = -2ω, whose cube is -8ω³ = -8. The sum is -16.

Question 20Mathematics

The value of ∫ from 0 to π of x sin x/(1 + cos²x) dx is

  1. π²/2
  2. π/2
  3. π²
  4. π²/4 Correct answer.

Explanation

Let I be the integral and substitute x → π - x. Adding the two expressions gives 2I = π ∫ from 0 to π of sin x/(1 + cos²x) dx. With u = cos x, this becomes π ∫ from -1 to 1 du/(1 + u²) = π(π/4 + π/4) = π²/2. Hence I = π²/4.

Question 21Mathematics

In the expansion of (1 + x)^7, the coefficient of x^3 is

  1. 84
  2. 56
  3. 35 Correct answer.
  4. 21

Explanation

In the binomial expansion of (1 + x)^n, the coefficient of x^r is C(n, r). Here n = 7 and r = 3, so the coefficient is C(7, 3) = 35. The other listed values are binomial coefficients for different pairs of n and r.

Question 22Mathematics

The first term of an arithmetic progression is 1 and the sum of its first 20 terms is 400. The common difference is

  1. 2 Correct answer.
  2. 1
  3. 3
  4. 4

Explanation

The sum of the first n terms of an arithmetic progression is S_n = n/2[2a + (n - 1)d]. Substituting a = 1, n = 20 and S_20 = 400 gives 10(2 + 19d) = 400, so 2 + 19d = 40 and d = 2.

Question 23Mathematics

The value of sin 15° cos 15° is

  1. √3/2
  2. 1/4 Correct answer.
  3. √3/4
  4. 1/2

Explanation

Using the double-angle identity sin 2θ = 2 sin θ cos θ, we get sin 15° cos 15° = (1/2) sin 30°. Since sin 30° = 1/2, the product equals 1/4.

Question 24Mathematics

From 5 distinct books, the number of ways to choose 3 books and arrange them in a row is

  1. 120
  2. 10
  3. 20
  4. 60 Correct answer.

Explanation

Choosing and arranging 3 books from 5 means taking ordered selections, so the count is P(5, 3) = 5 × 4 × 3 = 60. The value C(5, 3) = 10 would count only the choices, without arranging the selected books.

Question 25Mathematics

The radius of the circle x² + y² - 6x + 8y - 11 = 0 is

  1. 36
  2. 5
  3. 6 Correct answer.
  4. √11

Explanation

Completing the square gives (x - 3)² + (y + 4)² = 9 + 16 + 11 = 36. This is a circle with centre (3, -4) and radius √36 = 6.

Question 26Mathematics

The value of lim(x→4) (x² - 16)/(√x - 2) is

  1. 32 Correct answer.
  2. 16
  3. 0
  4. 8

Explanation

Factor x² - 16 = (x - 4)(x + 4) and rationalise √x - 2 = (x - 4)/(√x + 2). The quotient becomes (x + 4)(√x + 2), which tends to 8 × 4 = 32 as x tends to 4.

Question 27Mathematics

A bag contains 3 red and 4 blue balls. Two balls are drawn at random without replacement. The probability that both are red is

  1. 3/7
  2. 9/49
  3. 1/7 Correct answer.
  4. 2/7

Explanation

The total number of ways to draw 2 balls from 7 is C(7, 2) = 21. The number of ways to draw 2 red balls from 3 is C(3, 2) = 3, so the probability is 3/21 = 1/7.

Question 28Mathematics

If f(x) = x^x for x > 0, then f'(1) equals

  1. 0
  2. 1 Correct answer.
  3. e
  4. 2

Explanation

Write x^x = e^(x ln x). Differentiating gives f'(x) = x^x(ln x + 1). At x = 1 this equals 1 × (0 + 1) = 1.

Question 29Mathematics

Let A be a 2 × 2 matrix with trace(A) = 5 and det(A) = 3. Then det(A² - 5A + 3I) equals

  1. 5
  2. 3
  3. 0 Correct answer.
  4. 9

Explanation

By the Cayley-Hamilton theorem, A satisfies A² - (trace A)A + (det A)I = 0. Hence A² - 5A + 3I is the zero matrix, whose determinant is 0.

Question 30Mathematics

If z is a complex number such that |z| = 1 and z ≠ -1, then the point representing (z - 1)/(z + 1) lies on

  1. the imaginary axis Correct answer.
  2. the line y = x
  3. the unit circle
  4. the real axis

Explanation

Let w = (z - 1)/(z + 1). Since |z| = 1, we have conjugate(z) = 1/z. Taking conjugates gives conjugate(w) = (1/z - 1)/(1/z + 1) = (1 - z)/(1 + z) = -w, so the real part of w is zero. Therefore w lies on the imaginary axis.

Question 31Mathematics

A set A has 6 elements. The number of proper subsets of A is

  1. 32
  2. 63 Correct answer.
  3. 64
  4. 62

Explanation

A set with n elements has 2^n subsets in total. Proper subsets exclude the set itself, so the count is 2^6 − 1 = 63.

Question 32Mathematics

The equation of the straight line passing through the point (2, 3) and parallel to the line 4x − 3y = 7 is

  1. 3x + 4y − 18 = 0
  2. 4x − 3y + 1 = 0 Correct answer.
  3. 4x + 3y − 17 = 0
  4. 4x − 3y − 1 = 0

Explanation

Parallel lines have proportional x- and y-coefficients, so the required line has the form 4x − 3y + k = 0. Substituting (2, 3) gives 8 − 9 + k = 0, hence k = 1.

Question 33Mathematics

The value of sin 75° − sin 15° is

  1. √2/2 Correct answer.
  2. √3/2
  3. 1/2
  4. √6/2

Explanation

Using sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2), we get 2 cos 45° sin 30° = 2 × (√2/2) × (1/2) = √2/2.

Question 34Mathematics

The value of ∫ from 0 to 1 of x e^(x²) dx is

  1. (e − 1)/2 Correct answer.
  2. e − 1
  3. e/2
  4. (e − 1)/4

Explanation

Substitute u = x², so du = 2x dx. The integral becomes (1/2)∫ from 0 to 1 of e^u du = (1/2)(e − 1).

Question 35Mathematics

A card is drawn at random from a standard pack of 52 playing cards. The probability that the card is a king or a heart is

  1. 13/52
  2. 4/13 Correct answer.
  3. 17/52
  4. 1/13

Explanation

There are 4 kings and 13 hearts, but the king of hearts is counted twice. Thus the favourable count is 4 + 13 − 1 = 16, and the probability is 16/52 = 4/13.

Question 36Mathematics

The perpendicular distance from the point (1, 2, 3) to the plane 2x − y + 2z = 5 is

  1. 5/3
  2. 2/3
  3. 1/3 Correct answer.
  4. 1

Explanation

The distance from (x₁, y₁, z₁) to ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²). Here the numerator is |2 − 2 + 6 − 5| = 1 and the denominator is √(4 + 1 + 4) = 3.

Question 37Mathematics

If z = (3 + 4i)/(1 + i), then |z| equals

  1. 5/2
  2. 5/√2 Correct answer.
  3. 5
  4. √5

Explanation

The modulus of a quotient is the quotient of the moduli, so |z| = |3 + 4i| / |1 + i|. This gives √(9 + 16) / √(1 + 1) = 5/√2.

Question 38Mathematics

The order and degree of the differential equation (d²y/dx²)^3 + (dy/dx)^5 = 0 are respectively

  1. 5 and 3
  2. 2 and 3 Correct answer.
  3. 2 and 5
  4. 3 and 2

Explanation

The order is the highest derivative present, which is d²y/dx², so the order is 2. The degree is the exponent of that highest derivative when the equation is polynomial in derivatives, giving degree 3.

Question 39Mathematics

The number of ways to distribute 5 distinct balls into 3 distinct boxes so that no box is empty is

  1. 120
  2. 243
  3. 150 Correct answer.
  4. 180

Explanation

Using inclusion-exclusion, the number of onto functions from 5 balls to 3 boxes is 3^5 − C(3,1)2^5 + C(3,2)1^5 = 243 − 96 + 3 = 150.

Question 40Mathematics

If a, b and c are unit vectors such that a + b + c = 0, then a·b + b·c + c·a equals

  1. −3/2 Correct answer.
  2. 3/2
  3. 0
  4. −1

Explanation

Taking the squared magnitude of a + b + c = 0 gives 0 = |a|² + |b|² + |c|² + 2(a·b + b·c + c·a). Since each vector is a unit vector, 0 = 3 + 2S, so S = −3/2.

Question 41Mathematics

If one root of the equation x² − 7x + k = 0 is 2, then the value of k is

  1. 10 Correct answer.
  2. 14
  3. 12
  4. 5

Explanation

Substituting x = 2 into the equation gives 4 − 14 + k = 0, so k = 10. Equivalently, the sum of the roots is 7, so the other root is 5 and the product of the roots is 2 × 5 = 10.

Question 42Mathematics

The coefficient of x² in the expansion of (2x − 3)⁴ is

  1. 216 Correct answer.
  2. 108
  3. 54
  4. −216

Explanation

The term containing x² comes from choosing r = 2 in the general binomial term C(4,r)(2x)^r(−3)^(4−r). This gives C(4,2)(2x)²(−3)² = 6 × 4x² × 9 = 216x². Hence the required coefficient is 216.

Question 43Mathematics

The value of sin 30° cos 60° + cos 30° sin 60° is

  1. √3/2
  2. 0
  3. 1/2
  4. 1 Correct answer.

Explanation

The expression has the form sin A cos B + cos A sin B, which equals sin(A + B). Here A = 30° and B = 60°, so the value is sin 90° = 1.

Question 44Mathematics

The sum of the infinite geometric series 4 + 2 + 1 + 1/2 + ⋯ is

  1. 12
  2. 6
  3. 16
  4. 8 Correct answer.

Explanation

The first term is a = 4 and the common ratio is r = 1/2, with |r| < 1, so the series converges. Its sum is a/(1 − r) = 4/(1 − 1/2) = 8.

Question 45Mathematics

Two independent events A and B have P(A) = 0.4 and P(B) = 0.5. The value of P(A ∪ B) is

  1. 0.9
  2. 0.1
  3. 0.7 Correct answer.
  4. 0.2

Explanation

For independent events, P(A ∩ B) = P(A)P(B) = 0.4 × 0.5 = 0.2. Using the addition rule, P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7.

Question 46Mathematics

The eccentricity of the ellipse 9x² + 16y² = 144 is

  1. 3/4
  2. 4/3
  3. √7/4 Correct answer.
  4. √7/3

Explanation

Dividing by 144 gives x²/16 + y²/9 = 1, so a² = 16 and b² = 9. Thus c² = a² − b² = 7, and the eccentricity is e = c/a = √7/4.

Question 47Mathematics

The scalar projection of the vector a = 2i + 3j + k on the vector b = i + j + k is

  1. 6
  2. √3
  3. 4√3/3
  4. 2√3 Correct answer.

Explanation

The scalar projection is (a · b)/|b|. Here a · b = 2 + 3 + 1 = 6 and |b| = √3, so the projection is 6/√3 = 2√3.

Question 48Mathematics

The number of distinct arrangements of the letters of the word CUSAT in which the two vowels are together is

  1. 24
  2. 48 Correct answer.
  3. 12
  4. 120

Explanation

The word CUSAT has five distinct letters, with vowels U and A. Treat U and A as one block, giving four units: the block, C, S, and T. These can be arranged in 4! ways, and the two vowels can be arranged inside the block in 2! ways, giving 24 × 2 = 48.

Question 49Mathematics

The value of ∫₀¹ x² e^(−x) dx is

  1. 1 − 2/e
  2. 1 − 5/e
  3. 2 + 5/e
  4. 2 − 5/e Correct answer.

Explanation

Integrating by parts twice gives an antiderivative −e^(−x)(x² + 2x + 2). Evaluating from 0 to 1 gives −5/e − (−2) = 2 − 5/e.

Question 50Mathematics

If ω is a non-real cube root of unity, then (1 + ω − ω²)³ equals

  1. 8
  2. 1
  3. −8 Correct answer.
  4. −1

Explanation

Since 1 + ω + ω² = 0, we have 1 + ω = −ω². Therefore 1 + ω − ω² = −2ω², whose cube is (−2)³(ω²)³ = −8ω⁶ = −8, because ω³ = 1.