Karnavati University Engineering Mathematics Past Paper 2024-2025 PDF
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2024
Karnavati University
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This document contains a question bank for the first semester of engineering mathematics. The questions cover linear algebra concepts such as vector spaces and subspaces. The document is suitable for undergraduate engineering students at Karnavati University.
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Karnavati University, Gandhinagar First Semester 2024-2025, (Engineering Mathematics-I) Question Bank (Unit-1) 1. Let V denote the set of ordered triples (x, y, z) and define addition in V as in R3. For each of the following definitions of scalar multipl...
Karnavati University, Gandhinagar First Semester 2024-2025, (Engineering Mathematics-I) Question Bank (Unit-1) 1. Let V denote the set of ordered triples (x, y, z) and define addition in V as in R3. For each of the following definitions of scalar multiplication, decide whether V is a vector space. (i) a(x, y, z) = (ax, y, az) (ii) a(x, y, z) = (ax, 0, az) (iii) a(x, y, z) = (0, 0, 0) (iv) a(x, y, z) = (2ax, 2ay, 2az) 2. Are the following sets vector spaces with the indicated operations? If not, why not? (i) The set V of nonnegative real numbers; ordinary addition and scalar multiplication. (ii) The set V of all polynomials of degree ≥ 3, together with 0 ; operations of P. (iii) The set of all polynomials of degree ≤ 3; operations of P. (iv) The set {1, x, x2 ,...}; operations of P. a b (v) The set V of all 2 × 2 matrices of the form ; operations of M22. 0 c (vi) The set V of 2 × 2 matrices with equal column sums; operations of M22. (vii) The set V of 2 × 2 matrices with zero determinant; usual matrix operations. (viii) The set V of real numbers; usual operations. (ix) The set V of all ordered pairs (x, y) with the addition of R2 , but using scalar multiplication a(x, y) = (ax, −ay). (x) The set V of all ordered pairs (x, y) with the addition of R2 , but using scalar multiplication a(x, y) = (x, y) for all a in R. (xi) The set V of all 2 × 2 matrices whose entries sum to 0; operations of M22. (xii) The set V of all 2 × 2 matrices with the addition of M22 but scalar multiplication * defined by a ∗ X = aX T. 3. Let V be the set of all positive real numbers with vector addition being ordinary multi- plication, and scalar multiplication being a v = v a. Show that V is a vector space. 4. If V is the set of ordered pairs (x, y) of real numbers, show that it is a vector space with addition (x, y) ⊕ (x1 , y1 ) = (x + x1 , y + y1 + 1) and scalar multiplication a (x, y) = (ax, ay + a − 1). What is the zero vector in V. 5. Which of the following are subspaces of P3 ? Support your answer. (i) U = {f (x) | f (x) ∈ P3 , f (2) = 1} (ii) U = {xg(x) | g(x) ∈ P2 } (iii) U = {xg(x) | g(x) ∈ P3 } (iv) U = {xg(x) + (1 − x)h(x) | g(x) and h(x) ∈ P2 } (v) U = The set of all polynomials in P3 with constant term 0 (vi) U = {f (x) | f (x) ∈ P3 , deg f (x) = 3} 6. Which of the following are subspaces of M22 ? Support your answer. a b (i) U = a, b, and c in R} 0 c a b (ii) U = a + b = c + d; a, b, c, d in R} c d (iii) U = A | A ∈ M22 , A = AT (iv) U = {A | A ∈ M22 , AB = 0} , B a fixed 2 × 2 matrix (v) U = {A | A ∈ M22 , A2 = A} (vi) U = {A | A ∈ M22 , A is not invertible } 7. Which of the following are subspaces of F[0, 1] ? Support your answer. (i) U = {f | f (0) = 0} (ii) U = {f | f (0) = 1} (iii) U = {f | f (0) = f (1)} (iv) U = {f | f (x) ≥ 0 for all x in [0, 1]} (v) U = {f | f (x) = f (y) for all x and y in [0, 1]} (vi) U = {f | f (x + y) = f (x) + f (y) for all x and y in [0, 1]} R1 o (vii) U = {f | f is integrable and 0 f (x)dx = 0 8. Write each of the following as a linear combination of x + 1, x2 + x, and x2 + 2. (i) x2 + 3x + 2 (ii) 2x2 − 3x + 1 (iii) x2 + 1 (iv) x 9. Determine whether v lies in span {u, w} in each case. (i) v = 3x2 − 2x − 1; u = x2 + 1, w = x + 2 (ii) v = x, u = x2 + 1, w = x + 2 Page 2 1 3 1 −1 2 1 (iii) v = ;u = ,w = −1 1 2 1 1 0 1 −4 1 −1 2 1 (iv) v = ;u = ,w = 5 3 2 1 1 0 10. (i) Show that R3 is spanned by {(1, 0, 1), (1, 1, 0), (0, 1, 1)}. (ii) Show that P2 is spanned by {1 + 2x2 , 3x, 1 + x}. (iii) Show that M22 is spanned by 1 0 1 0 0 1 1 1 , , , 0 0 0 1 1 0 0 1 11. Is it possible that {(1, 2, 0), (1, 1, 1)} can span the subspace U = {(a, b, 0) | a and b in R} ? 12. Show that each of the following sets of vectors is independent. (i) {1 + x, 1 − x, x + x2 } in P2 (ii) {x2 , x + 1, 1 − x − x2 } in P2 1 1 1 0 0 0 0 1 (iii) , , , in M22. 0 0 1 0 1 −1 0 1 1 1 0 1 1 0 1 1 (iv) , , , in M22. 1 0 1 1 1 1 0 1 13. Which of the following subsets of V are independent? (i) V = P2 ; {x2 + 1, x + 1, x} (ii) V = P2 ; {x2 − x + 3, 2x2 + x + 5, x2 + 5x + 1} 1 1 1 0 1 0 (iii) V = M22 ; , , 0 1 1 1 0 1 −1 0 1 −1 1 1 0 −1 (iv) V = M2n ; , , , 0 −1 −1 1 1 1 −1 0 1 1 1 (v) V = F[1, 2]; x , x2 , x3 1 1 (vi) V = F[0, 1]; x2 +x−6 , x2 −5x+6 , x21−9 14. Find all values of a such that the following are independent in R3. (i) {(1, −1, 0), (a, 1, 0), (0, 2, 3)} (ii) {(2, a, 1), (1, 0, 1), (0, 1, 3)} 15. Show that the following are bases of the space V indicated. (i) {(1, 1, 0), (1, 0, 1), (0, 1, 1)}; V = R3 Page 3 (ii) {(−1, 1, 1), (1, −1, 1), (1, 1, −1)}; V = R3 1 0 0 1 1 1 1 0 (iii) , , , ; 0 1 1 0 0 1 0 0 (iv) {1 + x, x + x2 , x2 + x3 , x3 } ; V = P3 16. Exercise 7.1.1 Show that each of the following functions is a linear transformation. (i) T : R2 → R2 ; T (x, y) = (x, −y) (reflection in the x ax is) (ii) T : R3 → R3 ; T (x, y, z) = (x, y, −z) (reflection in the x − y plane) (iii) T : Mnn → Mnn ; T (A) = AT + A (iv) T : Pn → R; T [p(x)] = p(0) (v) T : Pn → R; T (r0 + r1 x + · · · + rn xn ) = rn (vi) T : Rn → R; T (x) = x · z, z a fixed vector in Rn (vii) T : Pn → Pn ; T [p(x)] = p(x + 1) (viii) T : Rn → V ; T (r1 , · · · , rn ) = r1 e1 + · · · + rn en where {e1 ,... , en } is a fixed basis of V (ix) T : V → R; T (r1 e1 + · · · + rn en ) = r1 , where {e1 ,... , en } is a fixed basis of V 17. In each case, show that T is not a linear transformation. (i) T : Mnn → R; T (A) = det A (ii) T : Mnm → R; T (A) = rank A (iii) T : R → R; T (x) = x2 (iv) T : V → V ; T (v) = v + u where u 6= 0 is a fixed vector in V (T is called the translation by u) 18. In each case, assume that T is a linear transformation. (i) If T : V → R and T (v1 ) = 1, T (v2 ) = −1, find T (3v1 − 5v2 ). (ii) If T : V → R and T (v1 ) = 2, T (v2 ) = −3, find T (3v1 + 2v2 ). 2 2 1 1 1 0 −1 (iii) If T : R → R and T = ,T = , find T. 3 1 1 1 3 2 2 1 0 1 1 1 (iv) If T : R → R and T = ,T = , find T. −1 1 1 0 −7 (v) If T : P2 → P2 and T (x + 1) = x, T (x − 1) = 1, T (x2 ) = 0, find T (2 + 3x − x2 ). (vi) If T : P2 → R and T (x + 2) = 1, T (1) = 5, T (x2 + x) = 0, find T (2 − x + 3x2 ). Page 4