Tiêu đề D.J. Science College - XI Mathematics Prelim Paper.pdf ✅

DJ Sindh Govt. Science College, Karachi Preliminary Examination 2024 Mathematics – I For Science Pre-Engineering & Science General Groups Time allowed: 3 hours Maximum Marks: 100 SECTION ‘A’ (MULTIPLE CHOICE QUESTIONS) – (M.C.Qs.) Time allowed: 20 minutes (Marks : 20) NOTE: (i). This section consists of 20 part questions and all are to be answered. Each question carries one mark. (ii). The correct answer bubble must be filled on OMR sheet (I) A B C D pasted in answer script. (iii). Use only blue / black ball point pen or pointer on OMR sheet. (iv). Avoid using pencil / White-o pen on OMR sheet. (v). All notations are used in their usual meanings. The use of Scientific Calculator is allowed. Q. 1. Choose the correct answer for each from the given options. 1. |x + 5i| = 3 then x is equal to: (a) ±4 (b) ±4i (c) ±22i (d) None of these. 2. ( 1+i 1−i ) 2 = _________________ (a) −1 (b) 1 (c) i (d) −i 3. A square matrix A = [aij] for which all aij = 0, i < j then A is called: (a) Upper triangular (b) Lower triangular (c) Symmetric (d) Hermitian 4. The number of non-zero rows in echelon form of a matrix is: (a) Order (b) Rank (c) Leading Row (d) Leading Column 5. The cofactor A22 of [ 1 2 4 −1 2 5 0 1 −1 ] is: (a) 0 (b) −1 (c) 1 (d) 2 6. If |a⃗ × b⃗⃗| = |a⃗ . b⃗⃗| then the angle between a⃗ and b⃗⃗ is: (a) 0 (b) π 2 (c) π 4 (d) π 7. The number of vectors of unit length perpendicular to the plane of vectors a⃗ = 2î+ ĵ+ 2k̂ and b⃗⃗ = ĵ+ k̂ is: (a) One (b) Two (c) Three (d) Infinite 8. The sum of infinite geometric series is a finite number if: (a) |r| > 1 (b) |r| = 1 (c) |r| ≥ 1 (d) |r| < 1 9. If a−b b−c = a b then a, b and c are in: (a) A.P (b) G.P (c) H.P (d) None of These 10. ∑ n 20 0 n=3 = _____________: (a) 1 (b) 19 (c) 20 (d) 18
11. Two teams A and B are playing a match, the probability that team A does not loose is: (a) 1 3 (b) 2 3 (c) 1 2 (d) 0 12. The number of terms in the expansion of(1 + x) 1 3 is: (a) 4 3 (b) 4 (c) ∞ (d) 2 13. The expansion of (1 − 2x) −2 is valid if: (a) |x| < 0 (b) |x| < 1 2 (c) |x| < 2 (d) |x| < 1 14. The graph of y = x n is symmetric to _________ if n is an odd integer. (a) x − axis (c) y − axis (c) Origin (d) None 15. A point of a solution region region where two of its boundary lines intersect is called_______. (a) Middle point (b) Origin (c) Corner point (d) Feasible point 16. 2sin 7θ sin 2θ is equal to: (a) cos 5θ − cos 9θ (b) cos 9θ − cos 5θ (c) sin 9θ + sin 5θ (d) sin 9θ − sin 5θ 17. cos 2θ is equal to: (a) cos2 θ − sin2 θ (b) 2 cos2 θ − 1(c) 1 − 2 sin2 θ (d) all of these 18. Circum radius of ∆ABC is: (a) ∆ S (c) ∆ S−b (c) ∆ S−a (d) abc 4∆ 19. Inverse function exists if and only if the function is: (a) Bijective (b) One-to-One (c) Onto (d) Into 20. tan−1 (−x) = _____________. (a) − tan−1 x (b) π − tan−1 x (c) cot−1 x (d) tan−1 x
DJ Sindh Govt. Science College, Karachi Preliminary Examination 2024 Mathematics – I For Science Pre-Engineering & Science General Groups Time allowed: 2 hours and 40 minutes Maximum Marks: 80 NOTE: (i). Attempt any ten parts from Section ‘B’ and any five questions from Section ‘C’. SECTION ‘B’ (SHORT-ANSWER QUESTIONS) (Marks : 40) NOTE: Answer Any ten parts of the questions from this section. All questions carry equal marks. (i). Find the real and imaginary part of ( 3i−2 2−3i ) −2 by using any method. (ii).Find the adjoint of the matrix A = [ 4 6 8 1 3 2 2 7 5 ]. , (iii). Without expanding determinant, prove that | 1 a b + c 1 b c + a 1 c a + b | = 0. (iv). Find a unit vector perpendicular to each of vector a⃗ + b⃗⃗ and a⃗ − b⃗⃗ where; a⃗ = 3î+ 5ĵ+ k̂ and b⃗⃗ = î+ 2ĵ− 5k̂. (v).Find three consecutive terms in G.P. whose sum is 39 and their product is 729. (vi). Find the sum of first forty-six natural numbers starting from the 10. (vii). How many diagonals and triangles can be drawn in a plane by joining the vertices of an octagon? (viii). Prove that 7 n − 4 n is divisible by 3, using mathematical induction. (ix). The coefficient of the fifth, sixth and seventh terms of the expansion (1 + x) n form an A.P. Find n. (x). Find the inverse of f and determine the domain and range of f −1 for the real valued function f(x) = x−1 x−3 , x ≠ 3. (xi). Find the equation of the graph of the function of the type y = 3x 2 + bx + c which crosses the x-axis at the points (−4, 0) and (5, 0). (xii). If A + B + C = 180o than prove that: tan A + tanB + tan C = tan A tanB tan C (xiii). Find the smallest angle in ΔABC, when: a = 25cm, b = 18cm and c = 21cm (xiv). Two hikers start from the same point; one walks 9km heading east, the other one 10 km heading 55o northeast. How far apart are they at the end of their walks? (xv). By using properties of graph of cosine, show that; cos(2π + θ) = cos θ. OR Show that: sin(cos−1 y) = √1 − y 2. Q.No. 2
SECTION ‘C’ (DETAILED-ANSWER QUESTIONS) (Marks : 40) NOTE: Answer any Five questions from this section. Question 3 is compulsory. All questions carry equal marks. (i. e. 8 marks of each question). 3. Use Gauss-Jordan method to solve the system of linear equations: x − y + 4z = 4 2x + 2y − z = 2 3x − 2y + 3z = −3 4. Find n so that x n−5+y n−5 x n−6+yn−6 may be H.M between x and y. 5. A marble is drawn at random from a box containing 20 red, 10 white, 25 orange and 15 blue marbles. Find the probability that it is: (i) orange or red (ii) not blue or red (iii) red, white or blue 6. If 1 x = 2 5 + 1.3 2! ( 2 5 ) 2 + 1.3.5 3! ( 2 5 ) 3 + ⋯, then use binomial theorem to show that: 4x 2 − 20x − 1 = 0. 7. In a parallelogram ABCD, X is the midpoint of AB⃗⃗⃗⃗⃗⃗ and Y divides BC⃗⃗⃗⃗⃗⃗ in 1: 2. Show that if Z divides DX⃗⃗⃗⃗⃗⃗ in 6: 1 then it also divides AY⃗⃗⃗⃗⃗⃗ in 3: 4. 8. A toy factory manufactures two types of toys A and B and sells for Rs.25 and Rs. 20 respectively. The toys A and B respectively, require 20 units and 12 units daily out of 2000 available resources units. Both require a production time of 5 minutes and total working hours are 9 per day. What should be quantity for each of the toys to maximize the selling amount? 9. Find general solution of trigonometric equation: sin 4θ − sin 2θ = cos 3θ and verify the solution. 10. Use trigonometric formulae to prove that: sin 2θ + sin 4θ + sin 6θ + sin 8θ cos 2θ + cos 4θ + cos 6θ + cos 8θ = tan 5θ