Mathematics SSC-II Model Question Paper PDF

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Version No. ROLL NUMBER   ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ⓪ ① ① ① ① ① ① ① ① ① ① ① ② ② ② ② ② ② ② ② ② ② ② ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ Answer Sheet No.____________ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ Sign. of Candidate___________ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ Sign. of ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ ⑨ Invigilator___________ MATHEMATICS SSC–II (Science Group) (Curriculum 2006) SECTION – A (Marks 15) Time allowed: 20 Minutes Section – A is compulsory. All parts of this section are to be answered on this page and handed over to the Centre Superintendent. Deleting/overwriting is not allowed. Do not use lead pencil. Q.1 Fill the relevant bubble for each part. All parts carry one mark. (1) Which one of the following types represents (𝑥 − 3)(𝑥 + 3) = 0? A. Quadratic equation ⃝ B. Linear equation ⃝ C. Cubic equation ⃝ D. Pure quadratic equation ⃝ (2) If 𝑏 2 − 4𝑎𝑐 of an equation is the discriminant than the equation would be of the form: A. 𝑎𝑥 2 − 𝑏𝑥 + 𝑐 = 0 ⃝ B. 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 ⃝ 2 2 C. 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 ⃝ D. 𝑎𝑥 − 𝑏𝑥 − 𝑐 = 0 ⃝ (3) Which one of the following cannot be factorized without using synthetic division method? A. 3𝑥 2 + 5𝑥 + 2 ⃝ B. 5𝑥 + 10 ⃝ 4 3 2 1 C. 3𝑥 + 3𝑥 − 2𝑥 + 6 ⃝ D. 𝑥 − 𝑥2 ⃝ (4) If 𝛼,𝛽 are the roots of 2𝑥 2 − 6𝑥 − 4 = 0, then what is value of 𝛼 2 𝛽 3 + 𝛼 3 𝛽 2 ? A. −12 ⃝ B. 12 ⃝ C. 6 ⃝ D. −6 ⃝ 𝑥3 (5) Which one of the following are the partial fractions of 𝑥 3 +1? 𝐴𝑥 3 𝐵𝑥+𝐶 𝐴 𝐵𝑥+𝐶 A. + 𝑥 2 −𝑥+1 ⃝ B. 1 + 𝑥−1 + 𝑥 2 +𝑥+1 ⃝ 𝑥+1 𝐴 𝐵𝑥+𝐶 𝐴 𝐵𝑥+𝐶 C. 1 + 𝑥+1 + 𝑥 2 −𝑥−1 ⃝ D. 1 + 𝑥+1 + 𝑥 2 −𝑥+1 ⃝ (6) Which one of the following expressions shows the shaded region? A. 𝐴 ∩ 𝐵′ ⃝ ′ B. 𝐴 ∩𝐵 ⃝ ′ C. 𝐴∪𝐵 ⃝ ′ D. 𝐴 ∪𝐵 ⃝ 𝑈 Page 1 of 2 (7) If 𝒙 = 7, ∑ 𝑓 = 30 and ∑ 𝑓𝑥 = 120 + 3𝑘 then value of k is A. 30 ⃝ B. −30 ⃝ C. −11 ⃝ D. 11 ⃝ (8) Which one of the following is NOT equal to tan 𝜃 for a unit circle? cos 𝜃 1 A. ⃝ B. ⃝ sin 𝜃 cot 𝜃 sec 𝜃 sin 𝜃 C. ⃝ D. ⃝ cos 𝜃 cos 𝜃 (9) Which one of following is the radius of a circle, if an arc of 10cm subtends an angle of 60° ? 30 𝜋 A. cm ⃝ B. cm ⃝ 𝜋 30 10800 1 C. cm ⃝ D. cm ⃝ 𝜋 6 𝑜 (10) What is the value of 𝑚∠𝐴𝑂𝐵 in the adjoining figure of a hexagon? A. 360° ÷ 45° ⃝ 𝐴 B. 360° ÷ 60° ⃝ C. 360° ÷ 30° ⃝ 𝐵 D. 360° ÷ 120° ⃝ (11) What is the elevation of Sun if a pole of 6m high casts a shadow of 2√3𝑚? A. 30° ⃝ B. 45° ⃝ C. 60° ⃝ D. 90° ⃝ (12) If ̅̅̅̅ 𝐴𝐵 = 6𝑐𝑚 is a chord of a circle with centre O and̅̅̅̅̅ 𝑂𝐶 ⊥ ̅̅̅̅ 𝐴𝐵 , then length of AC will be: A C B A. 3 ⃝ B. 2 ⃝ O C. 12 ⃝ D. 14 ⃝ (13) What is the value of x if 64, x and 1 are in continued proportion? A. 3 ⃝ B. ±√3 ⃝ C. √3 ⃝ D. ±3 ⃝ D (14) In the drawn figure, what is the value of 𝑚∠𝐵𝐶𝐷? C A. 165° ⃝ B. 155° ⃝ C. 80° ⃝ D. 130° ⃝ O 50° A B (15) If 𝑓: 𝐵 → 𝐴, then which one of the following represents a/an? f A. Onto function ⃝ A 𝐵 B. Bijective function ⃝ C. Injective function ⃝ 1 a 2 D. Into function ⃝ 3 b 4 c Page 2 of 2  Federal Board SSC-II Examination Mathematics Model Question Paper (Science Group) (Curriculum 2006) Time allowed: 2.40 hours Total Marks: 60 Note: Attempt any nine parts from Section ‘B’ and any three questions from Section ‘C’ on the separately provided answer book. Write your answers neatly and legibly. Log book will be provided on demand. SECTION – B (Marks 36) Q.2 Attempt any NINE parts from the following. All parts carry equal marks. (9  4 = 36) i. Solve the equation 3𝑥 2 + 4𝑥 − 5 = 5𝑥 2 + 2𝑥 + 1. ii. Product of two consecutive numbers is 132. a. If the smaller number is x then what is the larger number? b. Show that 𝑥 2 + 𝑥 − 132 = 0 c. Solve the equation 𝑥 2 + 𝑥 − 132 = 0 and hence find the numbers. iii. If P is directly proportional to Q and P = 12 when Q = 4.What is: a. the equation connecting P and Q. b. the value of P, when Q = 8 c. the value of Q, when P = 21 iv. Solve the system of equations: 4𝑥 2 + 3𝑦 2 = 37 ; 3𝑥 2 − 𝑦 2 = 5 v. If U= {1, 2, 3,..... , 10}, A = {2, 4, 6} and B = {1, 3, 5}, then find a. A’ b. B’ c. (A∩B)’ d. Verify that (A∩B)’ = A’ U B’ vi. Given that set A = {1, 2, 3} and B = {2, 4, 6}, then find: (i) A × B (ii) R = {(x, y) | y = 2x} (iii) Domain and Range of R vii. The table given below shows the number of goals scored by a soccer team in 10 matches: 4 1 2 1 0 0 3 2 3 3 Find: a. Mean b. Median c. Mode 4 viii. If 𝑡𝑎𝑛 𝜃 = 3 and 𝑠𝑖𝑛 𝜃 < 0 a. Find the quadrant in which the terminal side of the angle lies? b. Find the values of 𝑠𝑒𝑐 𝜃 and 𝑐𝑜𝑠𝑒𝑐𝜃. c. Show that 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑜𝑠𝑒𝑐 2 𝜃. 𝑠𝑖𝑛 𝜃 ix. Prove that 1+𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑡 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃. x. In ∆𝑃𝑄𝑅, 𝑚𝑄𝑅 = 6𝑐𝑚, 𝑚𝑃𝑅 = 2√2𝑐𝑚 and ∠𝑃𝑅𝑄 = 135˚. a. Draw perpendicular from P to 𝑄𝑅, to meet 𝑄𝑅 produced at S and find 𝑅𝑆. b. Find 𝑃𝑄 by using (𝑚𝑃𝑄)2 = (𝑚𝑄𝑅)2 + (𝑚𝑃𝑅)2 + 2(𝑚𝑄𝑅)(𝑚𝑅𝑆). Page 1 of 2  xi. In the given figure, 𝑚𝐴𝐵̅̅̅̅ = 10𝑐𝑚, 𝑚𝐶𝐷 ̅̅̅̅ = 8𝑐𝑚 𝑚𝑂𝐴 = 7𝑐𝑚 8cm D C Find: (i) ̅̅̅̅̅ 𝑚𝐴𝑀 (ii) ̅̅̅̅ 𝑂𝑃 (iii) ̅̅̅̅̅ 𝑚𝑂𝑀 P O 7cm M A B 10cm xii. Prove that if a line is drawn perpendicular to a radial segment of a circle at its outer end point, it is tangent to the circle at that point. xiii. A, B, C and P are four points on a circle with centre O. P Given that POC is a diameter of the circle. 𝒙 30° Find: O a. 𝑥 b.𝑦 c.∠𝐴𝑂𝐵 𝒚 Also write the reasons to justify your steps. A B C xiv. ̅̅̅̅ = 6𝑐𝑚, 𝐵𝐶 Circumscribe a circle about a triangle ABC with sides 𝐴𝐵 ̅̅̅̅ = 4𝑐𝑚, ̅̅̅̅ 𝐴𝐶 = 4𝑐𝑚 and measure its radius. SECTION – C (Marks 24) Note: Attempt any THREE questions. Each question carries equal marks. (3  8 = 24) Q.3 The area of a rectangle is 48cm2. If length and width of each is increased by 4cm. The area of larger rectangle is increased by 12cm2. Find the length and width of the original rectangle. Q.4 Prove that if two arcs of a circle (or of congruent circles) are congruent then the corresponding chords are equal. 𝑥−6𝑎 𝑥+6𝑏 Q.5 Using theorem of componendo-dividendo, find the value of − , if 𝑥+6𝑎 𝑥−6𝑏 12𝑎𝑏 𝑥= 𝑎−𝑏 𝑥2 Q.6 Resolve (1−𝑥)(1+𝑥 2)2 into partial fractions. Q.7 Find the range, variance and standard deviation for the following data set: 1245, 1255, 1654, 1547, 1245, 1255, 1547, 1737, 1989, 2011. Page 2 of 2  MATHEMATICS SSC-II Student Learning Outcomes Alignment Chart (Curriculum 2006) Q1 Contents and Scope Student Learning Outcomes Sec-A 1 8.1 Quadratic Equation Define quadratic equation. 9.1 Nature of the Roots of a i) Define discriminant (𝑏2 − 4𝑎𝑐) of 2 thequadratic expression ax2+bx +c. Quadratic Equation 3 9.6 Synthetic Division i) Describe the method of synthetic division. 9.4Symmetric Functions of Roots ii) Evaluate a symmetric Function of the roots of a 4 of a Quadratic Equation. quadratic equation in terms of its coefficients. 11.2 Resolution of Fraction Resolve an algebraic fraction into partial fractions 5 into Partial Fractions. when its denominator consists of non-repeated linear factors. 12.1.3 Venn Diagram i)Use Venn diagram to represent 6 union and intersection of sets, complement of a set. 13.3 Measures of Central i) Calculate the arithmetic mean by definition (for 7 Tendency ungrouped data) 16.3 Trigonometric Ratios iii) Define trigonometric ratios and their reciprocals 8 with the help of a unit circle. 16.2 Sector of a circle i) Establish the rule 𝑙 = 𝑟𝜃 , where 𝑟 is the radius of 9 the circle, 𝑙 the length of circular arc and 𝜃 the central angle measured in radians. 30.2 Circles attached to polygons viii) Circumscribe a regular hexagon about a given 10 circle. 16.5Angle of elevation and ii) Solve real life problems involving angle of 11 Depression. elevation and depression 25.1 Chords of a Circle Prove the following theorem along with corollaries and apply them to solve appropriate 12 problems. iii) Perpendicular from the centre of a circle on a chord bisects it. 10.1 Ratio, Proportions and ii) Find 3rd, 4th mean and continued proportion. 13 Variations 28.1 Angle in a Segment of a Prove the following theorem along with Circle corollaries and apply them to solve appropriate 14 problems. i) The measure of a central angle of a minor arc of a circle, is double that of the angle subtended by the corresponding major arc. 12.3 Function ii) To demonstrate the following: Into function 15 One-one function Injective function Surjective function  Bijective function Sec B Q2 8.2Solution of Quadratic i) Solve a quadratic equation in one variable by i Equations Factorization, Completing square 9.7 Simultaneous Equations Solve a system of two equations in two variables when ii one equation is linear and the other is quadratic, both the equations are quadratic. 10.1 Ratio, Proportion and i) Define ratio, proportions and variations iii Variation. (direct and inverse) 9.7 Simultaneous Equations Solve a system of two equations in two variables when iv one equation is linear and the other is quadratic, both the equations are quadratic. 12.1.2 Properties of Union and iv) Give formal proofs of the following fundamental Intersection properties of union and intersection of two or three sets. Commutative property of union, Commutative property of intersection, v Associative property of union, Associative property of intersection, Distributive property of union over intersection, Distributive property of intersection over union, De Morgan’s laws. 12.1.4 Ordered Pairs and viii) Recognize ordered pairs and Cartesian Cartesian Product product. vi 12.2 Binary relation Define binary relation and identify its domain and range. 13.3 Measures of Central i) Calculate Tendency (for ungrouped and grouped data) vii Arithmetic mean by definition and using deviations from assumed mean, Median, mode geometric mean and harmonic mean 16.3 Trigonometric Ratios v) Recognize the signs of trigonometric ratios in different quadrants viii vi) Find the values of remaining trigonometric ratios if one trigonometric ratio is given. 16.4 Trigonometric Identities Prove the trigonometric identities and apply them to ix show different trigonometric relations. 24.1 Projection of a side of a Prove the following theorem along with x triangle corollaries and apply them to solve appropriate problems.  i) In an obtuse-angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle together with twice the rectangle contained by one of the sides, and the projection on it of the other. 25.1 Chords of a Circle Prove the following theorem along with corollaries and apply them to solve appropriate xi problems. iii) Perpendicular from the centre of a circle on a chord bisects it. 26.1 Tangent to a Circle Prove the following theorems along with corollaries and apply them to solve appropriate problems. xii i) If a line is drawn perpendicular to a radial segment of a circle at its outer end point, it is tangent to the circle at that point. 28.1 Angle in a Segment of a Prove the following theorem along with Circle corollaries and apply them to solve appropriate problems. xiii i) The measure of a central angle of a minor arc of a circle, is double that of the angle subtended by the corresponding major arc. 30.2 Circles attached to i) Circumscribe a circle about a given triangle. xiv Polygons Sec C 9.7 Simultaneous Equations Solve the real life problems leading to quadratic Q3 equations. 27.1 Chords and Arcs Prove the following theorems along with corollaries and apply them to solve appropriate problems. Q4 i) If two arcs of a circle (or of congruent circles) are congruent then the corresponding chords are equal. 10.2 Theorems on Proportion Apply theorem of componendo-dividendo to find Q5 proportions. 11.2 Resolution of Fraction into Resolve an algebraic fraction into partial Q6 Partial Fractions fractions when its denominator consists of repeated quadratic factors. Q 7 13.4 Measures of Dispersion Measure range, variance and standard deviation.  MATHEMATICS SSC-II Table of Specifications 8. Quadratic Equations 9. Theory of Quadratic 10. Variations 11. Partial Fractions 12. Sets and Functions 13. Basic Statistics 16. Introduction to Triangle 24. Projection of a Side of a 25. Chords of a Circle 26. Tangent to a Circle 27. Chords and Arcs 28. Angle in a Segment 30. Practical Geometry assessment objective Total marks for each % age Trigonometry Equations Circles of a Circle Topics Knowledge based 1 (15) (1) 1 (8) (1) 1 (10) (1) 1 (1) (1) 1 (2) (1) 2 iii (4) 2 vi (4) 7 (4) 2 xii (4) 4 (8) 33 29.7% 2 viii (2) 2 v (2) Understanding 1 (3) (1) based 1 (4) (1) 1 (7) (1) 1 (5) (1) 1 (6) (1) 2 viii (2) 2 i (4) 2 ii (4) 1(13) (1) 2 vii (4) 2 x(a) (2) 1 (14) (1) 2 xiv (4) 55 49.5% 6 (8) 2 v (2) 2 ix (2) 2 iv (4) 7 (4) 3 (8) Application 1 (9) (1) based 1 (12) (1) 5 (8) 1 (11) (1) 2 x (b)(2) 2 xiii (4) 23 20.7% 2 xi (4) 2 ix (2) Total marks for each topic 05 19 13 09 10 13 11 04 05 04 08 05 05 111 100% KEY: 1(1)(1) Question No. (Part No.) (Allocated Marks)