Mathematics Standard Set-1 CBSE Examination Board Question Paper 30/4/1

Examination Documentation and Logistics\n\n* Examination Series: C4ABDI4\n* Set Number: SET-1\n* Question Paper Code: 30/4/1\n* Subject: MATHEMATICS (STANDARD)\n* Time Duration: 3 hours\n* Maximum Marks: 80\n* Question Paper Contents: 23 printed pages and 38 questions.\n* Reading Instructions:\n * 15 minutes of dedicated reading period starting from 10:15 a.m. to 10:30 a.m.\n * During this period, candidates are prohibited from writing any answers in the answer-book.\n * Candidate must write the Q.P. Code on the title page of the answer-book.\n\n# General Instructions\n\n1. Mandatory Completion: All 38 questions are compulsory.\n2. Paper Structure: The paper is divided into five Sections: A, B, C, D, and E.\n3. Section Details:\n * Section A: Questions 1 to 18 (MCQs) and Questions 19 to 20 (Assertion-Reason), totaling 20 questions of $1$ mark each.\n * Section B: Questions 21 to 25 (Very Short Answer Type), totaling 5 questions of $2$ marks each.\n * Section C: Questions 26 to 31 (Short Answer Type), totaling 6 questions of $3$ marks each.\n * Section D: Questions 32 to 35 (Long Answer Type), totaling 4 questions of $5$ marks each.\n * Section E: Questions 36 to 38 (Case Study Based), totaling 3 questions of $4$ marks each.\n4. Internal Choices: No overall choice is provided. Internal choices are provided in 2 questions of Section B, 2 in Section C, 2 in Section D, and 3 in Section E.\n5. Mathematical Constants: Use $\frac{22}{7}$ for $\pi$ wherever required unless otherwise stated.\n6. Supplementary Tools: Calculators are strictly prohibited. Diagrams must be drawn neatly.\n\n# Section A: Multiple Choice Questions (1 Mark Each)\n\n1. Linear Equations in Two Variables:\n * Question: If $ax + by = a - b$ and $bx + ay = 0$, then the value of $x + y$ is:\n * (A) $\frac{a - b}{a^2 + b^2}$ (B) $\frac{a - b}{a + b}$ (C) $\frac{a - b}{a - b}$ (D) $\frac{a + b}{a - b}$\n\n2. HCF and LCM Properties:\n * Question: The HCF of two numbers $65$ and $104$ is $13$. If the LCM of $65$ and $104$ is $40x$, then the value of $x$ is:\n * (A) $5$ (B) $8$ (C) $40$ (D) $13$\n\n3. Polynomial Evaluation:\n * Question: If a polynomial $p(x)$ is given by $p(x) = x^2 - 5x + 6$, then the value of $p(1) + p(4)$ is:\n * (A) $0$ (B) $2$ (C) $1$ (D) $-1$\n\n4. Quadratic Discriminant:\n * Question: If the discriminant of the quadratic equation $3x^2 - 2x + c = 0$ is $16$, then the value of $c$ is:\n * (A) $1$ (B) $4$ (C) $2$ (D) $-4$\n\n5. Circle Geometry (Arc Length):\n * Question: If an arc subtends an angle of 90^\\circ at the centre of a circle, then the ratio of its length to the circumference of the circle is:\n * (A) $1:2$ (B) $1:3$ (C) $1:4$ (D) $2:3$\n\n6. Sector of a Circle:\n * Question: The area of the sector of a circle of radius $12\\,\\text{cm}$ is $60\\pi\\,\\text{cm}^2$. The central angle of this sector is:\n * (A) 120^\\circ (B) 150^\\circ (C) 75^\\circ (D) 60^\\circ\n\n7. Empirical Relationship in Statistics:\n * Question: If the difference of mode and median of a data is $24$, then the difference of its median and mean is:\n * (A) $12$ (B) $24$ (C) $8$ (D) $36$\n\n8. Probability with Dice:\n * Question: Two dice are tossed simultaneously. The probability of getting odd numbers on both the dice is:\n * (A) $\\frac{1}{6}$ (B) $\\frac{1}{4}$ (C) $\\frac{1}{2}$ (D) $\\frac{3}{4}$\n\n9. Solid Hemisphere Geometry:\n * Question: The ratio of the total surface area of a solid hemisphere to the square of its radius is:\n * (A) $2\\pi:1$ (B) $3\\pi:1$ (C) $4\\pi:1$ (D) $1:4\\pi$\n\n10. Trigonometric Values:\n * Question: If $\sin(\\theta) = 1$, then the value of $\frac{1}{2}\\sin(\\frac{\\theta}{2})$ is:\n * (A) $1$ (B) $\\frac{1}{2}$ (C) $\\frac{1}{2\\sqrt{2}}$ (D) $\\frac{1}{\\sqrt{2}}$\n\n11. Arithmetic Progression (A.P.):\n * Question: Three numbers in A.P. have the sum $30$. What is its middle term?\n * (A) $4$ (B) $10$ (C) $16$ (D) $8$\n\n12. Triangle Similarity (Thales's Theorem):\n * Question: In $\triangle ABC$, $DE \\parallel BC$. If $AD = 4\\,\\text{cm}$, $AB = 9\\,\\text{cm}$ and $AC = 13.5\\,\\text{cm}$, then the length of $EC$ is:\n * (A) $6\\,\\text{cm}$ (B) $7.5\\,\\text{cm}$ (C) $9\\,\\text{cm}$ (D) $5.7\\,\\text{cm}$\n\n13. Parallel Lines in Algebra:\n * Question: Two lines are given to be parallel. The equation of one of these lines is $5x - 3y = 2$. The equation of the second line can be:\n * (A) $15x + 9y = 5$ (B) $-15x + 9y = 5$ (C) $-15x - 9y = 5$ (D) $9x - 15y = 6$\n\n14. Heights and Distances:\n * Question: At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude at that time is:\n * (A) 30^\\circ (B) 45^\\circ (C) 60^\\circ (D) 90^\\circ\n\n15. Circle Tangents:\n * Question: In the given figure, $AB$ and $AC$ are tangents to the circle. If \angle ABC = 42^\\circ, then the measure of $\angle BAC$ is:\n * (A) 96^\\circ (B) 42^\\circ (C) 106^\\circ (D) 86^\\circ\n\n16. Coordinate Geometry - Parallelograms:\n * Question: The fourth vertex $D$ of a parallelogram $ABCD$ whose three vertices are $A(-2, 3)$, $B(6, 7)$ and $C(8, 3)$ is:\n * (A) $(0, 1)$ (B) $(1, 0)$ (C) $(-1, 0)$ (D) $(0, -1)$\n\n17. Basic Probability Identities:\n * Question: For an event $E$, if $P(E) + P(\\bar{E}) = q$, then the value of $q^2 - 4$ is:\n * (A) $-3$ (B) $3$ (C) $5$ (D) $-5$\n\n18. Geometric Properties of Tangents:\n * Question: QR is a common tangent to two circles touching externally at $A$. The tangent at $A$ meets $QR$ at $P$. If $AP = 4.2\\,\\text{cm}$, then the length of $QR$ is:\n * (A) $4.2\\,\\text{cm}$ (B) $2.1\\,\\text{cm}$ (C) $8.4\\,\\text{cm}$ (D) $6.3\\,\\text{cm}$\n\n# Assertion and Reason Statements (1 Mark Each)\n\nCodes for Selection:\n(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).\n(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).\n(C) Assertion (A) is true, but Reason (R) is false.\n(D) Assertion (A) is false, but Reason (R) is true.\n\n19. Assertion (A): The ratio in which the point $(-3, k)$ divides the line segment joining the points $(-5, 4)$ and $(-2, 3)$ is $1:2$.\n Reason (R): Mid-point of a line segment divides the line segment in the ratio $1:1$.\n\n20. Assertion (A): If the circumference of a circle is $176\\,\\text{cm}$, then its radius is $28\\,\\text{cm}$.\n Reason (R): Circumference $= 2\\pi \\times \\text{radius}$ of a circle.\n\n# Section B: Very Short Answer Questions (2 Marks Each)\n\n21. Lowest Common Multiple (LCM) Application:\n Three bells toll at intervals of $9$, $12$, and $15$ minutes respectively. If they start tolling together, after what time will they next toll together?\n\n22. Sector and Arc Calculations:\n * (a) The minute hand of a clock is $14\\,\\text{cm}$ long. Find the area on the face of the clock described by the minute hand in $5$ minutes.\n * OR\n * (b) Find the length of the arc of a circle which subtends an angle of 60^\\circ at the centre of the circle of radius $42\\,\\text{cm}$.\n\n23. Trigonometric Ratios and Angles:\n * (a) Evaluate: \frac{5\\cos^2(60^\\circ) + 4\\sec^2(30^\\circ) - \\tan^2(45^\\circ)}{\\sin^2(30^\\circ) + \\sin^2(60^\\circ)}\n * OR\n * (b) If $\sin(A - B) = \\frac{1}{2}$ and $\cos(A + B) = \\frac{1}{2}$, where 0^\\circ < A + B \\le 90^\\circ and $A > B$, find $\angle A$ and $\angle B$.\n\n24. Circle Angles:\n In the given figure, $O$ is the centre of the circle. If \angle AOB = 145^\\circ, then find the value of $x$.\n\n25. Triangle Similarity:\n In the given figure, $\triangle AHK \\sim \\triangle ABC$. If $AK = 8\\,\\text{cm}$, $BC = 3.2\\,\\text{cm}$ and $HK = 6.4\\,\\text{cm}$, then find the length of $AC$.\n\n# Section C: Short Answer Questions (3 Marks Each)\n\n26. Probability with Three Coins:\n Three coins are tossed simultaneously. What is the probability of getting:\n (i) at least one head?\n (ii) at most one tail?\n (iii) exactly two tails?\n\n27. Trigonometric Proof:\n Prove that: $\frac{\\sin(\\theta) - \\cos(\\theta) + 1}{\\sin(\\theta) + \\cos(\\theta) - 1} = \\frac{1}{\\sec(\\theta) - \\tan(\\theta)}$\n\n28. Probability with Numbered Discs:\n A box contains $90$ discs numbered from $1$ to $90$. If one disc is drawn at random, find the probability that it bears a:\n (i) 2-digit number less than $40$.\n (ii) number divisible by $5$ and greater than $50$.\n (iii) a perfect square number.\n\n29. Linear Equations (Currency Context):\n Rehana went to a bank to withdraw $₹2000$. She asked the cashier to give her $₹50$ and $₹100$ notes only. Rehana received $25$ notes in total. Find how many notes of each denomination she received.\n\n30. Irrationality Proof:\n Prove that $5 + 2\\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.\n\n31. Polynomial Calculations:\n * (a) Find the zeroes of the polynomial $4x^2 + 4x - 3$ and verify the relationship between zeroes and coefficients of the polynomial.\n * OR\n * (b) If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 + x - 2$, then find the value of $\alpha^2 + \beta^2$.\n\n# Section D: Long Answer Questions (5 Marks Each)\n\n32. Heights and Distances (Pillars):\n Two pillars of equal lengths stand on either side of a road which is $100\\,\\text{m}$ wide, exactly opposite each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60^\\circ and 30^\\circ respectively. Find the length of each pillar and the distance of the point on the road from the pillars. (Use $\sqrt{3} = 1.732$).\n\n33. Geometric Similarity Proofs:\n * (a) $E$ is a point on side $AD$ produced of a parallelogram $ABCD$ and $BE$ intersects $CD$ at $F$. Show that $\triangle ABE \\sim \\triangle CFB$.\n * OR\n * (b) Sides $AB$, $BC$ and the median $AD$ of $\triangle ABC$ are respectively proportional to sides $PQ$, $QR$ and the median $PM$ of another $\triangle PQR$. Prove that $\triangle ABC \\sim \\triangle PQR$.\n\n34. Quadratic Equation Applications:\n * (a) A train travels a distance of $90\\,\\text{km}$ at a constant speed. Had the speed been $15\\,\\text{km/h}$ more, it would have taken $30$ minutes less for the journey. Find the original speed of the train.\n * OR\n * (b) Find the value of $c$ for which the quadratic equation $(c + 1)x^2 - 6(c + 1)x + 3(c + 9) = 0$ (where $c \\ne -1$) has real and equal roots.\n\n35. Statistics (Mode and Mean):\n The following table shows the ages of patients admitted in a hospital during a year:\n\n| Age (in years) | Number of Patients |\n| :--- | :--- |\n| 5 - 15 | 6 |\n| 15 - 25 | 11 |\n| 25 - 35 | 21 |\n| 35 - 45 | 23 |\n| 45 - 55 | 14 |\n| 55 - 65 | 5 |\n\nFind the mode and mean of the data given above.\n\n# Section E: Case Study Based Questions (4 Marks Each)\n\n36. Case Study 1: Rocket Science (Coordinate Geometry)\n Ryan, fascinated by stars, sketches a rocket design on a grid sheet with points marked A through G. Some points and lines are placed on a Cartesian coordinate system.\n * (i) Find the mid-point of the segment joining point $F$ and point $G$. (Reference grid shows coordinates for $F$ and $G$).\n * (ii) What is the distance between the points $A$ and $C$?\n * (iii) What are the coordinates of the point $D$?\n\n37. Case Study 2: Treasure Hunt (Arithmetic Progression)\n Treasure Hunt involves clues hidden in various spots forming an Arithmetic Progression (A.P.).\n * The problems likely involve finding specific terms ($a_n$), common difference ($d$), or the sum of terms ($S_n$) based on given clue values which were not explicitly listed in text but hinted at as forming a sequence.\n\n38. Case Study 3: Geometry/Statistics (Content varied by paper)\n The final section of Section E involves a third situational problem requiring mathematical modeling, typically involving a cylinder-top surface area or data interpretation.