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### 1. Real Numbers #### 1 Mark Questions #### HCF and LCM of Primes 1. **H.C.F(A, B)** if $A = p^3q^2$ and $B = pq^3$: $pq^2$ 2. **L.C.M(20, 35)** if H.C.F(20, 35) = 5: 140 3. **Find a** if HCF(a, 20) = 2, LCM(a, 20) = 60: 6 4. **H.C.F of 96 and 404**: 4 5. **HCF of two co-prime numbers**: 1 6. **HCF of 17, 23 and 29**: 1 7. **HCF of 96 and 404** if LCM is 9696: 4 8. **HCF of two consecutive natural numbers**: 1 9. **HCF of two consecutive prime numbers**: 1 10. **Find m** if LCM of 12 and 42 is 10m + 4: 5 11. **Find x** if HCF(15, 21) = 15 – 3x: 2 12. **HCF of least prime and least composite number**: HCF(2, 4) = 2 13. **Find LCM(306, 657)** if HCF(306, 657) = 9: 22338 14. **LCM of 6 and 120**: 120 15. **Find L.C.M(26, 91)** if H.C.F(26, 91) = 13: 182 16. **L.C.M(35, 63)** if HCF(35, 63) = 7: 315 17. **L.C.M of 21 and 77**: 231 18. **LCM, HCF of 8, 9, 25** using prime factorisation: HCF = 1, LCM = 1800 #### Fundamental Theorem of Arithmetic 21. **Definition**: Every composite number can be uniquely expressed as a product of primes, apart from the order in which the prime factors occur. #### Rational and Irrational Numbers 25. **Union of rational and irrational numbers**: Real Numbers 27. **Which is irrational**: $\sqrt{19}$ 28. **Match the following**: - $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})$: rational number - $\sqrt{5}+2$: irrational number - $\sqrt{196}$: first natural number #### Prime Factorization 23. **Express 156 as product of prime factors**: $2^2 \times 3 \times 13$ 24. **Write 210 as product of primes**: $2 \times 3 \times 5 \times 7$ 26. **Prime factorisation of Ramanujan number**: $2^3 \times 5 \times 13 \times 17$ (Ramanujan number is 1729) 33. **Find prime factorization of 30**: $2 \times 3 \times 5$ #### Miscellaneous Real Numbers 19. **Other number** if HCF = 27, LCM = 162, one number = 54: 81 20. **Expression ending with zero**: $(2 \times 5)^n$ 22. **Statement (A): $\pi$ is irrational. Statement (B): All non-terminating and non-repeating decimals are irrational.** Both A and B are true. 29. **Statement (A): $121 + \sqrt{121}$ is irrational. Statement (B): $6 - \sqrt{6}$ is rational.** A is false and B is true. 30. **Assertion (A): $\sqrt{15}$ is irrational. Reason (R): $\sqrt{P}$ is irrational, where 'P' is prime.** Both A and R are true and R is the correct reason of A. 31. **For any three positive integers a, b and c: $a \times b \times c = HCF(a, b, c) \times LCM(a, b, c)$**. False. 32. **Find m+n** if prime factorization of $2025 = 3^m \times 5^n$: 6 34. **Create a three digit number whose prime factorisation has three distinct prime numbers**: $2 \times 3 \times 5 = 30$ (not 3-digit, need to adjust) e.g., $2 \times 3 \times 17 = 102$ 35. **Unit digit in $5^{2025} + 6^{2026}$**: 1 ### 2. Polynomials #### 2 Marks Questions 1. **Quadratic polynomial** with sum of zeroes = 3, product of zeroes = -2: $x^2 - 3x - 2$ 2. **Quadratic polynomial** with sum of zeroes = product of zeroes: e.g., $x^2 + x + 1$ (many answers possible) 3. **General form of quadratic and cubic polynomial**: - Quadratic: $ax^2 + bx + c$ ($a \neq 0$) - Cubic: $ax^3 + bx^2 + cx + d$ ($a \neq 0$) - Example quadratic: $x^2+2x+1$ - Example cubic: $x^3+x^2+x+1$ 4. **Quadratic polynomial** with sum of zeroes = 1/4, product of zeroes = -1: $4x^2 - x - 4$ 5. **Quadratic polynomial** with sum of zeroes = 0, product of zeroes = $\sqrt{3}$: $x^2 + \sqrt{3}$ 6. **Rough diagram for $ax^2 + bx + c$ ($a > 0$)**: Parabola opening upwards. #### 1 Mark Questions 7. **Cubic polynomial which is a binomial**: e.g., $x^3+x$ 8. **Cubic binomial in x with constant term 5**: e.g., $x^3+5$ 9. **Cubic polynomial which is trinomial**: e.g., $x^3+x^2+5$ 10. **Standard form of polynomial of degree 3**: $ax^3 + bx^2 + cx + d$ ($a \neq 0$) 11. **Example for trinomial having degree 6**: e.g., $x^6+x^3+5$ 12. **Linear polynomial having zero '5'**: $x-5$ 13. **Polynomial of degree 2 with constant term 2026**: e.g., $x^2+2026$ 14. **Standard form of cubic polynomial**: $ax^3 + bx^2 + cx + d$ ($a \neq 0$) 15. **Polynomial of degree 4 with constant term -6**: e.g., $x^4-6$ 16. **Coefficient of x in $p(x) = 7x^3 – 6x^2 + 5x + 8$**: 5 17. **Coefficient of $x^3$ in $p(x) = 5x^4 + 4x^3 + 3x^2 + 2x + 1$**: 4 18. **Algebraic expression where exponent of any variable is a**: whole number 19. **Zero of $p(x) = 4x - 8$**: 2 20. **Degree of $p(x) = 5x^3 – 3x^2 + 4x + 7$**: 3 21. **Assertion (A): Zeroes of $p(x) = (x – 1)(x – 2)(x – 3)$ are 1, 2, 3. Reason (R): Zeroes are x-coordinates where graph intersects/touches X-axis.** Both A and R are true and R is the correct reason of A. 22. **Zeroes of $x^2 – 4x + 4$**: 2 23. **If $p(x) = x^3$, then $p(1/2)$**: 1/8 24. **Assertion (A): Zero of $p(x) = 5x – 10$ is 2. Reason (R): Zero of $ax + b$ is $-b/a$.** Both A and R are true and R is the correct reason of A. 25. **Quadratic polynomial with sum of zeroes = -3, product of zeroes = 2**: $x^2 + 3x + 2$ 26. **If $p(x) = ax^2 + bx + c$, then $b/a$ is equal to**: -(sum of zeroes) 27. **Gambhira draws a line cutting X-axis at (3/2, 0) and Y-axis at (0, 3). Standard polynomial form (ax + b)**: $2x + y - 3 = 0$ (or $2x+y=3$) 28. **Define zero of a polynomial**: A value of the variable for which the polynomial evaluates to zero. 29. **Zeroes of $x^2 - x - 6$**: 3, -2 30. **Value of m** if product of zeroes of $2x^2 + 6x + m$ is -1: -2 31. **If graph of polynomial does not intersect X-axis**: no real zeroes 32. **Sum of zeroes of $x^2 - 4x + 3$**: 4 33. **Product of zeroes of $x^2 - 3$**: -3 34. **Zeroes of $x^2 - 7$**: $\pm \sqrt{7}$ 35. **Zeroes of cubic polynomial $x^3 - 4x$**: 0, 2, -2 36. **Linear polynomial $ax + b$ intersects X-axis at**: $(-b/a, 0)$ 37. **Quadratic polynomial whose zeroes are $\sqrt{2}$ and $-\sqrt{2}$**: $x^2 - 2$ 38. **One of the zeroes of $p(x) = 6 - x - x^2$**: -3 39. **Match the following** for zeroes $\alpha, \beta, \gamma$ of $x^3 – 2x^2 + 3x – 4$: - $\alpha+\beta+\gamma$: 2 - $\alpha\beta + \beta\gamma + \gamma\alpha$: 3 - $\alpha\beta\gamma$: 4 40. **Statement (A): Graph of $y = x^2 – 5x + 6$ passes through (0, 6). Statement (B): Graph of $y = ax^2 + bx + c$ is a parabola.** Both A and B are true. 41. **Statement - I: Number of maximum zeroes of a cubic polynomial is 3. Statement - II: Zero of linear polynomial $x – 3$ is – 3.** Statement I is true and Statement II is false. 42. **If $-3/2$ is a zero of polynomial, then $p(x)$**: $2x+3$ is a factor. 43. **Zeroes of $x^2 - 1 = 0$**: $\pm 1$ 44. **If all coefficients of a polynomial are zero**: Zero polynomial 45. **Rough diagram of quadratic polynomial with two distinct real zeroes**: Parabola intersecting X-axis at two points. 46. **Rough diagram of quadratic polynomial with two real & equal zeroes**: Parabola touching X-axis at one point. 47. **Rough diagram of quadratic polynomial with no real zeroes**: Parabola not intersecting X-axis. 48. **General form of quadratic polynomial**: $ax^2 + bx + c$ ($a \neq 0$) 49. **Degree of a zero polynomial**: Undefined 50. **Degree '2025'**: 0 (constant polynomial) 51. **Polynomial in variable 'x' whose zero is $-b/a$**: $ax+b$ 52. **Rough graph of $y = x + 3$**: Straight line with positive slope, y-intercept 3. 53. **Other zero** if -1 is a zero of $x^2 – 7x – 8$: 8 54. **Find k** if 2 is a zero of $x^2 – 5x + k$: 6 55. **Graphs of quadratic polynomial**: - Who drawn correctly: Vani (parabola opening upwards, $a>0$) - Zeroes for Vamsi's graph: 0 - Zeroes for Muskan's graph: -1, 4 - Van's graph represents: Quadratic ### 3. Pair of Linear Equations in Two Variables #### 1 Mark Questions 1. **Linear equation parallel to $2x + 3y + 8 = 0$**: e.g., $2x + 3y + 5 = 0$ 2. **Pair of dependent linear equations**: e.g., $2x + 3y = 5$, $4x + 6y = 10$ 3. **Linear equation parallel to $3x – 2y + 4 = 0$**: e.g., $3x - 2y + 7 = 0$ 4. **Linear equation dependent to $2x + 3y – 8 = 0$**: e.g., $4x + 6y - 16 = 0$ 5. **Linear equation intersecting $3x + 2y – 7 = 0$**: e.g., $x - y = 0$ 6. **Pair of linear equations which are parallel**: e.g., $x+y=1$, $x+y=2$ 7. **Pair of linear equations which are independent**: e.g., $x+y=1$, $x-y=0$ 8. **Pair of linear equations for cricket coach**: $7b + 6a = 3800$, $3b + 5a = 1750$ 9. **Linear equation for textbook cost**: $y = 2x - 7$ (where y = cost of Math book, x = cost of English book) 10. **Pair of linear equations for pencils and pens**: $5p + 7e = 50$, $7p + 5e = 46$ 11. **Example of linear equation in two variables x and y**: $x+y=0$ 12. **Point of intersection of $x + y = 6$ and $x – y = 4$**: (5, 1) 13. **Line $2x – 3y = 8$ intersects X-axis at**: (4, 0) 14. **System of linear equation $x – 2y + 5 = 0$ and $3x - 6y + 15 = 0$**: Dependent (coincident lines) 15. **Pair of equation $2x + 3y + 5 = 0$ and $-3x – 6y + 1 = 0$ has**: Unique solution 16. **Value of 'k + 1'** if $x + y = 1$ and $kx + y = 2$ has no solutions: k = 1, so k+1 = 2 17. **Solution of $4x + 3y = 5$**: (–1, 3) 18. **Value of k** if (6, k) is a solution of $3x + y – 22 = 0$: 4 19. **Graph of linear equation in two variables represents a**: Straight line 20. **General form of linear equation in two variables**: $ax+by+c=0$ 21. **Value 'm'** for coincident lines $6x + 4y + 2 = 0$ and $3x + 2y + m = 0$: 1 22. **Find k** if (1, 3) is a solution of $2x – ky = 8$: -2 23. **Assertion (A): $2x + 3y + 6 = 0$ and $4x + 6y + 7 = 0$ have no solution. Reason (R): $a_1/a_2 = b_1/b_2 \ne c_1/c_2$ so, the linear equations are parallel.** Both A and R are true and R is the correct reason of A. ### 4. Quadratic Equations #### 2 Marks Questions 1. **Value of k** if both roots of $2x^2 + kx + 3 = 0$ are equal: $\pm 2\sqrt{6}$ 2. **Analyze whether $(x – 2)^2 + 1 = 2x – 3$ is quadratic**: Yes, $x^2-6x+8=0$ 3. **Check whether $(x + 1)^2 = 2(x – 3)$ is quadratic**: Yes, $x^2+2x+1=2x-6 \Rightarrow x^2+7=0$ 4. **Is $2x^2 − x + 1 = x(x + 1)$ quadratic**: Yes, $x^2-2x+1=0$ 5. **Quadratic equation** with sum of zeroes = $\tan 45^\circ$, product of zeroes = $\sec 60^\circ$: $x^2 - x + 2 = 0$ 6. **Check whether $(2x – 1)(x - 3) = (x + 5)(x – 1)$ is quadratic**: Yes, $2x^2-7x+3=x^2+4x-5 \Rightarrow x^2-11x+8=0$ #### 1 Mark Questions 7. **Quadratic equation with 1 as one of its roots**: e.g., $x^2-1=0$ 8. **Quadratic equation whose roots are reciprocal to each other**: e.g., $ax^2+bx+a=0$ 9. **Quadratic equation whose roots are 2 and 3**: $x^2-5x+6=0$ 10. **Quadratic equation with equal roots**: e.g., $x^2-2x+1=0$ 11. **Quadratic equation with -1 as one of its zeroes**: e.g., $x^2-1=0$ 12. **Quadratic equation with -2 as one of its zeroes**: e.g., $x^2-4=0$ 13. **Quadratic equation whose roots are reciprocal to each other**: e.g., $ax^2+bx+a=0$ 14. **Quadratic equation whose roots are numerically equal and opposite sign**: e.g., $x^2-4=0$ 15. **Quadratic equation with roots 3 and its additive inverse (-3)**: $x^2-9=0$ 16. **Quadratic equation with roots -2 and its additive inverse (2)**: $x^2-4=0$ 17. **Quadratic equation with roots 5 and 1**: $x^2-6x+5=0$ 18. **Situation as quadratic equation: "Product of two consecutive integers is 15"**: $x(x+1)=15 \Rightarrow x^2+x-15=0$ 19. **Quadratic equation with one root 0 and other -7**: $x(x+7)=0 \Rightarrow x^2+7x=0$ 20. **Value of 'k'** if $1/2$ is a root of $x^2 + kx - 5/4 = 0$: k = 2 21. **Other root** if one root of $x^2 – 7x + 12 = 0$ is 4: 3 22. **Standard form of quadratic equation**: $ax^2 + bx + c = 0$ ($a \neq 0$) 23. **Express $5x = 9x^2 + 1$ in standard form**: $9x^2 - 5x + 1 = 0$ 24. **Every quadratic equation has at most**: two zeroes 25. **Condition** if roots of quadratic equation are numerically equal and opposite sign: $b=0$ 26. **Condition** if roots of $ax^2 + bx + c = 0$ are reciprocal: $c=a$ 27. **Which of the following is a quadratic equation**: $(x+1)^2 = (x+2)(x-2)$ ($x^2+2x+1=x^2-4 \Rightarrow 2x+5=0$ (Not quadratic)) - Correct option: $3x + 1/x = 7$ ($3x^2-7x+1=0$) 28. **Match the following**: - $3x^2 – 5x + 2 = 0$: real and distinct roots - $16x^2 – 8x + 1 = 0$: real and equal roots - $2x^2 – 4x + 3 = 0$: not real roots 29. **Find k** if $x^2 + 4x + k = 0$ has real and distinct roots: $k ### 5. Arithmetic Progressions #### 1 Mark Questions 1. **Assertion (A): Common difference of A.P. 5, 9, 13..... 185 is 4. Reason (R): Common difference $d = a_n – a_{n-1}$.** Both A and R are true and R is the correct explanation of A. 2. **Statement (A): 3, 6, 9, 12,.... is an A.P. Statement (B): 2,2,2,2,.... is not an A.P.** Both A and B are true. 3. **Assertion (A): Sum of first 10 terms of AP : 4, 8, 12.... is 220. Reason (R): $n^{th}$ term of AP is $a + (n – 1)d$.** Assertion (A) is true but Reason (R) is false. (Sum is actually 220, but R is not the direct reason for A's truth) 4. **Match the following**: - Sum of first 10 natural numbers: 55 - Sum of first 10 odd natural numbers: 100 - Sum of first 10 even natural numbers: 110 5. **Assertion (A): 5 is the common difference of 100, 105, 110, ........ Reason (R): Common difference of an AP $d = a_n – a_{n-1}$.** Both A and R are true and R is the correct explanation of A. 6. **Match the following**: - $n^{th}$ term of AP $a, a+d....$: $a+(n-1)d$ - Sum of 'n' terms of AP whose first term 'a' and last term 'l': $n/2 (a+l)$ - Sum of 'n' terms of AP: $n/2 [2a+(n-1)d]$ 7. **Common difference** if first term is 6 and $n^{th}$ term is $6n$: 6 8. **Match the following** for AP 2, 4, 6, 8,...: - d: 2 - $a_3$: 6 - $a_4-a_2$: 4 9. **Assertion (A): In a finite AP first term is 1, last term is 20 and sum is 399. Then there are 38 terms. Reason (R): Sum of infinite AP = n/2 (first term + last term).** A is true, R is false. 10. **Compare 5th terms of A.P's**: $A = (3, 6, 9,...)$ and $B = (4, 8, 12,...)$. $A_5 = 15$, $B_5 = 20$. So B is greater. 11. **Formula for**: - $n^{th}$ term of an A.P: $a_n = a + (n-1)d$ - If 'a' is first term and '$a_n$' is last term sum of first 'n' terms: $S_n = n/2 (a+a_n)$ - Sum of first 'n' natural numbers: $S_n = n(n+1)/2$ - If a is first term and 'd' is common difference, then general form of an A.P: $a, a+d, a+2d, ...$ 12. **Common difference of A.P : p - q, p, p + q........**: q 13. **Value of x** if -3, x, 2 are three consecutive terms of an A.P: -0.5 14. **Find '$a_n$'** if $d = – 4, n = 7$ and $a_1 = 4$: -20 15. **Statement (A): If sum of n terms of an AP is $S_n$, then $n^{th}$ term is $S_n – S_{n-1}$. Statement (B): If $a_n$ is $n^{th}$ term of A.P, then n should be a positive integer.** Both A and B are true. 16. **Famous mathematician associated with finding sum of first 100 natural numbers**: Gauss 17. **Sum of first 50 even natural numbers**: 2550 18. **Arithmetic progression starting with 4**: e.g., 4, 6, 8, 10 19. **Match the following** if $n^{th}$ term of an AP is $a_n = 9 – 5n$: - $a_1$: 4 - $a_2$: -1 - d: -5 20. **If $S_n = 2n^2 + 5n$ then $a_2$**: 11 21. **If $a_1 = 2$ and $a_n = 26$, then find $a_2$**: (Need more info, like n or d to find $a_2$. Assuming it's an AP, $a_2$ cannot be determined uniquely without $n$ or $d$.) 22. **Find 'd'** if $a = – 18, n = 10, a_n = 0$: 2 23. **Which of the following list forms an A.P.**: B) 5, 2, – 1, – 4,..... 24. **Volume of a hemisphere of base radius 3 cm**: $18\pi$ $cm^3$ 25. **Match the following** if $n^{th}$ terms of an A.P is $a_n = 2n – 6$: - $a_1$: -4 - $a_2$: -2 - $a_3$: 0 26. **General form of an Arithmetic progression**: $a, a+d, a+2d, ...$ ### 6. Triangles #### 2 Marks Questions 1. **State Basic proportionality theorem (Thales theorem)**: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. 2. **State converse of basic proportionality theorem**: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 3. **Example for (i) Similar Figures and (ii) Non-Similar Figures**: - Similar: All circles, all squares. - Non-similar: A circle and a square, a triangle and a quadrilateral. 4. **Conditions for similarity of Triangles**: - AAA (Angle-Angle-Angle) - SSS (Side-Side-Side) - SAS (Side-Angle-Side) 5. **Define similar triangle**: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). 6. **State SAS criterion in similarities of triangles**: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. 7. **State SSS criterion in similarities of triangles**: If the corresponding sides of two triangles are in the same ratio, then their corresponding angles are equal and the two triangles are similar. 8. **State AAA criterion in similarities of triangles**: If the corresponding angles of two triangles are equal, then the two triangles are similar. 9. **Similarity criterion for similar triangle**: AAA, SSS, SAS. (Symbolic form needs an image) 10. **Two different examples of "pair of figures which are always similar"**: - Any two circles. - Any two squares. #### 1 Mark Questions 11. **Find EC** if AD = 2 cm, BD = 4 cm and AE = 5 cm, and DE || BC: 10 cm (using BPT) 12. **If $\triangle ABC \sim \triangle XYZ$, which may be true**: $\angle A = \angle X$ (Corresponding angles are equal) 13. **If DE // BC, then the length of EC**: 3 cm (using BPT, $AD/DB = AE/EC \Rightarrow 1.5/3 = 1/EC \Rightarrow EC=2$ cm from diagram given) 14. **Statement I: All similar triangles are congruent. Statement II: All right angled isosceles triangles are similar.** Statement I is false and Statement II is true. 15. **If $\triangle ABC$ and $\triangle PQR$, if $AB/QR = BC/PR = CA/PQ$, then which is true**: $\triangle CBA \sim \triangle PQR$ 16. **Perimeter of $\triangle ABC$** if $\triangle ABC \sim \triangle DEF$, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm: 8 cm ($P_{ABC}/P_{DEF} = AB/DE \Rightarrow P_{ABC}/(6+9+12) = 4/6 \Rightarrow P_{ABC}/27 = 2/3 \Rightarrow P_{ABC}=18$ cm) 17. **Statement (A): If two triangles are similar then their corresponding sides are equal. Statement (B): If two triangle are similar then their corresponding angles are equal.** A is false and B is true. 18. **Statement (A): Any two circles are similar. Statement (B): Any two equilateral triangles are similar.** Both A and B are true. 19. **Which of the following is not an identity**: $\sec^2 \theta + 1 = \tan^2 \theta$ (Correct is $\sec^2 \theta - \tan^2 \theta = 1$) 20. **If two triangles are equal angular what can you say about their corresponding sides**: They are proportional. 21. **Name the two figures that are always similar**: Circles, squares. 22. **Name the two figures that not always similar**: Rectangle and square, circle and ellipse. 23. **Which of the following are not always similar**: Line segments (can be similar), rectangles (not always similar), circles (always similar), squares (always similar). So, rectangles. 24. **Statement (A): If two triangles are congruent, then they are also similar triangles. Statement (B): All congruent triangles are similar but similar triangles need not be congruent.** Both A and B are true. 25. **Assertion (A): D and E are points of side AB and AC of $\triangle ABC$. $AD/DB = AE/EC$. Then DE || BC. Reason (R): If a line divides any two sides of a triangle in the same ratio, then it is parallel to the third side.** Both A and R are true and R is the correct reason of A. 26. **Match the following** if DE || BC and intersects AB in D and AC in E: - AD/DB: AE/EC - AB/AD: AC/AE - AB/DB: AC/EC 27. **If $\triangle ABC \sim \triangle DEF$, $\angle A = 50^\circ$, $\angle C = 70^\circ$, then $\angle D + \angle E = 120^\circ$**: True ($\angle B = 180 - 50 - 70 = 60$. $\angle D = \angle A = 50$, $\angle E = \angle B = 60$. So $\angle D + \angle E = 110^\circ$. So False) 28. **All equilateral triangles are always similar**: True 29. **Ratio between corresponding sides of two similar figures is called a scale factor**: True 30. **Find DY** if DE || YZ, XD = 2.4 cm, XE = 3.2 cm and EZ = 4.8 cm: 1.6 cm (using BPT, $XD/DY = XE/EZ \Rightarrow 2.4/DY = 3.2/4.8 \Rightarrow DY = 2.4 \times 4.8 / 3.2 = 3.6$ cm) (The diagram shows DY as 3.5, but the calculation with given values gives 3.6) 31. **Statement (A): In similarity of two polygons, either of the two conditions is not sufficient. Statement (B): In similarity of two triangles, it is not necessary to check both the conditions.** Both A and B are true. 32. **Criterion of similarity** for $\triangle ABC \sim \triangle PQR$: AA (Angle-Angle) 33. **Which makes triangles similar**: $\angle A = \angle D$ 34. **If $\triangle ABC \sim \triangle DEF$, then which is true**: $AB/DE = BC/EF = CA/FD$ 35. **Find BC** if $\triangle ABC \sim \triangle PQR$, AB = 3 cm, PQ = 2 cm, QR = 4 cm: 6 cm ($AB/PQ = BC/QR \Rightarrow 3/2 = BC/4 \Rightarrow BC = 6$) 36. **Two different examples of "pair of figures which are always similar"**: Any two equilateral triangles, any two regular hexagons. 37. **Perimeter of $\triangle ABC$** if $\triangle ABC \sim \triangle DEF$, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm: 18 cm (calculated in Q16) 38. **From figure, $AD/DB = AE/EC$ (True/False)**: True (Thales theorem) 39. **From figure x**: Needs specific values from figure to calculate x. 40. **From figure OA/OB**: Needs specific values from figure to calculate ratio. ### 7. Coordinate Geometry #### 2 Marks Questions 1. **Formulae for distance**: - Between $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ - Between origin (0,0) and $(x, y)$: $\sqrt{x^2+y^2}$ 2. **Coordinates of midpoint** of line segment joining $(-2, -2)$ and $(2, -4)$: $(0, -3)$ 3. **Coordinates of point A** if AB is diameter, center $(2, -3)$, B is $(1, 4)$: $(3, -10)$ 4. **Midpoint divides a line segment in which ratio**: 1:1 - **Coordinates of midpoint** of line segment joining $(x_1, y_1)$ and $(x_2, y_2)$: $((x_1+x_2)/2, (y_1+y_2)/2)$ 5. **Coordinates of point dividing line joining $(-1, 7)$ and $(4, -3)$ in ratio 2:3**: $(1, 3/5)$ 6. **Point on X-axis equidistant from $(2, -5)$ and $(-2, 9)$**: $(-7, 0)$ 7. **Distance between $(a \cos \theta, 0)$ and $(0, a \sin \theta)$**: a 8. **Rough figure of toy (cylinder mounted on hemisphere)**: (Needs image, but description is clear) ### 8. Introduction to Trigonometry #### 2 Marks Questions 1. **If $\tan(A + B) = \sqrt{3}$ and $\tan(A - B) = 1/\sqrt{3}$, find A and B**: A = 45°, B = 15° 2. **If $\sin(A-B)=1/2$, $\cos(A+B)=1/2$, find A and B**: A = 45°, B = 15° 3. **If $\sec A = 13/12$, find values of $\sin A$ and $\cos A$**: $\sin A = 5/13$, $\cos A = 12/13$ 4. **If $15 \cot A = 8$, determine $\sin A – \sec A$**: $15 \cot A = 8 \Rightarrow \cot A = 8/15 \Rightarrow \sin A = 15/17, \sec A = 17/8$. Then $\sin A - \sec A = 15/17 - 17/8 = (120 - 289)/136 = -169/136$ 5. **If $\cot \theta = 7/8$, evaluate $(1 + \sin \theta)(1 - \sin \theta) / ((1 + \cos \theta)(1 - \cos \theta))$**: $\cot^2 \theta = (7/8)^2 = 49/64$ 6. **Express $\tan A$ and $\cos A$ in terms of $\sec A$**: $\tan A = \sqrt{\sec^2 A - 1}$, $\cos A = 1/\sec A$ 7. **Express $(\operatorname{cosec} \theta – \cot \theta)^2$ in terms of $\cos \theta$**: $(1-\cos \theta)/(1+\cos \theta)$ 8. **Is it right to say $\cos(60^\circ + 30^\circ) = \cos 60^\circ \cos 30^\circ – \sin 60^\circ \sin 30^\circ$**: True 9. **Find acute angles A and B** if $\sin(A + 2B) = \sqrt{3}/2$ and $\cos(A + 4B) = 0$: A = 30°, B = 15° #### 1 Mark Questions 10. **Which trigonometric ratio is equal to adjacent side of $\theta$/hypotenuse**: $\cos \theta$ 11. **If $\tan A = 1$, then $\angle A$**: 45° 12. **If $\sin \theta = \cos \theta$, then $\tan \theta$**: 1 13. **If $\operatorname{cosec} \theta - \cot \theta = x$, then $\operatorname{cosec} \theta + \cot \theta$**: $1/x$ 14. **If 'A' is an acute angle, as value of A increases, then value of $\cos A$**: decreases 15. **From the adjacent figure, AB**: Needs figure. 16. **If $\cos A = 3/5$, then $\tan A$**: $4/3$ 17. **If $\sec \theta – \tan \theta = x$, then $\sec \theta + \tan \theta$**: $1/x$ 18. **Match the following**: - $\sin 90^\circ \times \cos 90^\circ$: 0 - $\cos \theta \times \sec \theta$: 1 - If $\operatorname{cosec} \theta + \cot \theta = 1/2$, then $\operatorname{cosec} \theta - \cot \theta$: 2 19. **Which of the following is not an identity**: $\sec^2 \theta + 1 = \tan^2 \theta$ 20. **If hypotenuse = 13 cm, adjacent side = 12 cm, then $\sin A$**: $5/13$ 21. **Define $\sin \theta, \cos \theta, \tan \theta$**: - $\sin \theta = \text{Opposite}/\text{Hypotenuse}$ - $\cos \theta = \text{Adjacent}/\text{Hypotenuse}$ - $\tan \theta = \text{Opposite}/\text{Adjacent}$ 22. **If $\sin \theta = 3/5$, then $\cos \theta$**: $4/5$ 23. **If $3 \cot \theta = 4$, then $\tan \theta$**: $4/3$ 24. **If $\sin A = \cos A$, then A**: 45° 25. **$3 \sin^2 \theta + 3 \cos^2 \theta$**: 3 26. **If $\tan \theta$ is not defined, then $\theta$**: 90° 27. **Value of $2 \tan^2 45^\circ + \cos^2 30^\circ – \sin^2 60^\circ$**: 2 28. **Value of $(1 - \tan^2 45^\circ) / (1 + \tan^2 45^\circ)$**: 0 29. **Value of $\sin A$ or $\cos A$ be always less than or equal to 1**: Because hypotenuse is the longest side in a right triangle. 30. **If $\sin \theta = \sqrt{3}/2$, then $\theta$**: 60° 31. **Match the following**: - $\sin^2 \theta + \cos^2 \theta$: 1 - $1 + \tan^2 \theta$: $\sec^2 \theta$ - $1 + \cot^2 \theta$: $\operatorname{cosec}^2 \theta$ 32. **If $x \tan 45^\circ \cos 60^\circ = \sin 60^\circ \cot 60^\circ$, then x**: $\sqrt{3}$ 33. **$( \sec A + \tan A)( \sec A – \tan A)$**: 1 34. **Match the following**: - $\sin \theta \times \cot \theta$: $\cos \theta$ - $\tan \theta \times \cos \theta$: $\sin \theta$ - $\sec \theta \times \sin \theta$: $\tan \theta$ 35. **Value of $\sin 0^\circ \sin 10^\circ \sin 20^\circ \sin 30^\circ ... \sin 90^\circ$**: 0 (due to $\sin 0^\circ = 0$) 36. **If $\sec \theta - \tan \theta = a/b$, then $\sec \theta + \tan \theta$**: $b/a$ 37. **If $\tan \theta = 1$, then $\theta$**: 45° 38. **If $\sin 2A = \cos 3A$, then $\cot 5A$**: 1 39. **First Indian mathematician used idea of "sine"**: Aryabhata 40. **Match the following**: - $\tan \theta$: $\sin \theta / \cos \theta$ - $\cot \theta$: $\cos \theta / \sin \theta$ - $\operatorname{cosec} \theta$: $\sqrt{1 + \cot^2 \theta}$ ### 9. Some Applications of Trigonometry #### 2 Marks Questions 1. **Define angle of elevation with rough diagram**: (Needs diagram) Angle formed by line of sight with horizontal when looking up. 2. **Define angle of depression with rough diagram**: (Needs diagram) Angle formed by line of sight with horizontal when looking down. 3. **Two real life situations where trigonometry is used**: Navigation, architecture, surveying. 4. **Height of tower** if 15m away, angle of elevation 60°: $15\sqrt{3}$ m 5. **Rough diagram** (player on tower, angle of depression 60°): (Needs diagram) 6. **Relation between angle of elevation and angle of depression from same point**: They are equal (alternate interior angles). 7. **Diagram for situation**: (Needs diagram) A person flying a kite at $\alpha^\circ$ with thread length 'l'. 8. **Trigonometric ratio to find height** if 30m away, angle of elevation 30°: $\tan 30^\circ$ 9. **Diagram for situation**: (Needs diagram) Clock tower, angle $\alpha$ from distance 'd'. 10. **Outline diagram for "ladder 10m touches wall at 5m"**: (Needs diagram) Right triangle with hypotenuse 10, height 5. 11. **Diagram for situation**: (Needs diagram) Tower on building, angles of elevation 45° and 60°. #### 1 Mark Questions 12. **Define "Line of sight"**: The line drawn from the eye of an observer to the object being viewed. 13. **In heights and distance problems, angles are always with respect to horizontal line**: True 14. **Which situation involves angle of depression**: A girl looking down at a car from the balcony. 15. **If height of tower = length of shadow, angle of elevation**: 45° (True) 16. **Angle of elevation of tower** if observer 1.5m tall, 28.5m away from 30m tower: 45° 17. **Angle of elevation** if length of shadow = height: 45° 18. **Angle of elevation** if height of pole/shadow = $\sqrt{3}:1$: 60° 19. **Angle of elevation of top of 100m tree from $100\sqrt{3}$m away**: 30° (False, it's 30 degrees not 60) 20. **Length of shadow of 8m tree when Sun's angle of elevation 45°**: 8m 21. **Angle of elevation of Sun** if shadow is $\sqrt{3}$ times height: 30° 22. **Identify and name (i) angle of elevation and (ii) angle of depression**: (Needs diagrams for identification) 23. **Real life situation where angle of elevation is used**: Looking up at a tall building. ### 10. Circles #### 2 Marks Questions 1. **Quadrilateral ABCD circumscribing a circle**: $AB + CD = BC + DA$ 2. **Length of PQ** if tangent PQ at P, radius 5cm, OQ = 12cm: $\sqrt{12^2 - 5^2} = \sqrt{144-25} = \sqrt{119}$ cm 3. **Radius of circle** if tangent length from Q is 24cm, distance from center OQ is 25cm: $\sqrt{25^2 - 24^2} = \sqrt{625-576} = \sqrt{49} = 7$ cm 4. **Find $\angle PTQ$** if TP and TQ are tangents, $\angle POQ = 110^\circ$: $180 - 110 = 70^\circ$ 5. **Parallelogram ABCD circumscribing a circle**: It is a rhombus. 6. **Length of chord of larger circle** if two concentric circles radii 5cm and 3cm: $2 \times \sqrt{5^2 - 3^2} = 2 \times 4 = 8$ cm 7. **How many tangents can be drawn**: - P outside: 2 - P on circle: 1 #### 1 Mark Questions 8. **Geometrical design (two tangents from external point)**: (Needs diagram) 9. **Rough sketch (tangent at end point of radius)**: (Needs diagram) Tangent perpendicular to radius. 10. **Rough sketch (parallelogram circumscribing circle)**: (Needs diagram) 11. **Geometrical design (tangents at ends of diameter are parallel)**: (Needs diagram) 12. **Draw a circle and two lines parallel to a given line, one tangent, one secant**: (Needs diagram) 13. **A circle can have**: two parallel tangents at the most. 14. **How many tangents exist from external point to a circle**: 2 15. **Define tangent to a circle**: A line that touches the circle at exactly one point. 16. **Define point of contact**: The common point of the tangent and the circle. 17. **Observe the figure. Equation of 'm'**: (Needs figure) Assuming l and m are parallel, m should be $2x+3y+k=0$. If A=(2,-3) is on l, $2(2)+3(-3)+5 = 4-9+5=0$. If B is on m, it should be $2x+3y-5=0$. 18. **Define secant**: A line that intersects a circle at two distinct points. 19. **Number of tangents that can be drawn to a circle**: Infinite (from points on the circle) / 2 (from external point) 20. **Number of tangents from a point outside a circle**: 2 21. **Number of tangents from a point on a circle**: 1 22. **Number of tangents from a point inside a circle**: 0 23. **Number of secants from a point outside a circle**: Infinite 24. **Number of secants from a point on a circle**: Infinite 25. **Number of secants from a point inside a circle**: Infinite 26. **Number of circles passing through a point**: Infinite 27. **Number of circles passing through two distinct points**: Infinite 28. **Number of circles passing through three non collinear points**: 1 29. **Angle between tangent and radius at point of contact**: 90° 30. **A circle can have**: two parallel tangents at the most. 31. **Define length of the tangent**: The length of the line segment from the external point to the point of contact on the circle. 32. **Lengths of tangents drawn from an external point to a circle are equal**: True 33. **Rough sketch (tangent at end points of diameter)**: (Needs diagram) 34. **If O is center, r is radius, P is a point, OP > r, then P lies in**: exterior of the circle. 35. **If TP and TQ are tangents, $\angle POQ = 110^\circ$, then $\angle PTO$**: 35° 36. **Tangents PA and PB from P to circle O. $\triangle PAB$ is a**: Isosceles triangle 37. **Name of parallelogram circumscribing circle**: Rhombus 38. **Name of rectangle circumscribing circle**: Square 39. **Length of AB** if ABCD circumscribes circle, BC = 8cm, CD = 7cm, DA = 5cm: 6 cm ($AB+CD=BC+DA \Rightarrow AB+7=8+5 \Rightarrow AB=6$) 40. **Statement A: Line intersecting circle at two points is secant. Statement B: Parallelogram circumscribing circle is rectangle.** A is true, B is false. 41. **Assertion (A): Tangents PA and PB from P to circle O then $\triangle APB$ is isosceles. Reason (R): Lengths of tangents from external point are equal.** Both A and R are true and R is the correct reason of A. 42. **Radius** if two tangents inclined at 60°, length of tangent 6cm: $2\sqrt{3}$ cm 43. **Which figure has part of parallel tangents**: (Needs figure) 44. **If $\angle APB = 60^\circ$ then**: - $\triangle OAP$ is: Right-angled triangle - $\angle AOP$: 60° - $\angle AOB$: 120° - $\angle OPB$: 30° ### 12. Surface Areas and Volumes #### 2 Marks Questions 1. **Volume of largest circular cone cut from cube of edge 7cm**: $1/3 \pi (3.5)^2 (7) = (1/3) \times (22/7) \times (3.5)^2 \times 7 = (1/3) \times 22 \times 3.5 \times 3.5 = 89.83$ cm$^3$ 2. **Surface area of adjacent solid**: (Needs diagram) 3. **Sum of surface areas of 2 hemispheres** from sphere radius r: $2 \times (2\pi r^2) = 4\pi r^2$ 4. **Volume of right circular cone** radius 6cm, height 7cm: $1/3 \pi (6)^2 (7) = (1/3) \times (22/7) \times 36 \times 7 = 264$ cm$^3$ 5. **Identify volume/area**: - Quantity of water in bottle: Volume - Canvas for tent: Area 6. **Total surface area of hemisphere** base radius 7cm: $3\pi r^2 = 3 \times (22/7) \times 7^2 = 462$ cm$^2$ #### 1 Mark Questions 7. **Surface area of hemisphere radius r**: $2\pi r^2$ (CSA) or $3\pi r^2$ (TSA) 8. **Identify correct relation**: - Curved surface area of a cone: $\pi rl$ - Total surface area of a cone: $\pi r(l+r)$ - Volume of a cone: $1/3 \pi r^2 h$ 9. **Identify correct relation**: - CSA of Hemisphere: $2\pi r^2$ - TSA of Hemisphere: $3\pi r^2$ - Volume of Hemisphere: $2/3 \pi r^3$ 10. **Side of cube** if volume is 125 cm$^3$: 5 cm 11. **Surface area of a top**: sum of curved surface areas of cylinder and cone. 12. **Rough figure of cone with base radius r, slant height l**: (Needs diagram) 13. **Rough figure of toy (cone mounted on hemisphere)**: (Needs diagram) 14. **Height of cone** if base radius 5cm, angle between radius and slant height 60°: $5\sqrt{3}$ cm 15. **Surface area of sphere radius 'x'**: $4\pi x^2$ 16. **Volume of prism**: Base Area $\times$ Height 17. **Ratio of volumes** if cone and cylinder have same radius and height: 1:3 18. **Surface area of hemisphere radius 'r'**: $3\pi r^2$ 19. **Match the following**: - Surface area of sphere: $4\pi r^2$ - Volume of sphere: $4/3 \pi r^3$ - CSA of hemisphere: $2\pi r^2$ - Volume of cylinder: $\pi r^2 h$ 20. **If radius of sphere doubled, surface area doubled**: False (Area becomes 4 times) 21. **If radius of hemisphere doubled, volume 8 times**: True 22. **Ratio of surface areas** if radii of two spheres are in ratio 3:2: 9:4 23. **Statement A: If radii of two spheres are in ratio 1:2, then volumes are in ratio 1:8. Statement B: CSA of cylinder is $\pi rh$.** A is true, B is false ($\pi rh$ is CSA of cylinder without top/bottom, CSA is $2\pi rh$). 24. **Vertical cross section of cylinder**: Rectangle 25. **Cross section of cube**: Square 26. **Side of cube** if volume is 64m$^3$: 4m 27. **Volume of cube** if total surface area is 54cm$^2$: 27 cm$^3$ 28. **Height of cuboid** if volume 440 cm$^3$, base area 88 cm$^2$: 5 cm 29. **Height of cone** if volume 462 cm$^3$, base radius 7 cm: 9 cm 30. **If volume and total surface area of cube are numerically equal, then its side is**: 6 units 31. **Rough diagram of combination of cone with hemisphere**: (Needs diagram) 32. **Rough diagram of combination of cone with cylinder**: (Needs diagram) 33. **Rough diagram of combination of cylinder and two hemispheres on each end**: (Needs diagram) 34. **Mistake if student calculated TSA using $\pi rl$**: Omitted base area of cone. ### 13. Statistics #### 8 Marks Questions 1. **Median age** from given literacy rate table: (Needs calculation) 2. **Mean daily expenditure** from given table: (Needs calculation) #### 1 Mark Questions 3. **Mean literacy rate**: (Needs calculation) 4. **Modal lifetimes of components**: (Needs calculation) 5. **Median monthly consumption**: (Needs calculation) 6. **Formula to find mode of grouped data**: $L + ((f_1 - f_0) / (2f_1 - f_0 - f_2)) \times h$. (Explain terms) 7. **Formula to find median of grouped data**: $L + ((N/2 - CF) / f) \times h$. (Explain terms) 8. **Formula to find mean by assumed mean method**: $Mean = A + (\sum f_i d_i / \sum f_i)$. (Explain terms) ### 14. Probability #### 1 Mark Questions 1. **Define sure event**: An event that is certain to happen. Example: Sun rising in the east. 2. **Define impossible event**: An event that cannot happen. Example: Rolling a 7 on a standard die. 3. **Define equally likely events**: Events that have the same probability of occurring. Example: Getting a head or tail in a coin toss. 4. **Define elementary events**: An event that has only one outcome. Example: Getting a 1 when rolling a die. 5. **Define complementary event**: Two events are complementary if one event occurs if and only if the other does not. Example: Getting a head and getting a tail in a coin toss. 6. **P(E) + P(E')**: 1 7. **Probability of getting head in fair coin toss**: 1/2 8. **Probability of getting number less than 4 (1,2,3) on die**: 3/6 = 1/2 9. **If P(E) = 0.05, P(E')**: 0.95 10. **Sum of probabilities of all elementary events**: 1 11. **Probability of certain event**: 1 12. **Probability that Raju gets 35 out of 35 in math exam**: 1 13. **If P(E) is probability of event 'E', then**: $0 \le P(E) \le 1$ 14. **Probability of an event that is certain to happen**: 1 15. **Which cannot be probability**: (Any value outside [0,1]) e.g., Negative value or >1. 16. **Probability that letter from "RAMANUJAN" is a vowel**: 4/9 (A,A,U,A) 17. **Statement A: Two coins tossed, P(no head) = 3/4. Statement B: Die thrown, P(composite) = 1/3.** Statement A is false (P(no head) = 1/4), B is true (Composite: 4,6). 18. **Assertion (A): P(win) = 5/12, P(lose) = 7/12. Reason (R): P(E) + P(not E) = 1.** Both A and R are true and R is the correct reason of A. 19. **Match the following**: - P(diamond card): 13/52 = 1/4 - P(black queen): 2/52 = 1/26 - P(number card): 36/52 = 9/13 20. **Probability of 3 heads in 3 coin tosses**: 1/8 21. **Sum of probabilities of all elementary events**: 1 22. **7/2 is not a probability**: True 23. **Letter of English alphabets chosen at random. P(consonant)**: 21/26 24. **In tossing a fair coin once the probability of getting head is**: 1/2 25. **Number chosen from 1 to 20. P(factor of 10)**: 4/20 = 1/5 (Factors of 10: 1, 2, 5, 10) 26. **Statement A: n coins tossed, outcomes = $2^n$. Statement B: n dice rolled, outcomes = $6^n$.** Both A and B are true. 27. **Two dice rolled, P(even numbers on both)**: 9/36 = 1/4 ( (2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6) ) 28. **Who wrote "The Book on Games of Chance"**: Gerolamo Cardano 29. **Who made significant contributions in theory of probability**: Pierre de Fermat and Blaise Pascal 30. **Who wrote "Theorie Analytique des probabilities"**: Pierre-Simon Laplace 31. **Assertion (A): Sindhu P(win) = 0.68, P(Sen win) = 0.32. Reason (R): P(E) + P(E') = 1.** Both A and R are true and R is the correct reason of A. 32. **Frame question on probability for rolling a die**: What is the probability of rolling an even number?