This compilation includes each and every question extracted from ICSE Mathematics examinations for the years 2017, 2018, 2019, 2020, 2023, 2024, and 2025, organized chapter-wise.
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q3a | Find the amount of sales tax (8% charged after two successive discounts of 10% and 5% on a catalogue price of ₹42,000) and the total price paid by the customer. |
| 2017 | Q8c | Find (i) VAT paid by the shopkeeper to the government, and (ii) the total amount paid by the customer inclusive of tax, given wholesaler/shopkeeper/customer chain and 12% sales tax (VAT) charged at every stage. |
| 2020 | Q4c | Calculate the total amount of GST paid and the total bill amount including GST paid by Mr. Bedi on medicines (5%), shoes (18%), and a discounted laptop bag (18%). |
| 2023 | Q1vii | If the printed price of an article is ₹3080 and the rate of GST is 10%, find the GST charged. |
| 2023 | Q5i | Calculate the total GST paid and the total bill amount, given prices, GST rates, and a discount on one item. |
| 2024 | Q1i | For an Intra-state sale, if the CGST paid is ₹120 and the marked price is ₹2000, find the rate of GST. |
| 2024 | Q10i | Calculate the Total GST paid and the Total bill amount including GST for two items (Laddu at 5% and Pastries at 18%). |
| 2025 | Q1xiii | If the marked price of an article is ₹1375 and CGST is 4%, find the price of the article including GST. |
| 2025 | Q7iii | Check if the total amount paid by a customer (₹2000) is correct for a bill involving two articles with different GST rates and a 5% discount applicable under certain conditions. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q2c | Find the amount Jaya must pay at the end of the second year to clear her debt after borrowing ₹50,000 for 2 years with successive interest rates of 12% and 15% and repaying ₹33,000 at the end of the first year. |
| 2017 | Q7a | Calculate the interest for 6 months (Jan. to June 2016) at 6% p.a. for a saving bank account, and find the amount received if the account is closed on 1st July 2016. |
| 2018 | Q1b | Sonia deposited ₹600 per month for 2½ years at 10% p.a. in a recurring deposit account. Find the maturity value. |
| 2018 | Q9a | Priyanka has a recurring deposit of ₹1000 per month at 10% p.a. If she gets ₹5550 as interest, find the total time for which the account was held. |
| 2019 | Q5b | Rekha opened a recurring deposit account for 20 months at 9% p.a. If she receives ₹441 as interest, find the amount deposited each month. |
| 2020 | Q8b | Mr. Sonu deposits ₹750 per month for 2 years. If he gets ₹19125 at maturity, find the rate of interest. |
| 2017 | Q11c | Mr. Richard has a recurring deposit account for 3 years at 7.5% p.a. simple interest. If he gets ₹8325 as interest, find the monthly deposit and maturity value. |
| 2023 | Q1xiv | Naveen deposits ₹800 every month for 6 months. If he receives ₹4884 at maturity, find the interest he earns. |
| 2023 | Q2ii | Salman deposits ₹1000 every month for 2 years. If he receives ₹26000 on maturity, find the total interest and the rate of interest. |
| 2024 | Q4i | Suresh deposits ₹2000 per month at 8% p.a. If he gets ₹1040 as interest at maturity, find the total time (in years) the account was held. |
| 2025 | Q1ii | If Mr. Anuj deposits ₹500 per month for 18 months and earns ₹570 interest, identify the correct expression for the matured amount. |
| 2025 | Q2ii | Mrs. Rao deposited ₹250 per month for 3 years and received ₹10,110 at maturity. Find the rate of interest, and the additional interest she would receive if she deposited ₹50 more per month. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q5b | Find the investment required in ₹50 shares selling at ₹60 to obtain an income of ₹450, given a 10% dividend rate. Also find the yield percent. |
| 2018 | Q5b | Calculate: (i) The number of shares purchased, (ii) The annual dividend received, and (iii) The rate of return he gets on his investment, if a man invests ₹22,500 in ₹50 shares available at 10% discount, with a 12% dividend. |
| 2019 | Q1b | Find the number of shares purchased and the annual income if a man invests ₹4500 in ₹100 shares available at 10% discount, paying 7.5% dividend. |
| 2019 | Q9c | Sachin invests ₹8500 in 10%, ₹100 shares at ₹170. He sells when the price rises by ₹30 and reinvests the proceeds in 12%, ₹100 shares at ₹125. Find the sale proceeds, the number of new shares, and the change in his annual income. |
| 2020 | Q5a | Calculate: (i) the total dividend paid by a company (500 shares, NV ₹120, 15% dividend), (ii) annual income of Mr. Sharma (80 shares), and (iii) the market value of each share if Mr. Sharma's return is 10%. |
| 2024 | Q1ix | Find the sum invested to purchase 15 shares of NV ₹75 available at a discount of 20%. |
| 2024 | Q9ii | Mr. Gupta invested ₹33000 in buying ₹100 shares at 10% premium. The dividend is 12%. Find the number of shares purchased and his annual dividend. |
| 2025 | Q1viii | Determine if shares are at par, below par, or above par, if a man invests in a company paying 12% dividend and gets a 10% return on his investment. |
| 2025 | Q8i | A man bought ₹200 shares at 25% premium. If he received a return of 5% on his investment, find the market value, dividend percent declared, and the number of shares purchased if the annual dividend is ₹1000. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q4c | Solve the inequation and represent the solution set on a number line: \(-8 \frac{1}{2} < -\frac{1}{2} - 4x \leq 7 \frac{1}{2}, x \in I\). |
| 2018 | Q4a | Solve the inequation, write the solution set, and represent it on the real number line: \(-2 + 10x \leq 13x + 10 < 24 + 10x, x \in Z\). |
| 2019 | Q1a | Solve the inequation, write the solution set (\(x \in W\)), and represent it on a real number line: \(11x - 4 < 15x + 4 \leq 13x + 14\). |
| 2020 | Q2b | Solve the inequation and represent the solution set on the number line: \(\frac{3x}{5} + 2 < x + 4 \leq \frac{x}{2} + 5, x \in R\). |
| 2023 | Q1xv | Find the solution set for the inequation \(2x + 4 \leq 14, x \in W\). |
| 2023 | Q5ii | Solve the inequation, write the solution set (\(x \in I\)), and represent it on the real number line: \(-5(x - 9) \geq 17 - 9x > x + 2\). |
| 2024 | Q7ii | Solve the inequation, write the solution set (\(x \in I\)), and represent it on the real number line: \(-3 + x \leq \frac{7x}{2} + 2 < 8 + 2x\). |
| 2025 | Q1xiv | Find the solution set for \(0 < -\frac{x}{3} < 2, x \in Z\). |
| 2025 | Q4i | Solve the inequation, write the solution set (\(x \in R\)), and represent it on the real number line: \(\frac{2x - 5}{3} < \frac{3x}{5} + 10 \leq \frac{4x}{5} + 11\). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q1b | Solve the equation \(4x^2 - 5x - 3 = 0\) and give the answer correct to two decimal places. |
| 2018 | Q4c | Solve \(x^2 + 7x = 7\) and give the answer correct to two decimal places. |
| 2018 | Q7a | Find the value of k for which the equation \(x^2 + 4kx + (k^2 - k + 2) = 0\) has equal roots. |
| 2017 | Q10a | The sum of the ages of Vivek and Amit is 47 years; their product is 550. Find their ages. |
| 2019 | Q4b | Solve \(x^2 - 4x - 8 = 0\) for x, giving the answer correct to three significant figures. |
| 2019 | Q11b | The product of two consecutive natural numbers which are multiples of 3 is 810. Find the two numbers. |
| 2020 | Q1a | Solve the Quadratic Equation \(x^2 - 7x + 3 = 0\), giving the answer correct to two decimal places. |
| 2020 | Q10b | The difference of two natural numbers is 7 and their product is 450. Find the numbers. |
| 2023 | Q1v | If 3 is a root of the quadratic equation \(x^2 - px + 3 = 0\), find the value of p. |
| 2023 | Q3i | Solve \(x^2 + 4x - 8 = 0\), giving the answer correct to one decimal place. |
| 2023 | Q7ii | A man covers 100 km at speed \(x\) km/hr. If speed were 5 km/hr more, time would be 1 hour less. Find \(x\). |
| 2024 | Q1iii | The roots of \(px^2 - qx + r = 0\) are real and equal if \(q^2 = 4pr\). |
| 2024 | Q6i | Solve \(2x^2 - 11x + 5 = 1\), giving the answer correct to three significant figures. |
| 2025 | Q1i | Determine the nature of the roots of the quadratic equation \(3x^2 + \sqrt{7}x + 2 = 0\). |
| 2025 | Q2i | Solve \(2x^2 - 5x - 4 = 0\), giving the answer correct to three significant figures. |
| 2025 | Q8iii | Find the original speed of a car and the time taken for the journey, given distance (350 km), reduction in speed (20 km/hr), and increase in time (2 hours). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q1a | If b is the mean proportion between a and c, show that \(\frac{a^4 + a^2b^2 + b^4}{b^4 + b^2c^2 + c^4} = \frac{a^2}{c^2}\). |
| 2017 | Q6c | Use properties of proportion to find \(m:n\) and \(\frac{m^2 + n^2}{m^2 - n^2}\), given \(\frac{7m + 2n}{7m - 2n} = \frac{5}{3}\). |
| 2018 | Q6a | Solve for x (x is positive) using properties of proportion: \(\frac{2x + \sqrt{4x^2 - 1}}{2x - \sqrt{4x^2 - 1}} = 4\). |
| 2019 | Q4a | The numbers \(K + 3, K + 2, 3K - 7\) and \(2K - 3\) are in proportion. Find K. |
| 2019 | Q9b | Using properties of proportion solve for x, given \(\frac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = 4\). |
| 2020 | Q7b | Find \(x:y\) using properties of proportion, given \(\frac{x^2 + 2x}{2x + 4} = \frac{y^2 + 3y}{3y + 9}\). |
| 2020 | Q9b | Prove that \(x^2 - 4ax + 1 = 0\), if \(x = \frac{\sqrt{2a + 1} + \sqrt{2a - 1}}{\sqrt{2a + 1} - \sqrt{2a - 1}}\). |
| 2023 | Q1xi | The mean proportional between 4 and 9 is ? |
| 2023 | Q8ii | What number must be added to each of the numbers 4, 6, 8, 11 in order to get the four numbers in proportion?. |
| 2023 | Q9i | Using Componendo and Dividendo solve for x: \(\frac{\sqrt{2x + 2} + \sqrt{2x - 1}}{\sqrt{2x + 2} - \sqrt{2x - 1}} = 3\). |
| 2024 | Q1vii | Find the values of x and y in a distance/time table assuming uniform speed. |
| 2024 | Q4iii | Given \(\frac{(a + b)^3}{(a - b)^3} = \frac{64}{27}\), find \(\frac{a + b}{a - b}\) and hence find \(a:b\). |
| 2025 | Q5i | Using properties of proportion, find the value of x: \(\frac{6x^2 + 3x - 5}{3x - 5} = \frac{9x^2 + 2x + 5}{2x + 5}; x \neq 0\). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q4a | What must be subtracted from \(16x^3 - 8x^2 + 4x + 7\) so that the resulting expression has \(2x + 1\) as a factor?. |
| 2018 | Q3a | If \((x + 2)\) and \((x + 3)\) are factors of \(x^3 + ax + b\), find the values of 'a' and 'b'. |
| 2019 | Q2a | Using the factor theorem, show that \((x - 2)\) is a factor of \(x^3 + x^2 - 4x - 4\). Hence factorise the polynomial completely. |
| 2018 | Q10a | Use factor theorem to factorise \(2x^3 + 3x^2 - 9x - 10\). |
| 2020 | Q2a | Use factor theorem to factorise \(6x^3 + 17x^2 + 4x - 12\) completely. |
| 2020 | Q8a | What must be added to the polynomial \(2x^3 - 3x^2 - 8x\), so that it leaves a remainder 10 when divided by \(2x + 1\)?. |
| 2019 | Q11a | Using the Remainder Theorem find k if the sum of the remainders (when \(x^3 + (kx + 8)x + k\) is divided by \(x + 1\) and \(x - 2\)) is 1. |
| 2023 | Q1ii | If \(x - 2\) is a factor of \(x^3 - kx - 12\), find the value of k. |
| 2023 | Q2i | Find the value of 'a' if \(x - a\) is a factor of \(3x^3 + x^2 - ax - 81\). |
| 2023 | Q10i | Factorize completely using factor theorem: \(2x^3 - x^2 - 13x - 6\). |
| 2024 | Q1ii | What must be subtracted from \(x^3 + x^2 - 2x + 1\), so that the result is exactly divisible by \((x - 3)\)?. |
| 2024 | Q8i | The polynomial \(3x^3 + 8x^2 - 15x + k\) has \((x - 1)\) as a factor. Find k. Hence factorize the resulting polynomial completely. |
| 2025 | Q1vii | Identify the factor common to \(x^2 - 4\) and \(x^3 - x^2 - 4x + 4\). |
| 2025 | Q5ii | If \((x - 2)\) is a factor of polynomial \(2x^3 - 7x^2 + kx - 2\), find k and hence factorise the resulting polynomial completely. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q2b | If \(A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} -2 & 1 \\ -3 & 2 \end{bmatrix}\) and \(A^2 - 5B^2 = 5C\). Find matrix C. |
| 2017 | Q5a | Find matrix X if \(X = B^2 - 4B\) where \(B = \begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}\). Hence solve for \(a\) and \(b\) given \(X \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 5 \\ 50 \end{bmatrix}\). |
| 2018 | Q1a | Find the value of 'x' and 'y' if: \(2 \begin{bmatrix} x & 7 \\ 9 & y - 5 \end{bmatrix} + \begin{bmatrix} 6 & -7 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} 10 & 7 \\ 22 & 15 \end{bmatrix}\). |
| 2018 | Q6b | Find \(AC + B^2 - 10C\) given matrices A, B, and C. |
| 2019 | Q3a | Simplify \(\sin A \begin{bmatrix} \sin A & -\cos A \\ \cos A & \sin A \end{bmatrix} + \cos A \begin{bmatrix} \cos A & \sin A \\ -\sin A & \cos A \end{bmatrix}\). |
| 2019 | Q7c | Given \(\begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix} M = 6 I\). State the order of M and find the matrix M. |
| 2020 | Q6a | Find \(A^2 - 2AB + B^2\) given matrices A and B. |
| 2020 | Q1b | Given \(A = \begin{bmatrix} x & 3 \\ y & 3 \end{bmatrix}\). If \(A^2 = 3I\), find x and y. |
| 2023 | Q1i | If \(\begin{bmatrix} 2 & 0 \\ 0 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 \\ -8 \end{bmatrix}\), find the value of x and y. |
| 2023 | Q4i | Find \(A(B + C) - 14I\) given matrices A, B, C, and I. |
| 2024 | Q1iv | If \(A = \begin{bmatrix} 2 & 2 \\ 0 & 2 \end{bmatrix}\) and \(A^2 = \begin{bmatrix} 4 & x \\ 0 & 4 \end{bmatrix}\), find the value of x. |
| 2024 | Q2i | Find the values of x and y, if \(A = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}\), \(B = \begin{bmatrix} 4 & 0 \\ y & 1 \end{bmatrix}\), and \(C = \begin{bmatrix} 4 & 0 \\ x & 1 \end{bmatrix}\), and \(AB = C\). |
| 2025 | Q1v | If \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\), then \(A^2\) is equal to.... |
| 2025 | Q3ii | Given matrices A, B, and C, find \(A + C\), \(B(A+C)\), \(5B\), and \(B(A+C) - 5B\). |
| Year | Q No. | Question |
|---|---|---|
| 2018 | Q2b | If \((k - 3), (2k + 1)\) and \((4k + 3)\) are three consecutive terms of an A.P., find the value of k. |
| 2018 | Q5a | The 4th term of a G.P. is 16 and the 7th term is 128. Find the first term and common ratio of the series. |
| 2018 | Q11a | The 4th term of an A.P. is 22 and 15th term is 66. Find the first term and common difference. Hence find the sum of the series to 8 terms. |
| 2019 | Q2c | In an A.P., the fourth and sixth terms are 8 and 14 respectively. Find the first term, common difference, and the sum of the first 20 terms. |
| 2019 | Q6b | The first and last term of a G.P. are 3 and 96 respectively. If the common ratio is 2, find 'n' (number of terms) and the sum of the n terms. |
| 2019 | Q8a | The sum of the first three terms of an A.P. is 42 and the product of the first and third term is 52. Find the first term and the common difference. |
| 2020 | Q5c | The 4th, 6th and the last term of a G.P. are 10, 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms of the series. |
| 2020 | Q10a | If the 6th term of an A.P. is equal to four times its first term and the sum of first six terms is 75, find the first term and the common difference. |
| 2023 | Q1x | The nth term of an A.P. is \(2n + 5\). Find the 10th term. |
| 2023 | Q6ii | The 5th term and the 9th term of an A.P. are 4 and –12 respectively. Find the first term, common difference, and sum of 16 terms of the AP. |
| 2023 | Q9ii | Which term of the A.P. 15, 30, 45, 60... is 300? Hence find the sum of all the terms. |
| 2024 | Q1viii | Find the 7th term of the A.P.: \(\frac{1}{a}, (\frac{1}{a} + 1), (\frac{1}{a} + 2)\dots\). |
| 2024 | Q2iii | Find the nth term of the G.P. 15, 30, 60, 120… in terms of n. Also find how many terms give the sum 945. |
| 2024 | Q6ii | The nth term of an A.P. is given by \(T_n = 6(7 - n)\). Find its first term, common difference, and sum of its first 25 terms. |
| 2025 | Q4ii | The first term of an A.P. is 5, the last term is 50 and their sum is 440. Find the number of terms and the common difference. |
| 2025 | Q10i | If 1701 is the nth term of the G.P. 7, 21, 63..., find the value of n and hence find the sum of the n terms. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q3b | P(1,–2) divides the line segment A(3,–6) and B(x, y) such that AP : PB is 2 : 3. Find the coordinates of B. |
| 2017 | Q7b | Plot A(0,4), B(2,3), C(1,1), D(2,0). Reflect points B, C, D on the y-axis (B', C', D'). Join the figure and write the equation of the line of symmetry. |
| 2018 | Q4b | If the straight lines \(3x - 5y = 7\) and \(4x + ay + 9 = 0\) are perpendicular, find the value of a. |
| 2018 | Q5c | Plot quadrilateral ABCD. Reflect it on the y-axis (A'B'CD). Write coordinates of A' and B'. Name two invariant points. Name the polygon A'B'CD. |
| 2018 | Q7c | M is a point on AB such that AM : MB = 1 : 2, where A(2, 5), B(–1, 2) and C(5, 8) are vertices. Find coordinates of M. Hence find the equation of the line passing through C and M. |
| 2019 | Q3b | P(3, 2) divides line segment MN (M and N on X and Y axes respectively) in ratio 2:3. Find coordinates of M and N, and the slope of MN. |
| 2019 | Q5c | Plot A(0,5), B(3,0), C(1,0), D(1,–5). Reflect B, C, D on y-axis (B', C', D'). Write coordinates and name the closed figure. |
| 2019 | Q8b | Find the slope of BC and the equation of a line perpendicular to BC and passing through A, given vertices A(3, 8), B(–1, 2) and C(6, –6). |
| 2020 | Q3c | In what ratio is the line joining P(5, 3) and Q(–5, 3) divided by the y-axis? Also find the coordinates of the point of intersection. |
| 2020 | Q7a | If the lines \(5x - 3y + 2 = 0\) and \(6x - py + 7 = 0\) are perpendicular, find p. Hence find the equation of a line passing through (–2, –1) and parallel to \(6x - py + 7 = 0\). |
| 2020 | Q8c | Plot A(–4, 0), B(–3, 2), C(0, 4), D(4, 1), E(7, 3). Reflect B, C, D, E on the x-axis (B', C', D', E'). Join the points and name the closed figure. |
| 2023 | Q1ix | Find the centroid G of \(\Delta ABC\) with vertices (–4, –2), (6, 2) and (4, 6). |
| 2023 | Q4ii | D is the midpoint of BC, where A(1, –1), B(0, 4) and C(–6, 4) are vertices. Find coordinates of D and the equation of the median AD. |
| 2023 | Q3iii | Plot A, B, C. Reflect A and B on the x-axis (E, D). Reflect B through the origin (F). Reflect B and C on the y-axis (H, G). Join points and name the closed figure. |
| 2023 | Q6iii | A and B are on the x-axis and y-axis. P is on AB such that AP : PB = 3 : 1. Find coordinates of P and the equation of a line passing through P and perpendicular to AB. |
| 2024 | Q1vi | Points A (x, y), B (3, -2) and C (4, -5) are collinear. Find the value of y in terms of x. |
| 2024 | Q3iii | Plot A(0, 3), B(2, 1) and C(4, -1). Reflect B and C in y-axis (B', C'). Reflect A in the line BB' (A'). Join \(ABA'B'\) and name the figure. |
| 2024 | Q7iii | Write down the coordinates of A and D. Find the centroid of \(\Delta ABC\). If D divides AC in ratio \(k:1\), find \(k\). Find the equation of the line BD. |
| 2025 | Q1iv | The equation of the line passing through origin and parallel to \(3x + 4y + 7 = 0\) is \(3x + 4y = 0\). |
| 2025 | Q3iii | Given parallelogram ABCD coordinates (from graph), calculate the coordinates of P (intersection of diagonals), find slopes of CB and DA to verify parallel lines, and find the equation of diagonal AC. |
| 2025 | Q6i | Plot A(1, 3), B(1, 2) and C(3, 0). Reflect A and B on the x-axis (E, D), reflect A and B through the origin (F, G), reflect A, B, C on the y-axis (J, I, H). Join all points and name the closed figure. |
| 2025 | Q7ii | Line PQ cuts off intercepts of 5 units (x-axis) and 3 units (y-axis). Line RS is perpendicular to PQ and passes through the origin. Find coordinates of P and Q, and the equation of line RS. |
| 2024 | Q10ii | If \(kx - y + 4 = 0\) and \(2y = 6x + 7\) are perpendicular, find \(k\). Find the equation of a line parallel to \(2y = 6x + 7\) and passing through (–1, 1). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q11b | Given \(\angle PSR = \angle QPR\). Prove \(\Delta PQR \sim \Delta SPR\). Find the length of QR and PS. Find \(\frac{area of \Delta PQR}{area of \Delta SPR}\). |
| 2018 | Q9b | In \(\Delta PQR\), MN is parallel to QR. Find \(MN:QR\). Prove \(\Delta OMN \sim \Delta ORQ\). Find \(Area of \Delta OMN : Area of \Delta ORQ\). |
| 2019 | Q6a | Given \(\angle PQR = \angle PST = 90^\circ\). Prove \(\Delta PQR \sim \Delta PST\). Find \(Area of \Delta PQR : Area of quadrilateral SRQT\). |
| 2020 | Q6b | Given chords AD and BC intersect at P. Prove \(\Delta PAB \sim \Delta PCD\). Find the length of CD. Find \(Area of \Delta PAB : Area of \Delta PCD\). |
| 2023 | Q1vi | Given \(\angle BAP = \angle DCP = 70^\circ\), PC = 6 cm, and CA = 4 cm, find the ratio PD : DB. |
| 2023 | Q5iii | Given AC // DE // BF. Prove \(\Delta GED \sim \Delta GBF\). Find DE and DB:AB. |
| 2024 | Q1xv | Given \(\Delta ABC \sim \Delta PQR\). If AD and PS are bisectors of \(\angle BAC\) and \(\angle QPR\), determine the valid similarity statement. |
| 2024 | Q6iii | Given right angled triangles \(\Delta ADB\) and \(\Delta ACB\). Prove \(\Delta APD \sim \Delta BPC\). Find BD and PB, PA, and area ratio \(\Delta APD : \Delta BPC\). |
| 2025 | Q1xi | If \(\Delta ABC \sim \Delta EFG\) and \(\angle ABC = \angle EFG = 60^\circ\), find the length of FG. |
| 2025 | Q2iii | In \(\Delta ABC\), \(\angle ABC = 90^\circ\). \(DE \perp AC\). Prove \(\Delta ABC \sim \Delta AED\). Find lengths BC, AD, AE. Find the scale factor of the map if the actual area of BCED is \(576 \text{ m}^2\). |
| 2025 | Q10ii | Chord SR produced meets tangent XTP at P. Prove \(\Delta PTR \sim \Delta PST\). Prove \(PT^2 = PR \times PS\). Find PT if PR = 4 cm and PS = 16 cm. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q6a | Construct the locus of points equidistant from AB and AC, and equidistant from BA and BC. Hence construct a circle touching the three sides of the triangle internally. |
| 2018 | Q8c | Construct a cyclic quadrilateral ABCD such that D is equidistant from AB and BC. |
| 2020 | Q10c | Construct a chord AB = 6 cm in a circle of radius 4.5 cm. Find the locus of points equidistant from A and B. Find the locus of points which are equidistant from AD and AB. |
| 2019 | Q8c | Locate a point A on the circumference of a semi-circle (diameter BC=7 cm) such that A is equidistant from B and C. Complete cyclic quadrilateral ABCD such that D is equidistant from AB and BC. |
| 2024 | Q1x | Define the circumcentre of a triangle (the point equidistant from the three vertices). |
| 2025 | Q1ix | Statement on circumcentre (equidistant from three non-collinear points) and incentre (intersection of angle bisectors). |
| 2024 | Q10iii | Construct the locus of points equidistant from B and C, equidistant from A and B. Mark O. Construct the locus of points keeping a fixed distance OA from O. Construct the locus of points equidistant from BA and BC. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q1c | AB and CD are two parallel chords of a circle. Radius is 13 cm. AB = 24 cm and CD = 10 cm. Find the distance between the two chords. |
| 2018 | Q2c | PQRS is a cyclic quadrilateral. Given \(\angle QPS = 73^\circ\), \(\angle PQS = 55^\circ\), and \(\angle PSR = 82^\circ\). Calculate \(\angle QRS\), \(\angle RQS\), and \(\angle PRQ\). |
| 2017 | Q8b | Prove that BD is a diameter of the circle and ABC is an isosceles triangle, given tangent PQ at A, and AB, AD as angle bisectors. |
| 2017 | Q9a | O is the centre. \(\angle DAE = 70^\circ\). Find \(\angle BCD\), \(\angle BOD\), and \(\angle OBD\). |
| 2018 | Q10b | O is the centre. If QR = OP and \(\angle ORP = 20^\circ\), find the value of x. |
| 2020 | Q3a | O is the centre. AB is a diameter. If AC = BD and \(\angle AOC = 72^\circ\). Find \(\angle ABC\), \(\angle BAD\), \(\angle ABD\). |
| 2020 | Q7c | TP and TQ are two tangents. If \(\angle BCQ = 55^\circ\) and \(\angle BAP = 60^\circ\), find \(\angle OBA\), \(\angle OBC\), \(\angle AOC\), and \(\angle ATC\). |
| 2019 | Q11c | AC is a diameter and side BC//AE in inscribed pentagon ABCDE. If \(\angle BAC = 50^\circ\), find \(\angle ACB\), \(\angle EDC\), \(\angle BEC\). Prove BE is a diameter. |
| 2019 | Q7a | AC is a tangent to the circle with centre O. If \(\angle ADB = 55^\circ\), find x and y. |
| 2023 | Q1iii | RT is a tangent touching the circle at S. If \(\angle PST = 30^\circ\) and \(\angle SPQ = 60^\circ\), find \(\angle PSQ\). |
| 2023 | Q2iii | O is the centre. CE is a tangent at A. If \(\angle ABD = 26^\circ\), find \(\angle BDA\), \(\angle BAD\), \(\angle CAD\), and \(\angle ODB\). |
| 2023 | Q4iii | PQ is a tangent at T. Chord AB produced meets tangent at P. Find length of PT, \(\angle BAT\), \(\angle BOT\), and \(\angle ABT\). |
| 2024 | Q1xii | PS and PT are the tangents. \(SQ \parallel PT\) and \(\angle SPT = 80^\circ\). Find \(\angle QST\). |
| 2024 | Q3ii | O is the centre. PR and PT are two tangents touching at Q and S. MN is a diameter. \(\angle PQM = 42^\circ\) and \(\angle PSM = 25^\circ\). Find \(\angle OQM\), \(\angle QNS\), \(\angle QOS\), and \(\angle QMS\). |
| 2024 | Q7i | \(\Delta ABC\) is isosceles and inscribed in a circle with centre O. PQ is a tangent at C. \(OM \perp AC\). \(\angle COM = 65^\circ\). Find \(\angle ABC\), \(\angle BAC\), and \(\angle BCQ\). |
| 2024 | Q9i | Diameter AB produced meets tangent PQ at P. Radius is 9 cm, PA = 24 cm. Find length of tangent PQ. |
| 2025 | Q1vi | Chords AC and BC are equal. If \(\angle ACB = 120^\circ\), find \(\angle ADB\). |
| 2025 | Q9iii | AB is a tangent at X. \(ZY = XY\). \(\angle OBX = 32^\circ\) and \(\angle AXZ = 66^\circ\). Find \(\angle BOX\), \(\angle CYX\), \(\angle ZYX\), and \(\angle OXY\). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q6a | Construct a triangle ABC in which AB = 7cm, \(\angle CAB=60^\circ\) and AC = 5cm. Construct the locus (equidistant from sides) and hence construct a circle touching the three sides internally. |
| 2018 | Q8c | Construct \(\Delta ABC\) such that BC = 5 cm, AB = 6.5 cm and \(\angle ABC = 120^\circ\). Construct a circum-circle of \(\Delta ABC\). Construct a cyclic quadrilateral ABCD (D equidistant from AB and BC). |
| 2020 | Q1c | Construct a triangle ABC where AB = 3 cm, BC = 4 cm and \(\angle ABC = 90^\circ\). Hence construct a circle circumscribing the triangle. Measure the radius. |
| 2019 | Q4c | Draw a circle of radius 4 cm. Mark center O. Mark P outside at 7 cm from O. Construct two tangents from P. Measure one tangent. |
| 2019 | Q8c | Construct a semi-circle with diameter BC = 7cm. Locate A on circumference equidistant from B and C. Complete cyclic quadrilateral ABCD, D equidistant from AB and BC. Measure \(\angle ADC\). |
| 2020 | Q10c | Construct a circle of radius 4.5 cm. Draw a chord AB = 6 cm. Find the locus of points equidistant from A and B, meeting the circle at D. Find the locus of points equidistant from AD and AB, meeting the circle at C. Measure CD. |
| 2023 | Q8iii | Construct \(\Delta ABC\) in which AB = 6 cm, \(\angle BAC = 120^\circ\) and AC = 5 cm. Construct a circle passing through A, B and C. Measure the radius. |
| 2024 | Q10iii | Construct \(\angle ABC = 90^\circ\), AB = 6 cm, BC = 8 cm. Construct loci: equidistant from B/C, equidistant from A/B (mark O). Construct locus with fixed distance OA from O. Construct locus equidistant from BA and BC. |
| 2025 | Q3i | Construct a \(\Delta ABC\), where AB = 6 cm, AC = 4.5 cm and \(\angle BAC = 120^\circ\). Construct a circle circumscribing the \(\Delta ABC\). Measure the radius. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q4b | Find the area of the shaded region in a rectangle consisting of a circle and two semi-circles, each of radius 5 cm (correct to three significant figures). |
| 2017 | Q6b | Find the height and curved surface area of a conical tent that accommodates 77 persons, each needing \(16 \text{ m}^3\) of air, given radius \(7 \text{ m}\). |
| 2018 | Q2a | Find the radius and volume of a cylinder given circumference of the base (132 cm) and height (25 cm). |
| 2018 | Q7b | Find the actual length of the diagonal AC (in km) and the actual area of the plot (in sq km) for a rectangular plot ABCD on a map drawn to a scale of 1:50,000. |
| 2018 | Q9c | Find the volume of a solid consisting of a right circular cylinder (H=4 cm) with a hemisphere (R=7 cm) at one end and a cone (H=4 cm, R=7 cm) at the other. |
| 2019 | Q3c | Find the radius and curved surface area of a cylinder formed when a solid metallic sphere (R=6 cm) is melted and made into a cylinder of height 32 cm (\(\pi = 3.1\)). |
| 2019 | Q6c | Find the volume of the remaining solid after a hemispherical and a conical hole are scooped out of a solid wooden cylinder. |
| 2019 | Q7b | If the scale factor of a building model is 1:30, find the actual height (in meters) for a model height of 80 cm. Find the volume of the tank on the model if the actual volume is \(27 \text{ m}^3\). |
| 2020 | Q4a | A spherical ball (R=6 cm) is melted and recast into 64 identical spherical marbles. Find the radius of each marble. |
| 2020 | Q11a | If the height of a model building is 0.8 m (scale 1:50), find the actual height. If the floor area of a flat is \(20 \text{ m}^2\), find the floor area of that in the model. |
| 2020 | Q11b | Find the volume of the remaining solid after two conical cavities are hollowed out of a wooden cylinder. |
| 2023 | Q1xiii | The volume of a cylinder of height 3 cm is \(48\pi\). Find the radius. |
| 2023 | Q7iii | A hemisphere (R=7 cm) surmounted by a cone (H=4 cm) is immersed in a cylinder (R=14 cm). Find the rise in the water level. |
| 2024 | Q1xiv | Find the ratio of the curved surface areas of two cylinders formed by rotating a rectangular sheet (\(11 \text{ cm} \times 7 \text{ cm}\)) first about the 11 cm side and then about the 7 cm side. |
| 2024 | Q2ii | Find the total surface area and total cost of painting two identical halves of a cylinder (diameter 7 cm, height 10 cm) cut along its height. |
| 2024 | Q8iii | Oil occupying \(3/4\) of a spherical vessel (R=28 cm) is poured into a cylindrical vessel (R=21 cm). Find the height of the oil in the cylindrical vessel. |
| 2025 | Q1xii | If the volume of two spheres is in the ratio \(27:64\), find the ratio of their radii. |
| 2025 | Q5iii | Find the sum of the total surface area of the three parts (cylindrical block and two hemispheres) of a capsule, given radius 3.5 cm and cylindrical length 14 cm. |
| 2025 | Q9i | A hollow sphere (external diameter 10 cm, internal diameter 6 cm) is melted and made into a solid right circular cone of height 8 cm. Find the radius of the cone. |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q2a | Evaluate without using trigonometric tables: \(\sin^2 28^\circ + \sin^2 62^\circ + \tan^2 38^\circ - \cot^2 52^\circ + \frac{1}{4 \sec^2 30^\circ}\). |
| 2017 | Q9c | Prove that \(\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta\). |
| 2018 | Q3b | Prove that \(\sqrt{\sec^2 \theta + \operatorname{cosec}^2 \theta} = \tan \theta + \cot \theta\). |
| 2018 | Q6c | Prove that \((1 + \cot \theta - \operatorname{cosec} \theta)(1 + \tan \theta + \sec \theta) = 2\). |
| 2019 | Q2b | Prove that \((\operatorname{cosec} \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta) = 1\). |
| 2020 | Q3b | Prove that \(\frac{\sin A}{1 + \cot A} - \frac{\cos A}{1 + \tan A} = \sin A - \cos A\). |
| 2020 | Q11c | Prove the identity \(\left(\frac{1 - \tan \theta}{1 - \cot \theta}\right)^2 = \tan^2 \theta\). |
| 2023 | Q1viii | \((1 + \sin A) (1 - \sin A)\) is equal to \(\cos^2 A\). |
| 2023 | Q3ii | Prove the identity: \((\sin^2 \theta - 1)(\tan^2 \theta + 1) + 1 = 0\). |
| 2024 | Q1xi | Identify the correct statement: \(\sin^2 \theta + \cos^2 \theta = 1\). |
| 2024 | Q3i | Factorize \(\sin^3 \theta + \cos^3 \theta\). Hence, prove the identity: \(\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} + \sin \theta \cos \theta = 1\). |
| 2025 | Q1x | Assertion (A): If \(\sin^2 A + \sin A = 1\) then \(\cos^4 A + \cos^2 A = 1\). Reason (R): \(1 - \sin^2 A = \cos^2 A\). |
| 2025 | Q4iii | Prove that \(\frac{(\operatorname{cosec} A + \cos A - 1)(\sin A + \cos A)}{\sin^3 A + \cos^3 A} = \operatorname{cosec} A \cdot \sec A\). |
| Year | Q No. | Question |
|---|---|---|
| 2017 | Q11a | Find the distance between two ships A and B, which are on opposite sides of a 60 m high lighthouse, given angles of depression \(60^\circ\) and \(45^\circ\) (correct to nearest whole number). |
| 2018 | Q10c | Find the height of tower PT, given the angle of elevation from P to QR (50m high) is \(60^\circ\) and from Q to PT is \(30^\circ\). |
| 2020 | Q6c | Find the height of the cliff and the distance between the cliff and the tower, given the angle of depression of the top and bottom of a 20 m tower from the top of the cliff are \(45^\circ\) and \(60^\circ\). |
| 2019 | Q10b | A man observes the angle of elevation of the top of a tower to be \(45^\circ\). Walking 20 m towards it, the angle changes to \(60^\circ\). Find the height of the tower (correct to 2 significant figures). |
| 2023 | Q9iii | From the top of a 100 m high tower, a man observes angles of depression of two ships A and B (on opposite sides) as \(45^\circ\) and \(38^\circ\). Find the distance between the ships (to the nearest metre). |
| 2024 | Q5ii | Find the distance AB between two points on opposite sides of a 100 m high tree, given the angles of elevation \(52^\circ\) and \(45^\circ\). |
| 2025 | Q6ii | Find the height of the tower AB and the building CD, given the angles of depression from C and E (mid-point of CD) to AB are \(35^\circ\) and \(14^\circ\), and the distance between them is 100 m. |
| Year | Q No. | Question | ||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2017 | Q3(c) | Marks of 10 students in ascending order are: \(13, 35, 43, 46, x, x+4, 55, 61, 71, 80\). If the median is 48, find the value of \(x\) and the mode. | ||||||||||||||||||||||
| 2017 | Q8(a) |
Calculate the mean using the step deviation method.
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| 2018 | Q3(c) |
Draw a histogram and estimate the mode.
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| 2018 | Q8(b) |
If the mean of the following distribution is 24, find the value of \(a\).
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| 2019 | Q1(c) |
Calculate the median and mode.
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| 2020 | Q5(b) |
The mean is 16. Find the value of frequency \(f\).
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| 2024 | Q1(v) | The median of the observations \(27, 31, 46, 52, x, x+4, 71, 79, 85, 90\) is 64. Find the value of \(x\). | ||||||||||||||||||||||
| 2025 | Q8(ii) |
Find the mean (nearest whole number) and median.
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| Year | Q No. | Question |
|---|---|---|
| 2017 | Q5c | Sixteen cards labelled a,b,c...p are shuffled. Find the probability that the card drawn is: (i) a vowel, (ii) a consonant, (iii) none of the letters of the word 'median'. |
| 2018 | Q1c | Cards bearing numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 are in a bag. Find the probability of getting: (i) a prime number, (ii) a number divisible by 4, (iii) a multiple of 6, (iv) an odd number. |
| 2019 | Q5a | There are 25 discs numbered 1 to 25. Find the probability that the number is: (i) an odd number, (ii) divisible by 2 and 3 both, (iii) a number less than 16. |
| 2020 | Q4b | Letters of the word 'AUTHORIZES' are on discs. Find the probability that the letter is: (i) a vowel, (ii) one of the first 9 letters of the English alphabet which appears in the word, (iii) one of the last 9 letters of the English alphabet which appears in the word. |
| 2023 | Q1iv | A letter is chosen at random from all the letters of the English alphabets. The probability that the letter chosen is a vowel is \(\frac{5}{26}\). |
| 2023 | Q7i | A bag contains 25 cards, numbered 1 to 25. Find the probability that the number is: (a) multiple of 5, (b) a perfect square, (c) a prime number. |
| 2024 | Q1xiii | Assertion (A): A die is thrown once and the probability of getting an even number is \(\frac{2}{3}\). |
| 2024 | Q8ii | Letters A, D, M, N, O, S, U, Y are on cards. Find the probability that the card drawn is a letter of the word: (a) MONDAY, (b) which does not appear in MONDAY, (c) which appears both in SUNDAY and MONDAY. |
| 2025 | Q1iii | Which of the following cannot be the probability of any event? (\(\frac{5}{4}, 0.25, \frac{1}{33}, 67%\)). |
| 2025 | Q9ii | Cards numbered 01 to 30. Find the probability of winning a Wall Clock (perfect square), a Water Bottle (even multiple of 3), or a Purse (prime number). |