AIMS TUTORIAL
Excellence in Mathematics Education
MATHEMATICS — 1B Model Papers
1
MODEL PAPER — 1
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Write the complex number (4 + 3i) / [(2 + 3i)(4 – 3i)] in the form a + ib.
2. Write z = −√7 + i√21 in the polar form.
3. If 1, ω, ω² are the cube roots of unity, then find the value of (1 − ω + ω²)⁵ + (1 + ω − ω²)⁵.
4. Form quadratic equation whose roots are (p − q)/(p + q), (p + q)/(p − q) (p ≠ ±q).
5. Find the algebraic equation whose roots are 2 times the roots of x⁵ − 2x⁴ + 3x³ − 2x² + 4x + 3 = 0.
6. Find the number of (i) 6 (ii) 7 letter palindromes that can be formed using the letters of the word EQUATION.
7. If ⁿPᵣ = 5040 and ⁿCᵣ = 210, find n and r.
8. Find the number of terms with non‑zero coefficients in (4x − 7y)⁴⁰ + (4x + 7y)⁴⁰.
9. Find the mean deviation about the median for the following data 4, 6, 9, 3, 10, 13, 2.
10. A Poisson variable satisfies P(X = 1) = P(X = 2). Find P(X = 5).
SECTION — B
Answer ANY FIVE questions (5 × 4 = 20)
11. If x + iy = 1/(1 + cos θ + i sin θ), then show that 4x² − 1 = 0.
12. Determine the range of the expression (x + 2)/(2x² + 3x + 6).
13. If the letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the rank of the word PRISON.
14. Prove that ⁴⁰C₂ₙ / ²⁰Cₙ = [1·3·5·…·(4n−1)] / [1·3·5·…·(2n−1)]².
15. Resolve x² / [(x − 1)(x − 2)] into partial fractions.
SECTION — C
Answer ANY FIVE questions (5 × 7 = 35)
18. Show that one value of [(1 + sin (π/8) + i cos (π/8)) / (1 + sin (π/8) − i cos (π/8))]⁸ is −1.
19. Solve the equation x⁴ + 2x³ − 5x² + 6x + 2 = 0 given that 1 + i is one of its roots.
20. If the 2nd, 3rd, 4th terms in the expansion of (a + x)ⁿ are respectively 240, 720, 1080, find a, x, n.
2
MODEL PAPER — 2
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line passing through (−4,5) and making equal intercepts on the axes.
2. Find the equation of the line parallel to 2x+3y+7=0 and passing through (5, 4).
3. Find the fourth vertex of the parallelogram whose consecutive vertices are (2,4, −1), (3,6, −1) and (4,5,1).
4. Find the intercepts of the plane 4x−3y−2z+2=0 on the coordinate axes.
5. Evaluate limx→∞ (e^{3x} − 1)/x.
6. Evaluate limx→∞ (x² + 5x + 2)/(2x² − 5x + 1).
7. Find the derivative of f(x) = x e^{x} sin x.
8. Find the derivative of y = e^{a\sin^{−1}x}.
9. If the increase in the side of a square is 4
10. Verify Lagrange’s mean value theorem for the function x²−1 on [2,3].
SECTION — B
Answer ANY FIVE questions (5 × 4 = 20)
11. A(1,2), B(2, −3) and C are 3 points. If P is a point such that PA² + PB² = 2PC², then show that the equation to the locus of P is 7x − 7y + 4 = 0.
12. When the origin is shifted to the point (2,3), the transformed equation of a curve is x² + 3xy − 2y² + 17x − 7y − 11 = 0. Find the original equation of the curve.
13. Transform the equation (x/a) + (y/b) = 1 into normal form.
SECTION — C
Answer ANY FIVE questions (5 × 7 = 35)
18. Find the orthocentre of the triangle formed by the vertices (5, −2), (−1, 2), (1, 4).
19. Prove that area of triangle formed by ax²+2hxy+by²=0, lx+my+n=0.
3
MODEL PAPER — 3
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line perpendicular to 5x − 3y + 1 = 0 and passing through (4, −3).
2. Find a, if the area of the triangle formed by x − 3y = 0, 3x + y = 4 is 6 sq. units.
3. Find the ratio in which the XZ‑plane divides line joining A(−2,3,4), B(1,2,3).
4. Find the equation of the plane which makes intercepts 1, 2, 4 on x, y, z‑axes.
5. Evaluate limx→0 (e^{\sin x} − 1)/\sin x.
6. Evaluate limx→∞ (11x³ − 3x + 4)/(13x³ − 5x² − 7).
7. Find the derivative of f(x) = \sin(\log x).
8. Find the derivative of 2x² − 3xy + y² + x + 2y − 8 = 0.
9. If y = x² + 3x + 6 then find Δy and dy when x = 10, Δx = 0.01.
10. State Lagrange’s mean value theorem.
4
MODEL PAPER — 4
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line passing through the point (−2,4) and making intercepts, whose sum is zero.
2. Find the distance between the parallel lines 5x−3y−4=0, 10x−6y−9=0.
3. Show that the points (1,2,3), (7, 0, 1), (−2, 3, 4) are collinear.
4. Find a triad of d.c’s of the normal to the plane x+2y+2z−4=0.
5. Evaluate limx→∞ (e^{x} − 1 − x)/(√(1+x) − 1).
6. Evaluate limx→0 (\cos x − \cos b x)/x².
7. Find the derivative of y = e^{\sin^{−1}x}.
8. Find the derivative of 7^{(x + 3x)}.
9. If the increase in the side of a square is 2
Back to top ↑
10. Verify Rolle’s theorem for the function x²−1 on [−1,1].
5
TS — MARCH 2020
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the value of p, if the straight lines x + p = 0, y + 2 = 0 and 3x + 2y + 5 = 0 are concurrent.
2. Find the length of the perpendicular drawn from the point (3,4) to the straight line 3x − 4y + 10 = 0.
3. Show that the points (1,2,3), (7, 0, 1), (−2, 3, 4) are collinear.
4. Find a triad of d.c’s of the normal to the plane x + 2y + 2z − 4 = 0.
5. Evaluate limx→0 (e^{10x} − 1)/x.
6. Find limx→∞ 1/(3|x| − 2x).
7. Find the derivative of y = x tan^{−1} x.
8. If y = a e^{mx} + b e^{−mx}, then prove that y” = m² y.
9. If y = 5x² + 6x + 6, then find Δy and dy when x = 2, Δx = 0.001.
10. Define the strictly increasing function and strictly decreasing function on an interval I.
6
TS — MARCH 2023
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the slope of the straight line passing through the points (3, 4), (7, −6).
2. Transform the following straight line equation into normal form 3x + 4y = 5.
3. Find the centroid of the tetrahedron whose vertices are (2,3, −4), (−3, 3, −2), (−1, 4, 2), (3, 5, 1).
4. Write the equation of the plane 4x − 4y + 2z + 5 = 0 in the intercept form.
5. Evaluate limx→−2 [1/(x² − 2) − 1/(x² − 4)].
6. Evaluate limx→0 (e^{7x} − 1)/x.
7. If y = \log (\sin(\log x)), find dy/dx.
8. Find the derivative of \sin^{−1}(3x − 4x³) w.r.t x.
9. Find Δy and dy for the function y = −x² + x, when x = 10, Δx = 0.1.
10. Verify Rolle’s theorem for the function y = f(x) = x² + 4 on [−3,3].
7
TS — MARCH 2024
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Show that the points (−5, 1), (5, 5), (10,7) are collinear.
2. Find the distance between the parallel lines 5x−3y−4=0, 10x−6y−9=0.
3. Find a 4th vertex of parallelogram whose consecutive vertices are (2,4,−1), (3,6,−1) and (4,5,1).
4. Write the equation of the plane 4x − 4y + 2z + 5 = 0 in the intercept form.
5. Compute limx→0 (e^{x} − 1)/(√(1+x) − 1).
6. Compute limx→π/2 (e^{π} − π/2).
7. If f(x) = 2x² + 3x − 5 then prove that f'(0) + 3f'(−1) = 0.
8. Find the derivative of \sin^{−1}\!(2x/(1 + x²)) w.r.t x.
9. Find the approximate value of \sqrt[3]{65}.
10. Verify Rolle’s theorem for the function y = f(x) = x² + 4 on [−3,3].
8
TS — MARCH 2025
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the slope of the line passing through the points (−3, 8), (10,5).
2. Transform the equation 3x + 4y = 5 into (i) slope–intercept form (ii) intercept form.
3. Find the distance of P(3, −2, 4) from the origin.
4. Find the equation of the plane which makes intercepts 1,2,4 on the x, y, z‑axes respectively.
5. Compute limx→∞ (x² + 2x + 3)/(3x² − 5x + 7).
6. Find limx→∞ (8|x| + 3x)/(3|x| − 2x).
7. Find the derivative of (4 + x²) e^{−2x}.
8. Find the derivative of \sin^{−1}(3x − 4x³) w.r.t x.
9. If y = x² + 3x + 6 then find Δy and dy when x = 10, Δx = 0.01.
10. Verify Rolle’s theorem for the function x² − 1 on [−1,1].
Elementor #2173
AIMS TUTORIAL
Excellence in Mathematics Education
MATHEMATICS — 1B Model Papers
1
MODEL PAPER — 1
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Write the complex number (4 + 3i) / [(2 + 3i)(4 – 3i)] in the form a + ib.
2. Write z = −√7 + i√21 in the polar form.
3. If 1, ω, ω² are the cube roots of unity, then find the value of (1 − ω + ω²)⁵ + (1 + ω − ω²)⁵.
4. Form quadratic equation whose roots are (p − q)/(p + q), (p + q)/(p − q) (p ≠ ±q).
5. Find the algebraic equation whose roots are 2 times the roots of x⁵ − 2x⁴ + 3x³ − 2x² + 4x + 3 = 0.
6. Find the number of (i) 6 (ii) 7 letter palindromes that can be formed using the letters of the word EQUATION.
7. If ⁿPᵣ = 5040 and ⁿCᵣ = 210, find n and r.
8. Find the number of terms with non‑zero coefficients in (4x − 7y)⁴⁰ + (4x + 7y)⁴⁰.
9. Find the mean deviation about the median for the following data 4, 6, 9, 3, 10, 13, 2.
10. A Poisson variable satisfies P(X = 1) = P(X = 2). Find P(X = 5).
SECTION — B
Answer ANY FIVE questions (5 × 4 = 20)
11. If x + iy = 1/(1 + cos θ + i sin θ), then show that 4x² − 1 = 0.
12. Determine the range of the expression (x + 2)/(2x² + 3x + 6).
13. If the letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the rank of the word PRISON.
14. Prove that ⁴⁰C₂ₙ / ²⁰Cₙ = [1·3·5·…·(4n−1)] / [1·3·5·…·(2n−1)]².
15. Resolve x² / [(x − 1)(x − 2)] into partial fractions.
SECTION — C
Answer ANY FIVE questions (5 × 7 = 35)
18. Show that one value of [(1 + sin (π/8) + i cos (π/8)) / (1 + sin (π/8) − i cos (π/8))]⁸ is −1.
19. Solve the equation x⁴ + 2x³ − 5x² + 6x + 2 = 0 given that 1 + i is one of its roots.
20. If the 2nd, 3rd, 4th terms in the expansion of (a + x)ⁿ are respectively 240, 720, 1080, find a, x, n.
2
MODEL PAPER — 2
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line passing through (−4,5) and making equal intercepts on the axes.
2. Find the equation of the line parallel to 2x+3y+7=0 and passing through (5, 4).
3. Find the fourth vertex of the parallelogram whose consecutive vertices are (2,4, −1), (3,6, −1) and (4,5,1).
4. Find the intercepts of the plane 4x−3y−2z+2=0 on the coordinate axes.
5. Evaluate limx→∞ (e^{3x} − 1)/x.
6. Evaluate limx→∞ (x² + 5x + 2)/(2x² − 5x + 1).
7. Find the derivative of f(x) = x e^{x} sin x.
8. Find the derivative of y = e^{a\sin^{−1}x}.
9. If the increase in the side of a square is 4
10. Verify Lagrange’s mean value theorem for the function x²−1 on [2,3].
SECTION — B
Answer ANY FIVE questions (5 × 4 = 20)
11. A(1,2), B(2, −3) and C are 3 points. If P is a point such that PA² + PB² = 2PC², then show that the equation to the locus of P is 7x − 7y + 4 = 0.
12. When the origin is shifted to the point (2,3), the transformed equation of a curve is x² + 3xy − 2y² + 17x − 7y − 11 = 0. Find the original equation of the curve.
13. Transform the equation (x/a) + (y/b) = 1 into normal form.
SECTION — C
Answer ANY FIVE questions (5 × 7 = 35)
18. Find the orthocentre of the triangle formed by the vertices (5, −2), (−1, 2), (1, 4).
19. Prove that area of triangle formed by ax²+2hxy+by²=0, lx+my+n=0.
3
MODEL PAPER — 3
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line perpendicular to 5x − 3y + 1 = 0 and passing through (4, −3).
2. Find a, if the area of the triangle formed by x − 3y = 0, 3x + y = 4 is 6 sq. units.
3. Find the ratio in which the XZ‑plane divides line joining A(−2,3,4), B(1,2,3).
4. Find the equation of the plane which makes intercepts 1, 2, 4 on x, y, z‑axes.
5. Evaluate limx→0 (e^{\sin x} − 1)/\sin x.
6. Evaluate limx→∞ (11x³ − 3x + 4)/(13x³ − 5x² − 7).
7. Find the derivative of f(x) = \sin(\log x).
8. Find the derivative of 2x² − 3xy + y² + x + 2y − 8 = 0.
9. If y = x² + 3x + 6 then find Δy and dy when x = 10, Δx = 0.01.
10. State Lagrange’s mean value theorem.
4
MODEL PAPER — 4
GUESS PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the equation of line passing through the point (−2,4) and making intercepts, whose sum is zero.
2. Find the distance between the parallel lines 5x−3y−4=0, 10x−6y−9=0.
3. Show that the points (1,2,3), (7, 0, 1), (−2, 3, 4) are collinear.
4. Find a triad of d.c’s of the normal to the plane x+2y+2z−4=0.
5. Evaluate limx→∞ (e^{x} − 1 − x)/(√(1+x) − 1).
6. Evaluate limx→0 (\cos x − \cos b x)/x².
7. Find the derivative of y = e^{\sin^{−1}x}.
8. Find the derivative of 7^{(x + 3x)}.
9. If the increase in the side of a square is 2
Back to top ↑
10. Verify Rolle’s theorem for the function x²−1 on [−1,1].
5
TS — MARCH 2020
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the value of p, if the straight lines x + p = 0, y + 2 = 0 and 3x + 2y + 5 = 0 are concurrent.
2. Find the length of the perpendicular drawn from the point (3,4) to the straight line 3x − 4y + 10 = 0.
3. Show that the points (1,2,3), (7, 0, 1), (−2, 3, 4) are collinear.
4. Find a triad of d.c’s of the normal to the plane x + 2y + 2z − 4 = 0.
5. Evaluate limx→0 (e^{10x} − 1)/x.
6. Find limx→∞ 1/(3|x| − 2x).
7. Find the derivative of y = x tan^{−1} x.
8. If y = a e^{mx} + b e^{−mx}, then prove that y” = m² y.
9. If y = 5x² + 6x + 6, then find Δy and dy when x = 2, Δx = 0.001.
10. Define the strictly increasing function and strictly decreasing function on an interval I.
6
TS — MARCH 2023
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the slope of the straight line passing through the points (3, 4), (7, −6).
2. Transform the following straight line equation into normal form 3x + 4y = 5.
3. Find the centroid of the tetrahedron whose vertices are (2,3, −4), (−3, 3, −2), (−1, 4, 2), (3, 5, 1).
4. Write the equation of the plane 4x − 4y + 2z + 5 = 0 in the intercept form.
5. Evaluate limx→−2 [1/(x² − 2) − 1/(x² − 4)].
6. Evaluate limx→0 (e^{7x} − 1)/x.
7. If y = \log (\sin(\log x)), find dy/dx.
8. Find the derivative of \sin^{−1}(3x − 4x³) w.r.t x.
9. Find Δy and dy for the function y = −x² + x, when x = 10, Δx = 0.1.
10. Verify Rolle’s theorem for the function y = f(x) = x² + 4 on [−3,3].
7
TS — MARCH 2024
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Show that the points (−5, 1), (5, 5), (10,7) are collinear.
2. Find the distance between the parallel lines 5x−3y−4=0, 10x−6y−9=0.
3. Find a 4th vertex of parallelogram whose consecutive vertices are (2,4,−1), (3,6,−1) and (4,5,1).
4. Write the equation of the plane 4x − 4y + 2z + 5 = 0 in the intercept form.
5. Compute limx→0 (e^{x} − 1)/(√(1+x) − 1).
6. Compute limx→π/2 (e^{π} − π/2).
7. If f(x) = 2x² + 3x − 5 then prove that f'(0) + 3f'(−1) = 0.
8. Find the derivative of \sin^{−1}\!(2x/(1 + x²)) w.r.t x.
9. Find the approximate value of \sqrt[3]{65}.
10. Verify Rolle’s theorem for the function y = f(x) = x² + 4 on [−3,3].
8
TS — MARCH 2025
PREVIOUS IPE PAPERS
SECTION — A
Answer ALL questions (10 × 2 = 20)
1. Find the slope of the line passing through the points (−3, 8), (10,5).
2. Transform the equation 3x + 4y = 5 into (i) slope–intercept form (ii) intercept form.
3. Find the distance of P(3, −2, 4) from the origin.
4. Find the equation of the plane which makes intercepts 1,2,4 on the x, y, z‑axes respectively.
5. Compute limx→∞ (x² + 2x + 3)/(3x² − 5x + 7).
6. Find limx→∞ (8|x| + 3x)/(3|x| − 2x).
7. Find the derivative of (4 + x²) e^{−2x}.
8. Find the derivative of \sin^{−1}(3x − 4x³) w.r.t x.
9. If y = x² + 3x + 6 then find Δy and dy when x = 10, Δx = 0.01.
10. Verify Rolle’s theorem for the function x² − 1 on [−1,1].