Practice questions
88551 published questions with worked solutions.
- If the system of equations x + 4y − z = λ, 7x + 9y + μz = −3, 5x + y + 2z = −1 has infinitely many solutions, then (2μ + 3λ) is...
- If the shortest distance between the lines x - 2 = y - 4 3 = z - 3 4 and x - 2 4 = y - 4 6 = z - 7 8 is 13 29 , then a value of λ...
- If the value of 3 36° + 5 18° 5 36° - 3 18° is a 5 - b c , where a, b, c are natural numbers and gcd(a, c) = 1, then a + b + c is...
- Let y = y(x) be the solution curve of the differential equation sec y dy dx + 2x sin y = x³ cos y, y(1) = 0. Then y(√3) is equal...
- The area of the region in the first quadrant inside the circle x² + y² = 8 and outside the parabola y² = 2x is equal to:
- If the line segment joining the points (5, 2) and (2, a) subtends an angle 4 at the origin, then the absolute value of the...
- Let a = 4 i - j + k, b = 11 i - j + k and c be a vector such that ( a + b) c = c (-2 a + 3 b). If (2 a + 3 b) c = 1670, then |...
- If the function f(x) = 2x³ − 9ax² + 12a²x + 1, a > 0 has a local maximum at x = α and a local minimum x = α², then α and α² are...
- There are three bags X, Y and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5...
- Let _ ^ _e4 dx e^x - 1 = 6. Then eᵅ and e⁻ᵅ are the roots of the equation:
- Let A = 2, 3, 6, 8, 9, 11 and B = 1, 4, 5, 10, 15 . Let R be a relation on A × B define by (a, b) R (c, d) if and only if 3ad −...
- For a, b > 0, let f(x) = cases ((a + 1)x) + b x x , & x < 0 3, & x = 0 ax + b^2x^2 - ax b a ,x x , & x > 0 cases be a continuous...
- If the term independent of x in the expansion of ( a ,x^2 + 1 2x^3 )^ 10 is 105, then a² is equal to:
- Let A be the region enclosed by the parabola y² = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the...
- If α = _ x 0^+ ( e^ x - e^ x x - x ) and β = _ x 0 (1 + x)^ 12 x are the roots of the quadratic equation ax² + bx − √e = 0, then...
- Let S be the focus of the hyperbola x^2 3 - y^2 5 = 1, on the positive x-axis. Let C be the circle with its centre at A(√6, √5)...
- The number of distinct real roots of the equation |x + 1||x + 3| − 4|x + 2| + 5 = 0, is ________ .
- Let a ray of light passing through the point (3, 10) reflects on the line 2x + y = 6 and the reflected ray passes through the...
- Let a, b, c ∈ N and a < b < c. Let the mean, the mean deviation about the mean and the variance of the 5 observations 9, 25, a,...
- Let α|x| = |y|e^ xy − β , α, β ∈ N be the solution of the differential equation x dy − y dx + xy(x dy + y dx) = 0, y(1) = 2. Then...