951 |
Let O be the vertex and Q be any point on the parabola x2 = 8y. If the point P divides the line segment internally in the ratio 1 : 3 then the locus of P is a) x2 = y b) y2 = x c) y2 = 2x d) x2 = 2y
Let O be the vertex and Q be any point on the parabola x2 = 8y. If the point P divides the line segment internally in the ratio 1 : 3 then the locus of P is a) x2 = y b) y2 = x c) y2 = 2x d) x2 = 2y
|
IIT 2015 |
|
952 |
Let S be the focus of the parabola y2 = 8x and PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of △QPS is a) 2 sq. units b) 4 sq. units c) 6 sq. units d) 8 sq. units
Let S be the focus of the parabola y2 = 8x and PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of △QPS is a) 2 sq. units b) 4 sq. units c) 6 sq. units d) 8 sq. units
|
IIT 2012 |
|
953 |
Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)If st = 1 then the tangent at P and normal at S to the parabola meet at a point whose ordinate is a) b) c) d)
Let a, r, s, t be non-zero real numbers. Let P(at2, 2at), Q, R(ar2, 2ar and S(as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose PQ is the focal chord and QR and PK are parallel, where K is point (2a, 0)If st = 1 then the tangent at P and normal at S to the parabola meet at a point whose ordinate is a) b) c) d)
|
IIT 2014 |
|
954 |
The tangent PT and the normal PN of the parabola y2 = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose a) Vertex is b) Directrix is x = 0 c) Latus rectum is d) Focus is (a, 0)
The tangent PT and the normal PN of the parabola y2 = 4ax at the point P on it meet its axis at the points T and N respectively. The locus of the centroid of the triangle PTM is a parabola whose a) Vertex is b) Directrix is x = 0 c) Latus rectum is d) Focus is (a, 0)
|
IIT 2009 |
|
955 |
Consider the following linear equations ax + by + cz = 0 bx + cy + az = 0 cx + ay + bz = 0 Match the statements/expressions in column 1 with column 2 Column 1 | Column2 | i. a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca | A. Equations represent planes meeting at only one single point | ii. a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca | B. The equations represent the line x = y = z | iii. a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca | C. The equations represent identical planes | iv. a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca | D.The equations represent the whole of the three dimensional space |
Consider the following linear equations ax + by + cz = 0 bx + cy + az = 0 cx + ay + bz = 0 Match the statements/expressions in column 1 with column 2 Column 1 | Column2 | i. a + b + c ≠ 0 and a2 + b2 + c2 = ab + bc + ca | A. Equations represent planes meeting at only one single point | ii. a + b + c = 0 and a2 + b2 + c2 ≠ ab + bc + ca | B. The equations represent the line x = y = z | iii. a + b + c ≠ 0 and a2 + b2 + c2 ≠ ab + bc + ca | C. The equations represent identical planes | iv. a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca | D.The equations represent the whole of the three dimensional space |
|
IIT 2007 |
|
956 |
Suppose , , are the vertices of an equilateral triangle inscribed in the circle = 2. If = 1 + i, then find and . a) b) c) d) None of the above
|
IIT 1994 |
|
957 |
For any positive integers m, n (with n ≥ m), we are given that Deduce that
For any positive integers m, n (with n ≥ m), we are given that Deduce that
|
IIT 2000 |
|
958 |
The value of the integral is equal to a) b) c) d)
The value of the integral is equal to a) b) c) d)
|
IIT 2012 |
|
959 |
Let F : ℝ → ℝ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ε (1, 3). Let f(x) = x F(x) for all x ε ℝ.If and , then the correct expression is/are a) b) c) d)
Let F : ℝ → ℝ be a thrice differentiable function. Suppose that F(1) = 0, F(3) = −4 and F′(x) < 0 for all x ε (1, 3). Let f(x) = x F(x) for all x ε ℝ.If and , then the correct expression is/are a) b) c) d)
|
IIT 2015 |
|
960 |
One or more than one correct options The options with the values of α and L that satisfy the equation is/are a) b) c) d)
One or more than one correct options The options with the values of α and L that satisfy the equation is/are a) b) c) d)
|
IIT 2010 |
|
961 |
The number of points in the interval in which attains its maximum value is a) 8 b) 2 c) 4 d) 0
The number of points in the interval in which attains its maximum value is a) 8 b) 2 c) 4 d) 0
|
IIT 2014 |
|
962 |
If Where takes only principal values then the value of is a) 6 b) 9 c) 8 d) 11
If Where takes only principal values then the value of is a) 6 b) 9 c) 8 d) 11
|
IIT 2015 |
|
963 |
The intercept on X axis made by the tangent to the curve which is parallel to the line y = 2x are equal to a) ±1 b) ±2 c) ±3 d) ±4
The intercept on X axis made by the tangent to the curve which is parallel to the line y = 2x are equal to a) ±1 b) ±2 c) ±3 d) ±4
|
IIT 2013 |
|
964 |
The common tangent to the curve x2 + y2 = 2 and the parabola y2 = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area (in square units) of the quadrilateral PQRS is a) 3 b) 6 c) 9 d) 15
The common tangent to the curve x2 + y2 = 2 and the parabola y2 = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area (in square units) of the quadrilateral PQRS is a) 3 b) 6 c) 9 d) 15
|
IIT 2014 |
|
965 |
The area (in square units) of the region described by (x, y) : y2 < 2x and y ≥ 4x – 1 is a) b) c) d)
The area (in square units) of the region described by (x, y) : y2 < 2x and y ≥ 4x – 1 is a) b) c) d)
|
IIT 2015 |
|
966 |
Let f: [−1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 –x), Ɐ x ∈ [−1, 2]. If and are the area of the region bounded by y = f(x), x = −1, x = 2 and the X- axis. Then a) R1 = 2R2 b) R1 = 3R2 c) 2R1 = R2 d) 3R1 = R2
Let f: [−1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 –x), Ɐ x ∈ [−1, 2]. If and are the area of the region bounded by y = f(x), x = −1, x = 2 and the X- axis. Then a) R1 = 2R2 b) R1 = 3R2 c) 2R1 = R2 d) 3R1 = R2
|
IIT 2011 |
|
967 |
If , then is equal to a) b) c) d)
If , then is equal to a) b) c) d)
|
IIT 2017 |
|
968 |
If , i = 1, 2, 3 are polynomials in x such that and F(x) = then (x) at x = a is equal to a) – 1 b) 0 c) 1 d) 2
If , i = 1, 2, 3 are polynomials in x such that and F(x) = then (x) at x = a is equal to a) – 1 b) 0 c) 1 d) 2
|
IIT 1985 |
|
969 |
If then f (x) increases in a) (−2, 2) b) No value of x c) (0, ∞) d) (−∞, 0)
If then f (x) increases in a) (−2, 2) b) No value of x c) (0, ∞) d) (−∞, 0)
|
IIT 2003 |
|
970 |
If n is a positive integer and 0 ≤ v < π then show that
If n is a positive integer and 0 ≤ v < π then show that
|
IIT 1994 |
|
971 |
for every 0 < α, β < 2.
for every 0 < α, β < 2.
|
IIT 2003 |
|
972 |
The value of where x > 0 is a) 0 b) – 1 c) 1 d) 2
The value of where x > 0 is a) 0 b) – 1 c) 1 d) 2
|
IIT 2006 |
|
973 |
The value of a) 5050 b) 5051 c) 100 d) 101
The value of a) 5050 b) 5051 c) 100 d) 101
|
IIT 2006 |
|
974 |
Multiple choices Let g (x) = x f (x), where at x = 0 a) g is but is not continuous b) g is while f is not c) f and g are both differentiable d) g is and is continuous
Multiple choices Let g (x) = x f (x), where at x = 0 a) g is but is not continuous b) g is while f is not c) f and g are both differentiable d) g is and is continuous
|
IIT 1994 |
|
975 |
A five digit number divisible by 3 is formed using the numerals 0, 1, 2, 3, 4, and 5 without repetition. Total number of ways this can be done is a) At least 30 b) At most 20 c) Exactly 25 d) None of these
A five digit number divisible by 3 is formed using the numerals 0, 1, 2, 3, 4, and 5 without repetition. Total number of ways this can be done is a) At least 30 b) At most 20 c) Exactly 25 d) None of these
|
IIT 1989 |
|