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2026 34 questions

Q112026 · 8 Apr 2026 · Shift 2
Let O be the vertex of the parabola $y^{2} = 4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :
1. A.1
2. B.2
3. C.4
4. D.8
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Answer: B
Q202026 · 8 Apr 2026 · Shift 2
Which of the following amino acid will give violet coloured complex with neutral ferric chloride solution?
1. A.Threonine
2. B.Serine
3. C.Tyrosine
4. D.Cysteine
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Answer: B
Q22026 · 6 Apr 2026 · Shift 1
Let one root of the quadratic equation in $x$ : $(k^{2} - 15 k + 27) x^{2} + 9 (k - 1) x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^{2} = 6 k x$ is equal to:
1. A.4
2. B.6
3. C.8
4. D.12
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Answer: D
Q32026 · 6 Apr 2026 · Shift 1
Let $e_{1}$ and $e_{2}$ be two distinct roots of the equation $x^{2} - a x + 2 = 0$ . Let the sets ${a \in R : e_{1} and e_{2} are the eccentricities of hyperbolas} = (α, β)$ , and ${a \in R : e_{1} and e_{2} are the eccentricities of an ellipse and a hyperbola, respectively} = (γ, \infty)$ . Then $α^{2} + β^{2} + γ^{2}$ is equal to:
1. A.18
2. B.22
3. C.26
4. D.34
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Answer: C
Q112026 · 6 Apr 2026 · Shift 1
If the eccentricity $e$ of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , passing through $(6, 43)$ , satisfies $15 (e^{2} + 1) = 34 e$ , then the length of the latus rectum of the hyperbola $\frac{x ^{2}}{b ^{2}} - \frac{y ^{2}}{2 ( a ^{2} + 1 )} = 1$ is:
1. A.10
2. B.20
3. C.25
4. D.30
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Answer: A
Q122026 · 6 Apr 2026 · Shift 1
Let chord PQ of length $313$ of the parabola $y^{2} = 12 x$ be such that the ordinates of points P and Q are in the ratio 1:2. If the chord PQ subtends an angle $α$ at the focus of the parabola, then $sin α$ is equal to:
1. A. $\frac{3}{5}$
2. B. $\frac{4}{5}$
3. C. $\frac{5}{13}$
4. D. $\frac{12}{13}$
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Answer: A
Q112026 · 6 Apr 2026 · Shift 2
The eccentricity of an ellipse E with centre at the origin O is $\frac{3}{2}$ and its directrices are $x = \pm \frac{4 6}{3}$ . Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :
1. A. $\frac{4 2}{7}$
2. B. $\frac{4 2}{7}$
3. C. $\frac{4}{7}$
4. D. $\frac{8}{7}$
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Answer: D
Q122026 · 6 Apr 2026 · Shift 2
Let $x = 9$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $\frac{1}{3}$ . Let $P (α, 0)$ , $α > 0$ , be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :
1. A. $9 y^{2} = 8 x (1 - x)$
2. B. $3 y^{2} = 4 x (1 - x)$
3. C. $9 y^{2} = 8 x (x - 1)$
4. D. $3 y^{2} = 4 x (x - 1)$
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Answer: A
Q92026 · 5 Apr 2026 · Shift 1
Let a focus of the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ be $S (4, 0)$ and its eccentricity be $\frac{4}{5}$ . If the point $P (3, α)$ lies on $E$ and $O$ is the origin, then the area of $Δ POS$ is equal to:
1. A.12/5
2. B.14/5
3. C.24/5
4. D.48/5
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Answer: C
Q122026 · 5 Apr 2026 · Shift 2
Let the directrix of the parabola $P : y^{2} = 8 x$ , cut x-axis at the point $A$ . Let $B (α, β)$ , $α > 1$ , be a point on $P$ such that the slope of $AB$ is $3/5$ . If $BC$ is a focal chord of $P$ , then six times the area of $Δ ABC$ is :
1. A. $80$
2. B. $160$
3. C. $174$
4. D. $192$
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Answer: B
Q132026 · 5 Apr 2026 · Shift 2
Let the eccentricity $e$ of a hyperbola satisfy the equation $6 e^{2} - 11 e + 3 = 0$ . If the foci of the hyperbola are $(3, 5)$ and $(3, - 4)$ , then the length of its latus rectum is :
1. A. $\frac{11}{3}$
2. B. $\frac{17}{3}$
3. C. $\frac{15}{2}$
4. D. $\frac{17}{2}$
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Answer: C
Q222026 · 4 Apr 2026 · Shift 1· Numerical answer
Consider the parabola $P : y^{2} = 4 k x$ and the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ . Let the line segment joining the points of intersection of P and E, be their latus rectums. If the eccentricity of E is $e$ , then $e^{2} + 22$ is equal to _________.
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Answer: 3
Q112026 · 4 Apr 2026 · Shift 2
Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\frac{8}{3}$ . If the line $x = α$ intersects the hyperbola H at the points A and B such that the area of the triangle AOB is $415$ , where O is the origin, then $α^{2}$ equals
1. A.12
2. B.16
3. C.24
4. D.25
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Answer: B
Q242026 · 4 Apr 2026 · Shift 2· Numerical answer
Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^{2} = 16 x$ . Let the vertex B containing the right angle be $(4, 8)$ and the locus of the centroid of $Δ ABC$ be a conic $C_{o}$ . Then three times the length of latus rectum of $C_{o}$ is ____.
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Answer: 16
Q102026 · 2 Apr 2026 · Shift 1
Let an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a < b$ , pass through the point $(4, 3)$ and have eccentricity $\frac{5}{3}$ . Then the length of its latus rectum is :
1. A. $\frac{4 5}{3}$
2. B. $25$
3. C. $\frac{7 5}{3}$
4. D. $\frac{8 5}{3}$
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Answer: D
Q112026 · 2 Apr 2026 · Shift 2
Let O be the origin, and P and Q be two points on the rectangular hyperbola $x y = 12$ such that the mid point of the line segment PQ is $(\frac{1}{2}, - \frac{1}{2})$ . Then the area of the triangle OPQ equals :
1. A. $\frac{3}{2}$
2. B. $\frac{5}{2}$
3. C. $\frac{7}{2}$
4. D. $\frac{9}{2}$
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Answer: C
Q122026 · 2 Apr 2026 · Shift 2
Let the parabola $y = x^{2} + p x + q$ passing through the point $(1, - 1)$ be such that the distance between its vertex and the $x$ -axis is minimum. Then the value of $p^{2} + q^{2}$ is :
1. A.2
2. B.4
3. C.5
4. D.8
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Answer: B
Q212026 · 28 Jan 2026 · Shift 1· Numerical answer
For some $θ \in (0, \frac{π}{2})$ , let the eccentricity and the length of the latus rectum of the hyperbola $x^{2} - y^{2} sec^{2} θ = 8$ be $e_{1}$ and $l_{1}$ , respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^{2} sec^{2} θ + y^{2} = 6$ be $e_{2}$ and $l_{2}$ , respectively. If $e_{1}^{2} = e_{2}^{2} (sec^{2} θ + 1)$ , then $(\frac{l _{1} l _{2}}{e _{1} e _{2}}) tan^{2} θ$ is equal to
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Answer: 8
Q92026 · 28 Jan 2026 · Shift 2
An ellipse has its center at $(1, - 2)$ , one focus at $(3, - 2)$ and one vertex at $(5, - 2)$ . Then the length of its latus rectum is :
1. A.6
2. B. $63$
3. C. $\frac{16}{3}$
4. D. $43$
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Answer: A
Q122026 · 28 Jan 2026 · Shift 2
Let A be the focus of the parabola $y^{2} = 8 x$ . Let the line $y = m x + c$ intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is $(\frac{7}{3}, \frac{4}{3})$ , then $(B C)^{2}$ is equal to :
1. A.89
2. B.80
3. C.32
4. D.41
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Answer: B
Q162026 · 28 Jan 2026 · Shift 2
Let the ellipse $E : \frac{x ^{2}}{144} + \frac{y ^{2}}{169} = 1$ and the hyperbola $H : \frac{x ^{2}}{16} - \frac{y ^{2}}{λ ^{2}} = - 1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$ , then the value of $24 (e + L)$ is :
1. A.296
2. B.126
3. C.67
4. D.148
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Answer: A
Q182026 · 28 Jan 2026 · Shift 2
Let $P_{1} : y = 4 x^{2}$ and $P_{2} : y = x^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y = α x, α > 0$ and $P_{1}$ , then $α$ is equal to :
1. A.12
2. B.15
3. C.8
4. D.6
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Answer: A
Q192026 · 24 Jan 2026 · Shift 1
Let each of the two ellipses $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, (a > b)$ and $E_{2} : \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1, (A < B)$ have eccentricity $\frac{4}{5}$ . Let the lengths of the latus recta of $E_{1}$ and $E_{2}$ be $l_{1}$ and $l_{2}$ , respectively, such that $2 l_{1}^{2} = 9 l_{2}$ . If the distance between the foci of $E_{1}$ is 8 , then the distance between the foci of $E_{2}$ is
1. A. $\frac{96}{5}$
2. B. $\frac{8}{5}$
3. C. $\frac{16}{5}$
4. D. $\frac{32}{5}$
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Answer: D
Q12026 · 23 Jan 2026 · Shift 1
Let the domain of the function $f (x) = lo g_{3} lo g_{5} lo g_{7} (9 x - x^{2} - 13)$ be the interval $(m, n)$ . Let the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ have eccentricity $\frac{n}{3}$ and the length of the latus rectum $\frac{8 m}{3}$ . Then $b^{2} - a^{2}$ is equal to :
1. A.7
2. B.9
3. C.11
4. D.5
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Answer: A
Q152026 · 23 Jan 2026 · Shift 1
Let the line $y - x = 1$ intersect the ellipse $\frac{x ^{2}}{2} + \frac{y ^{2}}{1} = 1$ at the points A and B . Then the angle made by the line segment AB at the center of the ellipse is :
1. A. $π - tan^{- 1} (\frac{1}{4})$
2. B. $\frac{π}{2} + tan^{- 1} (\frac{1}{4})$
3. C. $\frac{π}{2} + 2 tan^{- 1} (\frac{1}{4})$
4. D. $\frac{π}{2} - tan^{- 1} (\frac{1}{4})$
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Answer: B