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Ques 1 GATE 2026 SET-1
In the given figure, PQ̅ is the diameter of a circle with center O. Two points R and S are chosen on the circle such that ∠ROS = 80°. When PR̅ and QS̅ are extended, they meet at T. The value of ∠RTS is _________________
(40°) is the correct answer.
Ques 2 GATE 2025 SET-2
Three villages P, Q, and R are located in such a way that the distance PQ=13 km, QR=14 km and RP=15 km as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of this connecting road?
Note: The figure shown is representative.
(c) is the correct answer.
Ques 3 GATE 2025 SET-2
In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm2) of the rectangle PLMN?
Note: The figure shown is representative.
(c) is the correct answer.
The correct answer is Option C — 8 cm².
PQRS is a square with side 2 cm. PLMN is a rectangle where corner L lies on side QR, and side MN passes through corner S of the square.
Let PL = a (the length of the rectangle along PQ extended) and LM = b (the width). Since L is on QR and MN passes through S, using the geometric constraint that the rectangle''s diagonal or side must align with corner S, we can set up the relationship.
P is at origin (0,0), Q at (2,0), R at (2,2), S at (0,2). L is on QR so L = (2, h) for some h. N is on PS extended, so N = (0, h). M = (2+d, h) for some extension d, and MN passes through S=(0,2).
The line MN passes through S(0,2) and M. Since N=(0,h) and MN is horizontal (rectangle), MN passes through S means h=2. So L=(2,2)=R, which means PL = PQ extended to R, giving PL = 2√2 (diagonal of square) and LM = 2√2.
Area = PL × LM = 2√2 × 2√2 = 8 cm².
Ques 4 GATE 2025 SET-1
A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
(a) is the correct answer.
Ques 5 GATE 2025 SET-1
A circle with center at (x,y)=(0.5,0) and radius =0.5 intersects with another circle with center at (x,y)=(1,1) and radius =1 at two points. One of the points of intersection (x, y) is:
(a) is the correct answer.
Ques 6 GATE 2025 SET-1
An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of 2π/n, is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen?
Note: The figures shown are representative.
(a) is the correct answer.
Ques 7 GATE 2025 SET-1
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative.
(a) is the correct answer.
Explanation:
1. Understand the Given Distance Conditions:
• The shortest distance from a point to a line is the perpendicular height.
• Let the total height of the large triangle PQR (perpendicular distance from P to QR) be H.
• The problem states that the distance between parallel lines ST and QR is half of this total distance. Therefore, the height of the trapezium SQRT is H/2.
• This leaves the remaining height for the small triangle PST to be: H - H/2 = H/2.
2. Use the Property of Similar Triangles:
• Since ST is parallel to QR, triangle PST is similar to triangle PQR (ΔPST ∼ ΔPQR).
• The ratio of the heights of these two similar triangles is:
Height of ΔPST / Height of ΔPQR = (H/2) / H = 1/2
• Since the ratio of their corresponding heights is 1/2, the ratio of their areas must be the square of the scale factor:
Area(ΔPST) / Area(ΔPQR) = (1/2)2 = 1/4
3. Determine the Area Breakdown:
• Let the total area of the large triangle PQR be 4 units.
• Then, the area of the smaller top triangle PST is exactly 1 unit.
• The area of the bottom trapezium SQRT is the remaining area:
Area(Trapezium SQRT) = Area(ΔPQR) - Area(ΔPST) = 4 - 1 = 3 units.
4. Calculate the Required Ratio:
• The question asks for the ratio of the area of triangle PST to the area of trapezium SQRT:
Ratio = Area(ΔPST) / Area(Trapezium SQRT) = 1/3
Ques 8 GATE 2025
A rectangle has a length L and a width W, where L>W. If the width, W, is increased by 10%, which one of the following statements is correct for all values of L and W?
(c) is the correct answer.
Ques 9 GATE 2025
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius r cm as shown in the figure. The side of the dodecagon is d cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side r cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
(a) is the correct answer.
Ques 10 GATE 2025
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative.
(a) is the correct answer.
Ques 11 GATE 2025
A stick of length one meter is broken at two locations at distances of b1 and b2 from the origin (0), as shown in the figure. Note that 0
Note: All lengths are in meter. The figure shown is representative.
(d) is the correct answer.
Ques 12 GATE 2025
A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
(a) is the correct answer.
Ques 13 GATE 2025
A circle with center at (x,y)=(0.5,0) and radius =0.5 intersects with another circle with center at (x,y)=(1,1) and radius =1 at two points. One of the points of intersection (x,y) is:
(a) is the correct answer.