Secondary Examination — 2025

Factors of 3x² – x – 4 are:

If 2p + 1, 13 and 5p – 3 are in A.P., then the value of p is:

A person bought an almirah for ₹3250 and spent ₹750 on its repair. If he sold it for ₹5,000, his gain percent is:

A point both of whose x and y coordinates are negative lies in:

If the y coordinate of a point is zero, then the point lies:

If an arc of a circle subtends an angle of x° at the centre and y° at any point on the remaining part of the circle, then the relation between x and y is:

TP and TQ are two tangents from an external point T to a circle with centre O. If ∠POQ = 110°, then ∠PTQ is:

If PQ is a chord of a circle with centre O and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is:

The curved surface area (in cm²) of a right circular cone of slant height 10 cm and base radius 7 cm is:

Total surface area (in cm²) of a solid hemisphere of radius 10 cm, when π = 3.14, is:

The curved surface area of a cylinder of height 14 cm is 88 cm². The diameter (in cm) of the cylinder is:

The value of (1 – tan² 45°) / (1 + tan² 45°) is:

The value of (sin²22° + sin²68°)/(cos²22° + cos²68°) + sin²63° + cos63° sin27° is:

If tan(A – B) = 1/√3 and tan(A + B) = √3, then the values of A and B respectively are:

The mean of first five multiples of 7 is:

The median of 10, 12, 14, 16, 18, 20 is:

A die is thrown once. The probability of getting a number between 2 and 6 is:

Fill in the blanks: (i) If the 1st term and common difference of an A.P. are 6 and 5 respectively, then the 11th term is _____. (ii) The sum of first ten terms of the A.P. 7, 14, 21, 28, … is _____.

Match Column-I with Column-II: (i) The coefficient of y in 5(2x–4) + 3x + 4y – 7 = 0 is: [Options: 0, –3, 4] (ii) When y = 3 is written as ax + by + c = 0, the value of a is: [Options: 0, –3, 4]

Match Column-I with Column-II: (i) If the roots of (α–3)x² + 4(α–3)x + 4 = 0 are equal and real, then α = ? (ii) If x = 1/2 is a root of x² + kx – 5/4 = 0, then k = ?

Fill in the blanks: (i) The 20th term from the end of the A.P. 3, 8, 13, … 253 is _____. (ii) If the first term is 5, last term is 45 and sum of all terms is 400, then the number of terms is _____.

Write 'True' or 'False': (i) The graph of 2x – 3y = 6 intersects the y-axis at the point (0, 2). (ii) The graph of x = 2 is a line parallel to x-axis.

Fill in the blanks: (i) If the distance between A(0, 0) and B(x, 3) is 5 units, then x = _____. (ii) If the midpoint of the segment joining (x, 4) and (5, 12) is (4, 8), then x = _____.

Write 'True' or 'False': (i) If a chord of a circle equals the radius, the angle subtended at a point on the minor arc is 30°. (ii) If tangents PA and PB from external point P are inclined at 80°, then ∠POA = 60°.

Fill in the blanks: (i) PAB is a secant and PT is a tangent. PT = x cm, PA = 4 cm, AB = 5 cm. Then x = _____. (ii) AB is a diameter, XPY is tangent at P. If ∠PBA = 30°, then ∠BPY = _____.

Fill in the blanks: (i) In △ABC right-angled at C, AC = 4 cm, AB = 8 cm. Then ∠A = _____. (ii) In △ABC right-angled at B, BC = 5 cm, ∠BAC = 30°. Then AB = _____.

Write 'True' or 'False': (i) Two dice are thrown. Probability of getting same number on both is 1/9. (ii) Two coins are tossed. Probability of getting at least one tail is 1/2.

Find the LCM of P(x) = (x–2)(x²–3x+2) and Q(x) = x²–4.

A man invested ₹5,00,000. Year 1: 4% loss; Year 2: 5% profit; Year 3: 10% profit. Find his net profit over 3 years.

Find the centroid of a triangle whose vertices are A(5, –2), B(9, 6) and C(4, 5).

ABC is an isosceles triangle with AB = AC and XAY is a tangent to the circumcircle at A. Show that XY ∥ BC.

ABCD is a cyclic quadrilateral with ∠A = (x+2y)°, ∠B = (5y–x)°, ∠C = 2x°, ∠D = (x+y)°. Find x and y.

Find the perimeter and area of a sector of a circle of radius 14 cm and central angle 30°.

A solid metallic sphere of radius 21 cm is melted and recast into smaller cones each of radius 7 cm and height 3 cm. Find the number of cones formed.

Find the median of the following data: xᵢ: 5, 15, 25, 35, 45, 55 fᵢ: 8, 10, 16, 24, 15, 7

A bag has 5 red, 4 black and 3 green balls. One ball is drawn at random. Find the probability of: (i) a red or green ball (ii) a ball that is not red.

Solve graphically: x + y = 5, x – y = 1.

The length of a rectangular garden is 7 m more than its breadth. If area = 144 m², find the dimensions.

Prove that the tangents drawn from an external point to a circle are of equal length.

Two perpendicular paths of width 10 m each run through the middle of a rectangular park 200 m × 150 m. Find the cost of constructing the paths at ₹50 per m².

If the mean of the following data is 8, find the value of p: Xᵢ: 3, 5, 7, 9, 11, 13 Fᵢ: 6, 8, 15, p, 8, 4

A washing machine costs ₹19,400. Due to Diwali Sale it is available for ₹4,200 cash down + 3 equal monthly instalments at 16% p.a. Let each instalment = ₹x. (i) Total interest paid = ? (ii) Amount owed for 1st month? (iii) Amount owed for 3rd month? (iv) Total amount paid by buyer? (v) Amount of each instalment?

Construct a triangle ABC in which AB = 5 cm, BC = 4.2 cm and median CD = 3.8 cm.

The shadow of a tower at 30° elevation is 10 m longer than at 45°. Find the height of the tower. (Use √3 = 1.732)