Class 10 Maths Sample Paper 5 (CBSE 2026-27 pattern)

If \(a \gt b\) are natural numbers, the LCM of \(2^a \times 3^b\) and \(2^b \times 3^a\) is

1. (a) \(6^b\)
2. (b) \(6^a\)
3. (c) \(2^a \times 3^b\)
4. (d) \(6^{a+b}\)

If \(\alpha, \beta\) are the zeroes of \(x^2 - px + q\) \((q \ne 0)\), then \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}\) equals

1. (a) \(\dfrac{p^2 - 2q}{q}\)
2. (b) \(\dfrac{p^2 + 2q}{q}\)
3. (c) \(\dfrac{p^2 - 2q}{p}\)
4. (d) \(\dfrac{q^2 - 2p}{q}\)

The lines \(3x + 2ky = 2\) and \(2x + 5y + 1 = 0\) are parallel (distinct) when \(k\) equals

1. (a) \(\dfrac{4}{15}\)
2. (b) \(-\dfrac{15}{4}\)
3. (c) \(\dfrac{15}{4}\)
4. (d) \(\dfrac{5}{3}\)

The quadratic equation \(x^2 - kx + 8 = 0\) has real roots, one of which is the square of the other. The value of \(k\) is

1. (a) \(4\)
2. (b) \(6\)
3. (c) \(8\)
4. (d) \(12\)

A quadratic equation whose roots are \(2 + \sqrt3\) and \(2 - \sqrt3\) is

1. (a) \(x^2 - 4x + 1 = 0\)
2. (b) \(x^2 + 4x + 1 = 0\)
3. (c) \(x^2 - 4x - 1 = 0\)
4. (d) \(x^2 - 4x + 7 = 0\)

An AP has \(21\) terms and its middle term is \(12\). The sum of all its terms is

1. (a) \(126\)
2. (b) \(240\)
3. (c) \(252\)
4. (d) \(264\)

Which of the following points is equidistant from \(A(1, 3)\) and \(B(5, -1)\)?

1. (a) \((5, 3)\)
2. (b) \((3, 3)\)
3. (c) \((0, 0)\)
4. (d) \((4, 4)\)

The line \(x + y = 4\) divides the segment joining \((-1, 1)\) and \((5, 7)\) in the ratio

1. (a) \(1 : 2\)
2. (b) \(2 : 1\)
3. (c) \(1 : 3\)
4. (d) \(3 : 1\)

\(\triangle ABC \sim \triangle DEF\) with \(AB = 3x\) cm, \(DE = 6\) cm, \(BC = 12\) cm and \(EF = (2x + 2)\) cm. The value of \(x\) is

1. (a) \(2\)
2. (b) \(3\)
3. (c) \(4\)
4. (d) \(6\)

In \(\triangle ABC\), \(\angle B = 90^\circ\) and \(BD \perp AC\). If \(AD = 4\) cm and \(DC = 9\) cm, then \(BD\) is

1. (a) \(5\) cm
2. (b) \(6\) cm
3. (c) \(6.5\) cm
4. (d) \(\dfrac{36}{13}\) cm

A circle is inscribed in a right triangle with sides \(9\) cm, \(40\) cm and \(41\) cm. Its radius is

1. (a) \(3\) cm
2. (b) \(4\) cm
3. (c) \(4.5\) cm
4. (d) \(5\) cm

\(PA\) and \(PB\) are tangents from \(P\) to a circle with centre \(O\), and \(\angle AOB = 2\angle APB\). Then \(\angle APB\) is

1. (a) \(45^\circ\)
2. (b) \(60^\circ\)
3. (c) \(90^\circ\)
4. (d) \(120^\circ\)

If \(\tan\theta = \dfrac{a}{b}\), then \(\dfrac{a\sin\theta - b\cos\theta}{a\sin\theta + b\cos\theta}\) equals

1. (a) \(\dfrac{a^2 - b^2}{a^2 + b^2}\)
2. (b) \(\dfrac{a^2 + b^2}{a^2 - b^2}\)
3. (c) \(\dfrac{a - b}{a + b}\)
4. (d) \(1\)

If \(\sin\theta + \cos\theta = \sqrt2\) for acute \(\theta\), then \(\tan\theta + \cot\theta\) equals

1. (a) \(1\)
2. (b) \(\sqrt2\)
3. (c) \(2\)
4. (d) \(2\sqrt2\)

A cylinder and a cone have equal base radii and equal heights, and their curved surface areas are in the ratio \(8 : 5\). The ratio of radius to height is

1. (a) \(3 : 4\)
2. (b) \(4 : 3\)
3. (c) \(5 : 4\)
4. (d) \(8 : 5\)

The ratio of the total surface area of a solid hemisphere to the surface area of a sphere of the same radius is

1. (a) \(1 : 2\)
2. (b) \(2 : 3\)
3. (c) \(3 : 4\)
4. (d) \(1 : 1\)

A \(90^\circ\) sector is cut out of a circular sheet of radius \(14\) cm. The perimeter of the remaining (major) sector is \(\left(\pi = \dfrac{22}{7}\right)\)

1. (a) \(66\) cm
2. (b) \(80\) cm
3. (c) \(94\) cm
4. (d) \(116\) cm

A number \(x\) is chosen at random from \(1, 2, 3, 4\) and a number \(y\) at random from \(2, 3, 5\). The probability that \(x + y\) is a prime number is

1. (a) \(\dfrac13\)
2. (b) \(\dfrac{5}{12}\)
3. (c) \(\dfrac12\)
4. (d) \(\dfrac{7}{12}\)

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The equation \(x^2 + 3x + 5 = 0\) has no real roots.

Reason (R): If the sum of the roots of a quadratic equation is negative, its roots are not real.

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): For an acute angle \(\theta\), \(\tan\theta\) can be greater than \(100\).

Reason (R): \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\), and for acute \(\theta\) close to \(90^\circ\), \(\cos\theta\) is positive and as small as we like while \(\sin\theta\) is close to \(1\).

1. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. (c) Assertion (A) is true but Reason (R) is false.
4. (d) Assertion (A) is false but Reason (R) is true.

Let \(a = 2^3 \times 3\), \(b = 2 \times 3 \times 5\) and \(c = 3^n \times 5\), where \(n\) is a natural number. If \(\text{LCM}(a, b, c) = 2^3 \times 3^2 \times 5\), find \(n\) and \(\text{HCF}(a, b, c)\).

If \(\alpha\) and \(\beta\) are the zeroes of \(x^2 - 7x + k\) and \(2\alpha - 3\beta = 4\), find \(k\).

OR

Find all acute angles \(\theta\) satisfying \(2\sin^2\theta - 3\sin\theta + 1 = 0\).

An arc of a circle subtends an angle of \(72^\circ\) at the centre and has length \(8.8\) cm. Find the radius of the circle and the area of the sector. \(\left(\pi = \dfrac{22}{7}\right)\)

All two-digit numbers that can be formed from the digits \(2, 3, 5\) (repetition allowed) are written on identical cards, and one card is drawn at random. Find the probability that the number is (i) prime, (ii) divisible by \(5\).

OR

Let \(a\) and \(b\) be natural numbers with \(\text{HCF}(a, b) = 1\). Using prime factorisation, prove that \(\text{HCF}(a + b, ab) = 1\). Verify the result for \(a = 12\), \(b = 35\).

Find the values of \(k\) for which \((k + 1)x^2 - 2(k - 1)x + 1 = 0\) has equal roots. For each such \(k\), find the root.

In parallelogram \(ABCD\), \(E\) is a point on \(AD\) with \(AE : ED = 1 : 2\). \(BE\) meets the diagonal \(AC\) at \(F\). Prove that \(\triangle AFE \sim \triangle CFB\), and find \(AF : FC\) and \(EF : FB\). If \(AC = 16\) cm, find \(AF\).

\(PA\) and \(PB\) are tangents from \(P\) to a circle with centre \(O\), and \(\angle APB = 50^\circ\). \(C\) is a point on the minor arc \(AB\); the tangent at \(C\) meets \(PA\) at \(D\) and \(PB\) at \(E\). Prove that \(\angle DOE = \dfrac12\angle AOB\), and find \(\angle DOE\).

Prove that \((\sin\theta + \cos\theta + 1)(\sin\theta + \cos\theta - 1)\sec\theta\operatorname{cosec}\theta = 2\).

OR

The mode of the following distribution is \(36\). Find the missing frequency \(f\).

OR

Consider the AP \(40, 36, 32, \ldots\). (a) How many terms must be taken for the sum to be \(220\)? Explain why there are two answers. (b) What is the greatest possible value of the sum of the first \(n\) terms? Justify.

OR

In \(\triangle ABC\), \(D\) is a point on \(BC\) with \(BD : DC = 3 : 2\), and \(E\) is the mid-point of \(AD\). \(BE\) produced meets \(AC\) at \(F\). By drawing \(DG \parallel BF\) to meet \(AC\) at \(G\), find (a) \(AF : FC\), (b) \(BE : EF\).

OR

From a solid cylinder of radius \(6\) cm and height \(15\) cm, a conical cavity of the same radius and height \(8\) cm is hollowed out at one end. (a) Find the volume of the remaining solid. (b) Find its total surface area. (c) By what percentage is this surface area greater than that of the original solid cylinder? (Leave answers in terms of \(\pi\) where appropriate.)

The marks (out of 60) of \(80\) students in a class test are summarised in a 'less than' cumulative frequency table.

(a) Write the frequency distribution. (b) Find the median. (c) Find the mode. (d) Some students who took the test late all scored in the class \(50\text{–}60\). What is the least number of such students who must be added for the median to exceed \(40\)?

Kite festival. At the Uttarayan kite festival in Ahmedabad, Kabir flies a kite on a taut, straight string \(100\) m long, holding it at ground level. (Take \(\sqrt3 = 1.732\), \(\sqrt2 = 1.414\).)

1. The string makes \(30^\circ\) with the ground. How high is the kite? [1 mark]

2. At that moment, what is the horizontal distance of the kite from Kabir? [1 mark]

3. A gust lifts the kite so that the same string now makes \(60^\circ\) with the ground. By how much has the kite risen? [2 marks]

   OR

   Instead, from the \(30^\circ\) position, Kabir winds in string until the angle is \(45^\circ\) while the kite stays at the same height. How much string does he wind in? [2 marks]

Three friends' homes. On a town map with a coordinate grid (1 unit = 100 m), Asha lives at \(A(-3, 2)\), Bilal at \(B(5, 8)\) and Chitra at \(C(8, 4)\).

1. Find the distance between Asha's and Bilal's homes in metres. [1 mark]

2. Find the mid-point of \(AC\). [1 mark]

3. Show that \(\triangle ABC\) is right-angled, and verify that the mid-point of \(AC\) is equidistant from all three homes. [2 marks]

   OR

   Find the point \(D\) such that \(ABCD\) (in this order) is a rectangle, and verify one pair of equal opposite sides. [2 marks]

Juice counter. At a juice counter in Hyderabad, 2 glasses of mango juice and 3 glasses of orange juice cost ₹240, while 3 glasses of mango juice and 2 glasses of orange juice cost ₹235. Let a glass of mango juice cost ₹\(m\) and a glass of orange juice ₹\(o\).

1. Write the pair of linear equations. [1 mark]

2. Without finding \(m\) and \(o\) separately, find the total cost of one glass of mango juice and one glass of orange juice. [1 mark]

3. Find \(m\) and \(o\). [2 marks]

   OR

   A customer claims she paid ₹500 for 4 glasses of mango and 6 glasses of orange juice at these prices. Is this possible? Justify using the equations. [2 marks]