CBSE Class 10 Maths Standard 2023 Question Paper 30/6/2 with Solutions

If the zeroes of the quadratic polynomial $x^2+(a+1)x+b$ are 2 and –3, then

1. (A)a = –7, b = –1
2. (B)a = 5, b = –1
3. (C)a = 2, b = – 6
4. (D)a = 0, b = – 6

The number of quadratic polynomials having zeroes –5 and –3 is

1. (A)1
2. (B)2
3. (C)3
4. (D)more than 3

If the sum of the first n terms of an A.P be $3n^2+n$ and its common difference is 6, then its first term is

1. (A)2
2. (B)3
3. (C)1
4. (D)4

$\frac{{\cos }^2\theta }{{\sin }^2\theta }-\frac{1}{{\sin }^2\theta }$, in simplified form, is :

1. (A)${\tan }^2\theta$
2. (B)${\sec }^2\theta$
3. (C)1
4. (D)–1

In the given figure, DE$\parallel$BC. If AD = 3 cm, AB = 7 cm and EC = 3 cm, then the length of AE is

1. (A)2 cm
2. (B)2.25 cm
3. (C)3.5 cm
4. (D)4 cm

The LCM of smallest 2-digit number and smallest composite number is

1. (A)12
2. (B)4
3. (C)20
4. (D)40

The distance of the point (– 4, 3) from y-axis is

1. (A)– 4
2. (B)4
3. (C)3
4. (D)5

If one zero of the polynomial $x^2+3x+k$ is 2, then the value of k.

1. (A)– 10
2. (B)10
3. (C)5
4. (D)– 5

The point of intersection of the line represented by $3x-y=3$ and y-axis is given by

1. (A)(0, –3)
2. (B)(0, 3)
3. (C)(2, 0)
4. (D)(–2, 0)

If the quadratic equation $a{x}^2+bx+c=0$ has two real and equal roots, then 'c' is equal to

1. (A)$\frac{-b}{2a}$
2. (B)$\frac{b}{2a}$
3. (C)$\frac{-b^2}{4a}$
4. (D)$\frac{b^2}{4a}$

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability of getting a face card is

1. (A)$\frac{1}{2}$
2. (B)$\frac{3}{13}$
3. (C)$\frac{4}{13}$
4. (D)$\frac{1}{13}$

If $△PQR\sim △ABC$; PQ = 6 cm, AB = 8 cm and the perimeter of $△ABC$ is 36 cm, then the perimeter of $△PQR$ is

1. (A)20.25 cm
2. (B)27 cm
3. (C)48 cm
4. (D)64 cm

The volume of a right circular cone whose area of the base is 156 cm${}^2$ and the vertical height is 8 cm, is

1. (A)2496 cm${}^3$
2. (B)1248 cm${}^3$
3. (C)1664 cm${}^3$
4. (D)416 cm${}^3$

The circumferences of two circles are in the ratio 4 : 5. What is the ratio of their radii ?

1. (A)16 : 25
2. (B)25 : 16
3. (C)$2:\sqrt{5}$
4. (D)4 : 5

In the given figure, PA and PB are tangents from external point P to a circle with centre C and Q is any point on the circle. Then the measure of $\angle AQB$ is

1. (A)$62{\frac{1}{2}}^{\circ }$
2. (B)$125^{\circ }$
3. (C)$55^{\circ }$
4. (D)$90^{\circ }$

A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime number less than 23 is

1. (A)$\frac{7}{90}$
2. (B)$\frac{1}{9}$
3. (C)$\frac{4}{45}$
4. (D)$\frac{9}{89}$

The coordinates of the point where the line $2y=4x+5$ crosses $x$-axis is

1. (A)$\left(0,\frac{-5}{4}\right)$
2. (B)$\left(0,\frac{5}{2}\right)$
3. (C)$\left(\frac{-5}{4},0\right)$
4. (D)$\left(\frac{-5}{2},0\right)$

$({\cos }^{4}A-{\sin }^{4}A)$ on simplification, gives

1. (A)$2{\sin }^{2}A-1$
2. (B)$2{\sin }^{2}A+1$
3. (C)$2{\cos }^{2}A+1$
4. (D)$2{\cos }^{2}A-1$

Statement A (Assertion) : For $0<\theta \le 90^{\circ }$, $\text{cosec}\theta -\cot \theta$ and $\text{cosec}\theta +\cot \theta$ are reciprocal of each other.
Statement R (Reason) : ${\text{cosec}}^2\theta -{\cot }^2\theta =1$

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.

Statement A (Assertion) : If $5+\sqrt{7}$ is a root of a quadratic equation with rational co-efficients, then its other root is $5-\sqrt{7}$.
Statement R (Reason) : Surd roots of a quadratic equation with rational co-efficients occur in conjugate pairs.

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.

Find the length of the shadow on the ground of a pole of height 18 m when angle of elevation $\theta$ of the sun is such that $\tan \theta =\frac{6}{7}$.

In the given figure, ABC is a triangle in which DE$\parallel$BC. If AD = $x$, DB = $x-2$, AE = $x+2$ and EC = $x-1$, then find the value of $x$.

Diagonals AC and BD of trapezium ABCD with AB$\parallel$DC intersect each other at point O. Show that $\frac{OA}{OC}=\frac{OB}{OD}$.

Show that $6^n$ can not end with digit 0 for any natural number 'n'.

Find the LCM and HCF of 72 and 120.

Find the points on the $x$-axis, each of which is at a distance of 10 units from the point A(11, – 8).

In the given figure, PA is a tangent to the circle drawn from the external point P and PBC is the secant to the circle with BC as diameter. If $\angle AOC=130^{\circ }$, then find the measure of $\angle APB$, where O is the centre of the circle.

In the given figure, AB and CD are diameters of a circle with centre O perpendicular to each other. If OA = 7 cm, find the area of shaded region.

If $\sin \theta +\cos \theta =p$ and $\sec \theta +\text{cosec}\theta =q$, then prove that $q(p^2-1)=2p$.

The sum of two numbers is 15. If the sum of their reciprocals is $\frac{3}{10}$, find the two numbers.

If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2-7x+10=0$, find the quadratic equation whose roots are ${\alpha }^2$ and ${\beta }^2$.

Find the ratio in which the line segment joining the points A(6, 3) and B(–2, –5) is divided by $x$-axis.

In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB = AC. If AD $\perp$ BC and EF $\perp$ AC, then prove that $△ABD\sim △ECF$.

Find the HCF and LCM of 26, 65 and 117, using prime factorisation.

Prove that $\sqrt{2}$ is an irrational number.

250 apples of a box were weighed and the distribution of masses of the apples is given in the following table :
Mass (in grams): 80 – 100, 100 – 120, 120 – 140, 140 – 160, 160 – 180
Number of apples: 20, 60, 70, $x$, 60
(i) Find the value of $x$ and the mean mass of the apples. (3)
(ii) Find the modal mass of the apples (2)

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ=2\angle OPQ$.

A circle touches the side BC of a $△ABC$ at a point P and touches AB and AC when produced at Q and R respectively. Show that AQ = $\frac{1}{2}$ (Perimeter of $△ABC$).

A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 3.5 cm and the total height of the solid is 9.5 cm. Find the volume of the solid.

Find the sum of integers between 100 and 200 which are (i) divisible by 9 (ii) not divisible by 9.

Solve the equation :
$-4+(-1)+2+5+\dots +x=437$.

“Eight Ball” is a game played on a pool table with 15 balls numbered 1 to 15 and a “cue ball” that is solid and white. Of the 15 numbered balls, eight are solid (non-white) coloured and numbered 1 to 8 and seven are striped balls numbered 9 to 15.
The 15 numbered pool balls (no cue ball) are placed in a large bowl and mixed, then one ball is drawn out at random.
Based on the above information, answer the following questions :
(i) What is the probability that the drawn ball bears number 8 ?
(ii) What is the probability that the drawn ball bears an even number ?
OR What is the probability that the drawn ball bears a number, which is a multiple of 3 ?
(iii) What is the probability that the drawn ball is a solid coloured and bears an even number ?

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two Sections A and B. Tower is supported by wires from a point O.
Distance between the base of the tower and point O is 36 cm. From point O, the angle of elevation of the top of the Section B is $30^{\circ }$ and the angle of elevation of the top of Section A is $45^{\circ }$.
Based on the above information, answer the following questions :
(i) Find the length of the wire from the point O to the top of Section B. (1)
(ii) Find the distance AB. (2)
OR Find the area of $△OPB$.
(iii) Find the height of the Section A from the base of the tower. (1)

A coaching institute of Mathematics conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, there are 20 poor and 5 rich children, whereas in batch II, there are 5 poor and 25 rich children. The total monthly collection of fees from batch I is ₹ 9000 and from batch II is ₹ 26,000. Assume that each poor child pays ₹ $x$ per month and each rich child pays ₹ y per month.
Based on the above information, answer the following questions :
(i) Represent the information given above in terms of $x$ and y. (1)
(ii) Find the monthly fee paid by a poor child. (2)
OR Find the difference in the monthly fee paid by a poor child and a rich child.
(iii) If there are 10 poor and 20 rich children in batch II, what is the total monthly collection of fees from batch II ? (1)