Class 10 Mathematics • Full Test Paper Set 1
Overview
This page provides comprehensive Class 10 Mathematics – Full Test Board Paper Set 1. Class 10 Mathematics • Full Test Paper Set 1 - Free study material for Class 10 Maths. NCERT Solutions, Notes, and PYQs.
- All questions are compulsory.
- Use of calculator is not permitted.
- Write answers in your answer sheet.
SECTION A (MCQs & Assertion–Reason)
QQ1
If $a = 2^2 \times 5^x$, $b = 2^2 \times 5 \times 7$, $c = 2^2 \times 5 \times 11$ and $\text{LCM}(a,b,c) = 7700$, then the value of $x$ is
QQ2
The shortest distance (in units) of the point $(-6,4)$ from the $y$-axis is
QQ3
If the lines given by $5x+ky=7$ and $10x+6y-9=0$ are parallel, then the value of $k$ is
QQ4
A quadrilateral $PQRS$ is drawn to circumscribe a circle. If $PQ=8$ cm, $QR=6$ cm and $RS=5$ cm, then the length of $SP$ is
QQ5
If $\sec\theta - \tan\theta = x$, then $\sec\theta + \tan\theta$ is equal to
QQ6
Which one of the following is not a quadratic equation?
QQ7
The diameter of a circle is $14$ cm. Find its area. $(\pi=\frac{22}{7})$
QQ8
A card is drawn at random from a well-shuffled deck of $52$ playing cards. The probability of getting a red card is
QQ9
If $\cos x=\frac{\sqrt{3}}{2}$ and $0^\circ\le x\le90^\circ$, then $x$ is
QQ10
The HCF of two numbers is $12$ and their sum is $180$. The number of possible pairs is
QQ11
The base area of a right circular cone is $154\text{ cm}^2$ and its height is $6$ cm. Its volume is [Image of right circular cone volume]
QQ12
If the zeroes of $x^2+px+9$ are equal, then $p$ is
QQ13
The area of a sector of radius $21$ cm subtended by $60^\circ$ is [Image of area of sector of a circle]
QQ14
If $\triangle ABC \sim \triangle DEF$, $AB=5$ cm, $DE=10$ cm and $EF=8$ cm, then $BC$ is
QQ15
A letter is chosen at random from the word \textit{PROBABILITY}. The probability that it is a vowel is
QQ16
The points $A(1,2)$, $B(1,6)$, $C(5,6)$ and $D(5,2)$ are the vertices of a
QQ17
The median of a set of observations is $18$. If each observation is decreased by $3$, then the new median is
QQ18
The length of a tangent drawn from a point $26$ cm away from the centre of a circle of radius $10$ cm is
QQ19
QQ20
SECTION B
QQ21
A. The A.P $6, 10, 14, \dots$ has $50$ terms. Find the sum of last $10$ terms.
OR. Find the middle term of A.P $7, 14, 21, \dots, 287$
QQ22
If $\sin(A + B) = 1$ and $\cos(A - B) = \frac{1}{2}$, where $0^\circ < A, B < 90^\circ$, find the measure of angles $A$ and $B$.
QQ23
If $AP$ and $DQ$ are medians of triangles $ABC$ and $DEF$ respectively, where $\triangle ABC \sim \triangle DEF$, then prove that $\frac{AB}{DE} = \frac{AP}{DQ}$.
QQ24
A. A horse, a cow and a goat are tied, each by ropes of length $21$ m, at the corners $A$, $B$ and $C$ respectively, of a grassy triangular field $ABC$ with sides of lengths $42$ m, $56$ m and $70$ m. Find the area of grass field that can be grazed by them.
OR. Find the area of the major segment (in terms of $\pi$) of a circle of radius $7$ cm, formed by a chord subtending an angle of $60^\circ$ at the centre.
QQ25
A $\triangle ABC$ is drawn to circumscribe a circle of radius $5$ cm such that the segments $BD$ and $DC$ are of lengths $12$ cm and $10$ cm respectively. Find the lengths of the sides $AB$ and $AC$, if it is given that $\text{ar}(\triangle ABC) = 110\text{ cm}^2$.
For Visually Impaired: A circle is inscribed in a right-angled triangle $ABC$, right angled at $B$. If $BC = 15$ cm and $AB = 20$ cm, find the radius of the circle.
SECTION C
QQ26
In Figure, $XY$ and $X'Y'$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X'Y'$ at $B$. Prove that $\angle AOB = 90^\circ$. [Image of parallel tangents to a circle]
For Visually Impaired: Two tangents $PA$ and $PB$ are drawn to a circle with centre $O$ from an external point $P$. Prove that $\angle APB = 2(\angle OAB)$.
QQ27
In a workshop, the number of teachers of English, Hindi and Science are $48$, $72$ and $96$ respectively. Find the minimum number of rooms required, if in each room the same number of teachers are to be seated and all of them being of the same subject.
QQ28
Find the zeroes of the quadratic polynomial $x^2 - 9x + 20$ and verify the relationship between the zeroes and coefficients of the polynomial.
QQ29
A. If $\sin \theta + \cos \theta = \sqrt{2}$, then prove that $\tan \theta + \cot \theta = 2$.
OR. Prove that $\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \sec A + \tan A$.
QQ30
On a particular day, Riya and Neha couldn’t decide on who would get to drive the car. They had one coin each and flipped their coin exactly two times. The following was agreed upon: (1) If Riya gets two heads, she would drive the car. (2) If Neha gets a head followed by a tail, she would drive the car. Who has greater probability to drive the car that day? Justify your answer.
QQ31
A. The monthly income of Aman and Rohit are in the ratio $5 : 6$ and their monthly expenditures are in ratio $4 : 5$. If each saves ₹$12,000$ per month, find their monthly incomes.
OR. Solve the following system of equations graphically: $x + y = 8$, $x - y = 4$. Find the area of the triangle so formed by two lines and $x$-axis.
For Visually Impaired: Five years hence, father’s age will be four times the age of son. Five years ago, father was six times as old as his son. Find their present ages.
SECTION D
QQ32
A bus travels a distance of $120$ km at a certain speed. It then travels another distance of $180$ km at a speed $10$ km/h more than the original speed. If the total time taken for the journey is $5$ hours, find the original speed of the bus.
QQ33
Prove that if the diagonals of a parallelogram bisect each other, then it is a rectangle. Hence, in parallelogram $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. If $AC = 26$ cm and one side of the parallelogram is $10$ cm, find the area of the parallelogram.
QQ34
A. A solid is composed of a cylinder surmounted by a hemisphere. The radius of both the cylinder and hemisphere is $7$ cm and the height of the cylindrical part is $10$ cm. Find the total surface area of the solid.
OR. A solid right circular cone of height $14$ cm and base radius $7$ cm is melted and recast into a sphere. Find the radius of the sphere.
QQ35
A. The following table shows the distribution of marks obtained by $100$ students in an examination:
| Class Interval | $0–20$ | $20–40$ | $40–60$ | $60–80$ | $80–100$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Frequency | $12$ | $28$ | $30$ | $20$ | $10$ |
Find the mean marks using step deviation method.
OR. The mean of the following frequency distribution is $50$. Find the missing frequency $x$.
| Class Interval | $0–20$ | $20–40$ | $40–60$ | $60–80$ | $80–100$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Frequency | $5$ | $10$ | $x$ | $15$ | $10$ |
SECTION E (Case Study Based)
QQ36
Two students Ramesh and Suresh write two arithmetic progressions as:
Ramesh: $4, 9, 14, 19, \dots$
Suresh: $200, 195, 190, \dots$
i. Find the common difference of both A.P.s.
ii. Find the $25^{th}$ term of the A.P. written by Suresh.
iii_A. Find the sum of first $12$ terms of the A.P. written by Ramesh.
OR. Which term of both the A.P.s has the same value?
QQ37
The positions of three points $A$, $B$ and $C$ on a coordinate plane are given by $A(-2, 4)$, $B(2, 0)$ and $C(6, -4)$. [Image of coordinate geometry points]
i. Find the distance between $A$ and $C$.
ii. Find the coordinates of the midpoint of $AC$.
iii_A. Find the point on the $x$-axis which is equidistant from $A$ and $B$.
OR. Find the coordinates of the point which divides $AB$ internally in the ratio $1 : 3$.
QQ38
A flagstaff stands on the top of a tower. The height of the tower is $20$ m and the height of the flagstaff is $5$ m.
i. Find the angle of elevation of the top of the flagstaff from a point on the ground $15$ m away from the foot of the tower.
ii. If the angle of elevation of the top of the tower from a point is $45^\circ$, find the distance of the point from the foot of the tower.
iii_A. Find the distance of the point from the foot of the tower if the angle of elevation of the top of the flagstaff is $60^\circ$.
OR. Find the height of the flagstaff if the angle of elevation of its top from a point $20$ m away from the foot of the tower is $45^\circ$.