MATHS MCQ For Exam 2026
CHAPTER 1 : REAL NUMBERS
1. The prime factorisation of natural number 288 is:
2. If the HCF of 360 and 64 is 8, then their LCM is:
3. If two positive integers \( A \) and \( B \) can be expressed as \( A = xy^3 \) and \( B = x^4y^2z \) ; \( x, y \) being prime numbers then \( \text{HCF}(A, B) \) is :
4. The LCM of two numbers is 1200. Which of the following cannot be their HCF?
5. If \( \text{HCF}(26, 169) = 13 \), then \( \text{LCM}(26, 169) = ? \)
6. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
7. The HCF and LCM of 12, 21, 15 respectively are :
8. The ratio of LCM and HCF of the least composite number and the least prime number is :
9. If \( \text{LCM}(x, 18) = 36 \) and \( \text{HCF}(x, 18) = 2 \), then \( x = \)
10. If \( (a \times 5)^n \) ends with the digit zero for every natural number \( n \), then \( a \) is
11. There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students:
12. Three bells ring at intervals of 4, 7 and 14 minutes. All the three rang at 7 AM. When will they ring together again?
13. The product of a non-zero rational number and an irrational number is
14. The smallest irrational number by which \( \sqrt{18} \) should be multiplied so as to get a rational number is
15. If two positive integers \( a \) and \( b \) are written as \( a = p^3q^2 \) and \( b = pq^3 \); \( p, q \) are prime numbers, then \( \text{HCF}(a, b) \) is:
16. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
17. Three farmers have 490 kg, 588 kg and 882 kg of wheat respectively. Find the maximum capacity of a bag so that the wheat can be packed in exact number of bags
18. \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \) is an example of :
19. L.C.M of two numbers is 60 times of their H.C.F. Sum of H.C.F and L.C.M is 366. If one number is 72, then find the other number.
20. Two numbers are in the ratio 15:11 their HCF is 13 and LCM is 2145 then find the number.
21. The LCM of the two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find their HCF.
22. Assertion: The H.C.F. of two numbers is 16 and their product is 3072. Then their L.C.M. = 162.
Reason: If \( a \) and \( b \) are two positive integers, then \( \text{H.C.F.} \times \text{L.C.M.} = a \times b \).
Reason: If \( a \) and \( b \) are two positive integers, then \( \text{H.C.F.} \times \text{L.C.M.} = a \times b \).
23. Assertion: ‘2’ is an example of a rational number.
Reason: The square roots of all positive integers are irrational numbers.
Reason: The square roots of all positive integers are irrational numbers.
24. Assertion: If the HCF of two numbers is 5 and their product is 150, then their LCM is 30.
Reason: For any two positive integers \( p \) and \( q \), \( \text{HCF}(p, q) + \text{LCM}(p, q) = p \times q \)
Reason: For any two positive integers \( p \) and \( q \), \( \text{HCF}(p, q) + \text{LCM}(p, q) = p \times q \)
25. Assertion: (18, 25) is a pair of co-primes.
Reason: Pair of co-prime has a common factor 2.
Reason: Pair of co-prime has a common factor 2.
26. Assertion: \( \sqrt{x} \) is an irrational number, where \( x \) is a prime number.
Reason: Square root of any prime number is an irrational number.
Reason: Square root of any prime number is an irrational number.
27. Assertion: \( 3 \times 5 \times 7 + 7 \) is a composite number.
Reason: A composite number has factors one, itself and any other natural number.
Reason: A composite number has factors one, itself and any other natural number.
28. Assertion: \( (2-\sqrt{5}) \) is an irrational number.
Reason: The sum or difference of a rational and an irrational number is irrational.
Reason: The sum or difference of a rational and an irrational number is irrational.
29. Assertion: \( 12^n \) ends with the digit zero, where \( n \) is any natural number.
Reason: Any number ends with digit zero, if its prime factor is of the form \( 2^m \times 5^n \), where \( m \) and \( n \) are natural numbers.
Reason: Any number ends with digit zero, if its prime factor is of the form \( 2^m \times 5^n \), where \( m \) and \( n \) are natural numbers.
30. Assertion: HCF of (11,17) is 1.
Reason: If \( p \) and \( q \) are prime then HCF of \((p,q)\) is always 1.
Reason: If \( p \) and \( q \) are prime then HCF of \((p,q)\) is always 1.
CHAPTER 2 - POLYNOMIALS
1. Which of the following is not a polynomial?
2. Which are the zeroes of \( p(x) = 6x^2 – 7x – 3 \)
3. The number of zeroes of the polynomial from the graph is
4. Find the quadratic polynomial whose zeros are -3 and 4.
5. Which are the zeroes of \( p(x) = x^2 – 8x + 15 \)
6. Find the sum and product of the zeroes of polynomial \( x^2 - 3x + 5 \)
7. If one of the zeroes of quadratic polynomial \( (k + 3)x^2 + 2kx + 6 \) is -3, then find value of \( k \).
8. A quadratic polynomial whose sum and product of zeroes are –5 and 6 is
9. If the product of the zeroes of the quadratic polynomial \( 3x^2 + 5x + k \) is \( \frac{2}{3} \) then the value of \( k \) is
10. If one zero of the polynomial \( 6x^2 + 37x - (k - 2) \) is reciprocal of the other, then, what is the value of \( k \)?
11. The zeroes of the polynomial \( p(x) = x^2 + 4x + 3 \) are given by
12. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( f(x) = px^2 - 2x + 3p \) and \( \alpha + \beta = \alpha\beta \) then the value of \( p \)
13. The zeroes of the quadratic polynomial \( f(x) = x^2 + 99x + 127 \) are
14. The maximum number of zeroes a cubic polynomial can have, is
15. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( f(x) = x^2 - ax – b \), then the value of \( \alpha^2 + \beta^2 \) is
16. The number of polynomials having zeroes -3 and 5 is
17. If \( x+ 2 \) is factor of \( x^2 + ax + 2b \) and \( a + b = 4 \), then
18. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( f(x) = 4x^2 - 3x – 7 \), then the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is
19. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( f(x) = x^2 - ax – b \), then the value of \( \alpha^2 + \beta^2 \)
20. A quadratic polynomial, the sum of whose zeroes is - 5 and their product is 6, is
21. If the zeroes of the quadratic polynomial \( x^2 + (a + 1) x + b \) are 2 and –3, then
22. If one zero of the quadratic polynomial \( x^2 + 3x + k \) is 2, then the value of \( k \) is
23. If 2 and \( \frac{1}{2} \) are two zeroes of \( px^2 + 5x + r \), then
24. What should be subtracted from the polynomial \( x^2 – 16x + 30 \), so that 15 is the zero of the resulting polynomial?
25. Statement-1 (A): The polynomial \( f(x) = x^2 - 2x + 2 \) has two real zeros.
Statement-2 (R): A quadratic polynomial can have at most two real zeroes.
Statement-2 (R): A quadratic polynomial can have at most two real zeroes.
26. Statement-1 (A): A quadratic polynomial having \( \frac{1}{2} \) and \( \frac{1}{3} \) as its zeroes is \( 6x^2 - 5x + 1 \)
Statement-2 (R): Quadratic polynomial having \( \alpha \) and \( \beta \) as zeroes are given by \[ f(x) = k\{x^2 - (\alpha + \beta) x + \alpha\beta\} \] where \( k \) is a non-zero constant.
Statement-2 (R): Quadratic polynomial having \( \alpha \) and \( \beta \) as zeroes are given by \[ f(x) = k\{x^2 - (\alpha + \beta) x + \alpha\beta\} \] where \( k \) is a non-zero constant.
27. Statement-1 (A): If one root of the quadratic polynomial \( f(x) = (k-1)x^2 – 10x + 3, k \neq 1 \) is reciprocal of the other, then \( k = 4 \)
Statement-2 (R): The product of roots of the quadratic polynomial \( ax^2 + bx + c, a \neq 0 \) is \( \frac{c}{a} \)
Statement-2 (R): The product of roots of the quadratic polynomial \( ax^2 + bx + c, a \neq 0 \) is \( \frac{c}{a} \)
28. Statement-1 (A): If \( \alpha \) and \( \beta \) are zeroes of the quadratic polynomial \( x^2 + 7x + 12 \), then \( \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-7}{12} \)
Statement-2(R): If \( \alpha \) and \( \beta \) are zeroes of the quadratic polynomial \( ax^2 + bx + c \), then \( \alpha + \beta = \frac{-b}{a} \) and \( \alpha\beta = \frac{c}{a} \)
Statement-2(R): If \( \alpha \) and \( \beta \) are zeroes of the quadratic polynomial \( ax^2 + bx + c \), then \( \alpha + \beta = \frac{-b}{a} \) and \( \alpha\beta = \frac{c}{a} \)
29. Statement-1 (A): If \( \alpha, \beta \) and \( \gamma \) are zeroes of the polynomial \( 6x^3 + 3x^2 – 5x + 1 \), then \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} = 5 \)
Statement-2(R): If \( \alpha, \beta \) and \( \gamma \) are zeroes of the cubic polynomial \( ax^3 + bx^2 + cx + d \), then \( \alpha + \beta + \gamma = \frac{-b}{a} \)
Statement-2(R): If \( \alpha, \beta \) and \( \gamma \) are zeroes of the cubic polynomial \( ax^3 + bx^2 + cx + d \), then \( \alpha + \beta + \gamma = \frac{-b}{a} \)
30. Statement-1 (A): The polynomial \( p(x) = x^2 + 3x + 3 \) has two real zeroes.
Statement-2(R): A quadratic polynomial can have at most two real zeroes.
Statement-2(R): A quadratic polynomial can have at most two real zeroes.
CHAPTER 3 - PAIR OF LINEAR EQUATION IN TWO VARIABLES
Q1. The value of \( K \) for which the system of equation \( kx – y = 2 \), and \( 6x - 2y = 3 \) has a unique solution is.
Q2. If the system of equations \( kx – 5y = 2 \) and \( 4x + my = 10 \) has infinitely many solution then the value of \( k \) and \( m \) are.
Q3. 8 chairs and 5 tables cost Rs 10,500, while 5 chairs and 3 tables cost Rs 6,450. The cost of each chair will be.
Q4. The pair of linear equation \( 3x + 5y = 3 \) and \( 6x + ky = 8 \) do not have a solution, if \( k \) is
Q5. The pair of equation \( x = a \) and \( y = b \) graphically represents the lines which are.
Q6. The value of \( c \) for which the pair of equation \( cx – y = 2 \) and \( 6x – 2y = 3 \) will have no solution.
Q7. The pair of equation \( 5x - 15y = 8 \) and \( 3x - 9y = \frac{24}{5} \) has.
Q8. \( 19x – 17y = 55 \) and \( 17x - 19y = 53 \) then the value of \( (x – y) \) is.
Q9. If \( \frac{x}{a} + \frac{y}{b} = a + b \) and \( \frac{x}{a^2} - \frac{y}{b^2} = 0 \), then the value of \( (x – y) \).
Q10. If \( 2x + 3y = 0 \) and \( 4x – 3y = 0 \) then the value of \( (x + y) \) is.
Q11. If \( (6, k) \) is a solution of equation \( 3x + y – 22 = 0 \) then the value of \( k \) is.
Q15. If \( x = a, y = b \) is the solution of the equations \( x – y = 2 \) and \( x + y = 4 \), then the values of \( a \) and \( b \) are, respectively.
Q16. The value of \( k \) for which the system of equations \( x + 2y = 3 \) and \( 5x + ky + 7 = 0 \) has no solution is.
Q21. If \( 2x – 3y = 7 \) and \( (a + b)x – (a + b – 3)y = 4a + b \) have an infinite number of solutions, then.
Q24. The solution of the linear pair \( px + qy = p – q \) and \( qx – py = p + q \) is .
Q25. If \( 2x – 3y = 7 \) and \( (a + b)x – (a + b - 3)y = 4a + b \) represent coincident lines, then \( a \) and \( b \) satisfy the equation
Q27. If the system of equations \( 3x + y = 1 \) and \( (2k-1)x + (k-1)y = 2k+1 \) is inconsistent, then \( k \) equals to.
Q28. If one equation of a pair of dependent linear equations is \( -3x + 5y – 2 = 0 \). The second equation will be:
Q29. A fraction becomes \( \frac{1}{3} \) when 1 is subtracted from the numerator and it becomes \( \frac{1}{4} \) when 8 is added to its denominator. The fraction obtained is:
Q30. The angles of cyclic quadrilaterals \( ABCD \) are: \( A = (6x+10)^\circ \), \( B = (5x)^\circ \), \( C = (x+y)^\circ \) and \( D = (3y-10)^\circ \). The value of \( x \) and \( y \) is:
CHAPTER 4: QUADRATIC EQUATIONS
1. Which one of the following is not a quadratic equation?
2. Which of the following equations has 2 as a root?
3. If \( \frac{1}{2} \) is a root of the equation \( x^2 + kx – \frac{5}{4} = 0 \), then the value of \( k \) is
4. Which of the following equations has the sum of its roots as 3?
5. Values of \( k \) for which the quadratic equation \( 2x^2 – kx + k = 0 \) has equal roots is
6. The quadratic equation \( 2x^2 – \sqrt{5}x + 1 = 0 \) has
7. Which of the following equations has two distinct real roots?
8. Which of the following equations has no real roots?
9. The discriminant of the quadratic equation \( 3\sqrt{3}x^2 + 10x + \sqrt{3} = 0 \) is
10. A sum of ₹4000 was divided among \( x \) persons. Had there been 10 more persons, each would have got ₹80 less. Which of the following represents the above situation?
11. The product of two consecutive integers is equal to 6 times the sum of the two integers. If the smaller integer is \( x \), which of the following equations represent the above situation?
12. Consider the equation \( kx^2 + 2x = c (2x^2 + b) \). For the equation to be quadratic, which of these cannot be the value of \( k \)?
13. What is the smallest positive integer value of \( k \) such that the roots of the equation \( x^2 - 9x + 18 + k = 0 \) can be calculated by factoring the equation?
14. Rahul follows the below steps to find the roots of the equation \( 3x^2 – 11x - 20 = 0 \), by splitting the middle term.
Step 1: \( 3x^2 – 11x - 20 = 0 \)
Step 2: \( 3x^2 – 15x + 4x - 20 = 0 \)
Step 3: \( 3x (x - 5) + 4(x - 5) = 0 \)
Step 4: \( (3x - 4) (x - 5) = 0 \)
Step 5: \( x = \frac{4}{3} \) and \( 5 \)
In which step did Rahul make the first error?
Step 1: \( 3x^2 – 11x - 20 = 0 \)
Step 2: \( 3x^2 – 15x + 4x - 20 = 0 \)
Step 3: \( 3x (x - 5) + 4(x - 5) = 0 \)
Step 4: \( (3x - 4) (x - 5) = 0 \)
Step 5: \( x = \frac{4}{3} \) and \( 5 \)
In which step did Rahul make the first error?
15. The roots of \( ax^2 + bx + c = 0, a \neq 0 \) are real and unequal. Which of these is true about the value of discriminant, \( D \)?
16. Consider the equation \( px^2 + qx + r = 0 \). Which conditions are sufficient to conclude that the equation have real roots?
17. For what value of \( k \), the quadratic equation \( 3x^2 + 2kx + 27 = 0 \) has equal real roots?
18. If the equation \( x^2 - mx + 1 = 0 \) does not possess real roots, then
19. If \( \alpha \) and \( \beta \) are the roots of \( x^2 + 7x + 10 = 0 \), find the value of \( \alpha^2 + \beta^2 \)
20. If \( \alpha, \beta \) are the roots of the equation \( 2x^2 – x -1 = 0 \), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).
21. If one root of the equation \( 2y^2 – ay + 64 = 0 \) is twice the other, then find the values of \( a \).
22. If one root of the equation \( 3x^2 + kx + 81 = 0 \) (having real roots) is the square of the other, then value of \( k \)
23. A quadratic equation, the sum of whose roots is 0 and one root is 4, is
24. If the quadratic equation \( x^2 - 8x + k = 0 \) has real roots, then
25. If \( x = 3 \) is one of the roots of the quadratic equation \( x^2 – 2kx – 6 = 0 \), then the value of \( k \) is
26. Assertion(A): If one root of the quadratic equation \( 6x^2 – x – k = 0 \) is \( \frac{2}{3} \), then the value of \( k \) is 2.
Reason(R): The quadratic equation \( ax^2 + bx + c = 0, a \neq 0 \) has almost two roots.
Reason(R): The quadratic equation \( ax^2 + bx + c = 0, a \neq 0 \) has almost two roots.
27. Assertion(A): The roots of the quadratic equation \( x^2 + 2x + 2 = 0 \) are real
Reason(R): If discriminant \( D = b^2 – 4ac < 0 \) then the roots of quadratic equation \( ax^2 + bx + c = 0 \) are not real.
Reason(R): If discriminant \( D = b^2 – 4ac < 0 \) then the roots of quadratic equation \( ax^2 + bx + c = 0 \) are not real.
28. Assertion: \( (2x – 1)^2 – 4x^2 + 5 = 0 \) is not a quadratic equation.
Reason: An equation of the form \( ax^2 + bx + c = 0 \), (\( a \neq 0 \), where \( a, b \) and \( c \) are real numbers) is called a quadratic equation.
Reason: An equation of the form \( ax^2 + bx + c = 0 \), (\( a \neq 0 \), where \( a, b \) and \( c \) are real numbers) is called a quadratic equation.
29. Assertion: \( 3x^2 – 6x + 3 = 0 \) has equal real roots.
Reason: The quadratic equation \( ax^2 + bx + c = 0 \) have equal real roots if discriminant \( D > 0 \).
Reason: The quadratic equation \( ax^2 + bx + c = 0 \) have equal real roots if discriminant \( D > 0 \).
30. Assertion(A): The equation \( 9x^2 + 3kx + 4 = 0 \) has equal roots for \( k = 9 \).
Reason (R): If discriminant '\( D \)' of a quadratic equation is equal to zero, then roots of equation are real and equal.
Reason (R): If discriminant '\( D \)' of a quadratic equation is equal to zero, then roots of equation are real and equal.
CHAPTER 5: ARITHMETIC PROGRESSION
Q1. The 10th term of the AP: 5, 8, 11, 14, ... is
Q2. In an AP, if \( d = –4, n = 7, a_n = 4 \), then \( a \) is
Q3. The list of numbers – 10, – 6, – 2, 2,... is
Q5. If the 2nd term of an AP is 13 and the 5th term is 25, what is its 7th term?
Q7. If the first term of an AP is –5 and the common difference is 2, then the sum of the first 6 terms is
Q9. In an AP if \( a = –7.2, d = 3.6, a_n = 7.2 \), then \( n \) is
Q10. In an AP, if \( a = 3.5, d = 0, n = 101 \), then \( a_n \) will be
Q11. The 11th term of the AP: \( –5, –5/2, 0, 5/2, ... \) is
Q12. What is the common difference of an AP in which \( a_{18} – a_{14} = 32 \)?
Q17. In an AP if \( a = 1, a_n = 20 \) and \( S_n = 399 \), then \( n \) is
Q21. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, the sum of first \( N \) terms is?
Q22. In an AP, if \( S_n = 3n^2 + 5n \) and \( a_k = 164 \), then the value of \( k \) is
Q25. The houses of a row are numbered consecutively from 1 to 49. If there is a value of \( x \) such that the sum of the numbers of the houses preceding the house numbered \( x \) is equal to the sum of the numbers of the houses following it. Then the value of \( x \) is?
Q27. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
Q28. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
Q29. If the numbers \( n – 2, 4n – 1 \) and \( 5n + 2 \) are in AP, then the value of \( n \) is
Q30. Which term of the AP: 53, 48, 43,... is the first negative term?
CHAPTER 6 TRIANGLES
1. If \( \Delta ABC \sim \Delta PQR \), \( AB = 6.5 \text{ cm} \), \( PQ = 10.4 \text{ cm} \). Perimeter of \( \Delta ABC \) is \( 60 \text{ cm} \), then the perimeter of \( \Delta PQR \) is
2. \( XY \) is drawn parallel to the base \( BC \) of a \( \Delta ABC \) cutting \( AB \) at \( X \) and \( AC \) at \( Y \). If \( AB = 4 BX \) and \( YC = 2 \text{ cm} \), then \( AY \) is
3. From the below figure if \( \angle ACB = \angle CDA \), \( AD = 3 \text{ cm} \) and \( AC = 6 \text{ cm} \) then find the length of \( AB \)
4. In \( \Delta ABC \) and \( \Delta DEF \), \( \angle B = \angle E \), \( \angle F = \angle C \) and \( AB = 3DE \). Then the two triangles are
5. In \( \Delta ABC \), \( D \) and \( E \) are points on the sides \( AB \) and \( AC \) respectively such that \( DE \parallel BC \), if \( AD = 2.5 \text{ cm}, BD = 3.0 \text{ cm} \) and \( AE = 3.75 \text{ cm} \), find the length of \( AC \)
6. Assertion: If \( \Delta ABC \sim \Delta PQR \), then \( \angle A = \angle R \)
Reason: if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar
Reason: if in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar
7. In the below figure, if a line intersects sides \( AB \) and \( AC \) of \( \Delta ABC \) at \( D \) and \( E \) respectively and is parallel to \( BC \). Is \( \frac{AD}{AB} = \frac{AE}{AC} \) ?
8. In the figure given below, the two-line segments \( AC \) and \( BD \) intersect each other at the point \( P \) such that \( PA = 6 \text{ cm}, PB = 3 \text{ cm}, PC = 2.5 \text{ cm}, PD = 5 \text{ cm}, \angle APB = 50^\circ \) and \( \angle CDP = 30^\circ \), then \( \angle PBA \) is
9. \( \Delta ABC \) is such that \( AB = 3 \text{ cm}, BC = 2 \text{ cm}, CA = 2.5 \text{ cm} \). If \( \Delta ABC \sim \Delta DEF \) and \( EF = 4 \text{ cm} \), then perimeter of \( \Delta DEF \) is
10. \( ABCD \) is a trapezium with \( AD \) parallel \( BC \) and \( AD = 4 \text{ cm} \). If the diagonals \( AC \) and \( BD \) intersect each other at \( O \) such that \( \frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2} \), then \( BC = \)
11. In fig \( DE \parallel BC \), then the measure of \( AE \) is
12. The perimeter of two similar triangles \( ABC \) and \( LMN \) are 60cm and 48 cm respectively. If \( LM = 8 \text{ cm} \), then the length of \( AB \) is
13. Find the value of \( \angle BAD \) in \( \Delta ABC \), if \( D \) is a point on the side \( BC \) such that \( \frac{AB}{AC} = \frac{BD}{DC} \), \( \angle B = 70^\circ \) and \( \angle C = 50^\circ \)
15. Which of the following is true? From the figure \( \angle Q = \angle E = 80^\circ \) and \( \angle R = \angle D = 40^\circ \)
16. What is the value of \( x + y \), if \( \Delta ABC \sim \Delta PQR \)
17. Find the value of \( x \) for which \( DE \parallel BC \) in the adjoining figure
24. In a right angled triangle \( ABC \), \( \angle C = 35^\circ \) and in another right-angled triangle \( PQR \), \( \angle R = 35^\circ \). Then relation between the two triangles is:
25. In the given \( \Delta ABC \), line \( PQ \) is parallel to side \( BC \), then \( \angle B = \angle P \) because they are:
27. In the following figure \( LM \) is parallel to \( BC \) and \( LN \) is parallel to \( CD \) then which of the following relation is true:
29. \( E \) and \( F \) are the points on the sides \( PQ \) and \( PR \) respectively of \( \Delta PQR \)., \( PE = 4 \text{ cm} \), \( QE = 4.5 \text{ cm} \), \( PF = 8 \text{ cm} \) and \( RF = 9 \text{ cm} \).
A: Assertion: \( EF \) is not parallel to \( QR \)
R: Reason: In a triangle if two sides are divided proportionally by a line then the line is parallel to the third side.
A: Assertion: \( EF \) is not parallel to \( QR \)
R: Reason: In a triangle if two sides are divided proportionally by a line then the line is parallel to the third side.
CHAPTER-7 CO-ORDINATE GEOMETRY
1. The distance of a point \( P(x, y) \) from the origin is
2. The points on y-axis, whose ordinate is 3 and \( Q \) is a point (-5, 2), then the distance \( PQ \) is
3. The point on the x-axis which is equidistant from points (-1, 0) and (5, 0) is
4. The distance between \( A(1, 3) \) and \( B(x, 7) \) is 5. The possible values of \( x \) are
5. The perpendicular distance of \( A(5, 12) \) from the y-axis is
6. The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0)
7. The coordinates of a point \( A \), where \( AB \) is the diameter of a circle, whose centre is (2, -3) and \( B(1, 4) \) is:
8. If the points \( P(7, 3), Q(9, 4), R(8, k) \) and \( S(6, 1) \) taken in order, are the vertices of the rectangle, then the value of \( k \) is:
9. The number of points on x-axis which are at a distance \( k \), where \( k = 5 \), from the point (2, 3) are
10. The points (-5, 1), (1, p) and (4, -2) are collinear if the value of \( p \) is
11. The area of the triangle \( ABC \) with the vertices \( A(-5, 7), B(-4, -5) \) and \( C(4, 5) \) is
16. If \( O(p/3, 4) \) is the midpoint of the line segment joining the points \( P(-6, 5) \) and \( Q(-2, 3) \), the value of \( p \) is:
20. The distance of the point \( P(–6, 8) \) from the origin is
21. The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
23. If the points \( A(1, 2), O(0, 0) \) and \( C(a, b) \) are collinear, then
27. \( AOBC \) is a rectangle whose three vertices are \( A(0, 3), O(0, 0) \) and \( B(5, 0) \). The length of its diagonal is
Chapter 8 - INTRODUCTION TO TRIGONOMETRY
Q1. If \( \tan \theta = \frac{3}{4} \) then the value of \( \sin \theta \) is
Q2. If \( \sin (A + B) = \frac{\sqrt{3}}{2} \) and \( \tan (A – B) = 1 \). What are the values of \( A \) and \( B \)?
Q3. If \( \tan \alpha = \sqrt{3} \) and \( \text{cosec } \beta = 1 \), then the value of \( \alpha – \beta \)?
Q4. In triangle \( ABC \), right angled at \( C \), then the value of \( \text{cosec } (A + B) \) is
Q5. If \( \sin \theta - \cos \theta = 0 \) then the value of \( \sec \theta \)
Q6. What is the value of \( \sin 30^\circ + \cos 60^\circ \)?
Q7. If \( (1 + \cos A) (1 – \cos A) = 3/4 \), find the value of \( \sec A \).
Q8. If \( x \tan 60^\circ \cos 60^\circ = \sin 60^\circ \cot 60^\circ \), then \( x = \)
Q9. If \( \sin \theta + \cos \theta = \sqrt{2} \cos \theta \), then \( \tan \theta + \cot \theta = \)
Q10. If \( 2\sin^2 \beta - \cos^2 \beta = 2 \), then \( \beta \) is
Q11. If the angles of \( \Delta ABC \) are in ratio 1:1:2, respectively (the largest angle being angle \( C \)), then the value of \( \frac{\sec A}{\text{cosec } B} - \frac{\tan A}{\cot B} \) is
Q12. If \( 4 \tan A = 3 \), then \( \frac{4 \sin A - 3 \cos A}{4 \sin A + 3 \cos A} = \)
Q13. The value \( (5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ) / (\sin^2 30^\circ + \cos^2 30^\circ) \)
Q14. If \( \sin \theta = x \) and \( \sec \theta = y \) then value of \( \cot \theta \) is given by
Q15. If \( \cos x = \sqrt{\frac{5}{2}} \). Find the value of \( \tan x \)
Q16. \( \frac{2\tan 30^\circ}{(1 - \tan^2 30^\circ)} \) is equal to
Q17. If a triangle \( ABC \) is right-angled at \( C \). What will be the value of \( \cos(A+B) \)
Q18. What is the minimum value of \( \sin A, 0 \leq A \leq 90^\circ \)
Q19. If \( x \tan 45^\circ \sin 30^\circ = \cos 30^\circ \tan 30^\circ \), then \( x \) is equal to
Q20. If \( \sec A + \tan A = x \), then \( \tan A = \)
Q21. \( \frac{1 + \tan^2 A}{1 + \cot^2 A} = \)
Q22. If \( \sin A + \sin^2 A = 1 \), then find the value of \( \cos^2 A + \cos^4 A \).
Q23. If \( \sec \theta – \tan \theta = 1/3 \), then find the value of \( (\sec \theta + \tan \theta) \)
Q24. If \( x = a \cos \theta \) and \( y = b \sin \theta \), then \( b^2x^2 + a^2y^2 = \)
Q25. \( \sin 2A = 2 \sin A \) is true when \( A = \)
Q26. The value of the expression \( \sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta \) is
Q27. \( 5 \tan^2 A – 5 \sec^2 A + 1 \) is equal to
Q28. If \( 3\sec \theta – 5 = 0 \), then \( \cot \theta \) is
Q29. Assertion: The value of \( \text{cosec } 30^\circ + \cot 45^\circ \) is 3
Reason: \( \text{cosec } 30^\circ = 2, \cot 45^\circ = 1 \)
Reason: \( \text{cosec } 30^\circ = 2, \cot 45^\circ = 1 \)
Q30. Assertion: In a right \( \Delta ABC \), right angled at \( B \), if \( \tan A = 12/5 \), then \( \sec A = 13/5 \).
Reason: \( \cot A \) is the product of \( \cot \) and \( A \).
Reason: \( \cot A \) is the product of \( \cot \) and \( A \).
CHAPTER:- 9 SOME APPLICATIONS OF TRIGONOMETRY
Q1. If a tower 30 m high, casts a shadow \( 10\sqrt{3} \) m long on the ground, then what is the angle of elevation of the sun?
Q3. If the ratio of the height of a tower and the length of its shadow is \( 1:\sqrt{3} \), what is the angle of elevation of the Sun?
Q7. If the height and length of a shadow of a tower are the same, then the angle of elevation of Sun is:
Q9. A ladder makes an angle of \( 60^\circ \) with the ground, when placed along a wall. If the foot of ladder is 8 m away from the wall, the length of ladder is:
Q10. The angle of depression of an object on the ground, from the top of a 25 m high tower is \( 30^\circ \). The distance of the object from the base of tower is:
Q12. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of \( 30^\circ \) with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8 m. The height of the tree is
Q13. The angle of elevation of the top of a tower is \( 30^\circ \). If the height of the tower is tripled, then the angle of elevation of the top of a tower is:
Q14. An observer 1.5 m tall is 28.5 m away from a tower and the angle of elevation of the top of the tower from the eye of the observer is \( 45^\circ \). The height of the tower is:
Q16. The angle of elevation of the top of a building from a point on the ground, which is 30 m away from the foot of the building, is \( 30^\circ \). The height of the building is
Q18. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is \( 60^\circ \). Assuming that there is no slack in the string. The length of the string is:
Q19. A pole 6m high costs a shadow \( 2\sqrt{3} \)m long on the ground then the sun's elevation is:
Q21. From the top of a 120 m high tower, a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as \( 60^\circ \) and \( 45^\circ \). Then the distance between two cars is:
Q22. If two towers of height \( h_1 \) and \( h_2 \) subtends angles of \( 60^\circ \) and \( 30^\circ \) midpoint of the line joining their feet. Then what is \( h_1 : h_2 \) is:
Q23. If two poles are 25m and 15m high and the line joining their tops makes an angle \( 45^\circ \) with the horizontal. The distance between these poles is:
Q24. A vertical tower stands on horizontal plane and is surmounted by a vertical flag-staff of height 6 m. The angles at a point on the bottom and top of the flag-staff with the ground are \( 30^\circ \) and \( 45^\circ \) respectively. Then the height of the tower is:
Q25. The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is \( 30^\circ \) than when it is \( 60^\circ \). Then the height of tower is:
Q26. The angle of elevation of the top of a tower from certain point is \( 30^\circ \). If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by \( 15^\circ \). Find the height of the towe
Q27. The angle of elevation of the top of a vertical tower from a point on the ground is \( 60^\circ \). From another point 10 m vertically above the first, its angle of elevation is \( 45^\circ \). Then the height of the tower is:
Q28. The angle of elevation of an aeroplane from a point on the ground is \( 60^\circ \). After a flight of 30 seconds the angle of elevation becomes \( 30^\circ \). If the air plane is flying at a constant height of \( 3000\sqrt{3} \) m, Then the speed of the aeroplane is:
Q30. From a point on a bridge across a river the angle of depression of the banks on opposite sides of the river are \( 30^\circ \) and \( 45^\circ \) respectively. If the bridge is at the height of 30 m from the banks, the width of the river is
CHAPTER 10- Circle
8. In the given figure, if \( \angle RPS = 25^\circ \), the value of \( \angle ROS \) is
9. A tangent is drawn from a point at a distance of 17 cm of circle \( C(0, r) \) of radius 8 cm. The length of its tangent is
15. In the given figure, \( AB \) and \( AC \) are tangents to the circle with centre \( O \) such that \( \angle BAC = 40^\circ \), then \( \angle BOC \) is equal to
17. \( C_1 (O, r_1) \) and \( C_2(O, r_2) \) are two concentric circles with \( r_1 > r_2 \). \( AB \) is a chord of \( C_1(O, r_1) \) touching \( C_2(O, r_2) \) at point \( C \) then which one statement is true
18. Two parallel lines touch the circle at points \( A \) and \( B \) respectively. If area of the circle is \( 25\pi \), then \( AB \) is equal to
19. In figure \( AT \) is a tangent to the circle with centre \( O \) such that \( OT = 4 \text{ cm} \) and \( \angle OTA = 30^\circ \). Then \( AT \) is equal to
20. In figure if \( O \) is centre of a circle, \( PQ \) is a chord and the tangent \( PR \) at \( P \) makes an angle of \( 50^\circ \) with \( PQ \), then \( \angle POQ \) is equal to
21. In figure, \( O \) is the centre of a circle, \( AB \) is a chord and \( AT \) is the tangent at \( A \). If \( \angle AOB = 100^\circ \), then \( \angle BAT \) is equal to
22. If the angle between two radii of a circle is \( 110^\circ \), then the angle between the tangents at the ends of the radii is:
23. \( AB \) is a chord of the circle and \( AOC \) is its diameter such that angle \( \angle ACB = 50^\circ \). If \( AT \) is the tangent to the circle at the point \( A \), then \( \angle BAT \) is equal to
24. In the given figure, \( AB \) is a diameter of a circle with centre \( O \) and \( AT \) is a tangent. If \( \angle AOQ = 58^\circ \), find \( \angle ATQ \)
26. In figure, \( PQ \) is a chord of a circle with centre \( O \) and \( PT \) is a tangent. If \( \angle QPT = 60^\circ \), find \( \angle PRQ \).
27. If angle between two radii of a circle is \( 130^\circ \), the angle between the tangents at the ends of the radii is
28. A tangent \( PQ \) at a point \( P \) of a circle of radius 5 cm meets a line through the centre \( O \) at a point \( Q \) so that \( OQ = 12 \text{ cm} \). Length \( PQ \) is:
29. If two tangents inclined at an angle \( 60^\circ \) are drawn to a circle of radius 3 cm, then length of each tangent is equal to
30. In the figure below, \( PQ \) is a chord of a circle and \( PT \) is the tangent at \( P \) such that \( \angle QPT = 60^\circ \). Then \( \angle PRQ \) is equal to
CHAPTER 11 – AREA RELATED TO CIRCLES
Q1. Perimeter of sector of a circle having angle \( 90^\circ \) and radius 14 cm is
Q3. Area of clock swept by minute hand of diameter 42 cm from 12.00 to 3.00 is
Q5. Find the area of corresponding major sector of a circle of radius 14cm and central angle \( 90^\circ \).
Q7. A chord \( AB \) of a circle of radius 10cm subtends an angle of \( 60^\circ \) at the centre of the circle. The area of minor segment is
Q8. If the length of a circle subtending and angle of \( 60^\circ \) is 22 cm then the radius of circle is
Q11. If the sum of the circumference of two circles with radius \( r_1 \) and \( r_2 \) is equal to the circumference of a circle of radius \( R \) then
Q16. The area of a square that can be inscribed in a circle of radius 10 cm is
Q19. The area of circle that can be inscribed in a square of side 6cm is
Q21. The perimeter of a quadrant of a circle of radius \( r \) is
Q22. Circumferences of two circles are equal. Is it necessary that areas be equal? Why?
Q23. A car has two wipers which do not overlap each viper has a blade of length 21 cm sweeping through an angle of \( 120^\circ \). The total area cleaned at each sweep of the blades is
Q28. Assertion: The area of sector depends on the measure of the angle in the centre \( \theta \) and the square of the radius.
Reason: The measure of the angle at the centre is \( 180^\circ \) area of the sector = \( \pi r^2 \)
Reason: The measure of the angle at the centre is \( 180^\circ \) area of the sector = \( \pi r^2 \)
Q29. Assertion: If the ratio of the circumference of two circles is 3:1 then the ratio of their areas is 9:1.
Reason: If \( R_1 \) and \( R_2 \) are the radii of two circles then ratios of the areas is \( \frac{R_1^2}{R_2^2} \)
Reason: If \( R_1 \) and \( R_2 \) are the radii of two circles then ratios of the areas is \( \frac{R_1^2}{R_2^2} \)
Q30. Assertion: If the outer and inner diameter of a circular path is 10m and 6m then the area of the path is \( 16\pi \text{ m}^2 \)
Reason: if \( R \) and \( r \) be the radius of outer and inner circular path then the area of the path is \( \pi(R^2 - r^2) \) m^2
Reason: if \( R \) and \( r \) be the radius of outer and inner circular path then the area of the path is \( \pi(R^2 - r^2) \) m^2
CHAPTER 12 - SURFACE AREAS AND VOLUMES
1. A solid is of the form of a cone of radius ‘\( r \)’ surmounted on a hemisphere of the same radius. If the height of the cone is the same as the diameter of its base, then the volume of the solid is :
2. The curved surface area of a right circular cylinder of height 14 cm is \( 88 \text{ cm}^2 \). The diameter of its circular base is:
3. What is the total surface area of a solid hemisphere of diameter ‘\( d \)’ ?
4. If the area of the base of a cone is \( 51 \text{ cm}^2 \) and its volume is \( 85 \text{ cm}^3 \), then the vertical height of the cone is given as :
6. A medicine-capsule is in the shape of a cylinder of radius 0.25 cm with two hemispheres stuck to each of its ends. The length of the entire capsule is 2 cm. What is the total surface area of the capsule? (Take \( \pi \) as 3.14)
11. The volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is \( 12.8 \text{ m}^3 \). The breadth of the wall is
13. If a marble of radius 2.1 cm is put into a cylindrical cup full of water of radius 5cm and height 6 cm, then how much water flows out of the cylindrical cup?
16. A hollow cube of internal edge 22cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that \( \frac{1}{8} \) space of the cube remains unfilled. Then the number of marbles that the cube can accomodate is
18. A solid piece of iron in the form of a cuboid of dimensions \( 49\text{cm} \times 33\text{cm} \times 24\text{cm} \), is moulded to form a solid sphere. The radius of the sphere is
19. A right circular cylinder of radius \( r \) cm and height \( h \) cm (\( h>2r \)) just encloses a sphere of diameter
21. The area of the base of a rectangular tank is \( 6500 \text{ cm}^2 \) and the volume of water contained in it is \( 2.6 \text{ m}^3 \). The depth of water in the tank is
23. In a shower, 5 cm of rain falls. The volume of the water that falls on 2 hectares of ground, is
24. A mason constructs a wall of dimensions \( 270\text{cm} \times 300\text{cm} \times 350\text{cm} \) with the bricks each of size \( 22.5\text{cm} \times 11.25\text{cm} \times 8.75\text{cm} \) and it is assumed that \( \frac{1}{8} \) space is covered by the mortar. Then the number of bricks used to construct the wall is
26. How many bags of grain can be stored in a cuboidal granary (\( 8\text{m} \times 6\text{m} \times 3\text{m} \)), if each bag occupies a space of \( 0.64 \text{ m}^3 \)?
29. Assertion (A) : The surface area of largest sphere that can be inscribed in a hollow cube of side ‘\( a \)’ cm is \( \pi a^2 \text{ cm}^2 \).
Reason (R) : The surface area of a sphere of radius \( r \) is \( 4\pi r^2 \).
Reason (R) : The surface area of a sphere of radius \( r \) is \( 4\pi r^2 \).
CHAPTER 13- STATISTICS
2. If the mean of frequency distribution is 7.5 and \( \sum f_i x_i = 120 + 3k, \sum f_i = 30 \), then \( k \) is equal to:
6. If the mean of first \( n \) natural numbers is \( 3n/5 \), then the value of \( n \) is:
8. If mean of \( a, a+3, a+6, a+9 \) and \( a+12 \) is 10, then \( a \) is equal to;
11. The mean of following distribution is:
\[ \begin{matrix} X_i & 11 & 14 & 17 & 20 \\ F_i & 3 & 6 & 9 & 7 \end{matrix} \]
\[ \begin{matrix} X_i & 11 & 14 & 17 & 20 \\ F_i & 3 & 6 & 9 & 7 \end{matrix} \]
14. ... The sum of the lower limit of the modal class and upper limit of the median class is
22. The numbers are arranged in ascending order. If their median is 25, then \( x = ? \)
\( 5, 7, 10, 12, 2x-8, 2x+10, 35, 41, 42, 50 \)
\( 5, 7, 10, 12, 2x-8, 2x+10, 35, 41, 42, 50 \)
CHAPTER 14 – PROBABILITY
Q-1 What is the probability of getting a number less than 11?
Q-2 What is the probability of getting a multiple of 5?
Q-3 What is the probability of getting a number divisible by 3?
Q-4 What is the probability of getting a prime number?
Q-5 What is the probability of getting an even number?
Q-6 What is the probability of getting an even number as the sum?
Q-7 What is the probability of getting the sum greater than or equal to 10?
Q-8 What is the probability of getting a doublet of odd number?
Q-9 What is the probability that the difference of the numbers on the two dice is 2?
Q-10 What is the probability of getting a multiple of 5 as the sum?
Q-11 What is the probability of getting an ace card?
Q-12 What is the probability of getting a red card?
Q-13 What is the probability of getting either black or king card?
Q-14 What is the probability of getting red and a queen card?
Q-15 What is the probability of getting neither a heart nor a king card?
Q-16 The king, queen and jack of clubs are removed from a pack of 52 playing cards. One card is selected at random from the remaining cards. Find the probability that the card is neither a heart nor a king
Q-17 What is the probability of getting two heads?
Q-18 What is the probability of getting at least one head?
Q-19 What is the probability of getting no tail?
Q-20 What is the probability of getting at most one head?
Q-21 What is the probability that the candy taken out will be red?
Q-22 What is the probability that the candy taken out will be not green?
Q-23 What is the probability that the candy taken out will be red or green?
Q-24 What is the probability that it is not acceptable to Jimmy?
Q-25 What is the probability that it is acceptable to Sujatha?
Q-26 A letter is chosen at random from the letters of the word ‘ASSASSINATION’. The probability that the letter chosen is vowel
Q-27 The probability of getting 5 Sundays in the month of August.
Q-28 The probability of getting 53 Fridays in a leap year.
Q-29 A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.
Q-30 A bag contains 18 balls out of which \( x \) balls are red. If 2 more red balls are put in the bag, the probability of drawing a red ball will be \( \frac{9}{8} \) times the probability of drawing a red ball in the first case. Find the value of \( x \).
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