The value of $\sin ^2 \frac{5 \pi}{12}-\sin ^2 \frac{\pi}{12}$ is
- A$\sqrt{3} / 2$
- B$\frac{1}{2}$
- C$0$
- D1
Question types
Model Paper 5 question types
45 questions across 6 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
45
Questions
6
Question groups
5
Question types
01
M.C.Q (1 Marks)
18 Q→022 Marks Questions
7 Q→033 Marks Question
9 Q→045 Marks Questions
6 Q→05Assertion (A) & Reason (B) MCQ
2 Q→06Case study (4 Marks)
3 Q→Sample Questions
Model Paper 5 questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
Let R be set of points inside a rectangle of sides a and b (a, b > 1) with two sides along the positive direction of x-axis and y-axis. Then
- A$R=\{(x, y): 0 \leq x \leq a, 0 \leq y \leq b\}$
- B
- C
- D
If P(n, r) = C(n, r) then
- Ar = 0 or 2
- Br = 1 or n
- Cr = 0 or 1
- Dn = r
If $f(x)=\sqrt{1-x^2}, x \in(0,1)$, then $f^{\prime}(x)$, is equal to
- A$\sqrt{1-x^2}$
- B$\sqrt{x^2-1}$
- C$\frac{1}{\sqrt{1-x^2}}$
- D$\frac{-x}{\sqrt{1-x^2}}$
Solve: $3 x+5< x-7$, when $x$ is a real number
- A$x>-12$
- B$x<-12$
- C$x<-6$
- D$x>-6$
Find the equations of the lines which cut-off intercepts on the axes whose sum and product are 1 and -6 respectively.
What is represented by the shaded regions in each of the following Venn-diagrams.
A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.
Determine the probability p, for event. An odd number appears in a single toss of a fair die
Evaluate: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{x^2}$.
View full solution →Evaluate $\lim _{x \rightarrow 2}\left(\frac{x^2-4}{\sqrt{x+2}-\sqrt{3 x-2}}\right)$
View full solution →It is required to seat 5 men and 3 women in a row so that the women occupy the even places. How many such arrangements are possible?
For any sets A and B show that
i. $(A \cap B) \cup(A-B)=A$
ii. $A \cup(B-A)=A \cup B$
View full solution →i. $(A \cap B) \cup(A-B)=A$
ii. $A \cup(B-A)=A \cup B$
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is $\frac{9}{2}$, then write its first term and common difference.
View full solution →Evaluate $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$
View full solution →Show that the equation $x^2-2 y^2-2 x+8 y-1=0$ represents a hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latusrectum, coordinates of foci and vertices and equations of directrices of the hyperbola.
View full solution →From the frequency distribution consisting of 18 observations, the mean and the standard deviation were found to be 7 and 4, respectively. But on comparison with the original data, it was found that a figure 12 was miscopied as 21 in calculations. Calculate the correct mean and standard deviation.
Prove that: $\sin 5 x=5 \sin x-20 \sin ^3 x+16 \sin ^5 x$.
View full solution →Prove that $\cos 12^{\circ}+\cos 60^{\circ}+\cos 84^{\circ}=\cos 24^{\circ}+\cos 48^{\circ}$
View full solution →Solve for $x , \frac{|x+3|+x}{x+2}>1$zZ
View full solution →Assertion (A): The sum of first 6 terms of the GP 4, 16, 64, ... is equal to 5460 .
Reason (R): Sum of first n terms of the G.P is given by $S _{ n }=\frac{a\left(r^n-1\right)}{r-1}$, where $a =$ first term $r =$ common ratio and $|r|>1$.
Reason (R): Sum of first n terms of the G.P is given by $S _{ n }=\frac{a\left(r^n-1\right)}{r-1}$, where $a =$ first term $r =$ common ratio and $|r|>1$.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A.
- CA is true but R is false.
- DA is false but R is true.
Assertion (A): if A = set of letters in Alloy B = set of letters in LOYAL, then set A & B are equal sets.
Reason (R): If two sets have exactly the same elements, they are called equal sets.
Reason (R): If two sets have exactly the same elements, they are called equal sets.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A.
- CA is true but R is false.
- DA is false but R is true.
Read the following text carefully and answer the questions that follow:
A number of the form $Z=x+i y$, where $x$ and $y$ are real and $i=\sqrt{-1}$ is called a complex number. Consider the complex number $Z_1=2+3 i$ and $Z_2=4-3 i$.
i. Find the imaginary part of $Z_1 \overline{Z_1}$. (1)
ii. Find the real part of $\frac{z_1}{z_2}$. (1)
iii. Find the imaginary part of $Z_1-Z_2$. (2)
OR
Find the real part of $Z _1$. (2)
View full solution →A number of the form $Z=x+i y$, where $x$ and $y$ are real and $i=\sqrt{-1}$ is called a complex number. Consider the complex number $Z_1=2+3 i$ and $Z_2=4-3 i$.
i. Find the imaginary part of $Z_1 \overline{Z_1}$. (1)
ii. Find the real part of $\frac{z_1}{z_2}$. (1)
iii. Find the imaginary part of $Z_1-Z_2$. (2)
OR
Find the real part of $Z _1$. (2)
Read the following text carefully and answer the questions that follow: Representation of a Relation
A relation can be represented algebraically by roster form or by set-builder form and visually it can be represented by an arrow diagram which are given below
i. Roster form In this form, we represent the relation by the set of all ordered pairs belongs to R.
ii. Set-builder form In this form, we represent the relation $R$ from set $A$ to set $B$ as $R=\{(a, b): a \in A, b \in B$ and the rule which relate the elements of A and B \}.
iii. Arrow diagram To represent a relation by an arrow diagram, we draw arrows from first element to second element of all ordered pairs belonging to relation R.
Questions
i. If n(A) = 3 and B = {2, 3, 4, 6, 7, 8} then find the number of relations from A to B. (1)
ii. If A = {a, b} and B = {2, 3}, then find the number of relations from A to B. (1)
iii. If A = {a, b} and B = {2, 3}, write the relation in set-builder form. (2)
OR
Express of $R =\{( a , b ): 2 a + b =5 ; a , b \in W \}$ as the set of ordered pairs (in roster form). (2)
View full solution →A relation can be represented algebraically by roster form or by set-builder form and visually it can be represented by an arrow diagram which are given below
i. Roster form In this form, we represent the relation by the set of all ordered pairs belongs to R.
ii. Set-builder form In this form, we represent the relation $R$ from set $A$ to set $B$ as $R=\{(a, b): a \in A, b \in B$ and the rule which relate the elements of A and B \}.
iii. Arrow diagram To represent a relation by an arrow diagram, we draw arrows from first element to second element of all ordered pairs belonging to relation R.
Questions
i. If n(A) = 3 and B = {2, 3, 4, 6, 7, 8} then find the number of relations from A to B. (1)
ii. If A = {a, b} and B = {2, 3}, then find the number of relations from A to B. (1)
iii. If A = {a, b} and B = {2, 3}, write the relation in set-builder form. (2)
OR
Express of $R =\{( a , b ): 2 a + b =5 ; a , b \in W \}$ as the set of ordered pairs (in roster form). (2)
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